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Oscillation of functional differential equations
MATHEMATICAL AND
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OClIENOI C ~ . '
RRO'lr°
COMPUTER
MODELLING
RLgEVIER
Mathematical and Computer Modelling 41 (2005) 417-461 www.elsevier.com/locate/mcm
Oscillation of Functional Differential Equations R. P. AGARWAL Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901, U.S.A. agarwal@f it. edu
S. R. GRACE Department of Engineering Mathematics Faculty of Engineering, Cairo University Orman, Giza 12221, Egypt srgrace©eng, cu. edu. eg
I. KIGURADZE A. Razmadze Mathematical Institute of the Georgian Academy of Sciences Tbilisi, Georgia kig©rmi, acnet, ge
D. O'REGAN Department of Mathematics National University of Ireland Galway, Ireland donal, oregan@nuigalway, ie
(Received May 2004; accepted June 2004) A b s t r a c t - - S o m e new criteria for the oscillation of functional differential equations of the form, d[
1 d i d a d ]~ an--l(t) -~ an-2(t) d-t"'al(t) dt x(t) +Sq(t) f(x[g(t)])=O,
are presented in this paper. © 2005 Elsevier Ltd. All rights reserved.
Keywords--Oscillation, Nonoscillation, Functional, Nonlinear, Comparison.
1. I N T R O D U C T I O N In this paper, we are concerned with the oscillatory behavior of all solutions of the functional differential equation,
Lnx (t) + 5q (t) f (x [g (t)]) = 0,
0895-7177/05/$ - see front matter (~) 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm. 2004.06.018
(1.1; 5)
Typeset by A~ts-~X
R.P. AGARWAL et al.
418 where n >_ 2, ~ = -t-1, and
nox(t)
= x(t),
L k x (t) --
1
d
ak (t) dt
L k - l X (t)
'
k = 1, 2,
., n - 1, " "
(1.2)
d L n x (t) = ~ [L~-lX (t)] ~ . In what follows, we shall assume that
(i) ~ e c([t0, ~ ) , ~t+ = (0, ~ ) ) , to > 0, and
f (ii) (iii) (iv) (v)
q g f a
ds = ~ ,
~ai(s)
i
= 1,2,...,n-
(1.z)
1,
e C([to, c~), R+), ~ C([t0, co), R -- ( - o c , co)) is nondecreasing and limt__.= g(t) = ec, E C(R, ]R) is nondecreasing and x f ( z ) > O, for x ~ O, is a quotient of positive odd integers.
The domain :D(Ln) of Ln is defined to be the set of all functions x : [tz, c~) ~ R, t~ > to, such that Ljx(t), j = O, 1 , . . . , n exist and are continuous on [t~, c~). Our attention is restricted to those solutions x E 7)(Ln) of equation (1.1; 5) which satisfy sup{lx(t)l : t > T} > o, for every T > t~. We make the standing hypothesis that equation (1.1; 5) does possess such solutions. A solution of equation (1.1; 5) is called oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. Equation (1.1; 5) is called oscillatory if all its solutions are oscillatory. The problem of obtaining sufficient conditions to ensure that all solutions of certain higher-order nonlinear functional differential equations of type (1.1; 5) are oscillatory when a = 1 and/or a > 0 has been studied by a number of authors, see [1-29] and the references cited therein. Our main objective is to present a systematic study on the oscillation of equation (1.1; 5) and establish some new oscillation criteria. In Section 2, we shall give the proofs of some important lemmas which are useful throughout this paper. Section 3 is devoted to the study of equation (1.1; 5) when f satisfies the condition f±o~ d u / f l / ~ ( u ) < cc and in Section 4, we present some results for equation (1.1; 5) when f satisfies the condition f(x)sgn x > Ix[ ~, for x ¢ 0. In Section 5, we give results for equation (1.1; 5) when f satisfies a condition of the form f+o d u / f ( u l / ~ ) < ~ " Section 6 is devoted to the linearization of nonlinear oscillation results and via comparison with the oscillatory behavior of second-order linear equation, we present results for equation (1.1; 5) when f satisfies the condition f(x)sgn x >_ lxt ~, for x ¢ 0, where/3 is the quotient of positive odd integers, fl > a, = ~, and 13 < ~. In Section 7, we present some oscillation criteria for equation (1.1; 5) and its special case when ai = 1, i = 1, 2 , . . . , n - 1. The obtained results extend, improve and correlate a number of existing results.
2. P R E L I M I N A R I E S To formulate our results, we shall use the following notation from [15]: for a~ e C([t0, oc), R), i = 1 , 2 , . . . , we define I0 = 1,
/~(t,~;~,~-~,...,~0
=
//
a~(~)I~_~(~,~;~_~,...,~)
d~,
i=1,2,....
It is easy to verify from the definition of Ii that Ii ( t , s ; a l , . . . , a i ) = ( - 1 ) ' £ ( s , t ; a ~ , . . . , a l ) and
Ii(t,s;al,...,a~)
=
We shall need the following lemmas.
f
ai(u)Ii-l (t,u;al,...,ai-1)
du.
Functional Differential Equations
419
LEMMA 2.1. If X e D(Ln), where Ln is Ln defined by (1.2) with a = 1, then, the following formu/as hold, for 0 < i < k < n - 1 and t, s E [to, oo) k-1
Lix (t) = E Ij_~ (t, s; a,+l, . . . , ak-1) L~x (s)
+
j=i
/st
(2.1)
Ik-~-~ (t,~; a~+~,..., a~_~) a~ (u) Lkz (u) d~
and
k--1
L~x (t) = E ( - 1 ) J - ~ / j - i (s, t; a j , . . . , ai+l) L j x (s) J=~ + (-1)
-~
f'
(2.2)
- ~rk_~_l (u, t; ak_~,..., a~+~) ak (~) Ck~ (~) a~.
This lemma is a generalization of Taylor's formula with remainder encountered in calculus. The proof is immediate. LEMMA 2.2. Suppose condition (1.3) holds. I f x E ~)(L~) where L~ is defined as in Lemrna 2.1 is eventually of one sign, then, there exist a tz >_ to >_ 0 and an integer ~, 0 < e < n with n + ~, even for x(t)[mx(t) nonnegative eventually, or n + g odd, t'or x(t)L~x(t) nonpositive eventually and, such that, for every t >__t~,
e > 0 implies x(t)Lkx (t) > O,
k = O, 1,... , l
e < n - 1 implies ( - 1 ) e-k x (t) Lkx (t) > 0,
k = e, l + 1 , . . . , n.
(2.3)
This lemma generalizes the Lemma 1.1 from [15] and can be proved similarly. We note that if x(t) is a solution of equation (1.1; 5) which is eventually of one sign, then, d -dt- ( L ~ - l x ( t ) ) = a L ,~-1 _ l x (t) L , x (t), and since a satisfies (v), we see that Lnx and Lnx have the same sign. Moreover, one can easily see that Ljx(t), 0 <_j <_ n are eventually of one sign. It will be convenient to make use of the following notation in the remainder of this paper. For any T > to and all t _> T, we let
wt[t,T] =
Ii-l (t,s;al,...,al-1)ae(s)In-~-l
Wl[t,T] = a l (t)
( t , s ; a n - l , . . . , a t + l ) ds,
h_~(t,s;a2,...,al_l)at(s)I,~_l-l
N[t,T]=
rain N~[t,T] 2
and
(t,s;a,~-l,...,ae+l) ds, w[t,T]=
min
l
1
-
i,
2
-
i,
we[t,T].
For t > s > T, we let ~1 [t, s] = al (s)I,~-2 (t, s; a , , - 1 , . . . , as),
~
[t,T] = min{~l It, At], ~ [At, T], 2 < ~ < n - 1, and 0 < A < 1},
w , [t, s] ----In-1 (t, s; a l , . . . , a n - l ) ,
~ n [t,S] : al (t) In-2 (t, s ; a 2 , . . . , a n _ l ) , w__0 It, T] :
max
l<_i
I, ( t , T ; a l , . . . , a ~ ) ,
~o It, s] : In-1 (t, s; a , - 1 , . . . ,
al).
Now, we present the following lemmas.
t >_T/A,
420
R.P.
LEMMA
AGARWAL et al.
Let x E l)(Ln) be eventually positive, (-1)'~-~6 = - 1 and condition (1.3) hold.
2.3.
(il) Then, for 1 < g < n - 1 and a11 t > T > to,
x (t) > w~ [t, T] ( 6 L , _ l x (t)).
(2.4)
(i2) Then, for 2 < l < n - 1 and a11 t >_ T >_ to,
x' (t) >_~[t, T] ( 6 n , _ l x (t)).
(2.5)
(i3) For 1 < l < n - 1, let ), be a constant with 0 < ~ < 1. Then, there exists a T* >_ T/A, such that x' (At) _> ~ a [t, T] (65,_1x (t)), for t > T*. (2.6) PROOF. Let x E 7)(Ln) be eventually positive and ~, 1 < / < n - 2 be the integer assigned to the function x as in Lemma 2.2. From (2.2) with i, k, t, and s replaced b y / , n - 1, s, and t, respectively, we obtain in view of (2.3), n--2
Lex (s) = ~
( - 1 ) ~-~ Ij_~ (t, s; aj, . . . , a~+l) L j x (t)
+(1)
n-t-1
--
>_
it In_g_2(u,S;an_2,...,a~+l)an_l(COLn_lX(~Z)du
(2.7)
J,g
xn_~_s (~, s; an-s,..., a~+l) a~-i (~) (~L~_lx (~)) a~.
Using the fact that 6L~_lX is noninereasing on [tl, oe), for some tl _> to _> O, we obtain
Lex (s) >_ I,~-t-1 (t, s; a ~ - l , . . . , at+l) (6L,~_lx (t)),
for t > s > tl.
(2.8)
On the other hand, using (2.1) with i, k, and s replaced by 0,~ and t~, respectively, and we get t
~-1 X
(2.3),
Ij (t, tl; al,..., aj) Ljx (tl) +/~ I~-i (t, s;at,..., ai-1) a~(s) L~x(s) ds (2.9)
(t) j=O >
h - 1 ( t , s ; a l , . . . , a e _ l ) a ~ (s) L~x(s) ds,
for t > tl.
(2.10)
Substituting (2.8) in (2.10), we obtain x (t) _>
[/i
]
/~-1 (t, s; a l , . . . , a~-l) a~ (s) I , - ~ - 1 (t, s; a n - l , . . . , a~+l) ds ( S L y - i x (t)), for t >_ tl.
The case when g = n - 1 is contained in (2.10). This completes the proof of (il). To prove (is), we let 2 < Z < n - 1. As before, we obtain (2.8). On the other hand, using (2.1) with i, k, and s replaced by 1 , / a n d tl, respectively, and (2.3), we find
F£--1
x' (t) = al (t) [~-~"//(t, t l ; a 2 , . . . , a~) L~x (tl)
+
h-~ (t, ~; as,...,a~-~) a~ (s) L~x (s) as 1
]
h - s (t, s; a s , . . . , a~-l) at (s) L~x (s) ds,
> al (t) 1
Combining (2.11) with (2.8), we obtain the desired inequality.
(2.11) for t > tl.
Functional Differential Equations
421
To prove (i3), we only need to prove the case when g = 1. From (2.2) with i, k, t, and s replaced by 1, n - 1, s, and t, respectively, we have F
(--1) j'-I Ij-1 (t,s;aj,...,a2) Ljx(t)
X' (S)----al (s) / ~
Li=1
z'
+ ( - I F -~
&-a (~, s; a._~.,...,a2) a . _ l (~) L._~x (~) d~
] for t > s > tl.
Using the fact that 5Ln-tx is nonincreasing on [tl, e~), we have
x' (s) >__at (s) I,~-2 (t, s; a ~ - l , . . . , a2) (~L,_lz (t)) = ~ [t, s] (~L~_~z (t)),
for t > s > tl.
For a constant A with 0 < )~ < 1, there exists a T1 >__tl/A, such that
x' (at) > ~1 [t, At] (SL~_~x (t)),
for all t > T1,
(2.12)
for t > T1.
(2.13)
Combining (2.12) and (2.13), we obtain (2.6). This completes the proof.
|
and for 2 < g < n - 1, inequality (2.5) becomes z' (At) > ~ [At, tl] (SL~_~x (t)), LEMMA 2.4. Let x
E
Z)(L~) be eventually positive, condition (1.3) hold and ~ = 0. Then, for
t>s>T>to,
(2.14) and --X / (8) > Wl [t,$] ((--1) n - 1
nn_lX(t)).
(2.15)
PROOF. Let x c T)(L~) be eventually positive and g = 0. From Lemma 2.1, there exists a tl > to, such that (-1) y L j x ( t ) > 0 ,
O<_j
t>tl.
(2.16)
From (2.2) with i, k, t, and s replaced by 0, n - l , s, and t respectively, we obtain in view of (2.16) n--2
x (s) = ~
( - 1 ) j I~ (t, s; aj,..., al) L~x (t)
j=o + (-1) "-1
&-2 (~, s; a~-2,..., al) a._l (~) L~_lx (~) d~ In-2 (u, s; a , - 2 , . . . , al) an-1 (u) L,~-lX (u) du,
>_ ( - 1 ) "-1
t > s > tl.
Using the fact that ( - 1 ) ' ~ - l L n _ l x is nonincreasing on [tl, c¢), we have
x(slzI,_l(t,s;a~_~,...,~l)((-1F-1L~_l~(t)),
fort> s>tl.
Next, from (2.2) with i, k, t, and s replaced by 1, n - 1, s, and t, respectively, we have P
LJ=l
+ (--1) n-1
In-3 (u, s; a n - 2 , . . . , a2) an-1 (u) Ln-lX (u) d
>al(s)I,-2(t,s;a,_l,...,a2)((-1)"-lLn_lx(t)), This completes the proof.
for t > s > t l . |
1~. P. AGARWAL et al.
422
LEMMA 2.5. Let x E ~(L~) be eventually positive, condition (1.3) hold and £ = n. Then, for
t>s>T,
(2.17)
x (t) > w. It, s] L._lx (~) and
x' (t) _> @,~ [t, s] Ln-lx (s). PROOF. Let x E T)(Ln) be eventually positive and let ~ = n. By Lemma 2.2, there exists a tl _> to, such that Ljx (t) > O, 0 <_j <_n and t _> tl. (2.18) From (2.1) with i and k replaced by 0 and n - 1, respectively, and in view of (2.18), we obtain n--2
x (t) = E / J j=o
>_
t
(t, s; a l , . . . , a~-2) Ljx (s) + f
In-2 (t, u; al,..., a,_2) a,~-i (u) Ln_lx (u) du
J8
In-2(t,u;al,...,an-2)ar~-l(u)Ln-lX(U)
>In-l(t,s;al,...,a~_l)Ln-lX(s),
du
fort>s>tl.
(2.19) Once again, from (2.1) with i and k replaced by 1 and n - 1, respectively, we have
Fn--2 x' (t) = al (t) [ E Ij (t, s; a2,..., an-2) Ljx (s) L~=I +
;
___ax(t)
I,_3(t,u;a2,...,a,_2)an_l(u)L,-lX(U)
/'
du
]
I,,_~(t,~;~,...,~,_~)a~_~(~)L~_~(~)
>_al (t) I~-2 (t, s; a2,..., a,-1) L , _ l x (s),
(2.20)
e~
for t > s > tl.
This completes the proof.
|
Finally, we shall denote the sets of all proper solutions, all oscillatory solutions and all nonoscillatory solutions of equation (1.1; 6) by 3, (9, and Af, respectively. It is clear that 8 = O tAAf. It is easy to see that Af has a decomposition, such that JV" = .,V'l u Jr'3 u • • • u H,,_I,
if 6 = 1 and n is even,
Jr" = .A/'o u Jr'2 u • • • u H,_I,
if6=l
Jr" = .,V'o u N'2 u • • • u.,V',~,
if 6 = - 1 and n is even,
N
if 6 = - 1 and n is odd,
= JVl u.A/'3 u • • • u.~,~,
andnisodd,
where Af~ denotes the subset of Af consisting of all x satisfying (2.3). Next, for a function g E C([t0, c~), R), we put
A~ = (t e [to,oo): g(t) > t},
~ = {t e [to,oo): g(t) < t}.
Also, we shall introduce the notation, v(t) = max {min {s,g (s)} : 0 <_ to <_ s <_ t } ,
p(t) = min{max {s,g(s)} : s >_t}.
Note that the following inequalities hold,
g(s) <_T(t), f o r T ( t ) < s < t ,
and
g (s) > p ( t ) ,
for t < s < p ( t ) .
Functional Differential Equations 3. O S C I L L A T I O N
423
CRITERIA
I
In this section, we shall present some criteria for the oscillation of equation (1.1; 5) when the function f satisfies
f :t:c~ fl/adu(u-----~< oo.
(3.1)
According to the subsets of A: for equation (1.1; 6), we shall consider the two separate cases, (il) g E {1,2,..- , n (i2) g = n.
1} and
3.1. Af~ = 0, g E { 1 , 2 , . . . , n -
1}
In this section, we shall provide some sufficient conditions which ensure that Aft = 0. THEOREM 3.1. Let 1 < g < n - 1, ( - 1 ) n - t 5 = - 1 , a > 1, and Conditions (i)-(v), (1.3), (3.1) hold. Moreover, assume that there exist two functions ~ and ~ E Cl([to, c~),]~+), such that a (t) _< inf {t,g (t)},
l i m a (t) = 0%
and a' (t) > 0,
for t > to,
(3.2)
t -"'+ ~
and, for all large T >_to, ~' (t) >_ o where
{
and
~[a(t),T],
w* [a (t), T] =
~
[a (t), T],
/f
j
(
(~' (t))x/~
w* [a (t), T] a' (t)
)
< O,
when g > 2 and a (t ) > T, when g > 1 and, for some constant A, OT/A.
oo ~ (s) q (s) ds : 0%
(3.3)
(3.4)
(3.5)
then, N~ = O. PROOF. Let x E ~r, and assume that x(t) > 0, for t > to _> 0. By Lemma 2.2, there exists a tl > to, such that (2.3) holds, for all t _> tl. Now, we consider the following two cases, (I1) g > 2 a n d
(I2) g _> 1. CASE (I1). Assume g > 2. Define
5L~_lx (t)
w (t) = ¢ (t) f (x [o (t)]) '
for t > t2,
for some t2 >__tl.
(3.6)
Then,
W ' ( t ) = ~ ( t ) 5(L~_lx(t))' +~'(t) 5L~ _ ix (t) f (x [a (t)]) f (x [o" (t)])
-- ~ (t) 5 (L~_lX (t))
(3.7)
(x [a (t)]) x' [a (t)] a' (t)
f2 (X [O"(t)])
f (x [g (t)]) + ~, (t) ~ L ~ _ : (t) _< -~ (t) q (t) : (x [~ (t)]) f (x [~ (t)]) SLY_ix (t) ___-~ (t) q (t) + ~' (t) f (~ [~ (t)]) '
for t ___t2.
(3.8)
R.P. AGARWALet al.
424
By Lemma 2.3(i2), there exists a t3 _> t2, such that t >_ a(t) > t3 and
x' [a (t)] _> ~[c~(t) ,t3] (SLn_:(t)),
for t > ta.
(3.9)
Using (3.9) in (3.S), we have
~' (t)
: (~ [~ (t)])
w' (t) < -~ (t)q (t) +
=-((t)q(t)+
{ ~' [o (t)] ~' (t) ] ~ \~-5~~t)
/
p/~(x[~(t)])]'
~[gNit3--~' (t)
for t >_t4,
for some t4 > ta.
Integrating (3.10) from t4 to t, we obtain W (t) < W (t4) -
~ (s) q (s) ds +
\ ~
[a (s), t3] a' (s)
fl/(~ (x [a (s)])
ds
(3.11) _< W (t4) -
f:t, ((s) q (s) ds +
{~
x,lo-(,,)l,,,(s) "
[a (s), tz] a' (s)
fl/~ (x [(7 (s)])
However, by the Bonnet second mean value theorem, for a fixed t >_ t4 and, for some y E [t4,t], we have
~ [O"($), t3] O" (8)
= ~
[O"(t4), t3] O" (t4)
f ' / " (X [a (S)]) (3.12)
{ ((,(_t4))1%)fx[a
[~(t4)]
:/-7--
(u)
-- M,
where M is a positive constant. Using (3.12) in (3.11), we have
fi~(~)q (s) ds _< - W (t) + W (t,) + M%
(3.13)
Letting t ---+ cc in (3.13), we arrive at a contradiction to condition (3.5). CASE (I2). Assume e > 1. Define
w (t) = ~ (t)
5L~_ix (t) f (~ [~o (t)]) '
(3.14)
for some constant A,0 < A < 1 and, for t > t2, for some t2 _> tl//~. Then,
(L~ _ : (t))' + ~, (t) 5L~ - ix (t) W' (t) = ~ (t) f (x [~ (t)]) f (x [~ (t)])
(3.1s) (t) f ' (x [ia (t)]) x' [A(7it)] A~r' (t) -~(t) fi5L~_lx F[~v(T)]) Lg_lx (t)
_< -~ (t) q (t) - ~ (t) f (x bo (t)])"
(3.16)
By applying Lemma 2.303 ) and the procedure of Case ( I J we arrive at the desired contradiction. This completes the proof. | The following result is immediate and so the proof is omitted.
Functional Differential Equations
425
THEOREM 3.2. Let condition (3.3) in Theorem 3.1 be replaced by
(' (t) _> 0
and
f~
as < oo,
w* [a (s), T] a' (s)
(3.17)
where w*[~(t),T] is as in (3.4). Then, the conclusion of Theorem 3.1 holds. Next, we present the following results when Q (t) =
THEOREM 3.3.
q (s) ds < o0.
(3.18)
1, (-1)n-e5 = - 1 , Conditions (i)-(v), (1.3), and (3.1) hold, and suppose that there exists a function a 6 Cl([t0, oo),R+), such that (3.2) holds with a'(t) >_ O, for t >_to. if, for a11 large T _> to, Let
I < ~ < n -
f
oo w* [~r (s),T]
o.!
(s) QUa (s) ds = co,
(3.19)
where w*[a(t),T] is as in (3.4), then, Af~ -- 0. PROOF. Let x E Aft and assume that x(t) > 0, for t _> to _> 0. As in the proof of Theorem 3.1, we consider the cases, (I1) g _ > 2 a n d
(I2) e > 1. CASE (I1). Suppose l _> 2. Define W(t) as in
f*i q (s)
(3.6)
with ~(t) -- 1. Then, we obtain
6LT~_: (t~)
ds < f (x [a (t2)])'
and hence, for any t > t2,
Q (t) < 5L~_lX (t) : (~ [~ (t)]) or
5Ln-lx (t)
(3.20)
Q~/~ (t) <_f~/~ (~ [~ (t)])" Using (3.9) in (3.20), we have
~[a(t),t3]cr'(t)Q1/~(s) ds <_
x'[a(t)]cr'(t) t
fl/~ (~ [~ (t)]) '
_> t3 ~ t2.
(3.21)
Integrating (3.21) from t3 to t, we obtain
f,~
f,~ f,i x'[~(s)]o'(s) dE= [~t~c~)l dv ~3~ [° (s), t3] ~' (s) Q1/~ (s) ds < ix/. (x [~ (s)]) Jxt~c~3)l11/~ (~) ~
dv
< x[~(t3)] fl/"--""-Z~ (v) <
(x),
which contradicts condition (3.19). CASE (I2). Suppose £ > 1. Define W(t) as in (3.14) with ~(t) = 1. Then, as in the above proof one can easily see that, for all t > t2
QI/~ (t) <
5L,~_lx (t)
f~/~ (~ [:~ (t)])"
By applying Lemma 2.3(i3) and proceeding as in the proof of Case (I1) above, we arrive at the desired contradiction. This completes the proof. |
426
R . P . AGARWAL st al.
THEOREM 3.4. Let 1 < ~ < n - 1, (--1)n--t~ = --1, Conditions (i)-(v), (1.3), and (3.1) and (3.18) hold. Moreover, assume that there is a function ~ E Cl([t0,oe),R+), such that condition (3.2) holds with #(t) >_O, for t >_to and let
f' (~) f(~-~)/" (z) > k > o,
~o~ ~ ¢ o.
(3.22)
If, for all large T >_ to,
f
oo (w* [a (s), T] (7! (s))
(~,* [o (~), TI ~' (~)) Q(I+~)/~
× Q (s) +
(u) d~j]
(3.23)
1/or
as =
~,
where w*[o'(t),T] is as in (3.4), and ~={
k,
wheng_>2,
c,
c is any positive constant, when ~ _> 1,
then, A~ = 9. PROOF. Let x E Aft and assume that x(t) > O, for t >_ to >_ 0. As in the proof of Theorem 3.1, we consider the two cases, (I1) g _ > 2 a n d (I2) e _> 1. CASE (I1). Assume e >__2. Define W(t) as in (3.6) with ~(t) = 1 and obtain (3.7) which takes the form, (t) f, (x [a (t)]) x' [a (t)] a' (t) ' t -> t2. (3.24) w ' (t) < - q (t) /~5L~_lx (~ [~ (t)])
Again, as in the proof of Theorem 3.1, case (I1), we obtain (3.9), for t _> t3 >__t2. Using (3.9) in (3.24), we obtain W' (t) < - q (t) - k ~ [a (t), t3] a' (t) W (l+a)/a (t),
for t > t4 _> t3.
(3.25)
Integrating (3.25) from t to v > t and letting v -~ c~, we have
5 ( L , - l x (t)) a >_ f (x [a (t)]) × Q(t)+k
~ [ a ( s ) , t a ] a ' ( s ) W cl+~)/~(s) ds ,
fort>_t4,
and hence, W (t) ___Q (t),
for t >_ t4.
(3.27)
Using (3.9) and (3.27) in (3.26), we obtain
[
x' [a (t)] a' (t) > ~ [~ (t) ta] or' (t) Q (t) + k
/1/~ (x [~ (t)]) -
N [a (s), t3] ~' (s) Q(I+~)/~ (s) ds
Integrating this inequality from t4 to t and using condition (3.1), we obtain a contradiction to condition (3.23). CASE (I2). Assume g > 1. Define W(t) as in (3.14) with ¢(t) = 1. Then,
w ' (t) < - q (t)
(t) f, (x [Aa (t)]) x' [A~ (t)] ;ka' (t) ' /25g~_lx (x [~o (t)])
t >- t2.
By applying Lemma 2.3(i3) and proceeding as in the above case, we arrive at the desired conclusion. This completes the proof. We note that condition (3.22) can be replaced by
f' (x) fO-~)/~ (x) = "r (x), where V(x) is a positive nondeereasing function, for x # 0. Now, we have the following result.
| for x ¢ 0,
(3.28)
Functional Differential Equations
427
THEOREM 3.5. Let condition (3.22) in Theorem 3.4 be replaced by (3.28). If, for all large T > to
and all positive constant -k, condition (3.23) holds, then, Af~ = 0. PROOF. Let x C Af~ and let x(t) > 0, for t _> to _> 0. As in the proof of Theorem 3.4, Case (I1), we see that inequality (3.24) takes the form,
W'(t) <_ - q ( t )
-~[(x[o'(t)])~[cr(t),t3]o-I(t)w(lq-c~)/a(t),
for t _>~t 4 __>t 3.
(3.29)
Since x(t) is increasing on It1, co) and ~/(z) is a nondecreasing function, there exist a constant m > 0 and ts _> t4, such that x [cr (t)] _ m, for t >_ tb. (3.30) Using (3.30) in (3.29), we have W ' (t) _<~- q (t) - ~/(m) w [o" (t), t3] o-! (t) W (1-['°~)/Cz (t),
for t _> tb.
The rest of the proof is similar to that of Theorem 3.4, and hence, is omitted.
|
To prove our next linearization result, we need the following lemma which is given in [4]. LEMMA 3.1. The ha/f-//near equation,
(~ (t) (~' (t))~) ' + q (t) ~ (t) = 0, where ~ is as in (v), ~(t), q(t) e C([t0, co), R+) is nonoscillato~y if and only if there e~ist a number T >_to and a function v(t) E Cl([t0, co), ]~) which satisfies the inequality
~' (t) + ~ - 1 / ~ (t)Iv (t)[ (~+")/" + q (t) _< o,
on [T, co).
THEOREM 3.6. Let 1 < ~ < n - 1, ( - 1 ) n - t 6 = - 1 , Conditions (i)-(v), (1.3), (3.1), and (3.22) hold and suppose that there exists a function a 6 Cl([t0, oo),R+), such that condition (3.2) holds. If, for all large T > to, the second-order half-Iinear equation,
(w.[~(t):~r]~,(t))~
(W(t)) ~
+
q(t)y~(t)=o,
is oscillatory, where w*[o(t),T] is as in (3.4) and ~ is as in Theorem 3.4, then, ~
(3.31) = O.
PROOF. Let x 6 Af~ and assume that x(t) > 0, for t _> to _> 0. Proceeding as in the proof of Theorem 3.4 and considering the two cases, (I1) ~ _ > 2 a n d
(I2) ~ _> I. CASE (I1). Suppose ~ _> 2. Define the function W(t) as in (3.6) with ~(t) = 1 and proceed as in the proof of Theorem 3.4, Case (I1) to obtain (3.25). Now, applying Lemma 3.1 to inequality (3.25), we conclude that the equation (3.31) is nonoscillatory, which is a contradiction. CASE (I2). Suppose ~ > 1. Define the function W(t) as in (3.14) with ~(t) = 1. The rest of the proof is similar to that of Theorem 3.4, Case (I2) and that given above. We omit the details. Similarly, we can obtain the following result. THEOREM 3.7. Let condition (3.22) in Theorem 3.6 be replaced by condition (3.28). If, for all large T _> to and all positive constant k, the equation (3.31) is oscillatory, then, A;~ = 0. REMARK 3.1. According to the above results, if the hypotheses of any of these hold, then, we can easily see the following. (1) (2) (3) (4)
For For For For
even-order equation (1.1; 1), every solution x 6 0 . odd-order equation (1.1; 1), any solution x 6 0 tAAf0. even-order equation (1.1;-1), any solution x 6 0 tAAf0 UAf~. odd-order equation (1.1;-1), any solution x 6 0 U A/',~.
R.V. AGARWALet al.
428 Afn=O
3.2.
We note that this case occurred for equation (1.1; 5) when 5 = - 1 . We shall employ either one of the following conditions on the function f. (il) f satisfies for zy > 0,
--f (--xy) >_f (xy) > f (x) f (y),
(3.32)
or, we assume that 02) inf {f (~x)/f (~) : f # 0} > 0, for any x > 0. For this purpose, we need the function defined by
f*(x) = { sgn xinf{f(~lxl)0, f (5)
:"z
>0~, if x#0,
(3.33)
i f x = 0. It is easy to see that if(x) has the following properties. f* (x) is nondecreasing on R
and
If(~ [xl)l >--If(~)llf*(x)l, We set F(x) =
x f* (x) > 0,
for ~x > 0.
f (x),
if condition (3.32) holds,
f* (x),
if condition (3.35) holds,
and suppose that
+oo
for x # 0,
(3.34) (3.35)
(3.36)
du F(ul/")
< oo.
(3.37)
For more details see [25]. Now, we present the following results. THEOREM 3.8. Let 5 = -1, Conditions (i)-(v), (1.3), and either condition (3.32) or (3.35) hold
and -
X a -
F(x)
-+ 0,
as x --* oo,
(3.38)
where F as in (3.36). If lim sup t -~ o0
f
p(t) q(s)f(w~[g(s),p(t)]) ds >O,
(3.39)
J t
then, A/'. = O. PROOF. Assume that x C Afn and let x(t) > 0, for t _> to > 0. Then, there exists a tl > to, such that L j x (t) > O, 0 < i < n, on [tl,oO). (3.40) From (2.17) with t and s are replaced by g(s) and p(t), respectively, we have
x[g(s)]>_wn[g(s),p(t)]L~_lx[p(t)],
fortl <_t < s < p ( t ) .
(3.41)
Using (3.41) in equation (1.1;-1), we have d
ds (L~-lx (s)) = q (s) f (x [g (s)]) _> q (s) f (w,~ [g (s), p (t)] L,~-xx [p (t)]) > q (s) f (wn [g (s), p (t)]) F (L~_lx [p (t)]), fort1 ~ t < s < p ( t ) .
(3.42)
Functional Differential Equations
429
Integrating both sides of (3.42) from t to p(t), we obtain
L ' ~ _ l x [ p ( t ) ] - L ~ _ x x ( t ) >_
f
p(t)
q(s) f(w~[g(s),p(t)])F(Ln_lx[p(t)])
ds
,It
or
L~_lx [p (t)] > fp(t) q (s) f (Wn [g (S), p (t)]) ds. F (L~_lx [p(t)]) - j,
(3.43)
Taking lim sup of both sides of inequality (3.43) as t --* co and by (3.38), w e obtain a contradiction
to condition (3.39). This completes the proof.
|
THEOREM 3.9. Let 5 = -1, Conditions (i)-(v), (1.3), end either condition (3.32) or (3.35) and
(3.37) hold. A q (s) f (wn [g (s), s]) ds = OC,
(3.44)
g
then, N n = 0. PROOF. Let x e Af,, and assume that x(t) > 0, for t > to _> 0. From (2.17) w i t h t and s be replaced by g(t) and t, respectively, we have
x [g (t)] >_ w, [g (t), t] n~_ix (t),
for t • Ag n [tl, co),
(3.45)
where tl is as in the proof of Theorem 3.8. Set u(t) = LX_~x(t). Then, u(t) satisfies
du d---[= L~x (t) = q (t) f (x [g (t)])
_> q (t) I (w~ [g (t), t] L~_lZ (t)) > q (t) I [9 (t), t]) F (L _lx (t)) =q(t) f(wn[g(t),t])F(ul/~(t)),
fortEAgN[tl,co).
Dividing the above inequality by F(u~/a(t)) and integrating on Ag n arbitrary, we obtain
gn[~;1)T1]
q (s) f (w~ [g (s), s]) ds < -
-
F (,~I/G~ (S))
[t~,Tl],
ds = --
J.(tl)
where TI(_> tl) is
du F (V/~3t) "
Letting T1 ---* ec, we find gfl[tl,CO)
q (s) f (w~ [g (s), s]) ds < --
du
(tl) F (U l / a ) < CO.
This contradicts condition (3.44). This completes the proof.
I
THEOREM 3.10. Let 5 = -1, g(t) > t, for t >_to, Conditions (i)-(v), and (1.3) hold and X c~
f (x)
- * 0,
as x --* co.
(3.46)
If In-2 (g(t),u;al,... ,an-2)an-1 (u)
lira sup t ......+ Oo
q(s) ds
du > O,
J t
then, Afn = 0. PROOF. Let z E Af~ and assume that x(t) > 0, for t _> to _> 0. Then, there exists a tt >_to, such that (3.4o) holds, for t _> tl. Let T1 > tx be, such that inf{g(t) : t _> T1} _> tl. From (2.19) (see the proof of Lemma 2.5) with t and s replaced by g(t) and t, respectively, we have z [g (t)] >_
f
Jt
g(t) I , - 2 (g (t), u; a l , . . . , an-2) an-1 (u) L,_lX (u) du,
for t _> T1.
(3.47)
430
R . P . AGAI~WALet al.
Integrating equation (1.1;-1) from t to u, we obtain
L~_~ (u) - i~_~x (t) = or
L,~-lX (u) k
(I q (s) f (x [g
(s)])
q (~) f (x [g (~)l) d~
ds
,
(3.48)
for u > t > T1.
Using (3.48) in (3.47), one can easily see that x [g (t)]
I ~ l <" (z
[g (t)])
In-2
> -
(g (t), U;al,..., am-z) an-1 (u)
q (s) ds
ds.
.,~
The rest of the proof is similar to that of Theorem 3.8, and hence, is omitted.
I
In the following result, condition (3.32) or (3.35) is disregarded. THEOREM 3.11. Let 5 = - 1 ,
/A
g
Conditions (i)-(v), (1.3), f f (~) u
and
(3.1) hold. If
(3.49)
l/<~
×an_l(u)(/s q(r) dr)
duds=-cc,
then, dkfn -~ O. PROOF. Let x EAf, and suppose that x(t) > 0, for t > to _> 0. Then, there exists a tl _> to, such that (3.40) holds, for t _> tl. From (2.20), (see the proof of Lemma 2.5) with t and s replaced by g(t) and t, respectively, we get
f
x' [g (t)] > al [g (t)] ~t
g(t) 1~_3 (g (t), u; a 2 , . . . , a~-2)
×a~-l(u)n~_lx(u)du,
(3.50)
teAgA[tl,oc).
As in the proof of Theorem 3.10, we obtain (3.48). Using (3.48) in (3.50), one can easily see that
x' [g (t)] g' (t) > ~ [~ (t)] g' (t)
fl/~ (x [g (t)]) ~,"(') f I~-s (g (t), u; 32,... ,a~-2)a~-1 (u)
1/~
q (r) dr)
(3.51)
du.
Integrating (3.51) on A 9 N [tl,T1], where T1 ~ (tl) is arbitrary, we obtain
,n%,T~lal[g(s)lg'(s)S"(~)I,~-a(g(s),u;a2,...,a~-2)a,~-l(u)
q(rldr)l/"duds dx
/ ~[a(T~)l J~[g%)l
(x)
Letting T1 ~ co, we find
,nc<,,~) 31 [g (s)] g' (s)/g(') s~_3 (g (~),~; a~,..., ~_~)a~_1 (~)
(/; q(~) dr) 1/, ~ dx <- L fl/~ < oo. ~
[g(t~)]
This contradicts condition (3.49) and completes the proof.
(x)
I
Functional Differential Equations
431
REMARK 3.2.
By combining the hypotheses of any of the results of Section 3.1 for the equation (1.1;-1) with the appropriate ones of Section 3.2 for equation (1.1;-1), one can easily see the following. (1) For even-order equation (1.1;-1), any solution x E (_9 UAf0. (2) For odd-order equation (1.1;-1), every solution x E O. We can also conclude the following. (3) For odd-order equation (1.1;1) every unbounded solution x E O. (4) For even-order equation (1.1;-1) every unbounded solution x E O.
4.
OSCILLATION
CRITERIA
II
In this section, we shall give some oscillation criteria for equation (1.1; 5) when the function f satisfies f (x) sgn x _> Ix[ s , for x # 0. (4.1) Regarding the subsets of AY for equation (1.1; 5), we shall distinguish the following three separate cases,
(il) £ E { 1 , 2 , . . . , n - - 1}, (i2) / = 0, and (ia) f = n. 4.1. Ale = 0, £ E { 1 , 2 , . . . , n -
1}
In this section, we shall present some sufficient conditions which ensure that AYe = ~. For this, we shall need the following lemma due to [19]. LEMMA 4.1. I f X and Y are nonnegative numbers, then, X x - A X Y ~-1 + ( A - 1)Y~' k O,
A > 1,
X ~ - A X Y ~-1 - (1 - A) Y~ _< 0,
0 < A < 1.
and
In the above inequafities, the equality holds if and only if X = Y .
Now, we prove the following result. THEOREM 4.1. Let 1 _< e _< n - 1, (-1)n-e5 = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold. If there exist two functions a(t),[(t) E Cl([t0, oo),~[+), such that (3.2)holds and, for all large T >_ to,
lim sup
(~., (~))~+1 ~ (s) q (s) - 0
t---~O0
(e (s) ~, (s) ~. [~ (s), T])"
ds = co,
(4.2)
where w*[a(t),T] is as in (3.4) and
0 =
(a + 1) -(a+l) , A-~ ( a + 1)-(~+I),
if ~ _>2, O
ifg_>I,
then, AY~ = 0. PROOF. Let x E AYe and assume that x(t) > 0, for t > to _> 0. By Lemma 2.2, there exists a tl k to, such that (2.3) holds, for all t >_ tl. Now, we consider the following two cases,
(I1) t > z and (I2) t _> 1.
R.P. AGARWAL et at.
432 CASE (I1). Assume g > 2. Define
5Lg_lx (t)
w (t) = ~ (t) ~ [~ (t)] '
for t ___t~ _> t~.
(4.3)
Then, for t > t2, W' (t) : - ~ (t) q (t) f (x [g (t)]) + ~, (t) 5Lg-lx (t) ~- [o (t)] ~ [~ (t)] (4.4)
~L~_~x (t)~o_~ [~ (t)] x' [~ (t)] o' (t) [~ (t)]
- ~ (t) ~
< -~ (t) q (t) + ~' (t) W ('" - aa' (t) ~ (t) 5L~-lx (t) x' [a (t)]. (4.5) ~ (t) '~; ~+~ [~ (t)] By Lemma 2.3 (i2), there exists a t3 _> t2, such that t 2 a(t) >_t2 and (3.9) holds. Using (3.9) in (4.5), we have W' (t) < - ~ (t) q (t) q- ~' (t) W (t) -~ (t) --SO d (t) W [0" (t), t2] ~--l/a (t) W (a+l)/a (t),
(4.6)
for t >_ t3.
Fix t > t3 and set x = ( ~ [o (t), t21 ~' (t)) "/(~+~) ~-~/(~+~) (t) w (t)
,
~
= - -~> + I,1
and
y = ( a.~)a [~'~ (t) (t) ~/:W(~+l) (t) (a~[a(t),t2]a, Now by Lemma 4.1, for t _> t3, we obtain
~' (t) w (t) - ~ [~ (t) , t~] o' (t) w (~+~)/" (t) (t)
(
1 )a+l [
_< ~
(~,(t)~a+l ]-a ~ (t) \ ~ (t) ] ~ [a (t), t21 a' (t)
Thus, (4.6) reduces to
1 (~' (t))~+l (Or q- 1) a+l [~ ( t) w [ff (t),t2] 04 it)] a '
w ' (t) < - ~ (t) q (t) +
for t _> t3.
Integrating the above inequality from t3 to t, we have
0 < w it) _< w (t2) -
( i s ) q is) - 0 [((s) ~' (s) ~ [o (s), t2]] ~
Taking lim sup of both sides of this inequality as t - , c~, we obtain a contradiction to condition (4.2). CASE (I2). Assume g _> 1. Define
6L~_lx (t)
(4.7)
w (t) = ~ (t) x- [ ~ (t)]' for some constant )~, 0 < A < 1 and, for t _> t2, for some t2 _> tl/A. Then,
(t) w (t~ W'(t) _< - ~ ( t ) q(t) + ~' ~(t) 'J - ~'
5L~ ix (t)
(t) ~ (t) x ~+1 - [)~a (t)] z ' [ ~ (t)].
(4.s)
Next, by applying Lemma 2.303 ) and the procedure of Case (I1), we arrive at the desired contradiction. This completes the proof. | For each t > to, we let gi t) < t and define #(t) = sup{s >_ to : g(s) < t}. Clearly, #(t) >__t and g o/z(t) = t. Now, we give the following result.
Functional Differential Equations
433
THEOREM 4.2. Let 1 < ~ < n - 1, (-1)n-~5 = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold. If,
for all large T > to, limsupw~ [t,T]
q(s) ds > 1,
t-~
(4.9)
(t)
then, Af~ = O. PROOF. Let x E All and suppose that x(t) > 0, for t > to >_ 0. By Lemma 2.2, there exists a tl _> to, such that (2.3) holds, for t > tl. Integrating equation (1.1; 5) from t _> tl to u > t and letting u --* o% we obtain
5 ( L n - l x (t)) a ~
q (s) x a [g (s)] ds,
t > tl.
(4.10)
By Lemma 2.3 (il), there exists a t2 _> tl, such that
x (t) >_ wt It, t2] (hL~_lx (t)),
for t ~_ t2.
(4.11)
Thus,
x ~ (t) > w 7 [t, t2] (hLn-lx (t)) ~ >_ w~ [t, t2]
(S
q (s) x ~ [g (s)] ds
)
,
t > t2.
Now, by #(t) > t and the fact that x'(t) > 0 and g(s) > t, for s > #(t), it follows that q (s) x ~ [g (s)] ds
x ~ (t) > w~ [t, t2]
> w~ [t, t2] x ~ (t)
(t)
q (s) ds.
(4.12)
(t)
Dividing both sides of (4.12) by x~(t), we have
w~ [t, T]
(t)
q (s) ds < 1,
for t > t2.
(4.13)
Taking limsup of both sides of (4.13) as t ~ ~ , we o b t a i n a contradiction to condition (4.9). This completes the proof. | We note that inequality (2.4) may take the following form,
x (t) >_ w It, T] (hL~_lz (t)),
for all large T _> to and t _> T.
(4.14)
Clearly, condition (4.9) becomes l i m s u p w " [t,T
t--,oo
lff
(t)
q(s) ds > 1.
(4.15)
In the case of advanced equation (1.1; 5), i.e., g(t) _> t, for t _> to, Theorem 4.2 takes the following form. THEOREM 4.3. Let 1 < g < n - 1, (-1)'~-~5 ---- - 1 , g(t) > t, for t > to, Conditions (i)-(v), (1.3), and (4.1) hold. If, for all large T > to and t > T, lira sup w~ It, T]
q (s) ds > 1,
t---4OO
then, ~ = 0. Next, we present the following theorem when condition (3.18) holds.
(4.16)
434
R.P.
AGARWAL et al.
THEOREM 4.4. Let 1 < e < n - 1, ( - 1 ) ~ - t 5 = - 1 , Conditions (i)-(v), (1.3), (3.18), and (4.1) hold. Moreover, assume that there is a function a C Cl([to, oo), JR+), such that condition (3.2) holds with a'(t) >__O, for t >_ to. If, for all large T >_ to,
w* [~ (s), T] ~' (s) Q(~+I)/~ (s) ds
lim sup w [a (t), T] Q (s) + k t---+OO
> 1,
(4.17)
where w*[a(t),T] is as in (3.4) and
k = ~ a,
when g >_ 2,
[ aA, AE(O, 1), when~>_l, then,
= O.
PROOF. Let x E Aft and assume that x(t) > 0, for t > to >_ 0. As in the proof of Theorem 4.1, there exists a tl _> to, such that (2.3) holds, for t >_ tl and the two cases, (I1) ~ > _ 2 a n d
(h) e _ 1, are considered. CASE (I1). Assume ~ _> 2. Define W(t) as in (4.3) with ~(t) = 1 and as in the proof of Theorem 4.1, we obtain (4.6) which takes the form, W' (t) ~ - q (t) - c~#' (t) ~ [a (t), t2] W (a+1)/a (t),
(4.18)
for t > ta.
Integrating (4.18) from t to u k t and letting u --~ co, we find
5(L---n:-~x(--t)'~ > Q ( t ) + a
\
[o (t)])
~[a(s),t2]a'(s)W(~+l)/~(s)
ds
-
t>t2. '
(4.19)
-
Now, one can easily see that W (t) >__Q (t),
for t >_ t3.
(4.20)
Using (4.14) with T = t2, (4.20) in (4.19), we obtain
1 >w[cr(t),t2]
[
Q(t)+a
~[~(s),t2]a'(s)Q(~+l)/'~(s)
ds
],-
Taking lira sup of both sides of the above inequality as t --* co, we obtain a contradiction to condition (4.17). CASE (I2). Assume g _> 1. Define W(t) as in (4.7). Then, it is easy to see that
W' (t) <_ - q (t) - Aa~a [~ (t), t2] W (~+1)/~ (t),
for t > ta.
The rest of the proof is similar to that of Case (11) above, and hence, is omitted.
(4.21) |
Next, we have the following comparison results. THEOREM 4.5. Let 1 < g < n - 1, (-1)n-e(~ = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold and assume that there exists a function a E Cl([t0, co),]R+), such that condition (3.2) holds
with a'(t) >_ O, for t >_ to. If, for all large T >_ to with a(t) > T, every solution of the retarded first-order equation, y' (t) + q (t) wg [a (t), T l y [a (t)] = 0,
is oscillatory, then, Af~ = O.
(4.22)
Functional Differential Equations
435
PROOF. Let x e Afe and suppose that x(t) > 0, for t > to _> 0. There e x i s t s a t l _> to, such that (2.3) holds, for t > tl. By Lemma 2.3(il), there exist t3,t2 >_tl, such that a(t) > t2, for all t > t3, and x [a (t)] > we [a (t), t2l (SLy_ix [a (t)]),
for t > t3.
(4.23)
Using condition (4.1) and (4.23) in equation (1.1; 6), we obtain
_ s ddt (L~_lx(t))~=
d (SLn-lx (t) ) = q (t) f (x [g (t)]) > q (t) x ~ [a (t)] dt
> q (t)
(t), t:]
(t)]F,
for t _> t3.
Set y(t) = (SLn_lX(t)) ~ > O, t > t3, we get y' (t) + q (t) w~ [a (t), t2] y [a (t)] _< 0,
for t _> t3.
(4.24)
Integrating (4.24) from t > t3 to u and letting u --* c~, we have y (t) _>
q (s) w~ [a (s), ta] y [a (s)] ds,
t >>_ta.
As in [23], it is easy to conclude that there exists a positive solution y(t) of the equation (4.22) with l i m t - ~ y(t) = 0, which contradicts the fact that equation (4.22) is oscillatory. This completes the proof. 1 THEOREM 4.6. Let 1 < £ < n - 1, (-1)n-e5 = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold and assume that there exists a function a 6 Cl([t0, cc),N+), such that condition (3.2) holms. If, for 411 large T >_ to, the second-order equation,
(w*[a(t) i'T]cr'(t))" (y'(t))"
+
q(t)y"(t) = 0 ,
(4.25)
is oscillatory, where w*[a(t),T] is as in (3.4) and k is as in Theorem 4.4, then, Aft = O. PROOF. Let x 6 Are and assume that x(t) > 0, for t > to _> 0. As in the proof of Theorem 4.4, we consider the two cases, (I1) g _ > 2 a n d (I2) e > 1. CASE (I1). Let g >_ 2. Proceeding as in the proof of Theorem 4.4, Case (I1), we obtain inequality (4.18). By applying Lemma 3.1 to inequality (4.18), we conclude that the equation (4.25) is nonoscillatory which is a contradiction. CASE (I2). Let ~ >_ 1. Proceeding as in Theorem 4.4, Case (I2), to obtain inequality (4.21) and applying Lemma 3.1 to (4.21) to obtain the desired conclusion. This completes the proof. 1
4.2. X 0 = 0 This case occurs for equation (1.1; 5) when (1) 5 : 1 and n is odd, and (2) 5 : - 1 and n is even. We shall present some sufficient conditions which ensure that the subset Af0 for equation (1.1; 5) is empty.
436
R. P. AGARWAL et aL
THEOREM 4.7. Let (-1)n5 = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold. If lim sup f t
q (s) w~ [T (t), g (S)] ds > 1,
(4.26)
t-.-*oo J r ( t )
then, Ho = O. PROOF. Let x E No and assume that x(t) > 0, for t > to _> 0. By Lemma 2.2, there exists a t l _> to, such that ( - 1 ) i L~x (t) > 0 (0 < i < n), on [tl, oo). (4.27) Replacing t and s by T(t) and g(s), respectively, in (2.14), we have
x[g(s)]>_Wo[T(t),g(s)l((--1)~-lL~_lx[T(t)]),
fort>s>tl.
(4.28)
Using condition (4.1) and (4.28) in equation (1.1; 5), we find
d ((-1)'~-l L,~_lx(s)) ~ ds -- q (s) f (x [g (s)]) > q (s) x ~ [g (s)]
(4.29)
(
>_ q (s) w~' [r (t), g (s)l (-1) ~-1 g n - l z [r (t)l
,
for t > s > tl.
Integrating both sides of (4.29) from r(t) to t, we obtain
> ((-1) ~-1Ln_lX[r(t)])
o[;
or
((-1)'~-~L,~_~x[r(t)])
(t)
(t)
q(slw~[,(t),g(s)] ds,
]
q(s)w~[r(t),g(s)lds-1 <_0,
for t_> tl.
However, this is inconsistent with (4.26). This completes the proof.
|
REMARK 4.1. If g(t) < t, for t _> to, condition (4.26) becomes lim sup f t q (s) w~ [g (t), g (s)] ds > 1. t-~oo Jg(t)
(4.30)
THEOREM 4.8. Let (-1)n5 = - 1 , g(t) < t, for t >_ to, Conditions (i)-(v), (1.3), and (4.1) hold.
If t~oo Jg(t)
I,~-2(u,g(t);a,,_2,...,al)an_l
(u)
(L
q(~) d~-
du > 1,
(4.31)
then, JV'o = O. PROOF. Let x E JV'o and assume that x(t) > 0, for t :> to _> 0. There exists a t l >_ to, such that (4.27) holds, for all t > t~. Choose a t2 _> tl, such that inf{g(t) : t > t2} > t~. Replacing i, k, t, and s by 0, n - 1, g(t), and t, respectively, in (2.1), we can easily see that x [g (t)l __
((-1)
(t)
I n _ 2 (u, g (t) ; a,~-2, • • •, a~) a,~_~
(~) (4.32)
(u)) du,
for t > g(t) > t2.
Functional Differential Equations
437
Integrating equation (1.1; 5) from u to t, we get ((--1) n - 1 L , ~ _ l x ( u ) ) a >
((-1) n-1Ln_lx(t))a+
~utq(~-)f (x [g (7-)])d'r,
for t > u > t2,
or
((-1)n-iLk-ix(u))>
( ~ t q ( z ) f (x[g('r)]) dT) ~/~
(4.33)
Using (4.1) and (4.33) in (4.32), we obtain x [g (t)] > x [g (t)]
~gt (t)
In-2 (u, g (t) ; a ~ - 2 , . . . , al) a~-i (u)
(~uutq (T) dT)l/c~du,
or t
1>_ ~
t
(t)
In_2(u,g(t);an_2,...,al)an_l
1/a
(U) ( ~ q(T) d~')
du.
Taking limsup of both sides of the above inequality as t ~ ~ , we obtain a contradiction to condition (4.30). This completes the proof. 1 Next, we present the following comparison result. THEOREM 4.9. Let (-1)n5 = - 1 , g(t) < t, for t > to, Conditions (i)-(v), (1.3), and (4.1) hold and assume that there exists a function ~ E C([t0, c¢), R), such that ~'(t) > 0 and g(t) <<_~(t) <_t, for all t >>to. If the first-order delay equation,
y'(t)+q(t)w~ [y(t),g(t)]y[~(t)] = 0 ,
(4.34)
is oscillatory, then, No = 0. PROOF. Let x e No and assume that x(t) > 0, for t > to > 0. There exists a t 1 > to, such that (4.27) holds, for all t > tl. Choose a t2 > tl, such that inf{g(t) : t > t2} > tl. Next, replacing t and s by ~(t) and g(t), respectively, in (2.14), we have
x[g(t)]>_wo[~l(t),g(t)]((-1)n-lL,~-lx[~l(t)]),
fort>71(t)>g(t)>t2.
(4.35)
Using (4.1) and (4.35) in equation (1.1; 5), we have -d-t
(-1)
L~_lx(t)
= q ( t ) f ( x [ g ( t ) ] ) > q ( t ) x a[g(t)] > q(t)w~[~l(t),g(t)] ( ( - 1 ) n - 1 L n _ l x [ ~ ( t ) ] ) ~ , for t > ~/(t) > g (t) > t2.
Set u(t) =- ((-1)n-lL~_lx(t)) ~, t > t2, we have
du(t) - ~ > q(t)w~[~(t),g(t)]u[~(t)], dt
for t > v (t) > g (t) > t2.
The rest of the proof is similar to that of Theorem 4.5 and hence, is omitted. 4.3. A f t = 0 This is the case for equation (1.1; 5) when 5 -- - 1 . In this section, we shall present some sufficient conditions which ensure that the subset flfn for equation (1, 1; - 1 ) is empty.
R. P. AGARWALe t
438
al.
THEOREM 4.10. Let (i = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold. If limsup t --~ oo
[ p(t) q(s)w~ [g(s),p(s)] ds > 1,
(4.36)
dt
then, Afn = ~. PROOF. Let x E Af,, and let x(t) > 0, for t >_ to _> 0. There exists a tl >_ to, such that (3.40) holds, for t > tl. As in the proof of Theorem 3.8, we obtain (3.41). Using (3.41) and (4.1) in equation (1.1;-1), we have ,4 -~- (L~ 1x (8)) = q ($) f (X [g (8)]) > q ( S ) X a [g (S)] ds > q (s) wna [g (s), p (t)] L~_lX [p (t)],
(4.37) for t~ < t < s < p (t).
Integrating both sides of the inequality (4.37) from t to p(t), we obtain
g~,_lZ [0 (t)] _
[ p(t) q(s)w~ [g (s), ;(t)] ds. LT~_lZ[p (t)], Jt
or
;(t) q (s) w~ [g (s), p (s)] ds.
1 _> dt
Taking limsup of both sides of this inequality as t --* c~, we have a contradiction to condition (4.36). This completes the proof. | THEOREM 4.11. Let 5 -- - 1 , g(t) >__t, for t >_ to, Conditions (i)-(v), (1.3), and (4.1) hold. If limsup fg(t ) In-2 (g (t), u; a l , . . . , an-z) an-t (u) ( j u q (T) dr ) 1 / a du > 1, t - ~ oo
(4.38)
Jt
then, Afn = ~. PROOF. Let x cAfn and suppose that x(t) > O, for t > to ~ 0. There exists a t 1 >__to, such that (3.40) holds, for t _> tl. As in the proof of Theorem 3.10, we obtain the inequalities (3.47) and (3.48), for u > t > T1 >_ tl. Now, using (4.1) and (3.48) in (3.47), we can easily see that 1 _> limsup [ g ( O I n _ 2 ( g ( t ) , u ; a l , . . . , a n _ 2 ) a n - 1 t--* c~
J t
(u)
q(T)
tiT) 1/a
dB,
•.
which contradicts condition (4.38) and completes the proof.
|
Finally, we present the following comparison result. THEOREM 4.12. Let 5 = - 1 , g(t) ~ t, fort > to, Conditions (i)-(v), (1.3), and (4.1) and assume that there exists a function ~ e C([to, c~), •), such that ~'(t) >_ 0 and g(t) >_ ~(t) >_t, for t >_ to. If the first-order advanced equation,
y'(t)
=0,
(4.39)
is oscillatory, then, flfn = ~. PROOF. Let x C Af~ and assume that x(t) > 0, for t > to >_ 0. There exists a tl _> to, such that (3.40) holds, for t >__tl. Choose a t2 _> tl, such that inf{g(t) : t ~ t2} > tl. From (2.14) with t and s be replaced by g(t) and ~(t) respectively, we have x [g (t)] _> w,~ [g (t), ~ (t)] L n - l x [~ (t)],
for g (t) _> ~ (t) >__t >__t2.
(4.40)
Functional Differential Equations
439
Using (4.1) and (4.40) in equation (1.1;-1), we get d
ot
~ L ~ _ l x (t) = q (t) f (x [g (t)]) >_ q (t) x ~ [g (t)] :> q (t)wna [g (t),~ (t)] Lan_l x [~ (t)],
(4.41)
for g (t) >_ ~ (t) >_ t __ t2.
Set u(t) = L~_lx(t), t >_ t2, we have
(t)
- dt
> q (t) w~ [g (t) ~ (t)] u [~ (t)], '
for t > t2. -
However, in view of a similar result to Corollary 3.2.3 in [31], equation (4.39) has an eventually positive solution, a contradiction. This completes the proof. | REMARK 4.1. It is known that equation (4.34) is oscillatory if lim inf fv t q (s) w~ [7 (s), g (s)] ds > -1, t-*co (t) e
(4.42)
while equation (4.39) is oscillatory if
liminf
/
t .--.*o o
,It
((t)
1
q(s)wa~ [g(s),~(s)] ds > - .
(4.43)
e
REMARK 4.2. It is easy to see that appropriate combinations of the results of Sections 4.1-4.3 will imply the oscillation of equation (1.1; 5). In fact, one can easily see the following. (1) Any of the results of Section 4.1 with l • {1, 3 , . . . , n - 1} implies the oscillation of evenorder equation (1.1;1). (2) Any of the results of Section 4.1 with g • {2, 4 , . . . , n - 1} together with appropriate results of Section 4.2 implies the oscillation of odd-order equation (1.1;1). (3) Any of the results of Section 4.1 with g • {2, 4 , . . . , n - 2} together with suitable results from Sections 4.2 and 4.3 implies the oscillation of even-order equation (1.1;-1). (4) Any of the results of Section 4.1 with ~ • {1, 3 , . . . , n - 2} together with a suitable result of Section 4.3 implies the oscillation of odd-order equation (1.1;-1).
5. O S C I L L A T I O N
CRITERIA
III
In this section, we shall present some criteria for the oscillation of equation (1.1; 5) when the function f satisfies either condition (3.32), or (3.35) and f ~ o F du
<
where F is as in (3.36). According to the subsets of Af, for equation (1.1; 5), we shall consider the two cases, (il) g • { 1 , 2 , . . . , n 02) g -= 0. 5.1. ~
1} and
= ~,~ C { 1 , 2 , . . . , n -
1}
Here, we shall provide some sufficient conditions which ensure that Af~ = 0.
(s.1)
440
R. P. AGARWALeg al.
Conditions (i)-(v), (1.3), either (3.32) or (3.35) and (5.1) hold. If, for a11 large T >_to with g(t) >_T,
THEOREM 5.1. Let 1 < ~ _< n - 1, (-1)n-~6 = - 1 ,
i
¢~ q (s) f (wt [g (s), T]) ds = c~,
(5.2)
then, Hi = O. PROOF. Let x 6 Aft and assume that x(t) > 0, for t >_ to _> 0. There exists a t l that (2.3) holds, for t >_ tl. By Lemma 2.3 01), there exists a t2 _ tl, such that x [g (t)] >_we [g (t), tl] (6L,_tx (t)),
> to, such
for t >_ t2.
(5.3)
Using (3.32) or (3.35) and (5.3) in equation (1.1; 5), we have d
dt (6L~_lx (t) ) ~ = q (t) f (x [g (t)]) >_ q (t) f (w~ [g (t), tl] (6L~_lx (t))) > q (t) f (w~ [g (t), tl]) E ( ~ n . _ l x (t)),
for t > t~.
Substituting u(t), for (6L=_lx(t)) ~, t > t2, we have i du ~~ > q ( t ) f ( w l [ g ( t ) , t l l ) F ( u l / " ( t ) )
dt
-
for t > t2.
'
-
(5.4)
Dividing both sides of (5.4) by F(ul/a(t)) and integrating from t2 to t, we have q (s) f (w~ [g (s), tl]) ds <_
F (uX/~ (s))
J~(t)
f (-~T/~)"
Letting t --~ oc, we conclude that
/;
q(s) f (w~ [g(s),tl]) ds < 1 ~(~2) -- J0
du
F ( u l / ° 0 < (x),
which contradicts condition (5.1). This completes the proof. REMARK 5.1. By (3.35) with ~ = 1,
/0 s (x~io~
|
f(x) > f(1)F(x), for x > 0, it follows that
-<
F (~,i<,)'
Thus, (5.1) yields
S
for m > o.
du
-t-~
f (~1~) < o~.
THEOREM 5.2. Let 1 < ~ < n - 1, (-1)n-~6 = - 1 ,
Moreover, assume that there is a function a with a'(t) >_O, for t >_to and assume that
E
(5.5)
Conditions (i)-(v), (1.3), and (3.18) hold. such that condition (3.2) holds
Cl([to, oo), ]~+),
f (x) sgn x _> Ixl ~ ,
for ~ # o,
(5.6)
where/~ < a is a quotient of positive odd integers. If, for every constant c > O, and all large T >_to,
lim~_~supQ1/~ (t) w [a (t), T] 1 + ~ c
w* [a (s), T] a' (s) Q(l+~)ln (s) ds 11~ = oc,
(5.7)
where Q(t) > 0, fo~ t > to, ~* [Ht), T] is as i . (3A), then, ~ = ~. PROOF. Let x EAft and assume that x(t) > 0, for t > to >_ 0. There exists a t 1 _> to, such that (2.3) holds, for t _> tl. Now, there are two cases to consider, (I1) ~ _ > 2 a n d (I~) ~ > 1.
Functional Differential Equations
441
CASE (I~). Assume ~ k 2. Define
w (t) =
5L~_~x (t)
t>_tl.
• , [~ (t)] '
Then, for t > t2, we have
5L~_lx (t) , W' (t) <_ - q (t) - fla' (t) - ~ - ~ (-~)].x [a (t)],
for t k to.
As in the proof of Theorem 3.1, we obtain (3.9), for t >__t3 _> t2. Thus,
w ' (t) <_ - q (t) - ~-~ [~ (t) , t~] ,~' ( t) w ( ~+~)/"
(t) ~(~-")/~
[~ (t)],
for t k t4 k t3.
(5.8)
Integrating (5.8) from t k t4 to u and letting u -~ oo, we find
L'~_lX (t) ~ x ~ [(7 (t)] x Q (t) + ~
[a (s)]
~ [c~ (s), ta] ~' (s) w (~+~)/~ (s) x (~-~)/~
ds ,
for t >__t4,
(5.9)
and hence, W(t) >_ Q(t), for t _> t4. There exist a constant el > 0 and a ts _> t4, such that
5L,_lx (t) < el,
(5.10)
for t > ts.
For t _> ts, it follows from (5.9) and (5.10) that
x B/c~ [a (t)] < elQ -1/~ (t)
or
x [Cr(t)] _< Cl/~Q -1/~ (t),
and hence,
x (~-~)/~ [a (t)] __kc~-~)/~Q (~-~)/~ (t),
for t >__ts.
(5.11)
Using (5.11) in (5.9) yields
aLn_lX (a-~>/a [a (t)] ~ x [a (t)] [
× Q (t) + ~c~~-'~)/f~
/too
~ [a (s), t3] a' (s) Q(~+I)/~ (s) ds
]1/,
(5.12)
]~/,~
(5.13)
Using (4.14) with T = t3 in (5.12), we have • (--~)/- [~ (t)] _> ~ [o (t),
[
× Q (t) + ~c~ ~-~)/~ Again, using (5.11) in (5.13) yields
f~
t3]
~ [~ (s), t~] o' (s) Q(~+I)/~ (s) ds
[ ]~C~fl_,:x)/fl/o °
]l/a
Taking limsup of both sides of this inequality as t ~ c~, we obtain a contradiction to condition (5.7). CASE (I2). Assume g > 1. Define
(t) w (t) = 5L~_lx ~ [ ~ (t)]'
for t > t2
and proceed as in Case (11) we arrive at the desired conclusion.
|
442
R . P . AGARWAL et
al.
THEOREM 5.3. Let i < g < n - 1, ( - 1 ) " - t 5 = - 1 , Conditions (i)-(v), (1.3), and (5.6) hold. In addition, suppose that there exists a function (7 E Cl([t0, oo),]~+), such that condition (3.2) holds. If, for every constant c > 0 and all large T :> to, the second-order half-linear equation,
(w* [(7 (t),T] a' (t)) ~ (y' (t))"
+ cq (t) y" (t) = O,
(5.14)
is oscillatory, where w*[a(t), T] is as in (3.4) and
P=
I,
if t _> 2,
A E (0, i),
ifi>
i,
then, Af~ = 0. PROOF. Let x 6 Aft and assume that x(t) > 0, for t >_ to >_ 0. As in the proof of Theorem 5.2, we consider the two cases, (I1) ~_>2 and
t > 1. CASE (I~). Assume ~ _> 2. Proceeding as in the proof of Theorem 5.2, Case (I~), and obtain (5.8), for t > t 4. It is easy to see that there exist a t5 _> t4 and a constant cl > 0, such that
X [(7 (t)] _< ClW___0[(7 (t), t4],
(5.15)
for all t _> t5.
Using (5.15) in (5.8), we have
w ' (t) _< - q (t)
-pcl (~-~)1~ (~-~)1~ [a (t) , t4] ~ [a (t) , t4] (7' (t) W (1+~)1~ (t),
(5.16) for t _> ts.
By applying Lemma 3.1 to inequality (5.16), we conclude that the equation (5.14) is nonoscillatory, which is a contradiction. CASE (I2). Assume t >_ 1. Define W(t) as in the proof of Theorem 5.2, Case (I2) and proceed as in the proof of Case (I1), we arrive at the desired conclusion. This completes the proof. | 5.2. 2 5 0 = 0 This case occurs for equation (1.1; 5) when (1) 5 = 1 and n is odd, and (2) 5 = - 1 and n is even. We shall provide some sufficient conditions which ensure that the subset 25o for equation (1.1; 5) is empty. THEOREM 5.4. Let (-1)~5 -- - 1 , Conditions (i)-(v), (1.3), and either (3.32) or (3.35) hold and let ~c~ - -
F
If limsup t-~¢¢
~
--+ 0 ,
as u ~
oo.
(5.17)
t (t)
q(s) f ( w o [T(t) ,g (S)]) ds > O,
(5.18)
then, Afo = O. PROOF. Let x E 25o and assume that x(t) > 0, for t >_ to >_ 0. There exists a t 1 >_ to, such that (4.27) holds, for all t _> tl. Replacing t and s by ~'(t) and g(s), respectively, in (2.14), we obtain (4.28), for t > s > tl.
Functional Differential Equations
Using condition (3.32) or
(3.35)
443
and (4.2s) in equation (1.1; 5), we find
d ((_1),~_i L~_lx(s)) a =q(s)f(x[g(s)]) ds >__q (s) f (w0 [r (t), g (s)] ((--1) n Ln-lX [7- (t)])) _> q (s) f (w0 [T (t), g (s)]) F ( ( - 1 ) ~ Ln-lx Integrating (5.19) from 7-(t) to t and setting
u~[r(t)]
>
F (~ [7- (t)]) -
(t)
(5.19)
[7-(t)]).
u(t) = ((-1)'~L,~_lx(t)), we obtain q(s)f(wo[7-(t),g(s)]) ds.
Taking limsup of both sides of this inequality as t -~ e~, we obtain 0 _> lira sup
ft
t-*oo Jr(t) q (s) f
(w0 [7- (t),g (s)]) ds,
which contradicts condition (5.18). This completes the proof.
|
THEOREM 5.5. Let (-1)n6 = - 1 , Conditions (i)-(v), (1.3), (5.1), and either (3.32) or (3.35) hold. If
Te q (s) f (Wo [s,g (s)]) ds = oc,
(5.20 /
g
then, N o = @. PROOF. Let x 6 Af0 and assume that x(t) > 0, for t > to _> 0. There exists atl > to, such that (4.27) holds, for t >_ tl. Choose a t2 >_ tl, so large that t2 >_ tl and inf{g(t) : t >_ t2} > tl. Replacing t and s by t and g(t), respectively, in (2.14), we have
x[g(t)l>_wo[t,g(t)]((-1)n-lL,~_lx(t)),
for t e 7~g N [t2, c~).
Using (5.21), condition (3.32) or (3.35) in equation (1.1; (f) and letting > 0, for t > t2, we have
(5.21)
u(t) = ((-1)n-lL~_lx(t)) ~
du(t) at = q (t) S (x[g (t)]) _> q (t) f (~o [t,g (t)] u 1/" (t) ) >q(t) f(wo[t,g(t)])F(ul/"(t)),
forteUgn[t2,
oc ).
Choose a t3(> t2) arbitrarily. Dividing both sides of the above inequality by integrating over T~g n [tz, ta], we find
fn
q (s) f (wo [s,g (s)]) ds < ftl ~
u' (s)
ds = -f~(")
F(ul/~(t)) and
du
Letting t3 --* c~, we conclude that
fT¢.~nIt2,o~) q(s) f
(wo [s,g(s)]) ds < -
fu(tZ) F (ul/~) du
< c~,
.,o
which contradicts condition (5.20/. This completes the proof.
|
444
R.P. AGARWAL et al.
THEOREM 5.6. Let (-1)n(~ =
- 1 , Conditions (i)-(v), (1.3), and (5.17) with F be replaced by f
hold. H g(t) <_t, for t >_to and t
lira sup f~
t
i/c~
In_2(u,g(t);an_2,...,al)an_l (U) (~u q(T) dT)
du > O,
(5.22)
then, Afo = 0. PROOF. Let x EN'o and assume that x(t) > 0, for t_> to >_ 0. There exists a t l _> to, such that (4.27) holds, for all t >_ tl. Choose a t2 _> tl so large that t2 _ tl and inf{g(t) : t >_ t2} > tl. As in the proof of Theorem 4.8, we obtain (4.32) and (4.33). Now, by using (4.33) in (4.32) yields
x[g(t)] >_fl/~(x[g(t)])
/:
(t)
I~-2(u,g(t);an-2,...,al)an_l(u)
(//
q(T) dT/1/~ du.
The rest of the proof is similar to that of Theorem 5.4 and hence, is omitted.
|
THEOREM 5.7. Let (-1)~5 = - 1 , Conditions (i)-(v), and (1.3) hold and let g(t) <_t, for t > to,
and
(5.23)
0 fl/a (%~-------~ < 00. If
f ~ al [g(s)] g'(is)/ u S ) l / a
(~s (s)
I~-3 (u,g (s);an-2,.
,a2) a=-i (u)
q (T) dT
(5.24)
) du
ds = c~,
then, No = 0. PROOF. Let x E N0 and let x(t) > 0, for t > to _> 0. There exists a tl _> to, such that (4.27) holds, for t > tl. Choose a t 2 _> tl so large that t2 _> tl and inf{g(t) : t _> t2} > tl. Next, replacing i, k, t, and s by 1, n - 1, g(t), and t, respectively in (2.2), one can easily find
xt[g(t)] ~ax[g(t)]
(t)
In-3(u,g(t);an-2,...,a2)an-l(U)
(-1)n-lLn_lX(U)
du.
As in the proof of Theorem 4.8, we obtain (4.33). Thus, x' [g (~ (t)][gg'(t)]) (t)>:i/~
t t 1/a ax [g (t)] g' (t)fg (~) In-3 (u,g(t);an-2,..., a2)an-1 ( u ) ( f q ( r ) d ' r ) du.
The rest of the proof is similar to that of Theorem 5.5 and hence, is omitted.
|
Finally, we present the following comparison result. THEOREM 5.8. Let (-1)'~6 -----1, g(t) < t, for t >_to, Conditions (i)-(v), (1.3), and either (3.32) or (3.35) hold. Moreover, assume that there exists a function r/ E C([t0, oo),N), such that
r/(t) >_0 and g(t) <_~(t) <_t, for all t >_to. If the first-order delay equation,
y' (t) + q (t): (wo [,7 (t), g (t)]) F (y'/° [,7 (t)]) = o, is oscillatory, then, A[o
=
(5.25)
O.
PROOF. Let x E A/o and let x(t) > 0, for t _> to _> 0. As in the proof of Theorem 4.9, we obtain (4.35). Next, by using condition (3.32) or (3.35) and (4.35) in equation (1.1; 5), we have d ( ( _ l ) n _ 1 L n _ l x ( t ) ) ~ = q (t) f (x [g (t)])
dt
(5.26) > q (t) f (wo [7 (t), g (t)]) F ( ( - 1 ) ~-1 n n - l x [~?(t)]) ,
Functional Differential Equations
445
for t > y(t) > g(t) > t2. Set u(t) = ( ( - 1 ) n - l L n _ l x ( t ) ) % t >_ t2. Then, (5.26) takes the form,
_du(t) >q(t)f(wo[rl(t),g(t)])F(ul/=[rl(t)] ) dt '
for t > t2.
-
The rest of the proof is similar to that of Theorem 4.5, and hence, is omitted. This completes the proof. II REMARK 5.1. As before, appropriate combinations of the results of Sections 5.1 and 5.2 will imply the oscillation of equation (1.1; 6). Now, we have the following. (1) Any of the results of Section 5.1 with £ E {1, 3 , . . . , n - 1} implies the oscillation of evenorder equation (1.1; 1). (2) Any of the results of Section 5.1 with £ E {2, 4 , . . . , n - 1} together with appropriate results of Section 5.2 implies the oscillation of odd-order equation (1.1; 1). (3) Any of the results of Section 5.1 with £ E {1, 3 , . . . , n - 2} implies that any solution x(t) of odd-order equation (1.1;-1) belongs to the subset O UAfn. (4) Any of the results of Section 5.1 with ~ E {2, 4 , . . . , n - 2} together with appropriate ones of Section 5.2 implies that any solution x(t) of even-order equation (1.1;-1) belongs to the subset O U Afn.
6. L I N E A R I Z A T I O N
OF N O N L I N E A R
OSCILLATION RESULTS
By combining Theorems 3.6, 4.6, and 5.3, we obtain the following result. THEOREM 6.1. Let 1 < £ < n - 1, (-1)'~-~6 = - 1 , Conditions (i)-(v), and (1.3) hold and f (x) sgn x > [ x f ,
for x ~ 0,
(6.1)
where/3 is the quotient of positive odd integers. Moreover, suppose that there exists a function a E Cl([t0, oo),R+), such that condition (3.2) holds. If, for all large T >_ to >_ O, the half-linear differential equation,
(t), T]
(t)) @'
+ q (t)
it) ; o,
(6.2)
is oscillatory, where r (t) =
cl,
cl > 0 is any constant, when/3 > a,
(a/k) ~ ,
k is as in Theorem 4.4, when fl = a,
c2_~ -~ [va (t), T],
c2 > 0 is any constant, v is as in Theorem 5.3, when/3 < a,
then, Aft = 0. Next, we shall proceed further and obtain similar results as above, but instead of comparing with oscillation of half-linear equation (6.2), we shall compare with the oscillation of certain second-order linear ordinary differential equations. To obtain our next results, we shall need the following lemma given in [34]. LEMMA 6.1. Let r 6 C([T,c~),R+), T >_ to. If there exhsts a function w E CI([T,c~),R), such
that q (t) _< - w ' (t) - r (t) w 2 (t),
for every t _> T,
then, the second-order linear ordinary differential equation,
(x'
(t)
r (t))
is nonoscillatory. Now, we present the following theorem.
+ q (t) x (t) = o,
446
R . P . AGARWALe t
al.
THEOREM 6.2. Let a > 1, 1 < £ < n - 1, ( - 1 ) ~ - t 5 = - 1 , Conditions (i)-(v), (1.3), and (6.1) hold. Moreover, suppose there exists a function a E Cl([to, ce),R+), such that condition (3.2) holds. Then, Aft = 0, if there exists a constant A E (0, 1) when g >_ 1 and, for a//large T > to _> 0,
the second-order linear differential equation, !
i
(63)
is oscillatory, where ~t(t) = A a ' ( t ) w * [ o r ( t ) , T ] w ~ - l [ a ( t ) , T ] ,
a(t) > T,
w*[a(t),T] is as in Theorem 3.1 and A E (0,1), i[£ >_ 1, A = 1, iY£ ~ 2 and
(t) :
el,
cl > 0 is any constant, when fl > a,
1,
when fl = a,
c2_~ -~ [Aa (t), T],
c2 > 0 is any constant, when/3 < a.
PROOF. Let x E Af~ and assume that x(t) > 0, for t > to >_ 0. T h e r e exists a t l ~ to, such that (2.3) holds. Now, we consider the two cases, (il) l _ > l a n d
(i~) t ___2. CASE (I1). Assume g > 1. Define
5Lg_lx (t) w (t) - x~ [ ~ (t)]'
for t ~ tl and some A E (0, 1).
Then, for t _> tl, we have
~' [~or (t)] xB-1 [~or (t)].
w ' (t) ~_ - q (t) - A~or' (t) w 2 (t) - 5 L ~ _ l x (t)
(6.a)
From Lemma 2.3, it follows that there exist a constant A E (0, 1) and a t2 > tl/A, such that x' [Ao"(t)] ~ WX [o" (t), All (SLn-lx (t))
(6.5)
and
x [Aor (t)] > w [or (t), All ( S L n - l X
(t)).
(6.6)
Using (6.5) and (6.6) in (6.4), we have
W ! (t) __~ - q (t) - ~flort (t) wA [°r (t), All w (~-1 [(7 (t),tl] W 2 (t) x f~-a [~or (t)],
for t > t2,
or
w ' (t) _< - q (t) - ~ a (t) w 2 (t) x ~ - " [~or (t)],
for t >_ t2.
As in the proof of Theorem 3.6 when 13 > a and Theorem 5.3 when ~ < a, one can easily see that W' (t) _< - q (t) - f l ~ W 2 (t), for t _> t2. (6.7) Applying Lemma 6.1 to inequality we conclude that equation (6.3) is nonoscillatory, which is a contradiction. CASE (I2). Assume g _> 2. Here, we define W(t) as in Case (iz) with A = 1. The proof will be the same, and hence, we omit the details. | When a E (0, 1], we shall present the following criterion.
Functional Differential Equations
447
THEOREM 6.3. L e t 0 < a _< 1, 1 < ~ < n - l , ( - 1 ) ~ - t 5 ----- 1 , Conditions (i)-(v), (1.3), and (6.1) hold, Q(t) > 0, for t >_ to. Moreover, assume that there exists a function a(t) E Cl([t0,(:x)), ~+), such that (3.2) holds. Then, Af~ = 0 is there exists a constant A 6 (0, 1) when £ > 1 and, for all large T > to > O, the second-order linear equation,
(t)
(t)
+
(t) y (t) = o,
(6.8)
is oscillatory, where ~x (t) ----Aw* [a (t), T] Q(1-a)/~ (t) a' (t),
a (t) > T,
w*[a(t),T] is defined as in Theorem 3.1 and A C (0,1) if£ _> 1, A = 1 ifg > 2 and rl (t) =
Cl,
cl > 0 is any constant, when fl > a,
1,
when j3 = a,
c2w_~(~-~)/~ [Aa (t), T],
c2 > 0 is any constant, when ~ < a.
PROOF. Let x E Af~ and assume that x(t) > 0, for t > to > 0. As in the proof of Theorem 6.2, we obtain (6.4) which takes the form, W' (t) < - q (t) - A~a' (t) @:~ [a (t), tl] W 2 (t) W (1-a)/a (t) x (~-'~)1~ [Aa (t)],
t > tl.
(6.9)
As in the proof of Theorem 3.3, we have W (t) _> Q (t) > 0,
for t __ tl.
(6.10)
Using (6.10) in (6.9), we have W' (t) _~ - q (t) - After' (t) @x [~r(t), tl] Q ( 1 - , ) / , (t) W 2 (t) x (~-a)/`~ [Aa (t)],
t > tl.
The rest of the proof is similar to that of Theorem 6.2, and hence, is omitted. | Next, we shall provide proof of Theorem 4.1 when a > 1 and a C (0, 1] without employing Lemma 4.1. THEOREM 6.4. Let a > 1 and the hypotheses of Theorem 4.1 hold and condition (4.2) is replaced
by limsupft[~
(s) q (s) - ( 1 )
Jr
(~'(s))2
1 .]
a(s)
ds
=
(6.11)
where ~t(t) is defined as in Theorem 6.2. Then, Aft = 0. PROOF. Let x E Af~ and assume that x(t) > 0, for t _ to _> 0. Define 5Lan_l x (t) W (t) = ~ (t) x [Aa (t)] '
for t >_ t~ > to.
Proceeding as in the proof of Theorems 4.1 and 6.2, one can easily find W'(t) <_ -~(t)q(t) + - ~ W ( t )
..
- aA
~[cr(t),tl]w"-l[a(t),tl]W2(t),
for t > t2 >_T/A, for some A E (0, 1), or W' (t) _< - ~ (t) q (t) + ~ - - ~ W (~) - E - ~ t (t)
(t) 2
<-
[~(t) q(t)
4a
((t)
1 ] ' f°rt>-t.
a(t)
The rest of the proof is similar to that of previous results, and hence, is omitted.
|
R. P. AGAR,WAL
448
THEOREM 6.5. Let 0 < a _< 1 and the
et al.
hypotheses of Theorem 4.1 hold, Q(t) > O, and condi-
tion (4.2) is replaced by
1 (~'(s))2 limsupt_~ooJf T*[ ~(s) q ( s ) - 4a ~-(~) ~;l'(s) 1 ds = oo,
(6.12)
where f~l(t) is defined as in Theorem 6.3. Then, Aft = 0. PROOF. Let x e Ale and assume that x(t) > 0, for t > to >_ 0. Define the function W(t) as in the proof of Theorem 6.4 and proceed as in the proof of Theorems 4.1 and 6.3, to obtain
~' (t) w' (t) < -~ (t) q (t) + T s ~ w (t) - ~A~' (t) ~
[~ (t), tl] ¢-1/~
(t) w ~ (t) w (1-~)/" (t),
for t > t~
and
w (t) > ~ (t) Q (t),
for t > t2.
The rest of the proof is similar to that of Theorem 6.4, and hence, is omitted.
7. O S C I L L A T I O N
|
CRITERIA
From the previous results, we shall state as a sample some oscillation criteria for equation (1.1; 5). The formulations and statements of other results are left to the reader. THEOREM 7.1. Let n be even and assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function ~ E Cl([t0,oo),R+), such that (3.2) holds. (Sl). Suppose that
f 4-~ f~/~ du(~-----<---~o0 and there exists a function ~ E Cl([t0, co), R +) and a constant A E (0, 1), such that for all large T _> to >_ 0, ~' (t) > 0 and
(
(t))l/a ~;~ [a(~'(t), T] a' (t)
Then, the condition,
/
)'
_ O,
for t > cr (t) > T/A.
~ ~ (s) q (s) ds = co,
is sufficient for equation (1.1;1) to be oscillatory.
(s2). Suppose that f(x)sgn x >__Ixl", for x # O. If there exist a function ~(t) E Ct([t0, co), R +) and a constant A E (0, 1), such that for all large T > to with a(t) > T/A, either
r
lim sup f] [~ (s) q (s) t--*c~ JT
(," (s)F +:
~-~
(a-~- 1) a+l (~(S) O't(S)~X [a(s),T])"
ds = co,
or, a > 1 and
/:I ~(s)q(s)
limsuPt__.~
4aA ~(s)
1
a'(s)~x[a(s),Tlw~-l[a(s),T]
or, a < 1 and
limsupt~Jr [ ((s)q(s)-~
~,. -~-~
] ~'(s)~,[~(s),T]Q(l-~)/~(s)
where Q(t) > O, then equation (1.1;1) is oscillatory.
ds=co,
Functional Differential Equations
449
(S3). Suppose that either (3.32) or (3.35) holds and
f± o F (~1/~) du < oo, where F is as in (3.36). Then, for all large T _> to, the condition,
f
oo q (s)
f (w[g (s), T]) ds
0%
is sufficient for equation (1.1;1) to be oscillatory. (s4). Suppose that a > 1 and f(x)sgn x > [xlZ, for x # 0, where/~ is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if there exists a constant A E (0, 1), such that for all large T _> to _> 0, the second-order linear equation, !
r(t) h a ! (t)W~ [a (t), T] w ~-1 [a (t), T]
y' (t)
+ Zq (t) y (t) = 0,
is oscillatory, where r(t) is as in Theorem 6.2. (sh). Suppose that 0 < c~ < 1 and f(x)sgn x > Ix[~, for x ¢ 0, where fl is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if there exists a constant A 6 (0, 1), such that for all large T _> to _> 0, the second-order linear equation,
( ~ , (t) : ~ [~ (t), rl(t)T] Q(:-~)/~ (t)y,(t))' + ~q (t) y (t) = 0, is oscillatory, where rl(t) is as in Theorem 6.3 and Q(t) = f~o q(s)ds > O. THEOREM 7.2. Let n be odd and assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function a 6 Cl([t0, oo), R+), such that (3.2) holds. (ol). Suppose that g(t) <_t, for t > to, and
/:
o~
o
du f l / ~ (~----~< oo.
Then, for all large T >_ to with a(t) >_ T, the conditions,
/ where
ioo
al [g (s)] g' (s)
~
[o(s),T]
G !
(s) Q1/~ (s) ds = o o ,
(~. (~)
fn-3 (u, g (s) ;an-2,..., a2) an-1 (u)
(~s >l/a) q (T) dT
du
ds = c~
are sufficient for equation (1.1;1) to be oscillatory. (02). Suppose that f(x)sgn x _> Ix[~, for x ¢ 0. Then, for all large T > to with a(t) > T, the conditions,
[
lim sup w [a (t), T] Q (t) + a
~ [a (s), T] a' (s) Q(l+a)/~ (s) ds
$---~OO
where
lim sup t-~oo
//
(t)
q (s) w~ [~- (t), g (~)1 as > 1
axe sufficient for equation (1.1;1) to be oscillatory.
].°
> 1,
R.P. AGARWALet al.
450
(03). Suppose that either (3.32) or (3.35) holds and
du o F (~/") < o0, where F is as in (3.36). Then, for all large T > to with g(t) > T, the conditions, oo q (s) f (w[g (s), T]) ds = oo and limsup fl ~ q(s) f (wo [7(t),g(s)]) ds > 0, t ---*oo J r ( t )
are sufficient for equation (1.1;1) to be oscillatory.
(04).
Suppose that f(x)sgn x >_ Ix[~, for x ¢ 0, where fl is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if, for all large T >_ to with a(t) > T, the half-linear second-order equation,
(
(t)
[~ (t), T] ~' (t)
(y' (t))"
)'
+ q (t) y" (t) = 0,
is oscillatory, where r(t)=/
1,
[ c2w__c*--/3 0 [a (t), T], and lira sup t-~oo lira sup t-~oo
~
whenfl=a, c2 > 0 is any constant, when B < a,
t
(0
//
(t)
q (s) w~ [T (t), g (S)] ds > 1,
when fl = a,
q (s) wg [r (t), g (s)] ds > 0,
when fl < a.
(05). Suppose that a >__1 and f(x)sgn x > Ix]z, for x # O, where ~ is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if, for all large T > to with a(t) > T, the linear second-order equation,
)'
+Zq(t)y(t) =o,
q (s) w~ [r (t), g (s)] ds > 1,
when ~3 = a,
q(s)wg[r(t),g(s)J ds > O,
when ~ < a.
(~[o(t),T]w--l[o(t),T]o,(t) r (t)
y,
(t)
is oscillatory, where r(t) is as in (04) and t
lim sup t--*oo
Jr(t)
limsup f t t-~oo
Jr(t)
THEOREM 7.3. Let n be odd and assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function a e Cl([t0, cx~),R+), such that (3.2) holds. (31). Suppose that
f
±oo
du
f~l. (~---~< ~ '
Functional Differential Equations
451
and there exist a function ~ E Cl([t0, oo), R +) and a constant A E (0, 1), such that, for all large T _> to with ~(t) > T,
~' (t) _> O,
for t _> to,
N~ [a (s), T] a' (s)
and
Then, the conditions,
/
o~ ~ (s) q (s) ds = cx~
and
L al[g(s)]g'(s)~g(')In_a(g(s),u;a2,...,a~_2)an_l(U) (/ q('r) d'r)V~duds=oo, are sufficient for equation (1.1;-1) to be oscillatory. (J2). Suppose that f(x)sgn x > Ixl a, for x ¢ O. Equation (1.1;-1) is oscillatory if, for all large T > to with or(t) > T, the first-order delay equation,
y' (t) + q (t) w" [o (t) , T] y [0 (t)] = 0, is oscillatory and
p(O
q (s) w n [g (s), p (t)] ds > 1.
lim sup ,It
t--+ oo
(J3). Suppose that either (3.32) or (3.35) holds and
/ +~ ~o
du F (~1/.) < o~,
where F is as in (3.36). Then, for all large T >_ to with g(t) > T, the conditions,
/
~ q ( s ) f (w [g(s), T]) ds
oo,
/.4 q(s)f (w,~[g(s),s]) ds =
oo,
9
are sufficient for equation (1.1;-1) to be oscillatory. (J4). Suppose that f(x)sgn x > ]x]~, for x ~ 0, where fl is the quotient of positive odd integers. Equation (1.1;-1) is oscillatory if, for all large T >_ to with a(t) > T, the half-linear equation, I
r(t)
( ~ [a (t), T] o' (t)
(y' (t))"
+ q (t) y" (t) = 0,
for some A E (0, 1),
is oscillatory, where
I CI, cl > 0 is any constant, when ~ > a,
r(t)=
I,
when fl = a,
and limsup t'---~ O 0
f
p(t) q (s) w~ [g (s), s] ds > O,
when ~ > a,
q (s) w x [g (s), 0 (t)] as > 1,
when ~ = a.
Jt
p(t) lira sup t -.* oo
,It
R.P. AGARWALet al.
452
(is). Suppose that 0 < a < 1 and f(x)sgn x >_ Ix[~, for x ¢ 0, where f~ is the quotient of positive odd integers. Equation (1.1;-1) is oscillatory if, for all large T _> to with a(t) > T, the second-order equation,
~ (t)
A~ [cr (t), T] o' (t) Q(1-a)/a (t) y' (t)
)'
for some )~ E (0, 1),
if- ~q (t) y (t) = O,
is oscillatory, where r(t) is as in 04), Q(t) = f t q(s) ds > 0 and limsup fP(t) q(s)w~[g(s),s] ds > O,
when/3 > ~,
Jt
t -* c~
p(t) q (s) w~ [g (s), p (t)] ds > 1,
lim sup t --* o o
when j3 = a.
,It
THEOREM 7.4. Let n be even, assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function 0 E Cl([t0, oo), R+), such that (3.2) holds. (M1). Suppose that either (3.32) or (3.35) holds and
~oo o
du F (~1/.) < o~,
where F is as in (3.36). Then, for all large T > to with a(t) > T, the conditions,
ff
[o (s), T] 0' (s) QI/~ (s) ds = c¢,
f q(s) I ( ~ .
Q (t) =
ff
q (s) ds,
[g(s),s]) d8 = ~ ,
g
and limsup -[ t
t--+oo JT(t)
q(s)f(wo [~'(t),g(s)]) ds > 0,
are sufficient for equation (1.1;-1) to be oscillatory. (M2). Suppose that f(x)sgn x > [xl", for x ¢ 0. Then, for all large T > to with a(t) > T, the conditions,
limsupw[o(t),T]
Q(t)+a
~[o(s),T]o'(s)Q(l+a)/~(s) ds
t --+ O o
where
t
lira sup ,-~oo
and ~m sup t-~ oo
q i s) w~ [~ (t), g (s)]
(t)
f
ds > 1
p(t)
q (s) w~ [g (s), p (t)] ds > 1
J t
are sufficient for equation (1.1;--1) to be oscillatory. (M3). Suppose that either (3.32) or (3.35) holds and
~
oo o
du P (~1/~) < oo,
> 1,
Functional Differential Equations
453
where F is as in (3.36). Then, for all large T > toe the conditions,
/ lim sup f t t--,oo
oo q (s) f (w [g (s), T]) ds = c~,
q (s) f (Wo IT (t), g (S)]) ds = oo,
d~'(t)
and
f
q(s) f ( w n [g(s),s]) ds =
00,
g
are sufficient for equation (1.1;-1) to be oscillatory. (Ma). Suppose that c~ _> 1 and f(x)sgn x ~ Ixl ~, for x # O. Equation (1.1;-1) is oscillatory if, for all large T > to ~ O, the second-order equation,
(
1 y~ [(7 (t), T] W a - 1 [(7 it), T] (7' (t) (t)
)'
+ aq (t) y (t) = O,
is oscillatory,
i
lim sup
t
t---~oo J r ( t )
q (s) w~ IT (t), g (S)] ds > 1
and
f
lim sup t--coo
p(t) q (s) w,~ [g (s), p (t)] ds > 1.
.It
(Ms). Suppose that 0 < a _< 1 and f(x)sgn x > Ixl a, for x # O. Equation (1.1;-1) is oscillatory
if, for all large T > to with (7(t) > T, the second-order equation,
1 ' t +aq(t) y(t)=o, [(7(t) , T] (7' (t) Q(I--)/. (t)y ())' is oscillatory, where lim sup
f t
t--*oo
q (s) w~ [7- it), g (s)] ds > 1
(t)
and lira sup t--~
f
p(t) q (s) w~ [g (s), p (t)] ds > 1.
Jt
These theorems seem to be new even when specialized to the equation,
dt
x ("-1) (t)
+ ~q (t) f (x [g (t)]) = 0,
(7.1; 5)
R . P . AGARWALet al.
454
for which Conditions (ii)-(v) are satisfied. So, we state some of them below as corollaries, by noting that in this case, for T >_to and all t > T, (t - T) '~-1 wt [t,T] = ( n - 1 ) ( t - 1)! ( n - t w--t [t,T] = (n
-
1)!'
l
(t - T) n-2 2) (e - 2)! (n - e - 1)!'
2
~ [t, T] = [A (1 - A)]'~-2 t '~-2
(~ -
2)!
A C (0, 1), for all large t > T/A,
'
Atn-2 [t, s] = (,~ _ ~)!,
for some A E (0, 1), and all large t _> T,
Atn-1
for some A E (0, 1), and all large t >_ T,
w[a(t),T] = ( n - 1)!' WO [t ,S] -~ Wn [t,s] ---- ( t - - s ) n-1 (~ 1)! '
t>s>T.
-
Let n be even and assume that (ii)-(v) hold. Moreover, assume that there exists a function cr e Cl([to, oo),R+), such that (3.2) holds. COROLLARY 7.1.
(Sl). Suppose that
S
+~
du S~1~@------7< ~
and there exists a function ~(t) E Cl([t0, ec), R+), such that
~' (t)
> o
and
~a,_2(t)a,(t))
_<0,
fort>T,
for some T _> t0.
Then, the condition,
S
OO ~ (s) q (s) ds =
~¢,
is sufficient for equation (7.1;1) to be oscillatory. (s2). Suppose that f(x)sgn x >_ Ixl", for x 7~ 0. If there exists a function ~(t) E Cl([to, e~),R +) and a constant A E (0, 1), such that, for all large T _> to and t _> T either lim sup t---*oo
((n - 2)!)" A-" (~' (s)) ~+1 1 ] (a + 1) ~+1 (A (1 - A)) ('~-2)~ (~ (s) (a '~-2 (s) a' (s)) ~] ds = ec
~ (s) q (s) T
or, a > 1 and ( n - 2), ( ( n - 1)[) ~ - ' 4aA<~ (A ( i - A)) n-2(~('~-'iy~-~----~(s)~''(s)
l i m sup f t [ ~ ( s ) q ( s ) t--+~ .ST
or, 0 < a < _ l
((~' (s))2~l ~t ~ ' ~
,]J d s = o o
and lip_+sup
~T~'[
(n-- 2)! O(a-Za__ !S__) ] ~(s)q(s)-4aA(~Tf-_--~))n_2a.~_2(s)a,(s)j
Q(t) > 0, then, equation (7.1;1) is oscillatory.
(s3). Suppose that either (3.32), or (3.35) holds and
i+o F (ull du ")
< e~,
ds=o¢,
Functional Differential E q u a t i o n s
455
where F is as in (3.36). Then, the condition,
f =q(~) f (g"-~ (8)) ~s= ~¢, is sufficient for equation (7.1;1) to be oscillatory. (s4). Suppose that ~ > 1 and f(x)sgn x > Ixl ~, for x ¢ 0, where/3 is the quotient of positive odd integers. Equation (7.1;1) is oscillatory if there exists a constant A ~ (0, 1), such that for all large t > to, the second-order linear equation,
(~.-_?),_o ~(t) y' (t))' +/3q (t) u (t) = 0, (1 ~) -~ ~(~-~).+~-2 (t) ~' (t)
~.-~
-
is oscillatory, where
r (t) =
el '
cI > 0 is any constant, when/3 > a,
1,
when/3 = a,
c2a(n-1)(~-~) (t),
c2 > 0 is any constant, when/3 < a.
(ss). Suppose that 0 < a < 1 and f(x)sgn x > Ixl~, for x # 0, where/3 is the quotient of positive odd integers. Equation (7.1;1) is oscillatory if there exists a constant A ~ (0, 1), such that for all large t > to, the linear equation,
Q(a-1)/a (t) r l (t) (~ 2)! y' (t]~' +/3q (t) y (t) ----O, A~-1 (1 - A) '~-2 a n-2 (t) c' (t) " "] is oscillatory, where cl > 0 is any constant, when/3 > a,
Cl '
when/3 = a, 1, C2a (n-1)(a-jg)/a (t) , c2 > 0 is any constant, when/3 <
rl (t) =
ci.
COROLLARY 7.2. Let n be odd and assume that (ii)-(v) hold. Moreover, assume that there exists a function a C cl([to, oe), R+), such that (3.2) holds.
(oi).
Suppose that g(t) <_t, for t >_ to and
du
~
S±o
p i . (~-------5< ~"
Then, for all large t _> to, the condition,
i where
i
g'
(')
~
o.n--2
(~ _
(s; (.)
(s)
G!
g (,))..-~
VmV)'
Q1/. (s)
(s)
ds = o o ,
(L) q(r) d r
~/<.
d~
)
d8 = ~¢
are sufficient for equation (7.1;1) to be oscillatory. (02). Suppose that f(x)sgn x _> Ix[ ~, for x ~ 0. Then, for some constant A E (0, 1) and all large t, the conditions,
[
aA
lim._+s~p c,"-~ (t) Q (t) + 2)------'7. (n -
/-
a n-2
(s)
a'
(s)
Q(.+I)I.
(s) ds
]
11o~
> (n - 1)!,
456
R . P . AGARWALet
where limsup t--*oo
/;
(t)
al.
[~'(t)-g(s)](n-D~q(s) ds > ( ( n - 1)!) ~
are sufficient for equation (7.1;1) to be oscillatory. (03). Suppose that either (3.32) or (3.35) holds and
f~
du o F (~1.)
< ~'
where F is as in (3.36). Then, for all large t, the conditions,
7
~ q(8) f (gn--1 (S))
US = O0
and limsup
/'
q(s) f
( [r (t) -- g (s)] n-1 ) ds>O,
are sufficient for equation (7.1;1) to be oscillatory. (04). Suppose that a _> 1 and f(x)sgn x > [xl #, for x ¢ 0, where fl is the quotient of positive odd integers. Equation (7.1;1) is oscillatory if, for some constant ), E (0, 1) and all large t, the equation, //(n - 2)! ((n - 1)!)"-1 r (t) , ,.,'~' --,:-r t -~-~--~-~+-~--~(t)a it) y t~)) + n q ( t l Y ( t ) = O , is oscillatory, where r (t)
[ 1,
when fl = a,
t
c > 0 is any constant, when fl < a,
ca(n-1)(~-#) (t),
and t
limsup
fJr(t) IT (t) -- g (S)] (n-1)~ q (S) ds > ((n -
1)!) ~ ,
limsup f ' [r(t) -g(s)](n-1)Zq(s) ds > 0, t--*oo Jr(t)
when fl = a, when fl < a.
COROLLARY 7.3. Let n be odd and assume that Oi)-(v) hold. Moreover, assume that there exists a function a 6 Cl([to, c~), R+), such that (3.2) holds.
(J1)-
Suppose that
f
+~
du yli a (u-------~< c~,
and that there exists a function ~(t) E Cl([to, c~),R+), such that ~'(t)>0,
fort>to,
and
F (t.<,:-,
Then, the conditions,
f
~ ¢ (s) q (s) ds = ~¢
(s)<.. (,)) i
Functional
Differential
Equations
457
and ids) d
(g (s) (n-3)!
(4
thy3 (p)
d7)l/a
are sufficient for equation
(7.1; -1) to be oscillatory.
Suppose that f(z)sgn 2 2 ]zla, for 2 # 0. Equation and some constant X E (0, l), the first-order equation, (Jz).
(
Y’ (t> +
is oscillatory,
duds=o%
J s
L,
(n ! l)!
(I 1
d---l
(7.1; -1) is oscillatory if, for all large t
(9 4 w Y b WI = 07
and P(t) lim t--r00 J t
(53).
[g(s) - p (t)]‘“-l’”
q (s) ds > ((n - l)!)” .
Suppose that either (3.32) or (3.35) holds and
fm J rto
du F (da)
< O”’
where F is as in (3.36). Then, for all large t, the conditions,
s
O” q(s) f (g’+-l (s)) ds = oo
and
are sufficient for equation (J4),
(7.1; -1) to be oscillatory.
Suppose that 0 < o 5 1 and
f (~1w 2 2 14P, where p is the quotient of positive odd integers. constant X E (0,l) and all large t, the equation,
for 2 # 0, Equation
r (t) Q(a-1)/a (t)
(n - 2)! x [A (1 - A)]“-”
P--2 (t) o’ (t)
y’(t)
(7.1; -1)
is oscillatory
‘+Pdt)Y(t)
if, for some
=o,
is oscillatory, where r
(t) =
c,
c > 0 is any constant, when ,!9> o,
1, when ,L3= CY,
Q(t) = St” q(s) ds > 0 and p(t)
lim sup lim sup t--Km
[g (s) - s]@-l)
P(t) J
t+cc
t
st
[g (s) - p(t)]‘“-“”
q (s) ds > 0,
when p > cx,
q (s) ds > ((n - l)!)” ,
when 0 = (Y.
R, P. AGARWALet al.
458
COROLLARY 7.4. Let n be even and assume that (ii)-(iv) hold. Moreover, assume that there exists a function cre Cl([to, oo),R+), such that (3.2) holds. (M1). Suppose that either (3.32) or (3.35) holds and
f k~ o
du
F(u 1/~) < oo,
where F is as in (3.36). Then, for all large t, the conditions,
~ - 2 (s) a' (s) Q1/~ (s) ds = oo,
Q (t) =
q (s) ds,
/A q ( s ) f ( [ g ( s ) - s ] ~ - l ) d s = c % g
and
limsup t--~oo
/:
q (s) f (t)
([r (t) - g (s)] "-1 ) ds > O,
are sufficient for equation (7.1;-1) to be oscillatory. (M2). Suppose that f(x)sgn x > ]xl ~, for x ~ 0. Then, for some constant ), E (0, 1) and all large t, the conditions, limsupt~ ~----1~
Q(t)+~
(s)
(s)
where limsup ?/t [ r ( t ) - g ( s ) ] ( n - i ) a q ( s ) t--*oo
ds > ( ( n - 1)!) ~
Jr(t)
and limsup fp(t) [g (s) - p (t)] ('~-1)'~ q (s) ds > ((n - 1)!) '~ , t--+oo J t
are sufficient for equation (7.1;-1) to be oscillatory. (M3). Suppose that either (3.32) or (3.35) holds and
f
oo o
du F(ul/~)
where F is as in (3.36). Then, for all large t, the conditions,
f limsup t~oo
(t)
oo q (S) f (gn--1 iS))
ds
= ~,
q(s)f [r(t)-g(s)] ~-~ d s = ~ ,
and
£qls): are sufficient for equation (7.1;-1) to be oscillatory.
sl °-:) ds
_-_
(s) ds
> 1,
Functional
Differential
Equations
459
Suppose that cy 2 1 and f(z)sgn z 1 ]x]~, for z # 0. Equation for some constant X E (0,l) and all large t, the equation,
(7.1; -1) is oscillatory if,
(Md).
1
(t)cl’(t)Y’ (4 > ‘+ (n-
( &-lb++2
XOcY
2)!
((n- q!)“-1qWY@)= Ov
is oscillatory, t (7(t) -g (s)]+‘I”
lim sup
q (s) ds > ((n - 1))”
t-K%3 s r(t)
P(t) t-+co s
lim sup
[g(s) - p (t)]‘“-l’o
t
q (s) ds > ((n - l)!)O .
Suppose that 0 < o < 1 and f(z)sgn z 2 ]z]~, f or z # 0. Equation (7.1; -1) is oscillatory if, for some constant X E (0,l) and all large t, the equation, (MS).
I
(
y&--2
(t)
CT’ (tf
Q(l-a),cx
@)Y
(t)
>
+
$Y2)!
Q (t)
y (9
=
O,
is oscillatory, where t [I t-em J T(t)
lim sup
-g (s)](n-l)a
q (s) ds > ((n - I)!)”
and lim sup t--too
Jt
p(t)[g (s) - p (t)]‘“-l’*
8. GENERAL
* . q (s) ds > ((n - l)‘)Q
REMARKS
1. The results of this paper are presented in a form which is essentially new and are of a higher degree of generality. In fact, one can easily extract more criteria than those presented, for the oscillation of equation (1.1; S) and/or related equations as well as some special cases of equation (1.1; 6) and also for the mixed equations of the form,
Lx (t)+ 62 qjw fj (x1%@)I> = 0,
(8.1;6)
j=l
where L, is as in equation (1.1;6), qj E C([te,oo),R+), gj E C((to,co),R), and fj E C(n,a),zfj(z)>O,fj’(~)LO,forz#Oandlimt,,gj(t)=~,j=1,2,.,.,m. The formulation of such results are left to the reader. 2. The results of this paper can be extended to equations of type (1.1; S) where f need not be a monotonic function. As in [26, Lemma 31, the function f has the following decomposition (f is locally of bounded variation on [a, b] C R): f(x) = G(x)H(x), where G : (-oo, --cro]U [Qo,cQ> = RI, -+ Iw+, o. 2 0, nondecreasing on (-co, -QO] and nonincreasing on [WI, co) and H : B,, + R and nondecreasing on IF&. 3. The results of this paper can be extended to forced equations of the form,
Lx (t>+ h (t) f (x 19WI>= e0) , where e E C([te, oo), IR) as well as neutral functional
Ln(x (t>+ c(t>x [h(a>+ &Iw f where c and h E C([tc, oo),W).
differential
(8.2;6)
equations of the type
(x 19(a> = 07
(8.3;6)
460
R . P . AGARWALet al.
REFERENCES 1. R.P. Agarwal and S.R. Grace, Oscillation of forced functional differential equations generated by advanced arguments, Aequationes Math. 63, 26-45, (2002). 2. R.P. Agarwal and S.R. Grace, On the oscillation of higher-order differential equations with deviating arguments, Computers Math. Applic. 38 (3/4), 185-197, (1999). 3. R.P. Agarwal, S.R. Grace and D. O'Regan, Oscillation criteria for certain nth-order differential equations with deviating arguments, J. Math. Anal. Appl. 262, 601-622, (2001). 4. S.R. Grace, Oscillation of even order nonlinear functional differential equations with deviating arguments, Funkcialaj Ekvac. 32, 265-272, (1989). 5. S.R. Grace, Comparison theorems for forced functional differential equations, J. Math. Anal. Appl. 144, 168-182, (1989). 6. S.R. Grace, Oscillation criteria for forced functional differential equations with deviating arguments, J. Math. Anal. Appl. 145, 63-88, (1990). 7. S.R. Grace, Oscillation of functional differential equations with deviating arguments, J. Math. Anal. Appl. 149, 558-575, (1990). 8. S.R. Grace, Oscillatory properties of functional differential equations, J. Math. Anal. Appl. 160, 60-78,
(1991). 9. S.R. Grace, Oscillatory and asymptotic behavior of delay differential equations with a nonlinear damping term, J. Math. Anal. Appl. 168, 306-318, (1992). 10. S.R. Grace, Oscillation theorems of comparison type of delay differential equations with a nonlinear damping term, Math. Slovaca 44, 303-314, (1994). 11. S.R. Grace, Oscillation criteria for retarded differential equations with a nonlinear damping term, Aequationes Math. (to appear). 12. S.R. Grace and B.S. Lalli, Oscillation theorems for nth-order nonlinear differential equations with deviating arguments, Proc. Amer. Math. Soc. 90, 65-70, (1984). 13. S.R. Grace and B.S. Lalli, A comparison theorem for general nonlinear ordinary differential equations, J. Math. Anal. Appl. 120, 39-43, (1986). 14. M. K. Grammatikopoulos, R. Koplatadze and G. Kvinikadze, Linear functional differential equations with Property A, J. Math. Anal. Appl. 284 (1), 294-314, (2003). 15. I. Kiguradze and T. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Academic Publishers, Dordrecht, (1993). 16. I. Kiguradze and I.P. Stavroulakis, On existence of proper oscillatory solutions of advanced differential equations, (Russian), Differentsial'nye Uravneniya 34 (6), 751-757, (1998); Differ. Equations, (English translation) 34 (6), 748-754, (1998). 17. I. Kiguradze and I.P. Stavroulakis, On the oscillation of solutions of higher order Emden-Fowler advanced differential equations, Appl. Anal. 70 (1-2), 97-112, (1998). 18. I. Kiguradze, N. Partsvanla and I.P. Stavroulakis, On advanced functional differential equations with properties A and B, Mere. Differential Equations Math. Phys. 24, 146-150, (2001). 19. I. Kiguradze, N. Partsvania and I.P. Stavroulakis, On oscillatory properties of higher order advanced functional differential equations, (Russian), Differentsial'nye Uravneniya 38 (8), 1030-1041, (2002); Differ. Equations, (English translation) 38 (8), 1095-1107, (2002). 20. I. Kiguradze, N. Partsvanla and I.P. Stavroulakis, On oscillatory solutions of nonlinear differential equations with advanced arguments, Mere. Differential Equations Math. Phys. 25, 156-158, (2002). 21. R. Koplatadze and T. Chanturia, On Oscillatory Properties of Differential Equations with a Deviating Argument, (Russian), Tbilisi University Press, Tbilisi, (1977). 22. R. Koplatadze, On oscillatory properties of solutions of functional differential equations, Mere. Differential Equations Math. Phys. 3, 3-179, (1994). 23. R. Koplatadze, Comparison theorems for differential equations with several deviations. The case of property A, Mere. Differential Equations Math. Phys. 24, 115-124, (2001). 24. R. Koplatadze, Comparison theorems for differential equations with several deviations. The case of property B, Mere. Differential Equations Math. Phys. 26, 139-148, (2002). 25. Y. Kitamura, Oscillation of functional differential equations with general deviating arguments, Hiroshima Math. J. 15, 445-491, (1985). 26. T. Kusano and B.S. Lalli, On oscillation of half-linear functional differential equations with deviating arguments, Hiroshima Math. J. 24, 549-563, (1994). 27. W.E. Mahfoud, Oscillatory and asymptotic behavior of solutions of Nth-order nonlinear delay differential equations, J. Differential Equations 24, 75-98, (1979). 28. Ch.G. Philos, Bounded oscillation generated by retardation for differential equations of arbitrary order, Utilitas Math. 15, 161-182, (1979). 29. Ch.G. Philos, On the existence of nonoscillatory solutions tending to zero at ~ for differential equations with positive delays, Arch. Math. 36, 168-178, (1981). 30. G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, Second Edition, Cambridge Univ. Press, Cambridge, (1988). 31. I. GySri and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford Univ. Press, Oxford, (1991).
Functional Differential Equations
461
32. R.P. Agarwal, S.R. Grace and D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer, Dordrecht, (2000). 33. R.P. Agarwal, S.R. Grace and D. O'Regan, Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations, Kluwer, Dordrecht, (2002). 34. R.P. Agarwal, S.R. Grace and D. O'Regan, Oscillation Theory for Second Order Dynamic Equations, Taylor and Francis, United Kingdom, (2003). 35. Ch.G. Philos, Some comparison criteria in oscillation theory, J. Austral. Math. Soe. (Ser. A) 36, 176--186, (1984).
Available online at www.sciencedirect.com
OClIENOI C ~ . '
RRO'lr°
COMPUTER
MODELLING
RLgEVIER
Mathematical and Computer Modelling 41 (2005) 417-461 www.elsevier.com/locate/mcm
Oscillation of Functional Differential Equations R. P. AGARWAL Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901, U.S.A. agarwal@f it. edu
S. R. GRACE Department of Engineering Mathematics Faculty of Engineering, Cairo University Orman, Giza 12221, Egypt srgrace©eng, cu. edu. eg
I. KIGURADZE A. Razmadze Mathematical Institute of the Georgian Academy of Sciences Tbilisi, Georgia kig©rmi, acnet, ge
D. O'REGAN Department of Mathematics National University of Ireland Galway, Ireland donal, oregan@nuigalway, ie
(Received May 2004; accepted June 2004) A b s t r a c t - - S o m e new criteria for the oscillation of functional differential equations of the form, d[
1 d i d a d ]~ an--l(t) -~ an-2(t) d-t"'al(t) dt x(t) +Sq(t) f(x[g(t)])=O,
are presented in this paper. © 2005 Elsevier Ltd. All rights reserved.
Keywords--Oscillation, Nonoscillation, Functional, Nonlinear, Comparison.
1. I N T R O D U C T I O N In this paper, we are concerned with the oscillatory behavior of all solutions of the functional differential equation,
Lnx (t) + 5q (t) f (x [g (t)]) = 0,
0895-7177/05/$ - see front matter (~) 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm. 2004.06.018
(1.1; 5)
Typeset by A~ts-~X
R.P. AGARWAL et al.
418 where n >_ 2, ~ = -t-1, and
nox(t)
= x(t),
L k x (t) --
1
d
ak (t) dt
L k - l X (t)
'
k = 1, 2,
., n - 1, " "
(1.2)
d L n x (t) = ~ [L~-lX (t)] ~ . In what follows, we shall assume that
(i) ~ e c([t0, ~ ) , ~t+ = (0, ~ ) ) , to > 0, and
f (ii) (iii) (iv) (v)
q g f a
ds = ~ ,
~ai(s)
i
= 1,2,...,n-
(1.z)
1,
e C([to, c~), R+), ~ C([t0, co), R -- ( - o c , co)) is nondecreasing and limt__.= g(t) = ec, E C(R, ]R) is nondecreasing and x f ( z ) > O, for x ~ O, is a quotient of positive odd integers.
The domain :D(Ln) of Ln is defined to be the set of all functions x : [tz, c~) ~ R, t~ > to, such that Ljx(t), j = O, 1 , . . . , n exist and are continuous on [t~, c~). Our attention is restricted to those solutions x E 7)(Ln) of equation (1.1; 5) which satisfy sup{lx(t)l : t > T} > o, for every T > t~. We make the standing hypothesis that equation (1.1; 5) does possess such solutions. A solution of equation (1.1; 5) is called oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. Equation (1.1; 5) is called oscillatory if all its solutions are oscillatory. The problem of obtaining sufficient conditions to ensure that all solutions of certain higher-order nonlinear functional differential equations of type (1.1; 5) are oscillatory when a = 1 and/or a > 0 has been studied by a number of authors, see [1-29] and the references cited therein. Our main objective is to present a systematic study on the oscillation of equation (1.1; 5) and establish some new oscillation criteria. In Section 2, we shall give the proofs of some important lemmas which are useful throughout this paper. Section 3 is devoted to the study of equation (1.1; 5) when f satisfies the condition f±o~ d u / f l / ~ ( u ) < cc and in Section 4, we present some results for equation (1.1; 5) when f satisfies the condition f(x)sgn x > Ix[ ~, for x ¢ 0. In Section 5, we give results for equation (1.1; 5) when f satisfies a condition of the form f+o d u / f ( u l / ~ ) < ~ " Section 6 is devoted to the linearization of nonlinear oscillation results and via comparison with the oscillatory behavior of second-order linear equation, we present results for equation (1.1; 5) when f satisfies the condition f(x)sgn x >_ lxt ~, for x ¢ 0, where/3 is the quotient of positive odd integers, fl > a, = ~, and 13 < ~. In Section 7, we present some oscillation criteria for equation (1.1; 5) and its special case when ai = 1, i = 1, 2 , . . . , n - 1. The obtained results extend, improve and correlate a number of existing results.
2. P R E L I M I N A R I E S To formulate our results, we shall use the following notation from [15]: for a~ e C([t0, oc), R), i = 1 , 2 , . . . , we define I0 = 1,
/~(t,~;~,~-~,...,~0
=
//
a~(~)I~_~(~,~;~_~,...,~)
d~,
i=1,2,....
It is easy to verify from the definition of Ii that Ii ( t , s ; a l , . . . , a i ) = ( - 1 ) ' £ ( s , t ; a ~ , . . . , a l ) and
Ii(t,s;al,...,a~)
=
We shall need the following lemmas.
f
ai(u)Ii-l (t,u;al,...,ai-1)
du.
Functional Differential Equations
419
LEMMA 2.1. If X e D(Ln), where Ln is Ln defined by (1.2) with a = 1, then, the following formu/as hold, for 0 < i < k < n - 1 and t, s E [to, oo) k-1
Lix (t) = E Ij_~ (t, s; a,+l, . . . , ak-1) L~x (s)
+
j=i
/st
(2.1)
Ik-~-~ (t,~; a~+~,..., a~_~) a~ (u) Lkz (u) d~
and
k--1
L~x (t) = E ( - 1 ) J - ~ / j - i (s, t; a j , . . . , ai+l) L j x (s) J=~ + (-1)
-~
f'
(2.2)
- ~rk_~_l (u, t; ak_~,..., a~+~) ak (~) Ck~ (~) a~.
This lemma is a generalization of Taylor's formula with remainder encountered in calculus. The proof is immediate. LEMMA 2.2. Suppose condition (1.3) holds. I f x E ~)(L~) where L~ is defined as in Lemrna 2.1 is eventually of one sign, then, there exist a tz >_ to >_ 0 and an integer ~, 0 < e < n with n + ~, even for x(t)[mx(t) nonnegative eventually, or n + g odd, t'or x(t)L~x(t) nonpositive eventually and, such that, for every t >__t~,
e > 0 implies x(t)Lkx (t) > O,
k = O, 1,... , l
e < n - 1 implies ( - 1 ) e-k x (t) Lkx (t) > 0,
k = e, l + 1 , . . . , n.
(2.3)
This lemma generalizes the Lemma 1.1 from [15] and can be proved similarly. We note that if x(t) is a solution of equation (1.1; 5) which is eventually of one sign, then, d -dt- ( L ~ - l x ( t ) ) = a L ,~-1 _ l x (t) L , x (t), and since a satisfies (v), we see that Lnx and Lnx have the same sign. Moreover, one can easily see that Ljx(t), 0 <_j <_ n are eventually of one sign. It will be convenient to make use of the following notation in the remainder of this paper. For any T > to and all t _> T, we let
wt[t,T] =
Ii-l (t,s;al,...,al-1)ae(s)In-~-l
Wl[t,T] = a l (t)
( t , s ; a n - l , . . . , a t + l ) ds,
h_~(t,s;a2,...,al_l)at(s)I,~_l-l
N[t,T]=
rain N~[t,T] 2
and
(t,s;a,~-l,...,ae+l) ds, w[t,T]=
min
l
1
-
i,
2
-
i,
we[t,T].
For t > s > T, we let ~1 [t, s] = al (s)I,~-2 (t, s; a , , - 1 , . . . , as),
~
[t,T] = min{~l It, At], ~ [At, T], 2 < ~ < n - 1, and 0 < A < 1},
w , [t, s] ----In-1 (t, s; a l , . . . , a n - l ) ,
~ n [t,S] : al (t) In-2 (t, s ; a 2 , . . . , a n _ l ) , w__0 It, T] :
max
l<_i
I, ( t , T ; a l , . . . , a ~ ) ,
~o It, s] : In-1 (t, s; a , - 1 , . . . ,
al).
Now, we present the following lemmas.
t >_T/A,
420
R.P.
LEMMA
AGARWAL et al.
Let x E l)(Ln) be eventually positive, (-1)'~-~6 = - 1 and condition (1.3) hold.
2.3.
(il) Then, for 1 < g < n - 1 and a11 t > T > to,
x (t) > w~ [t, T] ( 6 L , _ l x (t)).
(2.4)
(i2) Then, for 2 < l < n - 1 and a11 t >_ T >_ to,
x' (t) >_~[t, T] ( 6 n , _ l x (t)).
(2.5)
(i3) For 1 < l < n - 1, let ), be a constant with 0 < ~ < 1. Then, there exists a T* >_ T/A, such that x' (At) _> ~ a [t, T] (65,_1x (t)), for t > T*. (2.6) PROOF. Let x E 7)(Ln) be eventually positive and ~, 1 < / < n - 2 be the integer assigned to the function x as in Lemma 2.2. From (2.2) with i, k, t, and s replaced b y / , n - 1, s, and t, respectively, we obtain in view of (2.3), n--2
Lex (s) = ~
( - 1 ) ~-~ Ij_~ (t, s; aj, . . . , a~+l) L j x (t)
+(1)
n-t-1
--
>_
it In_g_2(u,S;an_2,...,a~+l)an_l(COLn_lX(~Z)du
(2.7)
J,g
xn_~_s (~, s; an-s,..., a~+l) a~-i (~) (~L~_lx (~)) a~.
Using the fact that 6L~_lX is noninereasing on [tl, oe), for some tl _> to _> O, we obtain
Lex (s) >_ I,~-t-1 (t, s; a ~ - l , . . . , at+l) (6L,~_lx (t)),
for t > s > tl.
(2.8)
On the other hand, using (2.1) with i, k, and s replaced by 0,~ and t~, respectively, and we get t
~-1 X
(2.3),
Ij (t, tl; al,..., aj) Ljx (tl) +/~ I~-i (t, s;at,..., ai-1) a~(s) L~x(s) ds (2.9)
(t) j=O >
h - 1 ( t , s ; a l , . . . , a e _ l ) a ~ (s) L~x(s) ds,
for t > tl.
(2.10)
Substituting (2.8) in (2.10), we obtain x (t) _>
[/i
]
/~-1 (t, s; a l , . . . , a~-l) a~ (s) I , - ~ - 1 (t, s; a n - l , . . . , a~+l) ds ( S L y - i x (t)), for t >_ tl.
The case when g = n - 1 is contained in (2.10). This completes the proof of (il). To prove (is), we let 2 < Z < n - 1. As before, we obtain (2.8). On the other hand, using (2.1) with i, k, and s replaced by 1 , / a n d tl, respectively, and (2.3), we find
F£--1
x' (t) = al (t) [~-~"//(t, t l ; a 2 , . . . , a~) L~x (tl)
+
h-~ (t, ~; as,...,a~-~) a~ (s) L~x (s) as 1
]
h - s (t, s; a s , . . . , a~-l) at (s) L~x (s) ds,
> al (t) 1
Combining (2.11) with (2.8), we obtain the desired inequality.
(2.11) for t > tl.
Functional Differential Equations
421
To prove (i3), we only need to prove the case when g = 1. From (2.2) with i, k, t, and s replaced by 1, n - 1, s, and t, respectively, we have F
(--1) j'-I Ij-1 (t,s;aj,...,a2) Ljx(t)
X' (S)----al (s) / ~
Li=1
z'
+ ( - I F -~
&-a (~, s; a._~.,...,a2) a . _ l (~) L._~x (~) d~
] for t > s > tl.
Using the fact that 5Ln-tx is nonincreasing on [tl, e~), we have
x' (s) >__at (s) I,~-2 (t, s; a ~ - l , . . . , a2) (~L,_lz (t)) = ~ [t, s] (~L~_~z (t)),
for t > s > tl.
For a constant A with 0 < )~ < 1, there exists a T1 >__tl/A, such that
x' (at) > ~1 [t, At] (SL~_~x (t)),
for all t > T1,
(2.12)
for t > T1.
(2.13)
Combining (2.12) and (2.13), we obtain (2.6). This completes the proof.
|
and for 2 < g < n - 1, inequality (2.5) becomes z' (At) > ~ [At, tl] (SL~_~x (t)), LEMMA 2.4. Let x
E
Z)(L~) be eventually positive, condition (1.3) hold and ~ = 0. Then, for
t>s>T>to,
(2.14) and --X / (8) > Wl [t,$] ((--1) n - 1
nn_lX(t)).
(2.15)
PROOF. Let x c T)(L~) be eventually positive and g = 0. From Lemma 2.1, there exists a tl > to, such that (-1) y L j x ( t ) > 0 ,
O<_j
t>tl.
(2.16)
From (2.2) with i, k, t, and s replaced by 0, n - l , s, and t respectively, we obtain in view of (2.16) n--2
x (s) = ~
( - 1 ) j I~ (t, s; aj,..., al) L~x (t)
j=o + (-1) "-1
&-2 (~, s; a~-2,..., al) a._l (~) L~_lx (~) d~ In-2 (u, s; a , - 2 , . . . , al) an-1 (u) L,~-lX (u) du,
>_ ( - 1 ) "-1
t > s > tl.
Using the fact that ( - 1 ) ' ~ - l L n _ l x is nonincreasing on [tl, c¢), we have
x(slzI,_l(t,s;a~_~,...,~l)((-1F-1L~_l~(t)),
fort> s>tl.
Next, from (2.2) with i, k, t, and s replaced by 1, n - 1, s, and t, respectively, we have P
LJ=l
+ (--1) n-1
In-3 (u, s; a n - 2 , . . . , a2) an-1 (u) Ln-lX (u) d
>al(s)I,-2(t,s;a,_l,...,a2)((-1)"-lLn_lx(t)), This completes the proof.
for t > s > t l . |
1~. P. AGARWAL et al.
422
LEMMA 2.5. Let x E ~(L~) be eventually positive, condition (1.3) hold and £ = n. Then, for
t>s>T,
(2.17)
x (t) > w. It, s] L._lx (~) and
x' (t) _> @,~ [t, s] Ln-lx (s). PROOF. Let x E T)(Ln) be eventually positive and let ~ = n. By Lemma 2.2, there exists a tl _> to, such that Ljx (t) > O, 0 <_j <_n and t _> tl. (2.18) From (2.1) with i and k replaced by 0 and n - 1, respectively, and in view of (2.18), we obtain n--2
x (t) = E / J j=o
>_
t
(t, s; a l , . . . , a~-2) Ljx (s) + f
In-2 (t, u; al,..., a,_2) a,~-i (u) Ln_lx (u) du
J8
In-2(t,u;al,...,an-2)ar~-l(u)Ln-lX(U)
>In-l(t,s;al,...,a~_l)Ln-lX(s),
du
fort>s>tl.
(2.19) Once again, from (2.1) with i and k replaced by 1 and n - 1, respectively, we have
Fn--2 x' (t) = al (t) [ E Ij (t, s; a2,..., an-2) Ljx (s) L~=I +
;
___ax(t)
I,_3(t,u;a2,...,a,_2)an_l(u)L,-lX(U)
/'
du
]
I,,_~(t,~;~,...,~,_~)a~_~(~)L~_~(~)
>_al (t) I~-2 (t, s; a2,..., a,-1) L , _ l x (s),
(2.20)
e~
for t > s > tl.
This completes the proof.
|
Finally, we shall denote the sets of all proper solutions, all oscillatory solutions and all nonoscillatory solutions of equation (1.1; 6) by 3, (9, and Af, respectively. It is clear that 8 = O tAAf. It is easy to see that Af has a decomposition, such that JV" = .,V'l u Jr'3 u • • • u H,,_I,
if 6 = 1 and n is even,
Jr" = .A/'o u Jr'2 u • • • u H,_I,
if6=l
Jr" = .,V'o u N'2 u • • • u.,V',~,
if 6 = - 1 and n is even,
N
if 6 = - 1 and n is odd,
= JVl u.A/'3 u • • • u.~,~,
andnisodd,
where Af~ denotes the subset of Af consisting of all x satisfying (2.3). Next, for a function g E C([t0, c~), R), we put
A~ = (t e [to,oo): g(t) > t},
~ = {t e [to,oo): g(t) < t}.
Also, we shall introduce the notation, v(t) = max {min {s,g (s)} : 0 <_ to <_ s <_ t } ,
p(t) = min{max {s,g(s)} : s >_t}.
Note that the following inequalities hold,
g(s) <_T(t), f o r T ( t ) < s < t ,
and
g (s) > p ( t ) ,
for t < s < p ( t ) .
Functional Differential Equations 3. O S C I L L A T I O N
423
CRITERIA
I
In this section, we shall present some criteria for the oscillation of equation (1.1; 5) when the function f satisfies
f :t:c~ fl/adu(u-----~< oo.
(3.1)
According to the subsets of A: for equation (1.1; 6), we shall consider the two separate cases, (il) g E {1,2,..- , n (i2) g = n.
1} and
3.1. Af~ = 0, g E { 1 , 2 , . . . , n -
1}
In this section, we shall provide some sufficient conditions which ensure that Aft = 0. THEOREM 3.1. Let 1 < g < n - 1, ( - 1 ) n - t 5 = - 1 , a > 1, and Conditions (i)-(v), (1.3), (3.1) hold. Moreover, assume that there exist two functions ~ and ~ E Cl([to, c~),]~+), such that a (t) _< inf {t,g (t)},
l i m a (t) = 0%
and a' (t) > 0,
for t > to,
(3.2)
t -"'+ ~
and, for all large T >_to, ~' (t) >_ o where
{
and
~[a(t),T],
w* [a (t), T] =
~
[a (t), T],
/f
j
(
(~' (t))x/~
w* [a (t), T] a' (t)
)
< O,
when g > 2 and a (t ) > T, when g > 1 and, for some constant A, O
oo ~ (s) q (s) ds : 0%
(3.3)
(3.4)
(3.5)
then, N~ = O. PROOF. Let x E ~r, and assume that x(t) > 0, for t > to _> 0. By Lemma 2.2, there exists a tl > to, such that (2.3) holds, for all t _> tl. Now, we consider the following two cases, (I1) g > 2 a n d
(I2) g _> 1. CASE (I1). Assume g > 2. Define
5L~_lx (t)
w (t) = ¢ (t) f (x [o (t)]) '
for t > t2,
for some t2 >__tl.
(3.6)
Then,
W ' ( t ) = ~ ( t ) 5(L~_lx(t))' +~'(t) 5L~ _ ix (t) f (x [a (t)]) f (x [o" (t)])
-- ~ (t) 5 (L~_lX (t))
(3.7)
(x [a (t)]) x' [a (t)] a' (t)
f2 (X [O"(t)])
f (x [g (t)]) + ~, (t) ~ L ~ _ : (t) _< -~ (t) q (t) : (x [~ (t)]) f (x [~ (t)]) SLY_ix (t) ___-~ (t) q (t) + ~' (t) f (~ [~ (t)]) '
for t ___t2.
(3.8)
R.P. AGARWALet al.
424
By Lemma 2.3(i2), there exists a t3 _> t2, such that t >_ a(t) > t3 and
x' [a (t)] _> ~[c~(t) ,t3] (SLn_:(t)),
for t > ta.
(3.9)
Using (3.9) in (3.S), we have
~' (t)
: (~ [~ (t)])
w' (t) < -~ (t)q (t) +
=-((t)q(t)+
{ ~' [o (t)] ~' (t) ] ~ \~-5~~t)
/
p/~(x[~(t)])]'
~[gNit3--~' (t)
for t >_t4,
for some t4 > ta.
Integrating (3.10) from t4 to t, we obtain W (t) < W (t4) -
~ (s) q (s) ds +
\ ~
[a (s), t3] a' (s)
fl/(~ (x [a (s)])
ds
(3.11) _< W (t4) -
f:t, ((s) q (s) ds +
{~
x,lo-(,,)l,,,(s) "
[a (s), tz] a' (s)
fl/~ (x [(7 (s)])
However, by the Bonnet second mean value theorem, for a fixed t >_ t4 and, for some y E [t4,t], we have
~ [O"($), t3] O" (8)
= ~
[O"(t4), t3] O" (t4)
f ' / " (X [a (S)]) (3.12)
{ ((,(_t4))1%)fx[a
[~(t4)]
:/-7--
(u)
-- M,
where M is a positive constant. Using (3.12) in (3.11), we have
fi~(~)q (s) ds _< - W (t) + W (t,) + M%
(3.13)
Letting t ---+ cc in (3.13), we arrive at a contradiction to condition (3.5). CASE (I2). Assume e > 1. Define
w (t) = ~ (t)
5L~_ix (t) f (~ [~o (t)]) '
(3.14)
for some constant A,0 < A < 1 and, for t > t2, for some t2 _> tl//~. Then,
(L~ _ : (t))' + ~, (t) 5L~ - ix (t) W' (t) = ~ (t) f (x [~ (t)]) f (x [~ (t)])
(3.1s) (t) f ' (x [ia (t)]) x' [A(7it)] A~r' (t) -~(t) fi5L~_lx F[~v(T)]) Lg_lx (t)
_< -~ (t) q (t) - ~ (t) f (x bo (t)])"
(3.16)
By applying Lemma 2.303 ) and the procedure of Case ( I J we arrive at the desired contradiction. This completes the proof. | The following result is immediate and so the proof is omitted.
Functional Differential Equations
425
THEOREM 3.2. Let condition (3.3) in Theorem 3.1 be replaced by
(' (t) _> 0
and
f~
as < oo,
w* [a (s), T] a' (s)
(3.17)
where w*[~(t),T] is as in (3.4). Then, the conclusion of Theorem 3.1 holds. Next, we present the following results when Q (t) =
THEOREM 3.3.
q (s) ds < o0.
(3.18)
1, (-1)n-e5 = - 1 , Conditions (i)-(v), (1.3), and (3.1) hold, and suppose that there exists a function a 6 Cl([t0, oo),R+), such that (3.2) holds with a'(t) >_ O, for t >_to. if, for a11 large T _> to, Let
I < ~ < n -
f
oo w* [~r (s),T]
o.!
(s) QUa (s) ds = co,
(3.19)
where w*[a(t),T] is as in (3.4), then, Af~ -- 0. PROOF. Let x E Aft and assume that x(t) > 0, for t _> to _> 0. As in the proof of Theorem 3.1, we consider the cases, (I1) g _ > 2 a n d
(I2) e > 1. CASE (I1). Suppose l _> 2. Define W(t) as in
f*i q (s)
(3.6)
with ~(t) -- 1. Then, we obtain
6LT~_: (t~)
ds < f (x [a (t2)])'
and hence, for any t > t2,
Q (t) < 5L~_lX (t) : (~ [~ (t)]) or
5Ln-lx (t)
(3.20)
Q~/~ (t) <_f~/~ (~ [~ (t)])" Using (3.9) in (3.20), we have
~[a(t),t3]cr'(t)Q1/~(s) ds <_
x'[a(t)]cr'(t) t
fl/~ (~ [~ (t)]) '
_> t3 ~ t2.
(3.21)
Integrating (3.21) from t3 to t, we obtain
f,~
f,~ f,i x'[~(s)]o'(s) dE= [~t~c~)l dv ~3~ [° (s), t3] ~' (s) Q1/~ (s) ds < ix/. (x [~ (s)]) Jxt~c~3)l11/~ (~) ~
dv
< x[~(t3)] fl/"--""-Z~ (v) <
(x),
which contradicts condition (3.19). CASE (I2). Suppose £ > 1. Define W(t) as in (3.14) with ~(t) = 1. Then, as in the above proof one can easily see that, for all t > t2
QI/~ (t) <
5L,~_lx (t)
f~/~ (~ [:~ (t)])"
By applying Lemma 2.3(i3) and proceeding as in the proof of Case (I1) above, we arrive at the desired contradiction. This completes the proof. |
426
R . P . AGARWAL st al.
THEOREM 3.4. Let 1 < ~ < n - 1, (--1)n--t~ = --1, Conditions (i)-(v), (1.3), and (3.1) and (3.18) hold. Moreover, assume that there is a function ~ E Cl([t0,oe),R+), such that condition (3.2) holds with #(t) >_O, for t >_to and let
f' (~) f(~-~)/" (z) > k > o,
~o~ ~ ¢ o.
(3.22)
If, for all large T >_ to,
f
oo (w* [a (s), T] (7! (s))
(~,* [o (~), TI ~' (~)) Q(I+~)/~
× Q (s) +
(u) d~j]
(3.23)
1/or
as =
~,
where w*[o'(t),T] is as in (3.4), and ~={
k,
wheng_>2,
c,
c is any positive constant, when ~ _> 1,
then, A~ = 9. PROOF. Let x E Aft and assume that x(t) > O, for t >_ to >_ 0. As in the proof of Theorem 3.1, we consider the two cases, (I1) g _ > 2 a n d (I2) e _> 1. CASE (I1). Assume e >__2. Define W(t) as in (3.6) with ~(t) = 1 and obtain (3.7) which takes the form, (t) f, (x [a (t)]) x' [a (t)] a' (t) ' t -> t2. (3.24) w ' (t) < - q (t) /~5L~_lx (~ [~ (t)])
Again, as in the proof of Theorem 3.1, case (I1), we obtain (3.9), for t _> t3 >__t2. Using (3.9) in (3.24), we obtain W' (t) < - q (t) - k ~ [a (t), t3] a' (t) W (l+a)/a (t),
for t > t4 _> t3.
(3.25)
Integrating (3.25) from t to v > t and letting v -~ c~, we have
5 ( L , - l x (t)) a >_ f (x [a (t)]) × Q(t)+k
~ [ a ( s ) , t a ] a ' ( s ) W cl+~)/~(s) ds ,
fort>_t4,
and hence, W (t) ___Q (t),
for t >_ t4.
(3.27)
Using (3.9) and (3.27) in (3.26), we obtain
[
x' [a (t)] a' (t) > ~ [~ (t) ta] or' (t) Q (t) + k
/1/~ (x [~ (t)]) -
N [a (s), t3] ~' (s) Q(I+~)/~ (s) ds
Integrating this inequality from t4 to t and using condition (3.1), we obtain a contradiction to condition (3.23). CASE (I2). Assume g > 1. Define W(t) as in (3.14) with ¢(t) = 1. Then,
w ' (t) < - q (t)
(t) f, (x [Aa (t)]) x' [A~ (t)] ;ka' (t) ' /25g~_lx (x [~o (t)])
t >- t2.
By applying Lemma 2.3(i3) and proceeding as in the above case, we arrive at the desired conclusion. This completes the proof. We note that condition (3.22) can be replaced by
f' (x) fO-~)/~ (x) = "r (x), where V(x) is a positive nondeereasing function, for x # 0. Now, we have the following result.
| for x ¢ 0,
(3.28)
Functional Differential Equations
427
THEOREM 3.5. Let condition (3.22) in Theorem 3.4 be replaced by (3.28). If, for all large T > to
and all positive constant -k, condition (3.23) holds, then, Af~ = 0. PROOF. Let x C Af~ and let x(t) > 0, for t _> to _> 0. As in the proof of Theorem 3.4, Case (I1), we see that inequality (3.24) takes the form,
W'(t) <_ - q ( t )
-~[(x[o'(t)])~[cr(t),t3]o-I(t)w(lq-c~)/a(t),
for t _>~t 4 __>t 3.
(3.29)
Since x(t) is increasing on It1, co) and ~/(z) is a nondecreasing function, there exist a constant m > 0 and ts _> t4, such that x [cr (t)] _ m, for t >_ tb. (3.30) Using (3.30) in (3.29), we have W ' (t) _<~- q (t) - ~/(m) w [o" (t), t3] o-! (t) W (1-['°~)/Cz (t),
for t _> tb.
The rest of the proof is similar to that of Theorem 3.4, and hence, is omitted.
|
To prove our next linearization result, we need the following lemma which is given in [4]. LEMMA 3.1. The ha/f-//near equation,
(~ (t) (~' (t))~) ' + q (t) ~ (t) = 0, where ~ is as in (v), ~(t), q(t) e C([t0, co), R+) is nonoscillato~y if and only if there e~ist a number T >_to and a function v(t) E Cl([t0, co), ]~) which satisfies the inequality
~' (t) + ~ - 1 / ~ (t)Iv (t)[ (~+")/" + q (t) _< o,
on [T, co).
THEOREM 3.6. Let 1 < ~ < n - 1, ( - 1 ) n - t 6 = - 1 , Conditions (i)-(v), (1.3), (3.1), and (3.22) hold and suppose that there exists a function a 6 Cl([t0, oo),R+), such that condition (3.2) holds. If, for all large T > to, the second-order half-Iinear equation,
(w.[~(t):~r]~,(t))~
(W(t)) ~
+
q(t)y~(t)=o,
is oscillatory, where w*[o(t),T] is as in (3.4) and ~ is as in Theorem 3.4, then, ~
(3.31) = O.
PROOF. Let x 6 Af~ and assume that x(t) > 0, for t _> to _> 0. Proceeding as in the proof of Theorem 3.4 and considering the two cases, (I1) ~ _ > 2 a n d
(I2) ~ _> I. CASE (I1). Suppose ~ _> 2. Define the function W(t) as in (3.6) with ~(t) = 1 and proceed as in the proof of Theorem 3.4, Case (I1) to obtain (3.25). Now, applying Lemma 3.1 to inequality (3.25), we conclude that the equation (3.31) is nonoscillatory, which is a contradiction. CASE (I2). Suppose ~ > 1. Define the function W(t) as in (3.14) with ~(t) = 1. The rest of the proof is similar to that of Theorem 3.4, Case (I2) and that given above. We omit the details. Similarly, we can obtain the following result. THEOREM 3.7. Let condition (3.22) in Theorem 3.6 be replaced by condition (3.28). If, for all large T _> to and all positive constant k, the equation (3.31) is oscillatory, then, A;~ = 0. REMARK 3.1. According to the above results, if the hypotheses of any of these hold, then, we can easily see the following. (1) (2) (3) (4)
For For For For
even-order equation (1.1; 1), every solution x 6 0 . odd-order equation (1.1; 1), any solution x 6 0 tAAf0. even-order equation (1.1;-1), any solution x 6 0 tAAf0 UAf~. odd-order equation (1.1;-1), any solution x 6 0 U A/',~.
R.V. AGARWALet al.
428 Afn=O
3.2.
We note that this case occurred for equation (1.1; 5) when 5 = - 1 . We shall employ either one of the following conditions on the function f. (il) f satisfies for zy > 0,
--f (--xy) >_f (xy) > f (x) f (y),
(3.32)
or, we assume that 02) inf {f (~x)/f (~) : f # 0} > 0, for any x > 0. For this purpose, we need the function defined by
f*(x) = { sgn xinf{f(~lxl)0, f (5)
:"z
>0~, if x#0,
(3.33)
i f x = 0. It is easy to see that if(x) has the following properties. f* (x) is nondecreasing on R
and
If(~ [xl)l >--If(~)llf*(x)l, We set F(x) =
x f* (x) > 0,
for ~x > 0.
f (x),
if condition (3.32) holds,
f* (x),
if condition (3.35) holds,
and suppose that
+oo
for x # 0,
(3.34) (3.35)
(3.36)
du F(ul/")
< oo.
(3.37)
For more details see [25]. Now, we present the following results. THEOREM 3.8. Let 5 = -1, Conditions (i)-(v), (1.3), and either condition (3.32) or (3.35) hold
and -
X a -
F(x)
-+ 0,
as x --* oo,
(3.38)
where F as in (3.36). If lim sup t -~ o0
f
p(t) q(s)f(w~[g(s),p(t)]) ds >O,
(3.39)
J t
then, A/'. = O. PROOF. Assume that x C Afn and let x(t) > 0, for t _> to > 0. Then, there exists a tl > to, such that L j x (t) > O, 0 < i < n, on [tl,oO). (3.40) From (2.17) with t and s are replaced by g(s) and p(t), respectively, we have
x[g(s)]>_wn[g(s),p(t)]L~_lx[p(t)],
fortl <_t < s < p ( t ) .
(3.41)
Using (3.41) in equation (1.1;-1), we have d
ds (L~-lx (s)) = q (s) f (x [g (s)]) _> q (s) f (w,~ [g (s), p (t)] L,~-xx [p (t)]) > q (s) f (wn [g (s), p (t)]) F (L~_lx [p (t)]), fort1 ~ t < s < p ( t ) .
(3.42)
Functional Differential Equations
429
Integrating both sides of (3.42) from t to p(t), we obtain
L ' ~ _ l x [ p ( t ) ] - L ~ _ x x ( t ) >_
f
p(t)
q(s) f(w~[g(s),p(t)])F(Ln_lx[p(t)])
ds
,It
or
L~_lx [p (t)] > fp(t) q (s) f (Wn [g (S), p (t)]) ds. F (L~_lx [p(t)]) - j,
(3.43)
Taking lim sup of both sides of inequality (3.43) as t --* co and by (3.38), w e obtain a contradiction
to condition (3.39). This completes the proof.
|
THEOREM 3.9. Let 5 = -1, Conditions (i)-(v), (1.3), end either condition (3.32) or (3.35) and
(3.37) hold. A q (s) f (wn [g (s), s]) ds = OC,
(3.44)
g
then, N n = 0. PROOF. Let x e Af,, and assume that x(t) > 0, for t > to _> 0. From (2.17) w i t h t and s be replaced by g(t) and t, respectively, we have
x [g (t)] >_ w, [g (t), t] n~_ix (t),
for t • Ag n [tl, co),
(3.45)
where tl is as in the proof of Theorem 3.8. Set u(t) = LX_~x(t). Then, u(t) satisfies
du d---[= L~x (t) = q (t) f (x [g (t)])
_> q (t) I (w~ [g (t), t] L~_lZ (t)) > q (t) I [9 (t), t]) F (L _lx (t)) =q(t) f(wn[g(t),t])F(ul/~(t)),
fortEAgN[tl,co).
Dividing the above inequality by F(u~/a(t)) and integrating on Ag n arbitrary, we obtain
gn[~;1)T1]
q (s) f (w~ [g (s), s]) ds < -
-
F (,~I/G~ (S))
[t~,Tl],
ds = --
J.(tl)
where TI(_> tl) is
du F (V/~3t) "
Letting T1 ---* ec, we find gfl[tl,CO)
q (s) f (w~ [g (s), s]) ds < --
du
(tl) F (U l / a ) < CO.
This contradicts condition (3.44). This completes the proof.
I
THEOREM 3.10. Let 5 = -1, g(t) > t, for t >_to, Conditions (i)-(v), and (1.3) hold and X c~
f (x)
- * 0,
as x --* co.
(3.46)
If In-2 (g(t),u;al,... ,an-2)an-1 (u)
lira sup t ......+ Oo
q(s) ds
du > O,
J t
then, Afn = 0. PROOF. Let z E Af~ and assume that x(t) > 0, for t _> to _> 0. Then, there exists a tt >_to, such that (3.4o) holds, for t _> tl. Let T1 > tx be, such that inf{g(t) : t _> T1} _> tl. From (2.19) (see the proof of Lemma 2.5) with t and s replaced by g(t) and t, respectively, we have z [g (t)] >_
f
Jt
g(t) I , - 2 (g (t), u; a l , . . . , an-2) an-1 (u) L,_lX (u) du,
for t _> T1.
(3.47)
430
R . P . AGAI~WALet al.
Integrating equation (1.1;-1) from t to u, we obtain
L~_~ (u) - i~_~x (t) = or
L,~-lX (u) k
(I q (s) f (x [g
(s)])
q (~) f (x [g (~)l) d~
ds
,
(3.48)
for u > t > T1.
Using (3.48) in (3.47), one can easily see that x [g (t)]
I ~ l <" (z
[g (t)])
In-2
> -
(g (t), U;al,..., am-z) an-1 (u)
q (s) ds
ds.
.,~
The rest of the proof is similar to that of Theorem 3.8, and hence, is omitted.
I
In the following result, condition (3.32) or (3.35) is disregarded. THEOREM 3.11. Let 5 = - 1 ,
/A
g
Conditions (i)-(v), (1.3), f f (~) u
and
(3.1) hold. If
(3.49)
l/<~
×an_l(u)(/s q(r) dr)
duds=-cc,
then, dkfn -~ O. PROOF. Let x EAf, and suppose that x(t) > 0, for t > to _> 0. Then, there exists a tl _> to, such that (3.40) holds, for t _> tl. From (2.20), (see the proof of Lemma 2.5) with t and s replaced by g(t) and t, respectively, we get
f
x' [g (t)] > al [g (t)] ~t
g(t) 1~_3 (g (t), u; a 2 , . . . , a~-2)
×a~-l(u)n~_lx(u)du,
(3.50)
teAgA[tl,oc).
As in the proof of Theorem 3.10, we obtain (3.48). Using (3.48) in (3.50), one can easily see that
x' [g (t)] g' (t) > ~ [~ (t)] g' (t)
fl/~ (x [g (t)]) ~,"(') f I~-s (g (t), u; 32,... ,a~-2)a~-1 (u)
1/~
q (r) dr)
(3.51)
du.
Integrating (3.51) on A 9 N [tl,T1], where T1 ~ (tl) is arbitrary, we obtain
,n%,T~lal[g(s)lg'(s)S"(~)I,~-a(g(s),u;a2,...,a~-2)a,~-l(u)
q(rldr)l/"duds dx
/ ~[a(T~)l J~[g%)l
(x)
Letting T1 ~ co, we find
,nc<,,~) 31 [g (s)] g' (s)/g(') s~_3 (g (~),~; a~,..., ~_~)a~_1 (~)
(/; q(~) dr) 1/, ~ dx <- L fl/~ < oo. ~
[g(t~)]
This contradicts condition (3.49) and completes the proof.
(x)
I
Functional Differential Equations
431
REMARK 3.2.
By combining the hypotheses of any of the results of Section 3.1 for the equation (1.1;-1) with the appropriate ones of Section 3.2 for equation (1.1;-1), one can easily see the following. (1) For even-order equation (1.1;-1), any solution x E (_9 UAf0. (2) For odd-order equation (1.1;-1), every solution x E O. We can also conclude the following. (3) For odd-order equation (1.1;1) every unbounded solution x E O. (4) For even-order equation (1.1;-1) every unbounded solution x E O.
4.
OSCILLATION
CRITERIA
II
In this section, we shall give some oscillation criteria for equation (1.1; 5) when the function f satisfies f (x) sgn x _> Ix[ s , for x # 0. (4.1) Regarding the subsets of AY for equation (1.1; 5), we shall distinguish the following three separate cases,
(il) £ E { 1 , 2 , . . . , n - - 1}, (i2) / = 0, and (ia) f = n. 4.1. Ale = 0, £ E { 1 , 2 , . . . , n -
1}
In this section, we shall present some sufficient conditions which ensure that AYe = ~. For this, we shall need the following lemma due to [19]. LEMMA 4.1. I f X and Y are nonnegative numbers, then, X x - A X Y ~-1 + ( A - 1)Y~' k O,
A > 1,
X ~ - A X Y ~-1 - (1 - A) Y~ _< 0,
0 < A < 1.
and
In the above inequafities, the equality holds if and only if X = Y .
Now, we prove the following result. THEOREM 4.1. Let 1 _< e _< n - 1, (-1)n-e5 = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold. If there exist two functions a(t),[(t) E Cl([t0, oo),~[+), such that (3.2)holds and, for all large T >_ to,
lim sup
(~., (~))~+1 ~ (s) q (s) - 0
t---~O0
(e (s) ~, (s) ~. [~ (s), T])"
ds = co,
(4.2)
where w*[a(t),T] is as in (3.4) and
0 =
(a + 1) -(a+l) , A-~ ( a + 1)-(~+I),
if ~ _>2, O
ifg_>I,
then, AY~ = 0. PROOF. Let x E AYe and assume that x(t) > 0, for t > to _> 0. By Lemma 2.2, there exists a tl k to, such that (2.3) holds, for all t >_ tl. Now, we consider the following two cases,
(I1) t > z and (I2) t _> 1.
R.P. AGARWAL et at.
432 CASE (I1). Assume g > 2. Define
5Lg_lx (t)
w (t) = ~ (t) ~ [~ (t)] '
for t ___t~ _> t~.
(4.3)
Then, for t > t2, W' (t) : - ~ (t) q (t) f (x [g (t)]) + ~, (t) 5Lg-lx (t) ~- [o (t)] ~ [~ (t)] (4.4)
~L~_~x (t)~o_~ [~ (t)] x' [~ (t)] o' (t) [~ (t)]
- ~ (t) ~
< -~ (t) q (t) + ~' (t) W ('" - aa' (t) ~ (t) 5L~-lx (t) x' [a (t)]. (4.5) ~ (t) '~; ~+~ [~ (t)] By Lemma 2.3 (i2), there exists a t3 _> t2, such that t 2 a(t) >_t2 and (3.9) holds. Using (3.9) in (4.5), we have W' (t) < - ~ (t) q (t) q- ~' (t) W (t) -~ (t) --SO d (t) W [0" (t), t2] ~--l/a (t) W (a+l)/a (t),
(4.6)
for t >_ t3.
Fix t > t3 and set x = ( ~ [o (t), t21 ~' (t)) "/(~+~) ~-~/(~+~) (t) w (t)
,
~
= - -~> + I,1
and
y = ( a.~)a [~'~ (t) (t) ~/:W(~+l) (t) (a~[a(t),t2]a, Now by Lemma 4.1, for t _> t3, we obtain
~' (t) w (t) - ~ [~ (t) , t~] o' (t) w (~+~)/" (t) (t)
(
1 )a+l [
_< ~
(~,(t)~a+l ]-a ~ (t) \ ~ (t) ] ~ [a (t), t21 a' (t)
Thus, (4.6) reduces to
1 (~' (t))~+l (Or q- 1) a+l [~ ( t) w [ff (t),t2] 04 it)] a '
w ' (t) < - ~ (t) q (t) +
for t _> t3.
Integrating the above inequality from t3 to t, we have
0 < w it) _< w (t2) -
( i s ) q is) - 0 [((s) ~' (s) ~ [o (s), t2]] ~
Taking lim sup of both sides of this inequality as t - , c~, we obtain a contradiction to condition (4.2). CASE (I2). Assume g _> 1. Define
6L~_lx (t)
(4.7)
w (t) = ~ (t) x- [ ~ (t)]' for some constant )~, 0 < A < 1 and, for t _> t2, for some t2 _> tl/A. Then,
(t) w (t~ W'(t) _< - ~ ( t ) q(t) + ~' ~(t) 'J - ~'
5L~ ix (t)
(t) ~ (t) x ~+1 - [)~a (t)] z ' [ ~ (t)].
(4.s)
Next, by applying Lemma 2.303 ) and the procedure of Case (I1), we arrive at the desired contradiction. This completes the proof. | For each t > to, we let gi t) < t and define #(t) = sup{s >_ to : g(s) < t}. Clearly, #(t) >__t and g o/z(t) = t. Now, we give the following result.
Functional Differential Equations
433
THEOREM 4.2. Let 1 < ~ < n - 1, (-1)n-~5 = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold. If,
for all large T > to, limsupw~ [t,T]
q(s) ds > 1,
t-~
(4.9)
(t)
then, Af~ = O. PROOF. Let x E All and suppose that x(t) > 0, for t > to >_ 0. By Lemma 2.2, there exists a tl _> to, such that (2.3) holds, for t > tl. Integrating equation (1.1; 5) from t _> tl to u > t and letting u --* o% we obtain
5 ( L n - l x (t)) a ~
q (s) x a [g (s)] ds,
t > tl.
(4.10)
By Lemma 2.3 (il), there exists a t2 _> tl, such that
x (t) >_ wt It, t2] (hL~_lx (t)),
for t ~_ t2.
(4.11)
Thus,
x ~ (t) > w 7 [t, t2] (hLn-lx (t)) ~ >_ w~ [t, t2]
(S
q (s) x ~ [g (s)] ds
)
,
t > t2.
Now, by #(t) > t and the fact that x'(t) > 0 and g(s) > t, for s > #(t), it follows that q (s) x ~ [g (s)] ds
x ~ (t) > w~ [t, t2]
> w~ [t, t2] x ~ (t)
(t)
q (s) ds.
(4.12)
(t)
Dividing both sides of (4.12) by x~(t), we have
w~ [t, T]
(t)
q (s) ds < 1,
for t > t2.
(4.13)
Taking limsup of both sides of (4.13) as t ~ ~ , we o b t a i n a contradiction to condition (4.9). This completes the proof. | We note that inequality (2.4) may take the following form,
x (t) >_ w It, T] (hL~_lz (t)),
for all large T _> to and t _> T.
(4.14)
Clearly, condition (4.9) becomes l i m s u p w " [t,T
t--,oo
lff
(t)
q(s) ds > 1.
(4.15)
In the case of advanced equation (1.1; 5), i.e., g(t) _> t, for t _> to, Theorem 4.2 takes the following form. THEOREM 4.3. Let 1 < g < n - 1, (-1)'~-~5 ---- - 1 , g(t) > t, for t > to, Conditions (i)-(v), (1.3), and (4.1) hold. If, for all large T > to and t > T, lira sup w~ It, T]
q (s) ds > 1,
t---4OO
then, ~ = 0. Next, we present the following theorem when condition (3.18) holds.
(4.16)
434
R.P.
AGARWAL et al.
THEOREM 4.4. Let 1 < e < n - 1, ( - 1 ) ~ - t 5 = - 1 , Conditions (i)-(v), (1.3), (3.18), and (4.1) hold. Moreover, assume that there is a function a C Cl([to, oo), JR+), such that condition (3.2) holds with a'(t) >__O, for t >_ to. If, for all large T >_ to,
w* [~ (s), T] ~' (s) Q(~+I)/~ (s) ds
lim sup w [a (t), T] Q (s) + k t---+OO
> 1,
(4.17)
where w*[a(t),T] is as in (3.4) and
k = ~ a,
when g >_ 2,
[ aA, AE(O, 1), when~>_l, then,
= O.
PROOF. Let x E Aft and assume that x(t) > 0, for t > to >_ 0. As in the proof of Theorem 4.1, there exists a tl _> to, such that (2.3) holds, for t >_ tl and the two cases, (I1) ~ > _ 2 a n d
(h) e _ 1, are considered. CASE (I1). Assume ~ _> 2. Define W(t) as in (4.3) with ~(t) = 1 and as in the proof of Theorem 4.1, we obtain (4.6) which takes the form, W' (t) ~ - q (t) - c~#' (t) ~ [a (t), t2] W (a+1)/a (t),
(4.18)
for t > ta.
Integrating (4.18) from t to u k t and letting u --~ co, we find
5(L---n:-~x(--t)'~ > Q ( t ) + a
\
[o (t)])
~[a(s),t2]a'(s)W(~+l)/~(s)
ds
-
t>t2. '
(4.19)
-
Now, one can easily see that W (t) >__Q (t),
for t >_ t3.
(4.20)
Using (4.14) with T = t2, (4.20) in (4.19), we obtain
1 >w[cr(t),t2]
[
Q(t)+a
~[~(s),t2]a'(s)Q(~+l)/'~(s)
ds
],-
Taking lira sup of both sides of the above inequality as t --* co, we obtain a contradiction to condition (4.17). CASE (I2). Assume g _> 1. Define W(t) as in (4.7). Then, it is easy to see that
W' (t) <_ - q (t) - Aa~a [~ (t), t2] W (~+1)/~ (t),
for t > ta.
The rest of the proof is similar to that of Case (11) above, and hence, is omitted.
(4.21) |
Next, we have the following comparison results. THEOREM 4.5. Let 1 < g < n - 1, (-1)n-e(~ = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold and assume that there exists a function a E Cl([t0, co),]R+), such that condition (3.2) holds
with a'(t) >_ O, for t >_ to. If, for all large T >_ to with a(t) > T, every solution of the retarded first-order equation, y' (t) + q (t) wg [a (t), T l y [a (t)] = 0,
is oscillatory, then, Af~ = O.
(4.22)
Functional Differential Equations
435
PROOF. Let x e Afe and suppose that x(t) > 0, for t > to _> 0. There e x i s t s a t l _> to, such that (2.3) holds, for t > tl. By Lemma 2.3(il), there exist t3,t2 >_tl, such that a(t) > t2, for all t > t3, and x [a (t)] > we [a (t), t2l (SLy_ix [a (t)]),
for t > t3.
(4.23)
Using condition (4.1) and (4.23) in equation (1.1; 6), we obtain
_ s ddt (L~_lx(t))~=
d (SLn-lx (t) ) = q (t) f (x [g (t)]) > q (t) x ~ [a (t)] dt
> q (t)
(t), t:]
(t)]F,
for t _> t3.
Set y(t) = (SLn_lX(t)) ~ > O, t > t3, we get y' (t) + q (t) w~ [a (t), t2] y [a (t)] _< 0,
for t _> t3.
(4.24)
Integrating (4.24) from t > t3 to u and letting u --* c~, we have y (t) _>
q (s) w~ [a (s), ta] y [a (s)] ds,
t >>_ta.
As in [23], it is easy to conclude that there exists a positive solution y(t) of the equation (4.22) with l i m t - ~ y(t) = 0, which contradicts the fact that equation (4.22) is oscillatory. This completes the proof. 1 THEOREM 4.6. Let 1 < £ < n - 1, (-1)n-e5 = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold and assume that there exists a function a 6 Cl([t0, cc),N+), such that condition (3.2) holms. If, for 411 large T >_ to, the second-order equation,
(w*[a(t) i'T]cr'(t))" (y'(t))"
+
q(t)y"(t) = 0 ,
(4.25)
is oscillatory, where w*[a(t),T] is as in (3.4) and k is as in Theorem 4.4, then, Aft = O. PROOF. Let x 6 Are and assume that x(t) > 0, for t > to _> 0. As in the proof of Theorem 4.4, we consider the two cases, (I1) g _ > 2 a n d (I2) e > 1. CASE (I1). Let g >_ 2. Proceeding as in the proof of Theorem 4.4, Case (I1), we obtain inequality (4.18). By applying Lemma 3.1 to inequality (4.18), we conclude that the equation (4.25) is nonoscillatory which is a contradiction. CASE (I2). Let ~ >_ 1. Proceeding as in Theorem 4.4, Case (I2), to obtain inequality (4.21) and applying Lemma 3.1 to (4.21) to obtain the desired conclusion. This completes the proof. 1
4.2. X 0 = 0 This case occurs for equation (1.1; 5) when (1) 5 : 1 and n is odd, and (2) 5 : - 1 and n is even. We shall present some sufficient conditions which ensure that the subset Af0 for equation (1.1; 5) is empty.
436
R. P. AGARWAL et aL
THEOREM 4.7. Let (-1)n5 = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold. If lim sup f t
q (s) w~ [T (t), g (S)] ds > 1,
(4.26)
t-.-*oo J r ( t )
then, Ho = O. PROOF. Let x E No and assume that x(t) > 0, for t > to _> 0. By Lemma 2.2, there exists a t l _> to, such that ( - 1 ) i L~x (t) > 0 (0 < i < n), on [tl, oo). (4.27) Replacing t and s by T(t) and g(s), respectively, in (2.14), we have
x[g(s)]>_Wo[T(t),g(s)l((--1)~-lL~_lx[T(t)]),
fort>s>tl.
(4.28)
Using condition (4.1) and (4.28) in equation (1.1; 5), we find
d ((-1)'~-l L,~_lx(s)) ~ ds -- q (s) f (x [g (s)]) > q (s) x ~ [g (s)]
(4.29)
(
>_ q (s) w~' [r (t), g (s)l (-1) ~-1 g n - l z [r (t)l
,
for t > s > tl.
Integrating both sides of (4.29) from r(t) to t, we obtain
> ((-1) ~-1Ln_lX[r(t)])
o[;
or
((-1)'~-~L,~_~x[r(t)])
(t)
(t)
q(slw~[,(t),g(s)] ds,
]
q(s)w~[r(t),g(s)lds-1 <_0,
for t_> tl.
However, this is inconsistent with (4.26). This completes the proof.
|
REMARK 4.1. If g(t) < t, for t _> to, condition (4.26) becomes lim sup f t q (s) w~ [g (t), g (s)] ds > 1. t-~oo Jg(t)
(4.30)
THEOREM 4.8. Let (-1)n5 = - 1 , g(t) < t, for t >_ to, Conditions (i)-(v), (1.3), and (4.1) hold.
If t~oo Jg(t)
I,~-2(u,g(t);a,,_2,...,al)an_l
(u)
(L
q(~) d~-
du > 1,
(4.31)
then, JV'o = O. PROOF. Let x E JV'o and assume that x(t) > 0, for t :> to _> 0. There exists a t l >_ to, such that (4.27) holds, for all t > t~. Choose a t2 _> tl, such that inf{g(t) : t > t2} > t~. Replacing i, k, t, and s by 0, n - 1, g(t), and t, respectively, in (2.1), we can easily see that x [g (t)l __
((-1)
(t)
I n _ 2 (u, g (t) ; a,~-2, • • •, a~) a,~_~
(~) (4.32)
(u)) du,
for t > g(t) > t2.
Functional Differential Equations
437
Integrating equation (1.1; 5) from u to t, we get ((--1) n - 1 L , ~ _ l x ( u ) ) a >
((-1) n-1Ln_lx(t))a+
~utq(~-)f (x [g (7-)])d'r,
for t > u > t2,
or
((-1)n-iLk-ix(u))>
( ~ t q ( z ) f (x[g('r)]) dT) ~/~
(4.33)
Using (4.1) and (4.33) in (4.32), we obtain x [g (t)] > x [g (t)]
~gt (t)
In-2 (u, g (t) ; a ~ - 2 , . . . , al) a~-i (u)
(~uutq (T) dT)l/c~du,
or t
1>_ ~
t
(t)
In_2(u,g(t);an_2,...,al)an_l
1/a
(U) ( ~ q(T) d~')
du.
Taking limsup of both sides of the above inequality as t ~ ~ , we obtain a contradiction to condition (4.30). This completes the proof. 1 Next, we present the following comparison result. THEOREM 4.9. Let (-1)n5 = - 1 , g(t) < t, for t > to, Conditions (i)-(v), (1.3), and (4.1) hold and assume that there exists a function ~ E C([t0, c¢), R), such that ~'(t) > 0 and g(t) <<_~(t) <_t, for all t >>to. If the first-order delay equation,
y'(t)+q(t)w~ [y(t),g(t)]y[~(t)] = 0 ,
(4.34)
is oscillatory, then, No = 0. PROOF. Let x e No and assume that x(t) > 0, for t > to > 0. There exists a t 1 > to, such that (4.27) holds, for all t > tl. Choose a t2 > tl, such that inf{g(t) : t > t2} > tl. Next, replacing t and s by ~(t) and g(t), respectively, in (2.14), we have
x[g(t)]>_wo[~l(t),g(t)]((-1)n-lL,~-lx[~l(t)]),
fort>71(t)>g(t)>t2.
(4.35)
Using (4.1) and (4.35) in equation (1.1; 5), we have -d-t
(-1)
L~_lx(t)
= q ( t ) f ( x [ g ( t ) ] ) > q ( t ) x a[g(t)] > q(t)w~[~l(t),g(t)] ( ( - 1 ) n - 1 L n _ l x [ ~ ( t ) ] ) ~ , for t > ~/(t) > g (t) > t2.
Set u(t) =- ((-1)n-lL~_lx(t)) ~, t > t2, we have
du(t) - ~ > q(t)w~[~(t),g(t)]u[~(t)], dt
for t > v (t) > g (t) > t2.
The rest of the proof is similar to that of Theorem 4.5 and hence, is omitted. 4.3. A f t = 0 This is the case for equation (1.1; 5) when 5 -- - 1 . In this section, we shall present some sufficient conditions which ensure that the subset flfn for equation (1, 1; - 1 ) is empty.
R. P. AGARWALe t
438
al.
THEOREM 4.10. Let (i = - 1 , Conditions (i)-(v), (1.3), and (4.1) hold. If limsup t --~ oo
[ p(t) q(s)w~ [g(s),p(s)] ds > 1,
(4.36)
dt
then, Afn = ~. PROOF. Let x E Af,, and let x(t) > 0, for t >_ to _> 0. There exists a tl >_ to, such that (3.40) holds, for t > tl. As in the proof of Theorem 3.8, we obtain (3.41). Using (3.41) and (4.1) in equation (1.1;-1), we have ,4 -~- (L~ 1x (8)) = q ($) f (X [g (8)]) > q ( S ) X a [g (S)] ds > q (s) wna [g (s), p (t)] L~_lX [p (t)],
(4.37) for t~ < t < s < p (t).
Integrating both sides of the inequality (4.37) from t to p(t), we obtain
g~,_lZ [0 (t)] _
[ p(t) q(s)w~ [g (s), ;(t)] ds. LT~_lZ[p (t)], Jt
or
;(t) q (s) w~ [g (s), p (s)] ds.
1 _> dt
Taking limsup of both sides of this inequality as t --* c~, we have a contradiction to condition (4.36). This completes the proof. | THEOREM 4.11. Let 5 -- - 1 , g(t) >__t, for t >_ to, Conditions (i)-(v), (1.3), and (4.1) hold. If limsup fg(t ) In-2 (g (t), u; a l , . . . , an-z) an-t (u) ( j u q (T) dr ) 1 / a du > 1, t - ~ oo
(4.38)
Jt
then, Afn = ~. PROOF. Let x cAfn and suppose that x(t) > O, for t > to ~ 0. There exists a t 1 >__to, such that (3.40) holds, for t _> tl. As in the proof of Theorem 3.10, we obtain the inequalities (3.47) and (3.48), for u > t > T1 >_ tl. Now, using (4.1) and (3.48) in (3.47), we can easily see that 1 _> limsup [ g ( O I n _ 2 ( g ( t ) , u ; a l , . . . , a n _ 2 ) a n - 1 t--* c~
J t
(u)
q(T)
tiT) 1/a
dB,
•.
which contradicts condition (4.38) and completes the proof.
|
Finally, we present the following comparison result. THEOREM 4.12. Let 5 = - 1 , g(t) ~ t, fort > to, Conditions (i)-(v), (1.3), and (4.1) and assume that there exists a function ~ e C([to, c~), •), such that ~'(t) >_ 0 and g(t) >_ ~(t) >_t, for t >_ to. If the first-order advanced equation,
y'(t)
=0,
(4.39)
is oscillatory, then, flfn = ~. PROOF. Let x C Af~ and assume that x(t) > 0, for t > to >_ 0. There exists a tl _> to, such that (3.40) holds, for t >__tl. Choose a t2 _> tl, such that inf{g(t) : t ~ t2} > tl. From (2.14) with t and s be replaced by g(t) and ~(t) respectively, we have x [g (t)] _> w,~ [g (t), ~ (t)] L n - l x [~ (t)],
for g (t) _> ~ (t) >__t >__t2.
(4.40)
Functional Differential Equations
439
Using (4.1) and (4.40) in equation (1.1;-1), we get d
ot
~ L ~ _ l x (t) = q (t) f (x [g (t)]) >_ q (t) x ~ [g (t)] :> q (t)wna [g (t),~ (t)] Lan_l x [~ (t)],
(4.41)
for g (t) >_ ~ (t) >_ t __ t2.
Set u(t) = L~_lx(t), t >_ t2, we have
(t)
- dt
> q (t) w~ [g (t) ~ (t)] u [~ (t)], '
for t > t2. -
However, in view of a similar result to Corollary 3.2.3 in [31], equation (4.39) has an eventually positive solution, a contradiction. This completes the proof. | REMARK 4.1. It is known that equation (4.34) is oscillatory if lim inf fv t q (s) w~ [7 (s), g (s)] ds > -1, t-*co (t) e
(4.42)
while equation (4.39) is oscillatory if
liminf
/
t .--.*o o
,It
((t)
1
q(s)wa~ [g(s),~(s)] ds > - .
(4.43)
e
REMARK 4.2. It is easy to see that appropriate combinations of the results of Sections 4.1-4.3 will imply the oscillation of equation (1.1; 5). In fact, one can easily see the following. (1) Any of the results of Section 4.1 with l • {1, 3 , . . . , n - 1} implies the oscillation of evenorder equation (1.1;1). (2) Any of the results of Section 4.1 with g • {2, 4 , . . . , n - 1} together with appropriate results of Section 4.2 implies the oscillation of odd-order equation (1.1;1). (3) Any of the results of Section 4.1 with g • {2, 4 , . . . , n - 2} together with suitable results from Sections 4.2 and 4.3 implies the oscillation of even-order equation (1.1;-1). (4) Any of the results of Section 4.1 with ~ • {1, 3 , . . . , n - 2} together with a suitable result of Section 4.3 implies the oscillation of odd-order equation (1.1;-1).
5. O S C I L L A T I O N
CRITERIA
III
In this section, we shall present some criteria for the oscillation of equation (1.1; 5) when the function f satisfies either condition (3.32), or (3.35) and f ~ o F du
<
where F is as in (3.36). According to the subsets of Af, for equation (1.1; 5), we shall consider the two cases, (il) g • { 1 , 2 , . . . , n 02) g -= 0. 5.1. ~
1} and
= ~,~ C { 1 , 2 , . . . , n -
1}
Here, we shall provide some sufficient conditions which ensure that Af~ = 0.
(s.1)
440
R. P. AGARWALeg al.
Conditions (i)-(v), (1.3), either (3.32) or (3.35) and (5.1) hold. If, for a11 large T >_to with g(t) >_T,
THEOREM 5.1. Let 1 < ~ _< n - 1, (-1)n-~6 = - 1 ,
i
¢~ q (s) f (wt [g (s), T]) ds = c~,
(5.2)
then, Hi = O. PROOF. Let x 6 Aft and assume that x(t) > 0, for t >_ to _> 0. There exists a t l that (2.3) holds, for t >_ tl. By Lemma 2.3 01), there exists a t2 _ tl, such that x [g (t)] >_we [g (t), tl] (6L,_tx (t)),
> to, such
for t >_ t2.
(5.3)
Using (3.32) or (3.35) and (5.3) in equation (1.1; 5), we have d
dt (6L~_lx (t) ) ~ = q (t) f (x [g (t)]) >_ q (t) f (w~ [g (t), tl] (6L~_lx (t))) > q (t) f (w~ [g (t), tl]) E ( ~ n . _ l x (t)),
for t > t~.
Substituting u(t), for (6L=_lx(t)) ~, t > t2, we have i du ~~ > q ( t ) f ( w l [ g ( t ) , t l l ) F ( u l / " ( t ) )
dt
-
for t > t2.
'
-
(5.4)
Dividing both sides of (5.4) by F(ul/a(t)) and integrating from t2 to t, we have q (s) f (w~ [g (s), tl]) ds <_
F (uX/~ (s))
J~(t)
f (-~T/~)"
Letting t --~ oc, we conclude that
/;
q(s) f (w~ [g(s),tl]) ds < 1 ~(~2) -- J0
du
F ( u l / ° 0 < (x),
which contradicts condition (5.1). This completes the proof. REMARK 5.1. By (3.35) with ~ = 1,
/0 s (x~io~
|
f(x) > f(1)F(x), for x > 0, it follows that
-<
F (~,i<,)'
Thus, (5.1) yields
S
for m > o.
du
-t-~
f (~1~) < o~.
THEOREM 5.2. Let 1 < ~ < n - 1, (-1)n-~6 = - 1 ,
Moreover, assume that there is a function a with a'(t) >_O, for t >_to and assume that
E
(5.5)
Conditions (i)-(v), (1.3), and (3.18) hold. such that condition (3.2) holds
Cl([to, oo), ]~+),
f (x) sgn x _> Ixl ~ ,
for ~ # o,
(5.6)
where/~ < a is a quotient of positive odd integers. If, for every constant c > O, and all large T >_to,
lim~_~supQ1/~ (t) w [a (t), T] 1 + ~ c
w* [a (s), T] a' (s) Q(l+~)ln (s) ds 11~ = oc,
(5.7)
where Q(t) > 0, fo~ t > to, ~* [Ht), T] is as i . (3A), then, ~ = ~. PROOF. Let x EAft and assume that x(t) > 0, for t > to >_ 0. There exists a t 1 _> to, such that (2.3) holds, for t _> tl. Now, there are two cases to consider, (I1) ~ _ > 2 a n d (I~) ~ > 1.
Functional Differential Equations
441
CASE (I~). Assume ~ k 2. Define
w (t) =
5L~_~x (t)
t>_tl.
• , [~ (t)] '
Then, for t > t2, we have
5L~_lx (t) , W' (t) <_ - q (t) - fla' (t) - ~ - ~ (-~)].x [a (t)],
for t k to.
As in the proof of Theorem 3.1, we obtain (3.9), for t >__t3 _> t2. Thus,
w ' (t) <_ - q (t) - ~-~ [~ (t) , t~] ,~' ( t) w ( ~+~)/"
(t) ~(~-")/~
[~ (t)],
for t k t4 k t3.
(5.8)
Integrating (5.8) from t k t4 to u and letting u -~ oo, we find
L'~_lX (t) ~ x ~ [(7 (t)] x Q (t) + ~
[a (s)]
~ [c~ (s), ta] ~' (s) w (~+~)/~ (s) x (~-~)/~
ds ,
for t >__t4,
(5.9)
and hence, W(t) >_ Q(t), for t _> t4. There exist a constant el > 0 and a ts _> t4, such that
5L,_lx (t) < el,
(5.10)
for t > ts.
For t _> ts, it follows from (5.9) and (5.10) that
x B/c~ [a (t)] < elQ -1/~ (t)
or
x [Cr(t)] _< Cl/~Q -1/~ (t),
and hence,
x (~-~)/~ [a (t)] __kc~-~)/~Q (~-~)/~ (t),
for t >__ts.
(5.11)
Using (5.11) in (5.9) yields
aLn_lX (a-~>/a [a (t)] ~ x [a (t)] [
× Q (t) + ~c~~-'~)/f~
/too
~ [a (s), t3] a' (s) Q(~+I)/~ (s) ds
]1/,
(5.12)
]~/,~
(5.13)
Using (4.14) with T = t3 in (5.12), we have • (--~)/- [~ (t)] _> ~ [o (t),
[
× Q (t) + ~c~ ~-~)/~ Again, using (5.11) in (5.13) yields
f~
t3]
~ [~ (s), t~] o' (s) Q(~+I)/~ (s) ds
[ ]~C~fl_,:x)/fl/o °
]l/a
Taking limsup of both sides of this inequality as t ~ c~, we obtain a contradiction to condition (5.7). CASE (I2). Assume g > 1. Define
(t) w (t) = 5L~_lx ~ [ ~ (t)]'
for t > t2
and proceed as in Case (11) we arrive at the desired conclusion.
|
442
R . P . AGARWAL et
al.
THEOREM 5.3. Let i < g < n - 1, ( - 1 ) " - t 5 = - 1 , Conditions (i)-(v), (1.3), and (5.6) hold. In addition, suppose that there exists a function (7 E Cl([t0, oo),]~+), such that condition (3.2) holds. If, for every constant c > 0 and all large T :> to, the second-order half-linear equation,
(w* [(7 (t),T] a' (t)) ~ (y' (t))"
+ cq (t) y" (t) = O,
(5.14)
is oscillatory, where w*[a(t), T] is as in (3.4) and
P=
I,
if t _> 2,
A E (0, i),
ifi>
i,
then, Af~ = 0. PROOF. Let x 6 Aft and assume that x(t) > 0, for t >_ to >_ 0. As in the proof of Theorem 5.2, we consider the two cases, (I1) ~_>2 and
t > 1. CASE (I~). Assume ~ _> 2. Proceeding as in the proof of Theorem 5.2, Case (I~), and obtain (5.8), for t > t 4. It is easy to see that there exist a t5 _> t4 and a constant cl > 0, such that
X [(7 (t)] _< ClW___0[(7 (t), t4],
(5.15)
for all t _> t5.
Using (5.15) in (5.8), we have
w ' (t) _< - q (t)
-pcl (~-~)1~ (~-~)1~ [a (t) , t4] ~ [a (t) , t4] (7' (t) W (1+~)1~ (t),
(5.16) for t _> ts.
By applying Lemma 3.1 to inequality (5.16), we conclude that the equation (5.14) is nonoscillatory, which is a contradiction. CASE (I2). Assume t >_ 1. Define W(t) as in the proof of Theorem 5.2, Case (I2) and proceed as in the proof of Case (I1), we arrive at the desired conclusion. This completes the proof. | 5.2. 2 5 0 = 0 This case occurs for equation (1.1; 5) when (1) 5 = 1 and n is odd, and (2) 5 = - 1 and n is even. We shall provide some sufficient conditions which ensure that the subset 25o for equation (1.1; 5) is empty. THEOREM 5.4. Let (-1)~5 -- - 1 , Conditions (i)-(v), (1.3), and either (3.32) or (3.35) hold and let ~c~ - -
F
If limsup t-~¢¢
~
--+ 0 ,
as u ~
oo.
(5.17)
t (t)
q(s) f ( w o [T(t) ,g (S)]) ds > O,
(5.18)
then, Afo = O. PROOF. Let x E 25o and assume that x(t) > 0, for t >_ to >_ 0. There exists a t 1 >_ to, such that (4.27) holds, for all t _> tl. Replacing t and s by ~'(t) and g(s), respectively, in (2.14), we obtain (4.28), for t > s > tl.
Functional Differential Equations
Using condition (3.32) or
(3.35)
443
and (4.2s) in equation (1.1; 5), we find
d ((_1),~_i L~_lx(s)) a =q(s)f(x[g(s)]) ds >__q (s) f (w0 [r (t), g (s)] ((--1) n Ln-lX [7- (t)])) _> q (s) f (w0 [T (t), g (s)]) F ( ( - 1 ) ~ Ln-lx Integrating (5.19) from 7-(t) to t and setting
u~[r(t)]
>
F (~ [7- (t)]) -
(t)
(5.19)
[7-(t)]).
u(t) = ((-1)'~L,~_lx(t)), we obtain q(s)f(wo[7-(t),g(s)]) ds.
Taking limsup of both sides of this inequality as t -~ e~, we obtain 0 _> lira sup
ft
t-*oo Jr(t) q (s) f
(w0 [7- (t),g (s)]) ds,
which contradicts condition (5.18). This completes the proof.
|
THEOREM 5.5. Let (-1)n6 = - 1 , Conditions (i)-(v), (1.3), (5.1), and either (3.32) or (3.35) hold. If
Te q (s) f (Wo [s,g (s)]) ds = oc,
(5.20 /
g
then, N o = @. PROOF. Let x 6 Af0 and assume that x(t) > 0, for t > to _> 0. There exists atl > to, such that (4.27) holds, for t >_ tl. Choose a t2 >_ tl, so large that t2 >_ tl and inf{g(t) : t >_ t2} > tl. Replacing t and s by t and g(t), respectively, in (2.14), we have
x[g(t)l>_wo[t,g(t)]((-1)n-lL,~_lx(t)),
for t e 7~g N [t2, c~).
Using (5.21), condition (3.32) or (3.35) in equation (1.1; (f) and letting > 0, for t > t2, we have
(5.21)
u(t) = ((-1)n-lL~_lx(t)) ~
du(t) at = q (t) S (x[g (t)]) _> q (t) f (~o [t,g (t)] u 1/" (t) ) >q(t) f(wo[t,g(t)])F(ul/"(t)),
forteUgn[t2,
oc ).
Choose a t3(> t2) arbitrarily. Dividing both sides of the above inequality by integrating over T~g n [tz, ta], we find
fn
q (s) f (wo [s,g (s)]) ds < ftl ~
u' (s)
ds = -f~(")
F(ul/~(t)) and
du
Letting t3 --* c~, we conclude that
fT¢.~nIt2,o~) q(s) f
(wo [s,g(s)]) ds < -
fu(tZ) F (ul/~) du
< c~,
.,o
which contradicts condition (5.20/. This completes the proof.
|
444
R.P. AGARWAL et al.
THEOREM 5.6. Let (-1)n(~ =
- 1 , Conditions (i)-(v), (1.3), and (5.17) with F be replaced by f
hold. H g(t) <_t, for t >_to and t
lira sup f~
t
i/c~
In_2(u,g(t);an_2,...,al)an_l (U) (~u q(T) dT)
du > O,
(5.22)
then, Afo = 0. PROOF. Let x EN'o and assume that x(t) > 0, for t_> to >_ 0. There exists a t l _> to, such that (4.27) holds, for all t >_ tl. Choose a t2 _> tl so large that t2 _ tl and inf{g(t) : t >_ t2} > tl. As in the proof of Theorem 4.8, we obtain (4.32) and (4.33). Now, by using (4.33) in (4.32) yields
x[g(t)] >_fl/~(x[g(t)])
/:
(t)
I~-2(u,g(t);an-2,...,al)an_l(u)
(//
q(T) dT/1/~ du.
The rest of the proof is similar to that of Theorem 5.4 and hence, is omitted.
|
THEOREM 5.7. Let (-1)~5 = - 1 , Conditions (i)-(v), and (1.3) hold and let g(t) <_t, for t > to,
and
(5.23)
0 fl/a (%~-------~ < 00. If
f ~ al [g(s)] g'(is)/ u S ) l / a
(~s (s)
I~-3 (u,g (s);an-2,.
,a2) a=-i (u)
q (T) dT
(5.24)
) du
ds = c~,
then, No = 0. PROOF. Let x E N0 and let x(t) > 0, for t > to _> 0. There exists a tl _> to, such that (4.27) holds, for t > tl. Choose a t 2 _> tl so large that t2 _> tl and inf{g(t) : t _> t2} > tl. Next, replacing i, k, t, and s by 1, n - 1, g(t), and t, respectively in (2.2), one can easily find
xt[g(t)] ~ax[g(t)]
(t)
In-3(u,g(t);an-2,...,a2)an-l(U)
(-1)n-lLn_lX(U)
du.
As in the proof of Theorem 4.8, we obtain (4.33). Thus, x' [g (~ (t)][gg'(t)]) (t)>:i/~
t t 1/a ax [g (t)] g' (t)fg (~) In-3 (u,g(t);an-2,..., a2)an-1 ( u ) ( f q ( r ) d ' r ) du.
The rest of the proof is similar to that of Theorem 5.5 and hence, is omitted.
|
Finally, we present the following comparison result. THEOREM 5.8. Let (-1)'~6 -----1, g(t) < t, for t >_to, Conditions (i)-(v), (1.3), and either (3.32) or (3.35) hold. Moreover, assume that there exists a function r/ E C([t0, oo),N), such that
r/(t) >_0 and g(t) <_~(t) <_t, for all t >_to. If the first-order delay equation,
y' (t) + q (t): (wo [,7 (t), g (t)]) F (y'/° [,7 (t)]) = o, is oscillatory, then, A[o
=
(5.25)
O.
PROOF. Let x E A/o and let x(t) > 0, for t _> to _> 0. As in the proof of Theorem 4.9, we obtain (4.35). Next, by using condition (3.32) or (3.35) and (4.35) in equation (1.1; 5), we have d ( ( _ l ) n _ 1 L n _ l x ( t ) ) ~ = q (t) f (x [g (t)])
dt
(5.26) > q (t) f (wo [7 (t), g (t)]) F ( ( - 1 ) ~-1 n n - l x [~?(t)]) ,
Functional Differential Equations
445
for t > y(t) > g(t) > t2. Set u(t) = ( ( - 1 ) n - l L n _ l x ( t ) ) % t >_ t2. Then, (5.26) takes the form,
_du(t) >q(t)f(wo[rl(t),g(t)])F(ul/=[rl(t)] ) dt '
for t > t2.
-
The rest of the proof is similar to that of Theorem 4.5, and hence, is omitted. This completes the proof. II REMARK 5.1. As before, appropriate combinations of the results of Sections 5.1 and 5.2 will imply the oscillation of equation (1.1; 6). Now, we have the following. (1) Any of the results of Section 5.1 with £ E {1, 3 , . . . , n - 1} implies the oscillation of evenorder equation (1.1; 1). (2) Any of the results of Section 5.1 with £ E {2, 4 , . . . , n - 1} together with appropriate results of Section 5.2 implies the oscillation of odd-order equation (1.1; 1). (3) Any of the results of Section 5.1 with £ E {1, 3 , . . . , n - 2} implies that any solution x(t) of odd-order equation (1.1;-1) belongs to the subset O UAfn. (4) Any of the results of Section 5.1 with ~ E {2, 4 , . . . , n - 2} together with appropriate ones of Section 5.2 implies that any solution x(t) of even-order equation (1.1;-1) belongs to the subset O U Afn.
6. L I N E A R I Z A T I O N
OF N O N L I N E A R
OSCILLATION RESULTS
By combining Theorems 3.6, 4.6, and 5.3, we obtain the following result. THEOREM 6.1. Let 1 < £ < n - 1, (-1)'~-~6 = - 1 , Conditions (i)-(v), and (1.3) hold and f (x) sgn x > [ x f ,
for x ~ 0,
(6.1)
where/3 is the quotient of positive odd integers. Moreover, suppose that there exists a function a E Cl([t0, oo),R+), such that condition (3.2) holds. If, for all large T >_ to >_ O, the half-linear differential equation,
(t), T]
(t)) @'
+ q (t)
it) ; o,
(6.2)
is oscillatory, where r (t) =
cl,
cl > 0 is any constant, when/3 > a,
(a/k) ~ ,
k is as in Theorem 4.4, when fl = a,
c2_~ -~ [va (t), T],
c2 > 0 is any constant, v is as in Theorem 5.3, when/3 < a,
then, Aft = 0. Next, we shall proceed further and obtain similar results as above, but instead of comparing with oscillation of half-linear equation (6.2), we shall compare with the oscillation of certain second-order linear ordinary differential equations. To obtain our next results, we shall need the following lemma given in [34]. LEMMA 6.1. Let r 6 C([T,c~),R+), T >_ to. If there exhsts a function w E CI([T,c~),R), such
that q (t) _< - w ' (t) - r (t) w 2 (t),
for every t _> T,
then, the second-order linear ordinary differential equation,
(x'
(t)
r (t))
is nonoscillatory. Now, we present the following theorem.
+ q (t) x (t) = o,
446
R . P . AGARWALe t
al.
THEOREM 6.2. Let a > 1, 1 < £ < n - 1, ( - 1 ) ~ - t 5 = - 1 , Conditions (i)-(v), (1.3), and (6.1) hold. Moreover, suppose there exists a function a E Cl([to, ce),R+), such that condition (3.2) holds. Then, Aft = 0, if there exists a constant A E (0, 1) when g >_ 1 and, for a//large T > to _> 0,
the second-order linear differential equation, !
i
(63)
is oscillatory, where ~t(t) = A a ' ( t ) w * [ o r ( t ) , T ] w ~ - l [ a ( t ) , T ] ,
a(t) > T,
w*[a(t),T] is as in Theorem 3.1 and A E (0,1), i[£ >_ 1, A = 1, iY£ ~ 2 and
(t) :
el,
cl > 0 is any constant, when fl > a,
1,
when fl = a,
c2_~ -~ [Aa (t), T],
c2 > 0 is any constant, when/3 < a.
PROOF. Let x E Af~ and assume that x(t) > 0, for t > to >_ 0. T h e r e exists a t l ~ to, such that (2.3) holds. Now, we consider the two cases, (il) l _ > l a n d
(i~) t ___2. CASE (I1). Assume g > 1. Define
5Lg_lx (t) w (t) - x~ [ ~ (t)]'
for t ~ tl and some A E (0, 1).
Then, for t _> tl, we have
~' [~or (t)] xB-1 [~or (t)].
w ' (t) ~_ - q (t) - A~or' (t) w 2 (t) - 5 L ~ _ l x (t)
(6.a)
From Lemma 2.3, it follows that there exist a constant A E (0, 1) and a t2 > tl/A, such that x' [Ao"(t)] ~ WX [o" (t), All (SLn-lx (t))
(6.5)
and
x [Aor (t)] > w [or (t), All ( S L n - l X
(t)).
(6.6)
Using (6.5) and (6.6) in (6.4), we have
W ! (t) __~ - q (t) - ~flort (t) wA [°r (t), All w (~-1 [(7 (t),tl] W 2 (t) x f~-a [~or (t)],
for t > t2,
or
w ' (t) _< - q (t) - ~ a (t) w 2 (t) x ~ - " [~or (t)],
for t >_ t2.
As in the proof of Theorem 3.6 when 13 > a and Theorem 5.3 when ~ < a, one can easily see that W' (t) _< - q (t) - f l ~ W 2 (t), for t _> t2. (6.7) Applying Lemma 6.1 to inequality we conclude that equation (6.3) is nonoscillatory, which is a contradiction. CASE (I2). Assume g _> 2. Here, we define W(t) as in Case (iz) with A = 1. The proof will be the same, and hence, we omit the details. | When a E (0, 1], we shall present the following criterion.
Functional Differential Equations
447
THEOREM 6.3. L e t 0 < a _< 1, 1 < ~ < n - l , ( - 1 ) ~ - t 5 ----- 1 , Conditions (i)-(v), (1.3), and (6.1) hold, Q(t) > 0, for t >_ to. Moreover, assume that there exists a function a(t) E Cl([t0,(:x)), ~+), such that (3.2) holds. Then, Af~ = 0 is there exists a constant A 6 (0, 1) when £ > 1 and, for all large T > to > O, the second-order linear equation,
(t)
(t)
+
(t) y (t) = o,
(6.8)
is oscillatory, where ~x (t) ----Aw* [a (t), T] Q(1-a)/~ (t) a' (t),
a (t) > T,
w*[a(t),T] is defined as in Theorem 3.1 and A C (0,1) if£ _> 1, A = 1 ifg > 2 and rl (t) =
Cl,
cl > 0 is any constant, when fl > a,
1,
when j3 = a,
c2w_~(~-~)/~ [Aa (t), T],
c2 > 0 is any constant, when ~ < a.
PROOF. Let x E Af~ and assume that x(t) > 0, for t > to > 0. As in the proof of Theorem 6.2, we obtain (6.4) which takes the form, W' (t) < - q (t) - A~a' (t) @:~ [a (t), tl] W 2 (t) W (1-a)/a (t) x (~-'~)1~ [Aa (t)],
t > tl.
(6.9)
As in the proof of Theorem 3.3, we have W (t) _> Q (t) > 0,
for t __ tl.
(6.10)
Using (6.10) in (6.9), we have W' (t) _~ - q (t) - After' (t) @x [~r(t), tl] Q ( 1 - , ) / , (t) W 2 (t) x (~-a)/`~ [Aa (t)],
t > tl.
The rest of the proof is similar to that of Theorem 6.2, and hence, is omitted. | Next, we shall provide proof of Theorem 4.1 when a > 1 and a C (0, 1] without employing Lemma 4.1. THEOREM 6.4. Let a > 1 and the hypotheses of Theorem 4.1 hold and condition (4.2) is replaced
by limsupft[~
(s) q (s) - ( 1 )
Jr
(~'(s))2
1 .]
a(s)
ds
=
(6.11)
where ~t(t) is defined as in Theorem 6.2. Then, Aft = 0. PROOF. Let x E Af~ and assume that x(t) > 0, for t _ to _> 0. Define 5Lan_l x (t) W (t) = ~ (t) x [Aa (t)] '
for t >_ t~ > to.
Proceeding as in the proof of Theorems 4.1 and 6.2, one can easily find W'(t) <_ -~(t)q(t) + - ~ W ( t )
..
- aA
~[cr(t),tl]w"-l[a(t),tl]W2(t),
for t > t2 >_T/A, for some A E (0, 1), or W' (t) _< - ~ (t) q (t) + ~ - - ~ W (~) - E - ~ t (t)
(t) 2
<-
[~(t) q(t)
4a
((t)
1 ] ' f°rt>-t.
a(t)
The rest of the proof is similar to that of previous results, and hence, is omitted.
|
R. P. AGAR,WAL
448
THEOREM 6.5. Let 0 < a _< 1 and the
et al.
hypotheses of Theorem 4.1 hold, Q(t) > O, and condi-
tion (4.2) is replaced by
1 (~'(s))2 limsupt_~ooJf T*[ ~(s) q ( s ) - 4a ~-(~) ~;l'(s) 1 ds = oo,
(6.12)
where f~l(t) is defined as in Theorem 6.3. Then, Aft = 0. PROOF. Let x e Ale and assume that x(t) > 0, for t > to >_ 0. Define the function W(t) as in the proof of Theorem 6.4 and proceed as in the proof of Theorems 4.1 and 6.3, to obtain
~' (t) w' (t) < -~ (t) q (t) + T s ~ w (t) - ~A~' (t) ~
[~ (t), tl] ¢-1/~
(t) w ~ (t) w (1-~)/" (t),
for t > t~
and
w (t) > ~ (t) Q (t),
for t > t2.
The rest of the proof is similar to that of Theorem 6.4, and hence, is omitted.
7. O S C I L L A T I O N
|
CRITERIA
From the previous results, we shall state as a sample some oscillation criteria for equation (1.1; 5). The formulations and statements of other results are left to the reader. THEOREM 7.1. Let n be even and assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function ~ E Cl([t0,oo),R+), such that (3.2) holds. (Sl). Suppose that
f 4-~ f~/~ du(~-----<---~o0 and there exists a function ~ E Cl([t0, co), R +) and a constant A E (0, 1), such that for all large T _> to >_ 0, ~' (t) > 0 and
(
(t))l/a ~;~ [a(~'(t), T] a' (t)
Then, the condition,
/
)'
_ O,
for t > cr (t) > T/A.
~ ~ (s) q (s) ds = co,
is sufficient for equation (1.1;1) to be oscillatory.
(s2). Suppose that f(x)sgn x >__Ixl", for x # O. If there exist a function ~(t) E Ct([t0, co), R +) and a constant A E (0, 1), such that for all large T > to with a(t) > T/A, either
r
lim sup f] [~ (s) q (s) t--*c~ JT
(," (s)F +:
~-~
(a-~- 1) a+l (~(S) O't(S)~X [a(s),T])"
ds = co,
or, a > 1 and
/:I ~(s)q(s)
limsuPt__.~
4aA ~(s)
1
a'(s)~x[a(s),Tlw~-l[a(s),T]
or, a < 1 and
limsupt~Jr [ ((s)q(s)-~
~,. -~-~
] ~'(s)~,[~(s),T]Q(l-~)/~(s)
where Q(t) > O, then equation (1.1;1) is oscillatory.
ds=co,
Functional Differential Equations
449
(S3). Suppose that either (3.32) or (3.35) holds and
f± o F (~1/~) du < oo, where F is as in (3.36). Then, for all large T _> to, the condition,
f
oo q (s)
f (w[g (s), T]) ds
0%
is sufficient for equation (1.1;1) to be oscillatory. (s4). Suppose that a > 1 and f(x)sgn x > [xlZ, for x # 0, where/~ is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if there exists a constant A E (0, 1), such that for all large T _> to _> 0, the second-order linear equation, !
r(t) h a ! (t)W~ [a (t), T] w ~-1 [a (t), T]
y' (t)
+ Zq (t) y (t) = 0,
is oscillatory, where r(t) is as in Theorem 6.2. (sh). Suppose that 0 < c~ < 1 and f(x)sgn x > Ix[~, for x ¢ 0, where fl is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if there exists a constant A 6 (0, 1), such that for all large T _> to _> 0, the second-order linear equation,
( ~ , (t) : ~ [~ (t), rl(t)T] Q(:-~)/~ (t)y,(t))' + ~q (t) y (t) = 0, is oscillatory, where rl(t) is as in Theorem 6.3 and Q(t) = f~o q(s)ds > O. THEOREM 7.2. Let n be odd and assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function a 6 Cl([t0, oo), R+), such that (3.2) holds. (ol). Suppose that g(t) <_t, for t > to, and
/:
o~
o
du f l / ~ (~----~< oo.
Then, for all large T >_ to with a(t) >_ T, the conditions,
/ where
ioo
al [g (s)] g' (s)
~
[o(s),T]
G !
(s) Q1/~ (s) ds = o o ,
(~. (~)
fn-3 (u, g (s) ;an-2,..., a2) an-1 (u)
(~s >l/a) q (T) dT
du
ds = c~
are sufficient for equation (1.1;1) to be oscillatory. (02). Suppose that f(x)sgn x _> Ix[~, for x ¢ 0. Then, for all large T > to with a(t) > T, the conditions,
[
lim sup w [a (t), T] Q (t) + a
~ [a (s), T] a' (s) Q(l+a)/~ (s) ds
$---~OO
where
lim sup t-~oo
//
(t)
q (s) w~ [~- (t), g (~)1 as > 1
axe sufficient for equation (1.1;1) to be oscillatory.
].°
> 1,
R.P. AGARWALet al.
450
(03). Suppose that either (3.32) or (3.35) holds and
du o F (~/") < o0, where F is as in (3.36). Then, for all large T > to with g(t) > T, the conditions, oo q (s) f (w[g (s), T]) ds = oo and limsup fl ~ q(s) f (wo [7(t),g(s)]) ds > 0, t ---*oo J r ( t )
are sufficient for equation (1.1;1) to be oscillatory.
(04).
Suppose that f(x)sgn x >_ Ix[~, for x ¢ 0, where fl is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if, for all large T >_ to with a(t) > T, the half-linear second-order equation,
(
(t)
[~ (t), T] ~' (t)
(y' (t))"
)'
+ q (t) y" (t) = 0,
is oscillatory, where r(t)=/
1,
[ c2w__c*--/3 0 [a (t), T], and lira sup t-~oo lira sup t-~oo
~
whenfl=a, c2 > 0 is any constant, when B < a,
t
(0
//
(t)
q (s) w~ [T (t), g (S)] ds > 1,
when fl = a,
q (s) wg [r (t), g (s)] ds > 0,
when fl < a.
(05). Suppose that a >__1 and f(x)sgn x > Ix]z, for x # O, where ~ is the quotient of positive odd integers. Equation (1.1;1) is oscillatory if, for all large T > to with a(t) > T, the linear second-order equation,
)'
+Zq(t)y(t) =o,
q (s) w~ [r (t), g (s)] ds > 1,
when ~3 = a,
q(s)wg[r(t),g(s)J ds > O,
when ~ < a.
(~[o(t),T]w--l[o(t),T]o,(t) r (t)
y,
(t)
is oscillatory, where r(t) is as in (04) and t
lim sup t--*oo
Jr(t)
limsup f t t-~oo
Jr(t)
THEOREM 7.3. Let n be odd and assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function a e Cl([t0, cx~),R+), such that (3.2) holds. (31). Suppose that
f
±oo
du
f~l. (~---~< ~ '
Functional Differential Equations
451
and there exist a function ~ E Cl([t0, oo), R +) and a constant A E (0, 1), such that, for all large T _> to with ~(t) > T,
~' (t) _> O,
for t _> to,
N~ [a (s), T] a' (s)
and
Then, the conditions,
/
o~ ~ (s) q (s) ds = cx~
and
L al[g(s)]g'(s)~g(')In_a(g(s),u;a2,...,a~_2)an_l(U) (/ q('r) d'r)V~duds=oo, are sufficient for equation (1.1;-1) to be oscillatory. (J2). Suppose that f(x)sgn x > Ixl a, for x ¢ O. Equation (1.1;-1) is oscillatory if, for all large T > to with or(t) > T, the first-order delay equation,
y' (t) + q (t) w" [o (t) , T] y [0 (t)] = 0, is oscillatory and
p(O
q (s) w n [g (s), p (t)] ds > 1.
lim sup ,It
t--+ oo
(J3). Suppose that either (3.32) or (3.35) holds and
/ +~ ~o
du F (~1/.) < o~,
where F is as in (3.36). Then, for all large T >_ to with g(t) > T, the conditions,
/
~ q ( s ) f (w [g(s), T]) ds
oo,
/.4 q(s)f (w,~[g(s),s]) ds =
oo,
9
are sufficient for equation (1.1;-1) to be oscillatory. (J4). Suppose that f(x)sgn x > ]x]~, for x ~ 0, where fl is the quotient of positive odd integers. Equation (1.1;-1) is oscillatory if, for all large T >_ to with a(t) > T, the half-linear equation, I
r(t)
( ~ [a (t), T] o' (t)
(y' (t))"
+ q (t) y" (t) = 0,
for some A E (0, 1),
is oscillatory, where
I CI, cl > 0 is any constant, when ~ > a,
r(t)=
I,
when fl = a,
and limsup t'---~ O 0
f
p(t) q (s) w~ [g (s), s] ds > O,
when ~ > a,
q (s) w x [g (s), 0 (t)] as > 1,
when ~ = a.
Jt
p(t) lira sup t -.* oo
,It
R.P. AGARWALet al.
452
(is). Suppose that 0 < a < 1 and f(x)sgn x >_ Ix[~, for x ¢ 0, where f~ is the quotient of positive odd integers. Equation (1.1;-1) is oscillatory if, for all large T _> to with a(t) > T, the second-order equation,
~ (t)
A~ [cr (t), T] o' (t) Q(1-a)/a (t) y' (t)
)'
for some )~ E (0, 1),
if- ~q (t) y (t) = O,
is oscillatory, where r(t) is as in 04), Q(t) = f t q(s) ds > 0 and limsup fP(t) q(s)w~[g(s),s] ds > O,
when/3 > ~,
Jt
t -* c~
p(t) q (s) w~ [g (s), p (t)] ds > 1,
lim sup t --* o o
when j3 = a.
,It
THEOREM 7.4. Let n be even, assume that (i)-(v) and (1.3) hold. Moreover, assume that there exists a function 0 E Cl([t0, oo), R+), such that (3.2) holds. (M1). Suppose that either (3.32) or (3.35) holds and
~oo o
du F (~1/.) < o~,
where F is as in (3.36). Then, for all large T > to with a(t) > T, the conditions,
ff
[o (s), T] 0' (s) QI/~ (s) ds = c¢,
f q(s) I ( ~ .
Q (t) =
ff
q (s) ds,
[g(s),s]) d8 = ~ ,
g
and limsup -[ t
t--+oo JT(t)
q(s)f(wo [~'(t),g(s)]) ds > 0,
are sufficient for equation (1.1;-1) to be oscillatory. (M2). Suppose that f(x)sgn x > [xl", for x ¢ 0. Then, for all large T > to with a(t) > T, the conditions,
limsupw[o(t),T]
Q(t)+a
~[o(s),T]o'(s)Q(l+a)/~(s) ds
t --+ O o
where
t
lira sup ,-~oo
and ~m sup t-~ oo
q i s) w~ [~ (t), g (s)]
(t)
f
ds > 1
p(t)
q (s) w~ [g (s), p (t)] ds > 1
J t
are sufficient for equation (1.1;--1) to be oscillatory. (M3). Suppose that either (3.32) or (3.35) holds and
~
oo o
du P (~1/~) < oo,
> 1,
Functional Differential Equations
453
where F is as in (3.36). Then, for all large T > toe the conditions,
/ lim sup f t t--,oo
oo q (s) f (w [g (s), T]) ds = c~,
q (s) f (Wo IT (t), g (S)]) ds = oo,
d~'(t)
and
f
q(s) f ( w n [g(s),s]) ds =
00,
g
are sufficient for equation (1.1;-1) to be oscillatory. (Ma). Suppose that c~ _> 1 and f(x)sgn x ~ Ixl ~, for x # O. Equation (1.1;-1) is oscillatory if, for all large T > to ~ O, the second-order equation,
(
1 y~ [(7 (t), T] W a - 1 [(7 it), T] (7' (t) (t)
)'
+ aq (t) y (t) = O,
is oscillatory,
i
lim sup
t
t---~oo J r ( t )
q (s) w~ IT (t), g (S)] ds > 1
and
f
lim sup t--coo
p(t) q (s) w,~ [g (s), p (t)] ds > 1.
.It
(Ms). Suppose that 0 < a _< 1 and f(x)sgn x > Ixl a, for x # O. Equation (1.1;-1) is oscillatory
if, for all large T > to with (7(t) > T, the second-order equation,
1 ' t +aq(t) y(t)=o, [(7(t) , T] (7' (t) Q(I--)/. (t)y ())' is oscillatory, where lim sup
f t
t--*oo
q (s) w~ [7- it), g (s)] ds > 1
(t)
and lira sup t--~
f
p(t) q (s) w~ [g (s), p (t)] ds > 1.
Jt
These theorems seem to be new even when specialized to the equation,
dt
x ("-1) (t)
+ ~q (t) f (x [g (t)]) = 0,
(7.1; 5)
R . P . AGARWALet al.
454
for which Conditions (ii)-(v) are satisfied. So, we state some of them below as corollaries, by noting that in this case, for T >_to and all t > T, (t - T) '~-1 wt [t,T] = ( n - 1 ) ( t - 1)! ( n - t w--t [t,T] = (n
-
1)!'
l
(t - T) n-2 2) (e - 2)! (n - e - 1)!'
2
~ [t, T] = [A (1 - A)]'~-2 t '~-2
(~ -
2)!
A C (0, 1), for all large t > T/A,
'
Atn-2 [t, s] = (,~ _ ~)!,
for some A E (0, 1), and all large t _> T,
Atn-1
for some A E (0, 1), and all large t >_ T,
w[a(t),T] = ( n - 1)!' WO [t ,S] -~ Wn [t,s] ---- ( t - - s ) n-1 (~ 1)! '
t>s>T.
-
Let n be even and assume that (ii)-(v) hold. Moreover, assume that there exists a function cr e Cl([to, oo),R+), such that (3.2) holds. COROLLARY 7.1.
(Sl). Suppose that
S
+~
du S~1~@------7< ~
and there exists a function ~(t) E Cl([t0, ec), R+), such that
~' (t)
> o
and
~a,_2(t)a,(t))
_<0,
fort>T,
for some T _> t0.
Then, the condition,
S
OO ~ (s) q (s) ds =
~¢,
is sufficient for equation (7.1;1) to be oscillatory. (s2). Suppose that f(x)sgn x >_ Ixl", for x 7~ 0. If there exists a function ~(t) E Cl([to, e~),R +) and a constant A E (0, 1), such that, for all large T _> to and t _> T either lim sup t---*oo
((n - 2)!)" A-" (~' (s)) ~+1 1 ] (a + 1) ~+1 (A (1 - A)) ('~-2)~ (~ (s) (a '~-2 (s) a' (s)) ~] ds = ec
~ (s) q (s) T
or, a > 1 and ( n - 2), ( ( n - 1)[) ~ - ' 4aA<~ (A ( i - A)) n-2(~('~-'iy~-~----~(s)~''(s)
l i m sup f t [ ~ ( s ) q ( s ) t--+~ .ST
or, 0 < a < _ l
((~' (s))2~l ~t ~ ' ~
,]J d s = o o
and lip_+sup
~T~'[
(n-- 2)! O(a-Za__ !S__) ] ~(s)q(s)-4aA(~Tf-_--~))n_2a.~_2(s)a,(s)j
Q(t) > 0, then, equation (7.1;1) is oscillatory.
(s3). Suppose that either (3.32), or (3.35) holds and
i+o F (ull du ")
< e~,
ds=o¢,
Functional Differential E q u a t i o n s
455
where F is as in (3.36). Then, the condition,
f =q(~) f (g"-~ (8)) ~s= ~¢, is sufficient for equation (7.1;1) to be oscillatory. (s4). Suppose that ~ > 1 and f(x)sgn x > Ixl ~, for x ¢ 0, where/3 is the quotient of positive odd integers. Equation (7.1;1) is oscillatory if there exists a constant A ~ (0, 1), such that for all large t > to, the second-order linear equation,
(~.-_?),_o ~(t) y' (t))' +/3q (t) u (t) = 0, (1 ~) -~ ~(~-~).+~-2 (t) ~' (t)
~.-~
-
is oscillatory, where
r (t) =
el '
cI > 0 is any constant, when/3 > a,
1,
when/3 = a,
c2a(n-1)(~-~) (t),
c2 > 0 is any constant, when/3 < a.
(ss). Suppose that 0 < a < 1 and f(x)sgn x > Ixl~, for x # 0, where/3 is the quotient of positive odd integers. Equation (7.1;1) is oscillatory if there exists a constant A ~ (0, 1), such that for all large t > to, the linear equation,
Q(a-1)/a (t) r l (t) (~ 2)! y' (t]~' +/3q (t) y (t) ----O, A~-1 (1 - A) '~-2 a n-2 (t) c' (t) " "] is oscillatory, where cl > 0 is any constant, when/3 > a,
Cl '
when/3 = a, 1, C2a (n-1)(a-jg)/a (t) , c2 > 0 is any constant, when/3 <
rl (t) =
ci.
COROLLARY 7.2. Let n be odd and assume that (ii)-(v) hold. Moreover, assume that there exists a function a C cl([to, oe), R+), such that (3.2) holds.
(oi).
Suppose that g(t) <_t, for t >_ to and
du
~
S±o
p i . (~-------5< ~"
Then, for all large t _> to, the condition,
i where
i
g'
(')
~
o.n--2
(~ _
(s; (.)
(s)
G!
g (,))..-~
VmV)'
Q1/. (s)
(s)
ds = o o ,
(L) q(r) d r
~/<.
d~
)
d8 = ~¢
are sufficient for equation (7.1;1) to be oscillatory. (02). Suppose that f(x)sgn x _> Ix[ ~, for x ~ 0. Then, for some constant A E (0, 1) and all large t, the conditions,
[
aA
lim._+s~p c,"-~ (t) Q (t) + 2)------'7. (n -
/-
a n-2
(s)
a'
(s)
Q(.+I)I.
(s) ds
]
11o~
> (n - 1)!,
456
R . P . AGARWALet
where limsup t--*oo
/;
(t)
al.
[~'(t)-g(s)](n-D~q(s) ds > ( ( n - 1)!) ~
are sufficient for equation (7.1;1) to be oscillatory. (03). Suppose that either (3.32) or (3.35) holds and
f~
du o F (~1.)
< ~'
where F is as in (3.36). Then, for all large t, the conditions,
7
~ q(8) f (gn--1 (S))
US = O0
and limsup
/'
q(s) f
( [r (t) -- g (s)] n-1 ) ds>O,
are sufficient for equation (7.1;1) to be oscillatory. (04). Suppose that a _> 1 and f(x)sgn x > [xl #, for x ¢ 0, where fl is the quotient of positive odd integers. Equation (7.1;1) is oscillatory if, for some constant ), E (0, 1) and all large t, the equation, //(n - 2)! ((n - 1)!)"-1 r (t) , ,.,'~' --,:-r t -~-~--~-~+-~--~(t)a it) y t~)) + n q ( t l Y ( t ) = O , is oscillatory, where r (t)
[ 1,
when fl = a,
t
c > 0 is any constant, when fl < a,
ca(n-1)(~-#) (t),
and t
limsup
fJr(t) IT (t) -- g (S)] (n-1)~ q (S) ds > ((n -
1)!) ~ ,
limsup f ' [r(t) -g(s)](n-1)Zq(s) ds > 0, t--*oo Jr(t)
when fl = a, when fl < a.
COROLLARY 7.3. Let n be odd and assume that Oi)-(v) hold. Moreover, assume that there exists a function a 6 Cl([to, c~), R+), such that (3.2) holds.
(J1)-
Suppose that
f
+~
du yli a (u-------~< c~,
and that there exists a function ~(t) E Cl([to, c~),R+), such that ~'(t)>0,
fort>to,
and
F (t.<,:-,
Then, the conditions,
f
~ ¢ (s) q (s) ds = ~¢
(s)<.. (,)) i
Functional
Differential
Equations
457
and ids) d
(g (s) (n-3)!
(4
thy3 (p)
d7)l/a
are sufficient for equation
(7.1; -1) to be oscillatory.
Suppose that f(z)sgn 2 2 ]zla, for 2 # 0. Equation and some constant X E (0, l), the first-order equation, (Jz).
(
Y’ (t> +
is oscillatory,
duds=o%
J s
L,
(n ! l)!
(I 1
d---l
(7.1; -1) is oscillatory if, for all large t
(9 4 w Y b WI = 07
and P(t) lim t--r00 J t
(53).
[g(s) - p (t)]‘“-l’”
q (s) ds > ((n - l)!)” .
Suppose that either (3.32) or (3.35) holds and
fm J rto
du F (da)
< O”’
where F is as in (3.36). Then, for all large t, the conditions,
s
O” q(s) f (g’+-l (s)) ds = oo
and
are sufficient for equation (J4),
(7.1; -1) to be oscillatory.
Suppose that 0 < o 5 1 and
f (~1w 2 2 14P, where p is the quotient of positive odd integers. constant X E (0,l) and all large t, the equation,
for 2 # 0, Equation
r (t) Q(a-1)/a (t)
(n - 2)! x [A (1 - A)]“-”
P--2 (t) o’ (t)
y’(t)
(7.1; -1)
is oscillatory
‘+Pdt)Y(t)
if, for some
=o,
is oscillatory, where r
(t) =
c,
c > 0 is any constant, when ,!9> o,
1, when ,L3= CY,
Q(t) = St” q(s) ds > 0 and p(t)
lim sup lim sup t--Km
[g (s) - s]@-l)
P(t) J
t+cc
t
st
[g (s) - p(t)]‘“-“”
q (s) ds > 0,
when p > cx,
q (s) ds > ((n - l)!)” ,
when 0 = (Y.
R, P. AGARWALet al.
458
COROLLARY 7.4. Let n be even and assume that (ii)-(iv) hold. Moreover, assume that there exists a function cre Cl([to, oo),R+), such that (3.2) holds. (M1). Suppose that either (3.32) or (3.35) holds and
f k~ o
du
F(u 1/~) < oo,
where F is as in (3.36). Then, for all large t, the conditions,
~ - 2 (s) a' (s) Q1/~ (s) ds = oo,
Q (t) =
q (s) ds,
/A q ( s ) f ( [ g ( s ) - s ] ~ - l ) d s = c % g
and
limsup t--~oo
/:
q (s) f (t)
([r (t) - g (s)] "-1 ) ds > O,
are sufficient for equation (7.1;-1) to be oscillatory. (M2). Suppose that f(x)sgn x > ]xl ~, for x ~ 0. Then, for some constant ), E (0, 1) and all large t, the conditions, limsupt~ ~----1~
Q(t)+~
(s)
(s)
where limsup ?/t [ r ( t ) - g ( s ) ] ( n - i ) a q ( s ) t--*oo
ds > ( ( n - 1)!) ~
Jr(t)
and limsup fp(t) [g (s) - p (t)] ('~-1)'~ q (s) ds > ((n - 1)!) '~ , t--+oo J t
are sufficient for equation (7.1;-1) to be oscillatory. (M3). Suppose that either (3.32) or (3.35) holds and
f
oo o
du F(ul/~)
where F is as in (3.36). Then, for all large t, the conditions,
f limsup t~oo
(t)
oo q (S) f (gn--1 iS))
ds
= ~,
q(s)f [r(t)-g(s)] ~-~ d s = ~ ,
and
£qls): are sufficient for equation (7.1;-1) to be oscillatory.
sl °-:) ds
_-_
(s) ds
> 1,
Functional
Differential
Equations
459
Suppose that cy 2 1 and f(z)sgn z 1 ]x]~, for z # 0. Equation for some constant X E (0,l) and all large t, the equation,
(7.1; -1) is oscillatory if,
(Md).
1
(t)cl’(t)Y’ (4 > ‘+ (n-
( &-lb++2
XOcY
2)!
((n- q!)“-1qWY@)= Ov
is oscillatory, t (7(t) -g (s)]+‘I”
lim sup
q (s) ds > ((n - 1))”
t-K%3 s r(t)
P(t) t-+co s
lim sup
[g(s) - p (t)]‘“-l’o
t
q (s) ds > ((n - l)!)O .
Suppose that 0 < o < 1 and f(z)sgn z 2 ]z]~, f or z # 0. Equation (7.1; -1) is oscillatory if, for some constant X E (0,l) and all large t, the equation, (MS).
I
(
y&--2
(t)
CT’ (tf
Q(l-a),cx
@)Y
(t)
>
+
$Y2)!
Q (t)
y (9
=
O,
is oscillatory, where t [I t-em J T(t)
lim sup
-g (s)](n-l)a
q (s) ds > ((n - I)!)”
and lim sup t--too
Jt
p(t)[g (s) - p (t)]‘“-l’*
8. GENERAL
* . q (s) ds > ((n - l)‘)Q
REMARKS
1. The results of this paper are presented in a form which is essentially new and are of a higher degree of generality. In fact, one can easily extract more criteria than those presented, for the oscillation of equation (1.1; S) and/or related equations as well as some special cases of equation (1.1; 6) and also for the mixed equations of the form,
Lx (t)+ 62 qjw fj (x1%@)I> = 0,
(8.1;6)
j=l
where L, is as in equation (1.1;6), qj E C([te,oo),R+), gj E C((to,co),R), and fj E C(n,a),zfj(z)>O,fj’(~)LO,forz#Oandlimt,,gj(t)=~,j=1,2,.,.,m. The formulation of such results are left to the reader. 2. The results of this paper can be extended to equations of type (1.1; S) where f need not be a monotonic function. As in [26, Lemma 31, the function f has the following decomposition (f is locally of bounded variation on [a, b] C R): f(x) = G(x)H(x), where G : (-oo, --cro]U [Qo,cQ> = RI, -+ Iw+, o. 2 0, nondecreasing on (-co, -QO] and nonincreasing on [WI, co) and H : B,, + R and nondecreasing on IF&. 3. The results of this paper can be extended to forced equations of the form,
Lx (t>+ h (t) f (x 19WI>= e0) , where e E C([te, oo), IR) as well as neutral functional
Ln(x (t>+ c(t>x [h(a>+ &Iw f where c and h E C([tc, oo),W).
differential
(8.2;6)
equations of the type
(x 19(a> = 07
(8.3;6)
460
R . P . AGARWALet al.
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(1991). 9. S.R. Grace, Oscillatory and asymptotic behavior of delay differential equations with a nonlinear damping term, J. Math. Anal. Appl. 168, 306-318, (1992). 10. S.R. Grace, Oscillation theorems of comparison type of delay differential equations with a nonlinear damping term, Math. Slovaca 44, 303-314, (1994). 11. S.R. Grace, Oscillation criteria for retarded differential equations with a nonlinear damping term, Aequationes Math. (to appear). 12. S.R. Grace and B.S. Lalli, Oscillation theorems for nth-order nonlinear differential equations with deviating arguments, Proc. Amer. Math. Soc. 90, 65-70, (1984). 13. S.R. Grace and B.S. Lalli, A comparison theorem for general nonlinear ordinary differential equations, J. Math. Anal. Appl. 120, 39-43, (1986). 14. M. K. Grammatikopoulos, R. Koplatadze and G. Kvinikadze, Linear functional differential equations with Property A, J. Math. Anal. Appl. 284 (1), 294-314, (2003). 15. I. Kiguradze and T. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Academic Publishers, Dordrecht, (1993). 16. I. Kiguradze and I.P. Stavroulakis, On existence of proper oscillatory solutions of advanced differential equations, (Russian), Differentsial'nye Uravneniya 34 (6), 751-757, (1998); Differ. Equations, (English translation) 34 (6), 748-754, (1998). 17. I. Kiguradze and I.P. Stavroulakis, On the oscillation of solutions of higher order Emden-Fowler advanced differential equations, Appl. Anal. 70 (1-2), 97-112, (1998). 18. I. Kiguradze, N. Partsvanla and I.P. Stavroulakis, On advanced functional differential equations with properties A and B, Mere. Differential Equations Math. Phys. 24, 146-150, (2001). 19. I. Kiguradze, N. Partsvania and I.P. Stavroulakis, On oscillatory properties of higher order advanced functional differential equations, (Russian), Differentsial'nye Uravneniya 38 (8), 1030-1041, (2002); Differ. Equations, (English translation) 38 (8), 1095-1107, (2002). 20. I. Kiguradze, N. Partsvanla and I.P. Stavroulakis, On oscillatory solutions of nonlinear differential equations with advanced arguments, Mere. Differential Equations Math. Phys. 25, 156-158, (2002). 21. R. Koplatadze and T. Chanturia, On Oscillatory Properties of Differential Equations with a Deviating Argument, (Russian), Tbilisi University Press, Tbilisi, (1977). 22. R. Koplatadze, On oscillatory properties of solutions of functional differential equations, Mere. Differential Equations Math. Phys. 3, 3-179, (1994). 23. R. Koplatadze, Comparison theorems for differential equations with several deviations. The case of property A, Mere. Differential Equations Math. Phys. 24, 115-124, (2001). 24. R. Koplatadze, Comparison theorems for differential equations with several deviations. The case of property B, Mere. Differential Equations Math. Phys. 26, 139-148, (2002). 25. Y. Kitamura, Oscillation of functional differential equations with general deviating arguments, Hiroshima Math. J. 15, 445-491, (1985). 26. T. Kusano and B.S. Lalli, On oscillation of half-linear functional differential equations with deviating arguments, Hiroshima Math. J. 24, 549-563, (1994). 27. W.E. Mahfoud, Oscillatory and asymptotic behavior of solutions of Nth-order nonlinear delay differential equations, J. Differential Equations 24, 75-98, (1979). 28. Ch.G. Philos, Bounded oscillation generated by retardation for differential equations of arbitrary order, Utilitas Math. 15, 161-182, (1979). 29. Ch.G. Philos, On the existence of nonoscillatory solutions tending to zero at ~ for differential equations with positive delays, Arch. Math. 36, 168-178, (1981). 30. G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, Second Edition, Cambridge Univ. Press, Cambridge, (1988). 31. I. GySri and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford Univ. Press, Oxford, (1991).
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32. R.P. Agarwal, S.R. Grace and D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer, Dordrecht, (2000). 33. R.P. Agarwal, S.R. Grace and D. O'Regan, Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations, Kluwer, Dordrecht, (2002). 34. R.P. Agarwal, S.R. Grace and D. O'Regan, Oscillation Theory for Second Order Dynamic Equations, Taylor and Francis, United Kingdom, (2003). 35. Ch.G. Philos, Some comparison criteria in oscillation theory, J. Austral. Math. Soe. (Ser. A) 36, 176--186, (1984).