On the oscillatory and asymptotic behavior of even order nonlinear differential equations with retarded arguments

JOURNAL

OF MATHEMATICAL

ANALYSIS

AND

APPLICATIONS

137, 528-540 (1989)

On the Oscillatory and Asymptotic of Even Order Nonlinear Differential with Retarded Arguments S.

Behavior Equations

R. GRACE

Department of Mathematical Sciences, University of Petroleum and Minerals, P. 0. Box 1682, Dhahran, Saudi Arabia Submitted by E. Stanley Lee

Received June 9, 1987

Some new oscillation criteria for the retarded differential equations of the form

n is even, are established.

$2 1989 Academic

Press, Inc.

1. INTRODUCTION In this paper we are interested in-obtaining results on the oscillatory and asymptotic behavior of solutions of a broad class of even order nonlinear differential equations with deviating arguments. During the last 25 years there has been a great deal of work on the qualitative properties of solutions of equations of the type (a(t) x.(t)). + dt)fb(t))

=0

(

‘=z

d >

>

(6)

but, since we are interested in obtaining results for higher order differential equations, we chose not to list those papers. Considerably less is know about the behavior of solutions of equations of the form

(a(t) x’(t))’ + q(t)f(x(t) Kf(t)) = 09 528 0022-247X/89 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.

(E,)

529

RETARDED DIFFERENTIAL EQUATIONS

and as recent contributions to this study we cite the papers of Lalli [14] and Wong and Burton [21]. Recently, there has been interest in obtaining results on the oscillatory behavior of solutions of nonlinear equations with deviating arguments of the type

x”(t) +At) k(4 4th X’(f)) x.(t) + dt)fcG,(t)l)

O’Cg*(t)l) = 0.

(EA

Such results can be found in the papers of Grace and Lalli [S, 91 and the referencescited therein. In [13] Lalli and Grace extended the results in [8,9] to more general equations of the form (u(r) x+ y’(t))‘“’ + df)f(a!l(t)l)

Nx’Cg*(t)l) = 0,

n is even. (E4)

Here we consider the equation

Lx(t) + d~).f(-aI(~)l)

Mx’Cg2(t)l) =o,

n is even,

(E)

where for k = 1, 2, ,.., n

LJx(t) =x(t),

with a,(t) = 1. The oscillatory behavior of some special casesof Eq. (E) with h = 1 has been studied by many authors. As examples we refer the reader to the work of Grace and Lalli ([l-9], Graef et al. [lo], Kartsatos [ 123, Lalli and Grace [ 131, Philos [ 15, 161, SIicas and Staikos [ 171, Staikos [ 181, and Trench [ 19,201. However, when h & 1, nothing seemsto be known about the oscillatory and the asymptotic nature of the solutions of Eq. (E). In this work we give new criteria for all nontrivial solutions of Eq. (E) to be oscillatory and we establish some new sufficient conditions for any nontrivial solution x of Eq. (E) to be either oscillatory or satisfying &x(t) + 0 as t + GO,k = 1,2, .... n - 1. Some comparisons between our theorems and those of other authors are indicated, and some examples illustrating our results and the effect of the damping term h on the asymptotic and oscillatory character of Eq. (E) are also included.

2. MAIN RESULTS Consider the equation

L-4f) + cdf)f(xCs,(t)l) h(x’Cg,(t)l) = 0,

n is even,

(1)

530

S. R. GRACE

where &J(t) = x(t),

Lx(t)

= 4t)(L,-

IX(f))’

for k = 1, 2, .... n,

with a,(t) = 1, ai, q, gj: [to, m) -+ R = (-00, co), f, h: R -+ R are continuous, ai( t) > 0 (i = 1, 2, .... n - l), q(t) nonnegative and not identically zero on any ray of the form [t*, co) for some t* > t,, and lim,,, g,(t) = co (j= 1, 2). We assume that 02 1 s

-ds=

oo,

Pits)

where pLi(t) = max a,(s) for t 2 t* 2 t, and for i = 1, 2, .... n - 1, r*O, K,f(x)f(

f’(x)>0 for x#O and -f( - XY) 3f(XY) 2 y) for x, y > 0, where K, is a positive constant

(’ = d/dx),

h(x)>0 K,h(x)

(2)

(3)

for x#O, h’(x)30 for x>O, h(-xy)>h(xy)> h(y) for x, y > 0, where K2 is a positive constant.

(4)

We further assume that there exists a continuous function CT:[to, co)+ R

such that inf { gl(s), g2b) 1 da(t)r

lim a(t)=co,a.(t)aO ,-rClZ

for

t3t0.

(5)

The domain D(L,) of L, is defined to be the set of all functions x: [to, co) + R such that L,x(t) (j= 0, 1, .... n) exist and are continuous on [to, 03). By a solution of Eq. (1) we mean a function x E D(L,) which satisfies Eq. (1) on [I,, co). A nontrivial solution of Eq. (1) is called oscillatory if the set of its zeros is unbounded and it is called nonoscillatory otherwise. Equation (1) is called oscillatory if all its solutions are oscillatory. Remark 1. It follows from condition (3) that f(0) =0 and from condition (4) that either h(O) = 0 or h(O) > 0. Therefore, the functions x(t) - 0 and x(t) = constant when h(O) = 0 are solutions of Eq. (1). In this paper we consider such solutions as the “trivial solutions” of Eq. (1) and hence we exclude them from any further discussion.

RETARDED

DIFFERENTIAL

EQUATIONS

531

The following lemma generalizes a well-known lemmas of Kiguradze and can be proved similarly. LEMMA 1. Let condition (2) hold and let x If L,x(t) is of constant sign and not identically exist t, 2 t, and an integer 1 (I= 0, 1, .... n), negative or n + 1 odd for L,x nonpositive and

E D(L,) be a positive function. zero for all large t, then there with n + 1 even for L,x nonsuch that, for every t 2 t,,

(k=O, 1, ...) I- 1)

I> 0 implies L,x( t) > 0 and l
1 impZies (-l)‘+k

(k=l, I+ 1, .... n- 1).

L,x(t)>O

The following lemma appears in [4] and is needed in the sequel. LEMMA 2. Let n be even, x E D(L,) (2) hold. Zf

L .-,x(t)

with x(t) > 0 for t 3 to, and condition

for all

J+(t)<0

t > t, 2 t,,

t, is sufficiently large, then there exist T > t, and positive constants M, and M, such that, for each t 2 T, (i)

x(t)>Mlal(T

cc,t)Llx(t),

(ii)

x’(t) B M2u2(T, P, t) L

,x(t),

where

and

For convenience of notation for T> to and all t > T we let dt)f

(a,(T, /A g,(t))) h(dT,

~1,gdt))) = Q(T p>g)

and GK:f

(M, 1h(M,) = c > 0,

where M, and M, are as in Lemma 2.

532

S. R.GRACE

THEOREM

1. Let conditions (2~( 5) hold and

1x1

(6)

.!%f(,xl)h(x)‘“’

where M is a positive constant. If f or every large T with g,(t) > T (i = 1, 2) t>T>t,

lim sup ’ Q(T.mW+ I df) ,-CC

(7)

then Eq. (1) is oscillatory. Proof Let x(t) be a (nontrivial) nonoscillatory solution of Eq. (l), say x(t) > 0 for t > t,. Then there exists a t, > to so that x[g,(t)] > 0 for t 3 t,.

From Eq. (1) and conditions (3) and (4) we obtain L,x(t)

d0

for

tat,.

Moreover, q(t) & 0 on any ray [t*, co) for some t* B t, ensures that also has this property. Note next that the hypotheses of Lemma 1 are satisfied on [tl, co) which implies that there exists t, 3 t, such that

L,x(t)

x’(t)>0

and

L n-Ix(t)>0

for every t 3 t,.

(8)

Note next that the hypotheses of Lemma 2 are satisfied on [t2, co) which implies that there exists a t, 3 t, and positive constants M, and M, so that x(t)2 M1a,(t,, PL,t) L-Ix(t)

for

tat,

x’(t) 2 M*dt3,

for

t2 t,.

and PL,t) L,- ,x(t)

Choose a t4 2 t, so that g,(t) > t, for every t > t, (i= 1,2). Then

xCg1(t)l~ M,~I(t,> PL,g,(t)) L”V,xCg,(t)l

for

tat,

x’Cg*(t)l 3 M*Mt3, PYIT*(f)) L,xCs*(t)l

for

t> t,.

(9)

and (10)

Let z(t) = L,- 1x(t); then from Equation (1) and (3), (4), (9), and (10) we get

z.(t)+K:~f(M1)h(Mz)f(cc,(t~, ~L,g,(t)))h(az(t,, ~,gz(t))) for tat,. xf(zCgl(t)l)h(zCg,(t)l)~‘0,

RETARDED

DIFFERENTIAL

533

EQUATIONS

Or

z’(t) + cQ(t,, PL,g)f(dg,(t)l)

Wg,(t)l)

60

for

tat,.

(11)

Using conditions (3), (4), and (5) and the fact that z(t) is nonincreasing on [t4, cc) we obtain

z’(r) + cQ(t3, P, g)f(zC4t)l)

MzC4t))l) 60

for

t>fq.

(12)

Integrating (12) from a(t) to t, we have

z(t) -zC4t)l+

c j’ Q(w g)f(z[a(s)l) U(l)

h(zCa(s)l) ds~0.

Since a(t) 2 0 for t 3 s, we obtain

z(t) - zCdt)l+ cf(zC4t)l) WCdt)l)

I;,,, Q(h, p, g) ds GO. (13)

By using the fact that z’(t) < 0, we have lim z(t) = y,

1-m

(14)

where y 2 0 is a constant. Suppose y > 0; then, because of (7) and (13) it leads to a contradiction. Now, suppose y = 0. From (13) we have

o
zCdt)l S(zC4t)l) WCdt)l)’

(15)

If we take the limit superior of (15) as t -+ co, because of (6), we get a contradiction to (7). In the case where the assumption (6) does not hold, it is easy to verify the following. THEOREM 2. Suppose that the conditions (2)-(5) and (7) hold. Then every solution x(t) of Eq. (1) is oscillatory or lim, _ o. L,- Ix(t) = 0.

Remark 2. From Remark 1, it is understood that any constant solution of Eq. ( 1) when h(0) = 0 is excluded, and hence we mean by “Eq. (1) is oscillatory” that “all nonconstant solutions of Eq. (1) when h(0) = 0 are oscillatory and all solutions of Eq. (1) are oscillatory if h(0) > 0.”

The following theorem is concerned with the oscillatory and asymptotic behavior of the solutions of Eq. (1). 409/137/2-16

534

S.R.CiRACE

For any T > t, and all t > T we let

w(T,a, t)=

J’Ta,,-,;s,~,, J~-‘-..J~~ds,...ds,,

and

THEOREM

and

3. Let conditions (2)-(4) hold. Zf, in addition

a,gl(s))) hMT, a,gh))) ds= ~0, Jmds)f(UT, mw( T, a, s) P( T, a, g) ds = 00, J

(16)

(17)

for all sufficiently large T, with gi(t) > T (i= 1, 2) and t > T> t,,, then any solution x(t) of Eq. (1) is either oscillatory or satisfies lim, _ ooLkx(t) = 0 (k = 1, 2, .... n - 1). Proof Let x(t) be a nonoscillatory solution of Eq. (l), say x(t) > 0 for t 2 to. As in the proof of Theorem 1, we obtain LkX(t) (k = 0, 1, ...) n) are of one sign, x’(t)>O, L,-,x(t)>0 tat*.

for *

Now, we consider the following two cases: Case 1. L,x( t) > 0 for t 2 t,. Since n is even, the integer I assigned to the solution x(t) by Lemma 1 is odd and I> 3. A repeated application of I’Hopital’s rule leads to x.(t) x(t) = lim = lim &x(t)=& lim t-00 Yz(tz,a, t) I.+~ Y;(t*,a, t) r-+m Since lim, _ cogi(t) = co (i= 1, 2), there exist t, 2 t, and constants di> 0 (i= 1,2) such that

xCgl(t)l 2 ~1~2(~~Z~ 4 g,(t))

for

tZ3 t,

RETARDED

DIFFERENTIAL

535

EQUATIONS

and x’Cg2(f)l2 bY;(t*, a, g2(t))

for

t> t3.

Integrating Eq. (1) over [I~, t] we obtain

L-IX(~) = L

,x(t,) - j’ ds)f(-G~(s)l) (3

W’Cg,(s)l) ds

or L1x(t3)2

[’ q(s)f(6 1Y2(t 2, a>gl(s)))h(62~;(t,,

a,gAs)))ds

which contradicts (16). Thus L2x(t) cannot be positive for t > f2. Case 2. 15,x(t) t,. By Lemma 2, there exist t,> t2 and a positive constant N such that

Since lim r+m g,(t)= co, there exists a t,b t, such that

xCg,(t)l a Wl(Q4 gl(t)) hxCg,(t)l

for

t>t4.

(18)

Using (3), (4), and (18) in Eq. (1) we obtain

xf(b-a,(t)l

for

WI-a2(Ol)~O

tat,,

or

Lx(t) + c*P(h 4 ‘!Y)f(LxCg,(t)l) where c* = $K,f(N).

h(L-a2(~)1)

G0

for

tat,,

Setting u(t) = L,x(t) we obtain

R n-14t)+c*fYt3,

a,g)f(uCg,(t)l)h(uCg,(t)l)~0,

(19)

where R,-,u(t)=(u,-,(t)...(u,(t)(u,(t)u’)’)’...)’.

Now, u(t) is positive and decreasing solution of the odd inequality (19).

S. R.GRACE

536

Thus, by applying results in [4] and [16] we conclude that This comlim,, m u(t)=0 and hence L,x(t)-+O as t-+co,j=l,...,n-1. pletes the proof. For illustration we consider the following examples: EXAMPLE

1. Consider the equation

The conditions of Theorem 1 are satisfied and hence all nonconstant solutions of Eq. (20) are oscillatory. Next, consider the equation

(;(;(;xj.!)-g

(x[&p3 (x.[Ji],“= 0, tat?.(21)

The hypotheses of Theorem 3 are satisfied and hence every nonconstant solution x(t) of Eq. (21) is oscillatory or satisfies &x(t) + 0 as t + cc (k = 1, 2, ...) n - 1). Equation (21) has the nonoscillatory solution x(t) = In I. Indeed, Eq. (20) and (21) have some q, ai (i= 1,2, 3), gi (i= 1,2), and f’, hence the reason for the above conclusion is the effect of the damping term h. EXAMPLE

2. The equation

(e-t(e-‘(e-‘x.).).).+gfix e [;](x-[;])Lo,

t>o,

(22)

has the bounded nonoscillatory solution x(t) = 1 - e-’ satisfying the conclusion of Theorem 3. All conditions of Theorem 3 are satisfied. We may note that the differential equation (eC’(e-‘(e-lx’)‘)‘)’

+g

e

x

[Ii

= 0,

t>0,

is oscillaory by Theorem 1. Once again the reason is the effect of the damping term h. Remark 3. Since we impose no restriction on the functions g,, g,,f, and h other than those in conditions (3) and (4), we seethat the conditions

RETARDED

DIFFERENTIAL

537

EQUATIONS

(16) and (17) of Theorem 3 are not related. To see this, we consider the equation

+4(t) xCs1(t)l(l-f Cgz(t)l I) =o,

t>O.

(24)

If q(t) = t-14, g,(t) = t, and gz(t) = t3, then condition (16) is satisfied, while condition (17) is violated. Next, if q(t)= tp8, g,(t)=&, and g2(t) = J, then condition (17) holds and condition (16) is not satisfied. Hence, for both casesTheorem 3 fails to apply to Eq. (24). The following theorem is concerned with the comparison of Eq. (1) with the first order equation

y’(t) + cQ(T PL,g)fLYCg,(t)l) OCg*(t)l)

= 0,

(25)

where c and Q are as in Theorem 1. THEOREM 4. large T either:

Let g,(t) < t (i= 1,2), conditions (2)-(4)

hold, and for all

(i) every bounded solution y(t) of Eq. (25) is oscillatory, or (ii) every nonoscillatory solution y(t) of Eq. (25) satisfies lim 1--r03Y(t) z 0. Then Eq. (1) is oscillatory. Proof: Let x(t) be a (nontrivial) nonoscillatory solution of Eq. (1 ), say x(t) > 0 for t > to. As in the proof of Theorem 1, we get (11). Now, integrating (11) from t to U, t > t,, we have z(t) az(u) + c I ' Q(ts, PL,g)f

I

(zCsl(s)l) W&)1)

ds.

Since lim u_ m z(u) exists and is nonnegative we obtain

z(t) 2 c i m Q(t,, PL,g)f(zCsl(s)l) &Cgh)l) f

ds.

The function z(t) = L, _ I x(t) is strictly decreasing on [t2, co). Hence by Theorem 1 in [ 151 we conclude that there exists a positive solution y(t) of Eq. (25) with lim,,, y(t) = 0. This contradicts the assumption of the theorem and the proof is complete. The following result is concerned with the strongly sublinear equation of the form of Eq. (1); i.e., the function j7r satisfies du

I +ofWW-

and

du s-0 f(u) h(u) < co.

(26)

538

S.R. GRACE

THEOREM 5. Let conditions (2)-(5) and (26) hold. ufor large T with g,(t) > T (i= 1, 2), t > T> t,,, we have

all sufficiently

saQ(T,p,g,ds=m,

(27)

then Eq. (1) is oscillatory. Proof: Let x(t) be a nonoscillatory solution of Eq. (1). Assume x(t) > 0 for t 2 t,. As in the proof of Theorem 1, we obtain (12). Thus

z’(t) cQct3’ 14‘)’

-f(z[a(t)])

t2t4.

h(z[a(t)])’

Using conditions (3), (4), (5) and integrating from t, to t we get

a contradiction to (27). This completes the proof. For illustration we consider the following example. EXAMPLE

3. Consider the differential equation

(t(tx.)-)..+q

tr22i15(x(t))1/3 (x’(t))“‘=O,

t >O.

(28)

Here c~i(T, p, t) - O(t) and CI~(T, p, t) -constant for all large T, t + co, and f(x) h(x) = x 11”5.Equation (28) has the nonoscillatory solution x(t) = &. Only condition (27) of Theorem 5 is violated. We may note that Theorem 3 is not applicable to Eq. (28) since condition 16) is violated. Clearly, the conclusion of Theorem 3 fails (tx’ = L 1x = t J t f+Oas t-co). Next, consider the differential equation (t( tx.).)-

+

12)“’ 16

t-22/15(X(t))ll/15

=

0

3

t>

0.

(29)

The hypotheses of Theorem 5 are satisfied and hence Eq. (29) is oscillatory. It is easy to check that Eqs. (28) and (29) have the same q,fh, g,, and g,, and hence we conclude that the reason for that damage in the oscillatory character of Eq. (28) is the effect of the damping term h. Now, all the nonconstant solutions of the differential equation (t(tx’)‘)”

+ tr4’3(X(t))1’3 (Xjt))2’5 =o,

t>o

(30)

539

RETARDED DIFFERENTIALEQUATIONS

are oscillatory by Theorem 5 and also, the equation (?(fX')')"

+ tr4'3(X(t))11'15

=o,

f>O

(31)

is oscillatory by Theorem 5. Hence we conclude that if Eq. (1) is oscillatory, then the equation

Lx(t) + d~)f(xCg,(~)l) W[g2(t)l)

=o

(32)

is also oscillatory. The converse is obviously not true. Remark 4. (1) It is easy to check that our Theorems 1 and 5 are more general than Theorems 1 and 3 in [ 131. Also, Theorem 4 includes Theorem 2 in [4]. (2) The results of this paper are presented in a form which is essentially new. It includes as a special case some of the results in [8,9, 12, 161. (3) The “size” of the damping term h in Eq. ( 1) plays important roles in the study of the oscillatory behavior of Eq. (1). The presenceof the damping may preserve the oscillatory character of both damped and undamped (h s 1) equations as in Eqs. (30) and (31), or else make disruption in the oscillatory behavior of the undamped equation, as in the undamped oscillatory Eq. (29) and the damped Eq. (28). (4) In Theorem 3, we impose only condition (3) on h; therefore the effect of such damping may change the behavior of the solutions of Eq. (1) from oscillatory to nonoscillatory as illustrated in Examples 1 and 2. REFERENCES 1. S. R. GRACE, Oscillation theorems for nth order differential equations with deviating arguments, J. Math. Anal. Appl. 101 (1984), 268-296. 2. S. R. GRACE AND B. S. LALLI, Oscillation theorems for nth order delay differential equations, J. Math. Anal. Appl. 91 (1983), 352-366. 3. S. R. GRACE AND B. S. LALLI, A note on Ladas’ paper: Oscillatory effect of retarded actions, J. Math. Anal. Appl. 88 (1982) 2577264. 4. S. R. GRACE AND B. S. LALLI, Oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments, J. Math. Anal. Appl. 104 (1984), 79-94. 5. S. R. GRACE AND B. S. LALLI, Oscillatory behavior of nonlinear differential equations with deviating arguments, Bull. Austral. Math. Sot. 31 (1985), 127-136. 6. S. R. GRACE AND B. S. LALLI, Oscillation theorems for nth order nonlinear differential equations with deviating arguments, Proc. Amer. Marh. Sot. 90 (1984), 65-70. 7. S. R. GRACE AND B. S. LALLI, Oscillation theorems for damped differential equations of even order with deviating arguments, SIAM J. Math. Anal. 15 (1984), 308-316. 8. S. R. GRACE AND B. S. LALLI, Oscillatory behavior of nonlinear second order differential equations with deviating arguments, Bull. Insf. Mafh. Acad. Sinica 14 (1986), 187-196. 9. S. R. GRACE AND B. S. LALLI, An oscillation criterion for certain second order strongly sublinear differential equations, J. Math. Anal Appl. 123 (1987), 584-588.

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S. R. GRACE

10. J. R. GRAEF, M. K. GRAMMTIKOPOLJLOS,AND P. W. SPIKES, On the behavior of solutions of generalized Emden-Fowler equations with deviating arguments, Hiroshima Math. J. 12 (1982), l-10. 11. A. G. KARTSATOS, On positive solutions of perturbed nonlinear differential equations, J. Math. Anal. Appl. 47 (1974), 58-68. 12. A. G. KARTSATOS, Oscillation properties of solutions of even order differential equations, Bull. Fat. Sci. Ibaraki Univ. Math. 2 (1969), 9-14. 13. B. S. LALLI AND S. R. GRACE, Some oscillation criteria for delay differential equations of even order, J. Math. Anal. Appl. 119 (1986), 164-170. 14. B. S. LALLI, On boundedness of solutions of certain second order differential equations, J. Math. Anal. Appl. 25 (1969), 182-188. 15. CH. G. PHILOS, On the existence of nonoscillatory solutions tending to zero at co for differential equations with positive delays, Arch. Math. 36 (1981), 168-178. 16. CH. G. PHILOS, Oscillatory and asymptotic behavior of all solutions of differential equations with deviating arguments, Proc. Roy. Sot. Edinburgh Sect A 81 (1978), 195-210. 17. Y. G. SFICA.SAND V. A. STAIKOS, Oqcillation of differential equations with deviating arguments, Funkciul. Ekuac. 19 (1976). 3543. 18. V. A. STAIKOS, Basic results on oscillation for differential equations, Hiroshima Math. J. 10 (1980), 495-516. 19. W. F. TRENCH, Canonical forms and principal systems for general disconjugate equations, Trans. Amer. Math. Sot. 52 (1975), 147-155. 20. W. F. TRENCH, Asymptotic theory of perturbed general disconjugate equations, Hiroshima Math. J. 12 (1982), 43-58. 21. J. S. W. WONG AND T. A. BURTON, Some properties of u” + a(t)f(u) g(d) = 0, Monatsh. Math. 69 (1965), 364-374.