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Oscillations of second order neutral functional differential equations
N~IO'H- HOILAI~
Oscillations of Second Order N e u t r a l F u n c t i o n a l Differential E q u a t i o n s Jurang Yan
Department of Mathematics Shanxi University Taiyuan, Shanxi 030006 People's Republic of China
ABSTRACT Consider the second order neutral functional differential equation
"~
x(t)+ ~ , X ( t - 7 , )
+8~x(t-%)=0
i=1
(E)
j=l
where 8 = 5:1, ~?i, zi, ~, and at, i = 1, 2 , . . . , n, j = 1, 2 , . . . , m, are real numbers. Some new oscillation criteria are given for oscillation of (E). The results of this paper improve noticeably some known results. © Elsevier Science Inc., 1997
1.
INTRODUCTION Consider the second order neutral differential equation
,~t~ ~ ( t ) +
n , x ( t - ,-~1 + 8 i=l
~j~(t - ,~j) = o,
(El
j=l
where 8 = + 1 , ~7,, ri, ~j, a3, i - - 1 , 2 , . . . , n , j= 1,2,...,m axe real numbers, ~j >t 0. Our aim in this paper is to obtain some new sufficient conditions, involving the coefficients and the arguments only, under which
APPLIED MATHEMATICSAND COMPUTATION83:27-41 (1997) ©"Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010
0096-3003/97/$17.00 PI! S0096-3003(96)00043-4
28
J. YAN
all solutions of (E) are oscillatory. Our technique here is based on the study of the characteristic equation 12 1 +
~?~e-~" + 8
~je-"',--0.
(E*)
j=l
The advantage of working with these conditions rather than characteristic equation associated with the equations under consideration is that they are explicit and therefore easily verifiable. The oscillation criteria of this paper noticeably improve the results given in [1]. Necessary and sufficient conditions (in terms of the characteristic equation) for the oscillation of all solutions of second order neutral functional differential equations have been established [2-5]. Second order neutral differential equations were encountered in the study of vibrating masses attached to an elastic bar and also as the Euler equations in some variational problems (see Hale [6, p. 7]). As is customary, a solution of (E) is called oscillatory if it has arbitrarily large zeros on (0, oo). Otherwise, it is called nonoscillatory. The following result is extracted from [2].
LEMMA. Assume that tt = _ 1, ~?~, x,, aj ~ R and ~ >t 0, i = 1, 2 , . . . , n, j = 1, 2 , . . . , m. Then a necessary and sufficient condition for oscillation of all solutions of (E) is that its characteristic equation (E*) has no real roots. . MAIN RESULTS In this paper, we will study the oscillation of the following equations, as some special cases of (E),
d2[
r dt--~ x( t) + ]~- r~z(t Eqkx( t-
d2[
(E~)
L
dt---~ x( t) + F_,r~x(t
K
~,hjx(t+j a~)]
ak) + EP~X( t + El)],
K
~]qkx( t -
~'') -
~) + F_,hjx(t
ak) + ]~-,PlX( t + ill)I, L
,rj)] (E2)
Neutral Functional Equations
29
dt 2 x(t) + ~_,rix(t-'r~) + ~_~hjx(t + crj) I
J
= r , q k x ( t - ak) + r , p ~ z ( t + il~)], K
(E3)
L
and
dt 2 x(t) - E r , x ( t -
~,) - F_.hjx(t + o~j)
I
J
+r, qkx(t-ak) K
+ Ep~x(t + il0] =0.
(E4)
L
where ri, ri, i ~ I, hi, crj, j ~ J, are nonnegative constants and qk, ak, k ~ K, p~, ill, l ~ L are positive constants, while I = {1, 2 , . . . , rtl} , J = {1, 2 , . . . , n2}, K = {1, 2 , . . . , n3}, L = {1, 2 , . . . , n4}, 0 ~< T1 <~ ~'2 < " ' " < "rn, , 0 <<. ~r1 < ~ r 2 , . . . , < o'n2, 0 < ot 1 < a 2 < "'" < an3, and 0 < t l <
il2,--., < il~,.
For convenience, we use the following notations
R = Er,,
H = Ehj,
I
J
Q= Eqk,
P = EP~,
K
L
and the characteristic equations of (E1)-(E 4) are, respectively, FI(A ) - ~ 2 ( 1 +
r, rie - ~ , - r, hjeA~'J)--(~P~gqke-*=k+ I
J
]~p~e~*) = 0 , L
(ET) F2(A)-A2(I+
~_,rie ~ ' - ~_~hjeA~J)--( ~_~gqke-~k+ I
J
~Ple~)
=0,
L
(E~) + I
+
J
K
:
0
L
(E~)
30
J. YAN
and
+ ~.,qke -~'~ + ~_,pze ~ ' = O. I
J
K
L
(E~) First, we obtain sufficient conditions for the oscillation of (El).
THEOREM 1.
-e 4
Assume that a~ > ~'nl ,
]~plflt ~ + ]~hjexp L
~ %
(1)
> 1+ R
J
and
- e E q k ( a k - ~.)2 > 1 + ]~ r,exp 4
K
~
(7,-
I
• ~)).
(2)
Then all solutions of ( E l) oscillate.
PROOF. For A ~: 0, we have that
(3) I
J
and
- Fl( X) e~°'/x2 - [ E qke-'("'-'"') + EL
- [ e~'°' + E'~e-~("-'"l) hJe~(~J+'")]", - EJ
(4)
NeutralFunctionalEquations
31
Now our strategy is to prove that FI(A) < 0 for all h ~ ( - 0 % 0 0 in each of the following four cases
CASE 1.
0 < )t ~< ( P / ( 1 -4- R)) 1/2. F r o m (3) we find that
-FI(A)/~'2 >
>
( ~ qke-~"~+ ~-~Ple;~')/A2L- ( 1 ~
+R)
-(I+R)=0.
CASE 2. A > ( P / ( 1 "4- R)) 1/2. In view of (1) and the inequality e * t> for x I> 0, (3) yields
- r l ( ~ ) / ~ 2 > / ~1e 2 E p ~
2
- (1 +
ex
R)
L
CASE 3.
-(Q/(1 + R)) 1/2 ~< A <
0. From (4) we have that
-FI(A) e~nl/A2 > Q/(1--~) - ( I + R) =0. CASE4. h < - ( Q / ( 1 + R))W2. B y u s i n g ( 2 ) and eX>~ exfor x>~0, we obtian that
-
1 2 FI(A) e~%/A 2 >~ -~ e ~ qk( a~ - T.I)2 K
32
J. Y A N
Cases 1-4 and FI(0) < 0 imply that FI(A) < 0 for all h ~ (-~,oo), that is, (E*) has no real roots. By the lemma we conclue that all solutions of (E 1) oscillate. Next we present a theorem which describes the oscillatory character of
(E2). THEOREM 2.
Assume that fll > 7~,,
-le2~pl(flz - r~,)2 > 1 + ~ riexp 4
L
I
(i"
1 - - - ~ (~'i - •
(5)
and -~e= EqkakK + ~jhjexp -
~(r3
> (1 + R).
(6)
Then all solutions of ( E 2) oscillate. PROOF. From the characteristic equation (E*) of (E2), we have that for A¢0
K
L
I
and
J
33
Neutral Functional Equations Now, we consider the following four cases
CASE 1. 0 < A < ( / ) / ( 1 + R)) 1/2. Then, from (8), (5) and e ~ >I ex for x >t 0, we obtain that
>t
-
1 + •riexp
~
('i-
"-1)
1 E p,( 8 , -
> O.
I
CASE 2.
A > (P/(1 + R)) 1/2. From (8) it follows that
-F2(A) e-a%/A 2>
- (1 + R) = 0 .
-t- R)) 1/2 • A < 0. By using (7), (6) and e ~ >t ex for
CASE 3. - ( Q / ( 1 x >/0, we see that
-F2(A)//A2>/--~"~qka~ g
CASE 4.
~
)t < - ( Q / ( 1
I+R-
~hjexp j
-
--~rj I+R
>0.
+ R)) 1/2. Then, (6) and (7) yield
From Cases 1-4 and F2(0) < 0 we see that for all A ~ ( - ~ , ~)F2(A) < 0, which implies by Lemma that all solutions of (E 2) oscillate. The proof of the theorem is complete.
REMARK 1. The results in Theorems 1 and 2 not only depend on ~ and ri, also depend on hj and (rj (see Remark 1 in [2]).
34
J. YAN THEOREM 3. 1 e2~pl([3, 4 L
Assume that a I > T,~, [31 > ¢r,~, ~r~2)~> 1 + ~ r ~ e x p L
+
hjexp
('z + ~n,) I+R+H
1+ R + H
and
1
-~ e ~g qk( ak -- rn~)2 > 1 + ~] r~exp
[¢o
I
I+R+H
+ ~[~ hjexp j
( , , -- %,)
]
1 + R + H ( g J + %1) "
(lO)
Then all solutions of ( E3) oscillate. PROOF. From the characteristic equation (E~) of (E 3) we obtain that
(11)
-F3(h) ea"/h2=- [~"qke-"(~'"-%)+ n -[ea% + Y'~r~e-a(~'-%) + J Consider the following four cases
CASE 1.
0 < )t ~< ( P / ( 1
-F3(A) e - ~ " 2 / A 2 >
+ R + H)) 1/~. From (11), we have that 1+ R + H
(12)
35
Neutral Functional Equations
CASE 2. that
)t > ( P / ( 1 + R + H)) 1/2. In view of (11) and (9), we deduce
- r~(~)
1 ~- ~--~/z >~~ e 2 E
%)2
p,( ~, -
L
-
1+
r~exp -
I+/~+H
+ ~ hjexp j I + R +p
CASE 3.
(rj - q-2)
]
> 0.
- ( Q / ( 1 + R + H)) 1/2 .<< h < 0. From (12), we obtain that
/(
- F a ( h ) eX%/X 2 > Q
CASE 4.
H(
°
1 + R÷ H
)
0
h < - ( Q / ( 1 + R + H)) 1/2. Then (12) and (10) yield that
1
2
- F3(A) ea%/h 2 i> ~- e ~K qk( °tk - 7.,) 2
-
(
1 + ~riexp I
(i
+ ~ hjexp -J
I+R+H
Q
(z i - %~)
l+R+H(%+%~
)
)
l
>0.
From Cases 1-4 and Fa(0)< 0, we see that F3(A)< 0 for all A ( - 0%oo). By Lemma, all solutions of (E 3) oscillate. The proof of the theorem is complete.
36
J. YAN THEOREM 4.
Assume that R + H > 0, ot1 >
fll
~,2) > ~ / r i e x p -
Tn, ,
fll >
O'n2,
R + H
+ E hjexp j
( o'j - o'n2)
R+H
)
'
(13)
and
1
-e 4
Eqk(~ L
- %)
> E r , exp
R +-----H(~ - %)
+ ~j hjexp -
)
R + H ( qj + r~l) .
(14)
Then all solutions of ( E 4) oscillate.
PROOF.
From the characteristic equation (E*) of (E 4) we have that
+ e - A ~ . 2 - ~_,rie-A(~,+~.~ ) - ~_,hje~(~,-~.. ), (15) I
J
and
F4(a) ea~"l/a 2 -+ e'-,-
E r , e-~(~.-~ol)- E h j e ~ ÷ ~ o , ) . I
J
Now, we consider the following four cases
CASE 1.
0 < A ~< ( P / ( R
~(,~)~-"~-,/,~ I>
+ H)) W2 from (15) we get ~
+ e-"~-~ - R - H > O.
(16)
37
Neutral Functional Equations
CASE 2. A > ( P / ( R + H)) 1/2. In this case, from (15) and (13), it follows that e2
F~(~) ~-~'"/~ 2
>/~-Ep,(~,- %)' L
- ~hjexp
(¢"
R + H (°'~ - o'n,)
)
> 0.
J
CASE 3.
CASE 4. that
- - ( Q / ( R + H)) 1/2 ~< A < 0. Then from (16) we have
A < -(Q/(R
+ H)) 1/2. From (16), in view of (14), we have e2
F4(A) e ~ % / A 2 >/ ~ - E q~( ak - ,,,)2 K
-~hjexp y
[¢0 -
--(o'j+ R+H
]
~'nl) > 0.
From Cases 1-4, F4(0) > 0 and Lemma, we see that all solutions of (E 4) oscillate. The proof of the theorem is complete. 3.
APPLICATIONS AND REMARKS
Consider the following neutral differential equations which are special forms of (E1)-(E4) , respectively, d2 d t 2 [ x ( t ) + r x ( t - 7) - hx(t + (r)] = q x ( t - a) + px(t + [3), (E'I)
d2 d t 2 [ x ( t ) + rx(t + 7) - hx(t + cr)] -- q x ( t - a) + px(t + ~ ) ,
(E'2)
38
J. YAN d2 dt---7[x(t) + r x ( t -
7) + h x ( t + (r)] = q x ( t -
a ) + p x ( t + fl)
(E~)
and d2
dt 2 [ x ( t )
- r x ( t - 7) - h x ( t + o')] + qx( t - a ) + px( t + [3) -- O,
where r, 7, h and cr are nonnegative and q, a , p and fl are positive constants. T h e following oscillation results can be obtained from T h e o r e m s 1-4.
THEOREM 1'.
Assume that a > z,
4 pfl 2e2 + hexp
or
> 1+ r
and 1
~q(a
- 7 ) 2 e 2 > 1 + r.
(17)
Then all solutions of ( E 2) oscillate.
THEOREM 2'.
Assume that fl > 7, 1
-TP( f l 4-
7) :e2 > 1 + r
and -~ qa 2 e2 + hexp -
q)
l+r
Then all solutions of ( E~) oscillate.
v~ > l + r .
(18)
39
Neutral Functional Equations
THEOREM 3'.
Assume that a > v, [3 > cr,
-~p([3-~r)2e2>l+rexp
1 + r + h (~'+ or)
-
+h
and 1
"~q( a - ¢)2e 2 > 1 + r + hexp -
q l+r+h
(¢ + tr)].
Then all solutions of (E~) oscillate.
THEOREM 4'.
Assume that r + h > 0, a > • and [3 > (r,
-~ p( [3 - ~r)2e2 > rexp -
( , + ~r) + h
and
~q(~
- ~)2e2 > r+
hew
-
(~ + ~)].
Then all solutions of ( E~) oscillate.
In view of e * = 1 + ~ = 1 Xk/k!, X ~ (-- 0% ~), we now can obtain the following results from Theorems 1'-4'.
COROLLARY 1. Assume that a > r and (17) holds. If there exists a positive integer N such that
1p[32e2+h1
+ k_~1-~.TIV 1---~r ~
Then all solutions of ( E~) oscillate.
>l+r.
40
J. YAN
COROLLARY2. N such that
~q
Assume that [J > v and (18) holds. If there exists an odd
+hl+
E
q
k=l
Then all solutions of (E~) oscillate.
COROLLARY 3.
Assume that a > T and fl > ~r. If there exist an odd N
such that -1~ p ( f l - o " )2e2 > l + r
[1
(-1)k + k~= l
P
k!
+h
l+r+h
and N (_1) k -~q( a - r ) 2 e 2> 1 + r + h 1 +
kY~ =l
q
k!
(r+
~)
.
l+r+h
Then all solutions of ( E~) oscillate.
COROLLARY 4. Assume that r + h > O, a > exists an odd N such that 1 ~p(fl-
(r
)2e2
N (_1) k E k!
> 7" 1 +
• and fl > ~r. If there
P
(~'+ (r)
+ h
k=l
and
[
1 N (_1) k ~ q ( a -- v)2e2 > r + h 1 + • k! k=l
Then all solutions of ( E~) oscillate.
q
( v + (r)
.
Neutral Functional Equations REMARK 2.
41
Corollaries 1-4 improve, respectively, Theorems 1-4 in [1].
REMARK 3. By using technique of this paper, we can obtain more oscillation criteria for (E).
REMARK 4. Our technique can be extended to higher order neutral functional differential equations.
REMARK 5. It is easy to construct examples showing that our criteria for oscillation of (E'I)-(E'4) are essentially wider than the oscillation criteria in
[I]. REFERENCES 1 S. R. Grace, and B. S. Lalli, Oscillation theorems for second order neutral functional differential equations, Appl. Math. Comp. 51:119-133 (1992). 2 S.J. Bilchev, M. K. Grammatikopoulos, and I. P. Stavroulakis, Oscillations of higher order neutral differential equations, J. Austral. Math. Soc. (Series A) 52:261-284 (1992). 3 G. Ladas, E. C. Partheniadis, and Y. G. Sficas, Necessary and sufficient conditions for oscillations of second order neutral equations, J. Math. Anal. Appl. 138:214-231 (1989). 4 G. Ladas, E. C. Partheniadis, and Y. G. Sficas, Oscillations of second order neutral equations, Canad. J. Math. XI:1301-1314 (1988). 5 Z. Wang, A necessary and sufficient condition for the oscillation of higher order neutral equations, Tohuku Math. J. 41:575-588 (1989). 6 J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
Oscillations of Second Order N e u t r a l F u n c t i o n a l Differential E q u a t i o n s Jurang Yan
Department of Mathematics Shanxi University Taiyuan, Shanxi 030006 People's Republic of China
ABSTRACT Consider the second order neutral functional differential equation
"~
x(t)+ ~ , X ( t - 7 , )
+8~x(t-%)=0
i=1
(E)
j=l
where 8 = 5:1, ~?i, zi, ~, and at, i = 1, 2 , . . . , n, j = 1, 2 , . . . , m, are real numbers. Some new oscillation criteria are given for oscillation of (E). The results of this paper improve noticeably some known results. © Elsevier Science Inc., 1997
1.
INTRODUCTION Consider the second order neutral differential equation
,~t~ ~ ( t ) +
n , x ( t - ,-~1 + 8 i=l
~j~(t - ,~j) = o,
(El
j=l
where 8 = + 1 , ~7,, ri, ~j, a3, i - - 1 , 2 , . . . , n , j= 1,2,...,m axe real numbers, ~j >t 0. Our aim in this paper is to obtain some new sufficient conditions, involving the coefficients and the arguments only, under which
APPLIED MATHEMATICSAND COMPUTATION83:27-41 (1997) ©"Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010
0096-3003/97/$17.00 PI! S0096-3003(96)00043-4
28
J. YAN
all solutions of (E) are oscillatory. Our technique here is based on the study of the characteristic equation 12 1 +
~?~e-~" + 8
~je-"',--0.
(E*)
j=l
The advantage of working with these conditions rather than characteristic equation associated with the equations under consideration is that they are explicit and therefore easily verifiable. The oscillation criteria of this paper noticeably improve the results given in [1]. Necessary and sufficient conditions (in terms of the characteristic equation) for the oscillation of all solutions of second order neutral functional differential equations have been established [2-5]. Second order neutral differential equations were encountered in the study of vibrating masses attached to an elastic bar and also as the Euler equations in some variational problems (see Hale [6, p. 7]). As is customary, a solution of (E) is called oscillatory if it has arbitrarily large zeros on (0, oo). Otherwise, it is called nonoscillatory. The following result is extracted from [2].
LEMMA. Assume that tt = _ 1, ~?~, x,, aj ~ R and ~ >t 0, i = 1, 2 , . . . , n, j = 1, 2 , . . . , m. Then a necessary and sufficient condition for oscillation of all solutions of (E) is that its characteristic equation (E*) has no real roots. . MAIN RESULTS In this paper, we will study the oscillation of the following equations, as some special cases of (E),
d2[
r dt--~ x( t) + ]~- r~z(t Eqkx( t-
d2[
(E~)
L
dt---~ x( t) + F_,r~x(t
K
~,hjx(t+j a~)]
ak) + EP~X( t + El)],
K
~]qkx( t -
~'') -
~) + F_,hjx(t
ak) + ]~-,PlX( t + ill)I, L
,rj)] (E2)
Neutral Functional Equations
29
dt 2 x(t) + ~_,rix(t-'r~) + ~_~hjx(t + crj) I
J
= r , q k x ( t - ak) + r , p ~ z ( t + il~)], K
(E3)
L
and
dt 2 x(t) - E r , x ( t -
~,) - F_.hjx(t + o~j)
I
J
+r, qkx(t-ak) K
+ Ep~x(t + il0] =0.
(E4)
L
where ri, ri, i ~ I, hi, crj, j ~ J, are nonnegative constants and qk, ak, k ~ K, p~, ill, l ~ L are positive constants, while I = {1, 2 , . . . , rtl} , J = {1, 2 , . . . , n2}, K = {1, 2 , . . . , n3}, L = {1, 2 , . . . , n4}, 0 ~< T1 <~ ~'2 < " ' " < "rn, , 0 <<. ~r1 < ~ r 2 , . . . , < o'n2, 0 < ot 1 < a 2 < "'" < an3, and 0 < t l <
il2,--., < il~,.
For convenience, we use the following notations
R = Er,,
H = Ehj,
I
J
Q= Eqk,
P = EP~,
K
L
and the characteristic equations of (E1)-(E 4) are, respectively, FI(A ) - ~ 2 ( 1 +
r, rie - ~ , - r, hjeA~'J)--(~P~gqke-*=k+ I
J
]~p~e~*) = 0 , L
(ET) F2(A)-A2(I+
~_,rie ~ ' - ~_~hjeA~J)--( ~_~gqke-~k+ I
J
~Ple~)
=0,
L
(E~) + I
+
J
K
:
0
L
(E~)
30
J. YAN
and
+ ~.,qke -~'~ + ~_,pze ~ ' = O. I
J
K
L
(E~) First, we obtain sufficient conditions for the oscillation of (El).
THEOREM 1.
-e 4
Assume that a~ > ~'nl ,
]~plflt ~ + ]~hjexp L
~ %
(1)
> 1+ R
J
and
- e E q k ( a k - ~.)2 > 1 + ]~ r,exp 4
K
~
(7,-
I
• ~)).
(2)
Then all solutions of ( E l) oscillate.
PROOF. For A ~: 0, we have that
(3) I
J
and
- Fl( X) e~°'/x2 - [ E qke-'("'-'"') + EL
- [ e~'°' + E'~e-~("-'"l) hJe~(~J+'")]", - EJ
(4)
NeutralFunctionalEquations
31
Now our strategy is to prove that FI(A) < 0 for all h ~ ( - 0 % 0 0 in each of the following four cases
CASE 1.
0 < )t ~< ( P / ( 1 -4- R)) 1/2. F r o m (3) we find that
-FI(A)/~'2 >
>
( ~ qke-~"~+ ~-~Ple;~')/A2L- ( 1 ~
+R)
-(I+R)=0.
CASE 2. A > ( P / ( 1 "4- R)) 1/2. In view of (1) and the inequality e * t> for x I> 0, (3) yields
- r l ( ~ ) / ~ 2 > / ~1e 2 E p ~
2
- (1 +
ex
R)
L
CASE 3.
-(Q/(1 + R)) 1/2 ~< A <
0. From (4) we have that
-FI(A) e~nl/A2 > Q/(1--~) - ( I + R) =0. CASE4. h < - ( Q / ( 1 + R))W2. B y u s i n g ( 2 ) and eX>~ exfor x>~0, we obtian that
-
1 2 FI(A) e~%/A 2 >~ -~ e ~ qk( a~ - T.I)2 K
32
J. Y A N
Cases 1-4 and FI(0) < 0 imply that FI(A) < 0 for all h ~ (-~,oo), that is, (E*) has no real roots. By the lemma we conclue that all solutions of (E 1) oscillate. Next we present a theorem which describes the oscillatory character of
(E2). THEOREM 2.
Assume that fll > 7~,,
-le2~pl(flz - r~,)2 > 1 + ~ riexp 4
L
I
(i"
1 - - - ~ (~'i - •
(5)
and -~e= EqkakK + ~jhjexp -
~(r3
> (1 + R).
(6)
Then all solutions of ( E 2) oscillate. PROOF. From the characteristic equation (E*) of (E2), we have that for A¢0
K
L
I
and
J
33
Neutral Functional Equations Now, we consider the following four cases
CASE 1. 0 < A < ( / ) / ( 1 + R)) 1/2. Then, from (8), (5) and e ~ >I ex for x >t 0, we obtain that
>t
-
1 + •riexp
~
('i-
"-1)
1 E p,( 8 , -
> O.
I
CASE 2.
A > (P/(1 + R)) 1/2. From (8) it follows that
-F2(A) e-a%/A 2>
- (1 + R) = 0 .
-t- R)) 1/2 • A < 0. By using (7), (6) and e ~ >t ex for
CASE 3. - ( Q / ( 1 x >/0, we see that
-F2(A)//A2>/--~"~qka~ g
CASE 4.
~
)t < - ( Q / ( 1
I+R-
~hjexp j
-
--~rj I+R
>0.
+ R)) 1/2. Then, (6) and (7) yield
From Cases 1-4 and F2(0) < 0 we see that for all A ~ ( - ~ , ~)F2(A) < 0, which implies by Lemma that all solutions of (E 2) oscillate. The proof of the theorem is complete.
REMARK 1. The results in Theorems 1 and 2 not only depend on ~ and ri, also depend on hj and (rj (see Remark 1 in [2]).
34
J. YAN THEOREM 3. 1 e2~pl([3, 4 L
Assume that a I > T,~, [31 > ¢r,~, ~r~2)~> 1 + ~ r ~ e x p L
+
hjexp
('z + ~n,) I+R+H
1+ R + H
and
1
-~ e ~g qk( ak -- rn~)2 > 1 + ~] r~exp
[¢o
I
I+R+H
+ ~[~ hjexp j
( , , -- %,)
]
1 + R + H ( g J + %1) "
(lO)
Then all solutions of ( E3) oscillate. PROOF. From the characteristic equation (E~) of (E 3) we obtain that
(11)
-F3(h) ea"/h2=- [~"qke-"(~'"-%)+ n -[ea% + Y'~r~e-a(~'-%) + J Consider the following four cases
CASE 1.
0 < )t ~< ( P / ( 1
-F3(A) e - ~ " 2 / A 2 >
+ R + H)) 1/~. From (11), we have that 1+ R + H
(12)
35
Neutral Functional Equations
CASE 2. that
)t > ( P / ( 1 + R + H)) 1/2. In view of (11) and (9), we deduce
- r~(~)
1 ~- ~--~/z >~~ e 2 E
%)2
p,( ~, -
L
-
1+
r~exp -
I+/~+H
+ ~ hjexp j I + R +p
CASE 3.
(rj - q-2)
]
> 0.
- ( Q / ( 1 + R + H)) 1/2 .<< h < 0. From (12), we obtain that
/(
- F a ( h ) eX%/X 2 > Q
CASE 4.
H(
°
1 + R÷ H
)
0
h < - ( Q / ( 1 + R + H)) 1/2. Then (12) and (10) yield that
1
2
- F3(A) ea%/h 2 i> ~- e ~K qk( °tk - 7.,) 2
-
(
1 + ~riexp I
(i
+ ~ hjexp -J
I+R+H
Q
(z i - %~)
l+R+H(%+%~
)
)
l
>0.
From Cases 1-4 and Fa(0)< 0, we see that F3(A)< 0 for all A ( - 0%oo). By Lemma, all solutions of (E 3) oscillate. The proof of the theorem is complete.
36
J. YAN THEOREM 4.
Assume that R + H > 0, ot1 >
fll
~,2) > ~ / r i e x p -
Tn, ,
fll >
O'n2,
R + H
+ E hjexp j
( o'j - o'n2)
R+H
)
'
(13)
and
1
-e 4
Eqk(~ L
- %)
> E r , exp
R +-----H(~ - %)
+ ~j hjexp -
)
R + H ( qj + r~l) .
(14)
Then all solutions of ( E 4) oscillate.
PROOF.
From the characteristic equation (E*) of (E 4) we have that
+ e - A ~ . 2 - ~_,rie-A(~,+~.~ ) - ~_,hje~(~,-~.. ), (15) I
J
and
F4(a) ea~"l/a 2 -+ e'-,-
E r , e-~(~.-~ol)- E h j e ~ ÷ ~ o , ) . I
J
Now, we consider the following four cases
CASE 1.
0 < A ~< ( P / ( R
~(,~)~-"~-,/,~ I>
+ H)) W2 from (15) we get ~
+ e-"~-~ - R - H > O.
(16)
37
Neutral Functional Equations
CASE 2. A > ( P / ( R + H)) 1/2. In this case, from (15) and (13), it follows that e2
F~(~) ~-~'"/~ 2
>/~-Ep,(~,- %)' L
- ~hjexp
(¢"
R + H (°'~ - o'n,)
)
> 0.
J
CASE 3.
CASE 4. that
- - ( Q / ( R + H)) 1/2 ~< A < 0. Then from (16) we have
A < -(Q/(R
+ H)) 1/2. From (16), in view of (14), we have e2
F4(A) e ~ % / A 2 >/ ~ - E q~( ak - ,,,)2 K
-~hjexp y
[¢0 -
--(o'j+ R+H
]
~'nl) > 0.
From Cases 1-4, F4(0) > 0 and Lemma, we see that all solutions of (E 4) oscillate. The proof of the theorem is complete. 3.
APPLICATIONS AND REMARKS
Consider the following neutral differential equations which are special forms of (E1)-(E4) , respectively, d2 d t 2 [ x ( t ) + r x ( t - 7) - hx(t + (r)] = q x ( t - a) + px(t + [3), (E'I)
d2 d t 2 [ x ( t ) + rx(t + 7) - hx(t + cr)] -- q x ( t - a) + px(t + ~ ) ,
(E'2)
38
J. YAN d2 dt---7[x(t) + r x ( t -
7) + h x ( t + (r)] = q x ( t -
a ) + p x ( t + fl)
(E~)
and d2
dt 2 [ x ( t )
- r x ( t - 7) - h x ( t + o')] + qx( t - a ) + px( t + [3) -- O,
where r, 7, h and cr are nonnegative and q, a , p and fl are positive constants. T h e following oscillation results can be obtained from T h e o r e m s 1-4.
THEOREM 1'.
Assume that a > z,
4 pfl 2e2 + hexp
or
> 1+ r
and 1
~q(a
- 7 ) 2 e 2 > 1 + r.
(17)
Then all solutions of ( E 2) oscillate.
THEOREM 2'.
Assume that fl > 7, 1
-TP( f l 4-
7) :e2 > 1 + r
and -~ qa 2 e2 + hexp -
q)
l+r
Then all solutions of ( E~) oscillate.
v~ > l + r .
(18)
39
Neutral Functional Equations
THEOREM 3'.
Assume that a > v, [3 > cr,
-~p([3-~r)2e2>l+rexp
1 + r + h (~'+ or)
-
+h
and 1
"~q( a - ¢)2e 2 > 1 + r + hexp -
q l+r+h
(¢ + tr)].
Then all solutions of (E~) oscillate.
THEOREM 4'.
Assume that r + h > 0, a > • and [3 > (r,
-~ p( [3 - ~r)2e2 > rexp -
( , + ~r) + h
and
~q(~
- ~)2e2 > r+
hew
-
(~ + ~)].
Then all solutions of ( E~) oscillate.
In view of e * = 1 + ~ = 1 Xk/k!, X ~ (-- 0% ~), we now can obtain the following results from Theorems 1'-4'.
COROLLARY 1. Assume that a > r and (17) holds. If there exists a positive integer N such that
1p[32e2+h1
+ k_~1-~.TIV 1---~r ~
Then all solutions of ( E~) oscillate.
>l+r.
40
J. YAN
COROLLARY2. N such that
~q
Assume that [J > v and (18) holds. If there exists an odd
+hl+
E
q
k=l
Then all solutions of (E~) oscillate.
COROLLARY 3.
Assume that a > T and fl > ~r. If there exist an odd N
such that -1~ p ( f l - o " )2e2 > l + r
[1
(-1)k + k~= l
P
k!
+h
l+r+h
and N (_1) k -~q( a - r ) 2 e 2> 1 + r + h 1 +
kY~ =l
q
k!
(r+
~)
.
l+r+h
Then all solutions of ( E~) oscillate.
COROLLARY 4. Assume that r + h > O, a > exists an odd N such that 1 ~p(fl-
(r
)2e2
N (_1) k E k!
> 7" 1 +
• and fl > ~r. If there
P
(~'+ (r)
+ h
k=l
and
[
1 N (_1) k ~ q ( a -- v)2e2 > r + h 1 + • k! k=l
Then all solutions of ( E~) oscillate.
q
( v + (r)
.
Neutral Functional Equations REMARK 2.
41
Corollaries 1-4 improve, respectively, Theorems 1-4 in [1].
REMARK 3. By using technique of this paper, we can obtain more oscillation criteria for (E).
REMARK 4. Our technique can be extended to higher order neutral functional differential equations.
REMARK 5. It is easy to construct examples showing that our criteria for oscillation of (E'I)-(E'4) are essentially wider than the oscillation criteria in
[I]. REFERENCES 1 S. R. Grace, and B. S. Lalli, Oscillation theorems for second order neutral functional differential equations, Appl. Math. Comp. 51:119-133 (1992). 2 S.J. Bilchev, M. K. Grammatikopoulos, and I. P. Stavroulakis, Oscillations of higher order neutral differential equations, J. Austral. Math. Soc. (Series A) 52:261-284 (1992). 3 G. Ladas, E. C. Partheniadis, and Y. G. Sficas, Necessary and sufficient conditions for oscillations of second order neutral equations, J. Math. Anal. Appl. 138:214-231 (1989). 4 G. Ladas, E. C. Partheniadis, and Y. G. Sficas, Oscillations of second order neutral equations, Canad. J. Math. XI:1301-1314 (1988). 5 Z. Wang, A necessary and sufficient condition for the oscillation of higher order neutral equations, Tohuku Math. J. 41:575-588 (1989). 6 J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.