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Oscillation criteria for second order nonlinear neutral differential equations
Applied Mathematics and Computation 215 (2010) 4392–4399
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Oscillation criteria for second order nonlinear neutral differential equations q Mustafa Hasanbulli a, Yuri V. Rogovchenko a,b,* a b
Department of Mathematics, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey School of Pure and Applied Natural Sciences, University of Kalmar, SE-391 82 Kalmar, Sweden
a r t i c l e
i n f o
Keywords: Nonlinear neutral differential equations Second order Asymptotic behavior Oscillatory solutions Integral averaging
a b s t r a c t By refining the standard integral averaging technique, we obtain new oscillation criteria for a class of second order nonlinear neutral differential equations of the form
ðrðtÞðxðtÞ þ pðtÞxðt sÞÞ0 Þ0 þ qðtÞf ðxðtÞ; xðrðtÞÞÞ ¼ 0: Assumptions in our theorems are less restrictive, whereas the proofs are significantly simpler compared to those in the recent paper by Shi and Wang [18] and related contributions to the subject. Examples are provided to illustrate the relevance of new theorems. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction In this paper, we are concerned with the oscillatory behavior of second order nonlinear neutral delay differential equations
ðrðtÞðxðtÞ þ pðtÞxðt sÞÞ0 Þ0 þ qðtÞf ðxðtÞ; xðrðtÞÞÞ ¼ 0; 1
ð1Þ 2
where t P t 0 > 0; s P 0 is a constant, r; r 2 C ð½t 0 ; þ1Þ; ð0; þ1ÞÞ; p; q 2 Cð½t 0 ; þ1Þ; RÞ, and f 2 CðR ; RÞ. By a solution of Eq. (1) we mean a continuous function xðtÞ, defined on ½tx ; þ1Þ, such that rðtÞðxðtÞ þ pðtÞxðt sÞÞ0 is continuously differentiable and xðtÞ satisfies (1) for t P tx : In the sequel, we assume that solutions of Eq. (1) exist and can be continued indefinitely to the right. Recall that a nontrivial solution xðtÞ of Eq. (1) is called oscillatory if there exists a sequence of real numbers ftk g1 k¼1 , diverging to þ1, such that xðt k Þ ¼ 0. Neutral differential Eq. (1) is said to be oscillatory if all its solutions oscillate. Recently, an increasing interest in obtaining sufficient conditions for oscillatory or non-oscillatory behavior of different classes of differential and functional differential equations has been manifested. In particular, investigation of neutral differential equations is important since they are encountered in many applications in science and technology and are used, for instance, to describe distributed networks with lossless transmission lines, in the study of vibrating masses attached to an elastic bar, as well as in some variational problems, see [11]. It is well known [4] that the presence of a neutral term in a differential equation can cause oscillation, but it can also destroy oscillatory nature of a differential equation. In general, investigation of neutral differential equations is more difficult in comparison with ordinary differential equations, although certain similarities in the behavior of solutions of ordinary and neutral differential equations can be observed, see [1–20].
q
Research of the second author has been supported in part through the grant from the Faculty of Sciences and Technology of the University of Kalmar. * Corresponding author. Present address: University of Kalmar, School of Pure and Applied Natural Sciences, Famagusta, 391 82 Kalmar, SE, Sweden. E-mail addresses: [email protected], [email protected] (Yu.V. Rogovchenko).
0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.001
M. Hasanbulli, Yu.V. Rogovchenko / Applied Mathematics and Computation 215 (2010) 4392–4399
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In what follows, we briefly review several important oscillation results obtained for second order neutral differential equations. Grammatikopoulos et al. [9] established that condition
Z
þ1
qðsÞ½1 pðs rÞ ds ¼ þ1;
t0
ensures oscillation of a linear neutral differential equation
ðxðtÞ þ pðtÞxðt sÞÞ00 þ qðtÞxðt rÞ ¼ 0: Sufficient conditions for the oscillation of solutions of a slightly more general neutral differential equation
ðxðtÞ þ pðtÞxðt sÞÞ00 þ qðtÞxðrðtÞÞ ¼ 0; including the case when p ¼ 1, were obtained by Dzˇurina and Mihalı´ková [2]. By using the integral averaging method, Ruan [17] derived a number of general oscillation criteria for a nonlinear neutral differential equation
ðrðtÞðxðtÞ þ pðtÞxðt sÞÞ0 Þ0 þ qðtÞf ðxðt rÞÞ ¼ 0;
ð2Þ
whereas Li [14] provided classification of non-oscillatory solutions of Eq. (2) and established necessary and/or sufficient conditions for the existence of eventually positive solutions. Ruan’s results for Eq. (2) have been further improved by Li and Liu [15] who exploited a generalized Riccati transformation. Interesting applications of the integral averaging technique to oscillation of several classes of nonlinear neutral differential equations can be found in the papers by Dzˇurina and Lacková [1], Gai et al. [5], and Xu et al. [19]. In particular, the latter paper addresses the oscillation of a nonlinear neutral differential equation
ðrðtÞðxðtÞ þ pðtÞxðt sÞÞ0 Þ0 þ f ðt; xðtÞ; xðt rÞ; x0 ðtÞÞ ¼ 0; where
f ðt; xðtÞ; xðt rÞ; x0 ðtÞÞ P qðtÞf1 ðxðtÞÞf2 ðxðt rÞÞgðx0 ðtÞÞ; f1 ðxÞ P k1 > 0; f2 ðxÞ=x P k2 > 0; gðxÞ P k3 > 0. Recently, Shi and Wang [18] proved several oscillation criteria for Eq. (1), one of which we present below for the convenience of the reader. In what follows, we use the following notation:
D0 ¼ fðt; sÞ : t 0 6 s < t < þ1g and D ¼ fðt; sÞ : t 0 6 s 6 t < þ1g:
Theorem 1 [18, Theorem 2]. Let the following conditions hold: (A1) (A2) (A3) (A4)
for all t P t0 ; 0 6 pðtÞ 6 1; qðtÞ P 0, and qðtÞ is not identically zero for large t, R þ1 1 r ðsÞ ds ¼ þ1, 0 t P t0 ; rðtÞ 6 t; r ðtÞ > 0, and limt!þ1 rðtÞ ¼ þ1, for all f ðx;yÞ P K > 0, for y–0, and f ðx; yÞ has the sign of x and y if they have the same sign. y
Suppose further that there exist functions H 2 C 1 ðD; RÞ; h 2 CðD0 ; RÞ, and k; q 2 C 1 ð½t 0 ; þ1Þ; ð0; þ1ÞÞ satisfying
ðiÞ Hðt; tÞ ¼ 0; t P t0 ; Hðt; sÞ > 0; t > s P t0 ; @H ðiiÞ ðt; sÞ 6 0; ðt; sÞ 2 D0 ; @s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ q0 ðsÞ ðiiiÞ ½Hðt; sÞkðsÞ þ Hðt; sÞkðsÞ ¼ hðt; sÞ Hðt; sÞkðsÞ: @s qðsÞ Assume also that
Hðt; sÞ 6 þ1; 0 < inf lim inf t!þ1 sPt 0 Hðt; t 0 Þ
ð3Þ
and
lim supt!þ1
1 Hðt; t 0 Þ
Z
t
t0
rðrðsÞÞqðsÞ 2 h ðt; sÞ ds < þ1: r0 ðsÞ
If there exists a function B 2 Cð½t0 ; þ1Þ; RÞ such that
lim supt!þ1
Z
t
t0
r0 ðsÞB2þ ðsÞ ds ¼ þ1; kðsÞqðsÞrðrðsÞÞ
and
lim supt!þ1
1 Hðt; TÞ
Z t rðrðsÞÞqðsÞ 2 KHðt; sÞkðsÞqðsÞqðsÞð1 pðrðsÞÞÞ ðt; sÞ ds P BðTÞ; h 4r0 ðsÞ T
ð4Þ
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M. Hasanbulli, Yu.V. Rogovchenko / Applied Mathematics and Computation 215 (2010) 4392–4399
for any T P t0 , where Bþ ðtÞ ¼ maxðBðtÞ; 0Þ, then Eq. (1) is oscillatory. Very recently, Rogovchenko and Tuncay [16] established new oscillation criteria for a second order nonlinear differential equation with a damping term
ðrðtÞx0 ðtÞÞ0 þ pðtÞx0 ðtÞ þ qðtÞf ðxðtÞÞ ¼ 0; without an assumption that has been required in related results reported in the literature over the last two decades. The purpose of this paper is to strengthen oscillation results obtained for Eq. (1) by Shi and Wang [18] using a generalized Riccati transformation and developing ideas exploited by Rogovchenko and Tuncay [16]. In order to illustrate the relevance of our theorems, two examples are provided. 2. Main results We say that a continuous function H : D ! ½0; þ1Þ belongs to the class I if: (i) Hðt; tÞ ¼ 0 and Hðt; sÞ > 0 for ðt; sÞ 2 D0 ; (ii) H has a continuous partial derivative with respect to the second variable satisfying, for some locally integrable continuous function h,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ Hðt; sÞ ¼ hðt; sÞ Hðt; sÞ: @s
ð5Þ
Theorem 2. Let conditions ðA1 Þ ðA3 Þ of Theorem 1 hold with ðA4 Þ ðA4 Þ f ðx;yÞ y P j > 0, for y–0, and xf ðx; yÞ > 0, for xy > 0. Suppose that there exits a function q 2 C 1 ð½t 0 ; þ1Þ; RÞ such that, for some b P 1 and for some H 2 I,
lim supt!þ1
1 Hðt; t 0 Þ
Z t
Hðt; sÞwðsÞ
t0
bv ðsÞrðrðsÞÞ 2 ðt; sÞ ds ¼ þ1; h 4r0 ðsÞ
replaced
by
ð6Þ
where
wðtÞ ¼ v ðtÞ
jqðtÞ½1 pðrðtÞÞ þ r0 ðtÞ
r2 ðtÞq2 ðtÞ ðrðtÞqðtÞÞ0 rðrðtÞÞ
ð7Þ
and
Z
t
v ðtÞ ¼ expð2 r0 ðsÞ
rðsÞqðsÞ dsÞ: rðrðsÞÞ
ð8Þ
Then Eq. (1) is oscillatory. Proof. Let xðtÞ be a non-oscillatory solution of Eq. (1). Then, there exists a T 0 P t 0 such that xðtÞ–0 for all t P T 0 . Without loss of generality, we may assume that xðtÞ > 0 and xðrðtÞÞ > 0, for all t P T 0 P t0 . Define
zðtÞ ¼ xðtÞ þ pðtÞxðt sÞ;
t P T0:
Obviously, for all t P T 0 ; zðtÞ P xðtÞ > 0, and rðtÞz0 ðtÞ is non-increasing because
ðrðtÞz0 ðtÞÞ0 ¼ qðtÞf ðxðtÞ;
xðrðtÞÞÞ 6 0:
0
We claim that z ðtÞ P 0, for all t P T 0 . Otherwise, there should exist a T 1 P T 0 P t 0 such that z0 ðT 1 Þ < 0, which implies that rðT 1 Þz0 ðT 1 Þ < 0. Since rðtÞz0 ðtÞ is non-increasing and qðtÞ does not eventually vanish, there exists a T 2 P T 1 such that rðT 2 Þz0 ðT 2 Þ < 0 and rðtÞz0 ðtÞ 6 rðT 2 Þz0 ðT 2 Þ < 0, for all t P T 2 . Thus,
z0 ðtÞ 6 rðT 2 Þz0 ðT 2 Þ
1 : rðtÞ
Integration of the latter inequality from T 2 to t yields
zðtÞ 6 zðT 2 Þ þ rðT 2 Þz0 ðT 2 Þ
Z
t
T2
1 ds: rðsÞ
Passing in (9) to the limit as t ! þ1 and using ðA2 Þ, we conclude that
lim zðtÞ ¼ 1;
t!þ1
which contradicts the fact that zðtÞ > 0. Note that condition ðA4 Þ implies that
ð9Þ
M. Hasanbulli, Yu.V. Rogovchenko / Applied Mathematics and Computation 215 (2010) 4392–4399
ðrðtÞz0 ðtÞÞ0 þ jqðtÞxðrðtÞÞ 6 0:
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ð10Þ
On the other hand,
xðtÞ ¼ zðtÞ pðtÞxðt sÞ P zðtÞ pðtÞzðt sÞ P ð1 pðtÞÞzðtÞ: Since limt!þ1 rðtÞ ¼ þ1, there exists a T 3 P T 2 > 0 such that, for all t P T 3 ,
xðrðtÞÞ P ð1 pðrðtÞÞÞzðrðtÞÞ:
ð11Þ
It follows from (10) and (11) that, for all t P T 3 ,
ðrðtÞz0 ðtÞÞ0 6 jqðtÞð1 pðrðtÞÞÞzðrðtÞÞ:
ð12Þ
Introduce the generalized Riccati transformation by
uðtÞ ¼ v ðtÞrðtÞ
z0 ðtÞ þ qðtÞ ; zðrðtÞÞ
ð13Þ
where q is a C 1 function and v is defined by (8). Differentiating (13) and using (1), ðA3 Þ and ðA4 Þ, after some algebra we conclude that, for all t P T 3 ,
v 0 ðtÞ ðrðtÞz0 ðtÞÞ0 r0 ðtÞz0 ðrðtÞÞ z0 ðtÞ 2 þ v ðtÞðrðtÞqðtÞÞ0 uðtÞ þ v ðtÞ v ðtÞrðtÞ z0 ðtÞ zðrðtÞÞ v ðtÞ zðrðtÞÞ 2 v 0 ðtÞ r 2 ðtÞ uðtÞ 6 uðtÞ jv ðtÞqðtÞð1 pðrðtÞÞÞ v ðtÞr0 ðtÞ qðtÞ þ v ðtÞðrðtÞqðtÞÞ0 : rðrðtÞÞ v ðtÞrðtÞ v ðtÞ
u0 ðtÞ ¼
The latter inequality yields, for all t P T 3 ,
u0 ðtÞ 6 wðtÞ r0 ðtÞ
u2 ðtÞ ; v ðtÞrðrðtÞÞ
ð14Þ
where w is defined by (7). Multiplying (14) by Hðt; sÞ and integrating between T 3 and t, we have, for all b P 1 and for all t P T3,
Z t Z t bv ðsÞrðrðsÞÞ 2 ðb 1Þr0 ðsÞHðt; sÞ 2 Hðt; sÞwðsÞ h ðt; sÞ ds 6 Hðt; T 3 ÞuðT 3 Þ u ðsÞ ds 0 4r ðsÞ bv ðsÞrðrðsÞÞ T3 T3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi #2 Z t" r0 ðsÞHðt; sÞ bv ðsÞrðrðsÞÞ ds: uðsÞ þ hðt; sÞ bv ðsÞrðrðsÞÞ 4r0 ðsÞ T3
ð15Þ
Using monotonicity of H, we conclude that, for all t P T 3 ,
Z t bv ðsÞrðrðsÞÞ 2 Hðt; sÞwðsÞ ðt; sÞ ds 6 Hðt; T 3 ÞjuðT 3 Þj 6 Hðt; t 0 ÞjuðT 3 Þj; h 4r0 ðsÞ T3 and, correspondingly,
Z t
Hðt; sÞwðsÞ
t0
Z T3 bv ðsÞrðrðsÞÞ 2 ðt; sÞ ds 6 Hðt; t Þ juðT Þj þ jwðsÞj ds : h 0 3 4r0 ðsÞ t0
ð16Þ
By virtue of (16),
lim supt!þ1
1 Hðt; t 0 Þ
Z t Z T3 bv ðsÞrðrðsÞÞ 2 Hðt; sÞwðsÞ ðt; sÞ ds 6 juðT Þj þ jwðsÞj ds < þ1; h 3 4r0 ðsÞ t0 t0
which contradicts (6). Therefore, all solutions of Eq. (1) are oscillatory. h Efficient oscillation tests can be derived from Theorem 2 with the appropriate choice of the functions H and h. For instance, for ðt; sÞ 2 D, a Kamenev-type function H defined by Hðt; sÞ ¼ ðt sÞn1 , where n > 2 is an integer, belongs to the class I, and hðt; sÞ ¼ ðn 1Þðt sÞðn3Þ=2 : Thus, a consequence of Theorem 2 is the following oscillation criterion. Corollary 3. Suppose that there exists a function q 2 C 1 ð½t 0 ; þ1Þ; RÞ such that, for some integer n > 2 and for some b P 1,
lim supt!þ1 where w and
1 t n1
v ðsÞrðrðsÞÞ ðt sÞn3 ðt sÞ2 wðsÞ bðn 1Þ2 ds ¼ þ1; 4r0 ðsÞ t0
Z
t
v are as in Theorem 2. Then Eq. (1) is oscillatory.
Example 4. For t P 1, consider the second order neutral differential equation
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1 t ðxðtÞ þ xðt 1ÞÞ0 2t þ 1 t2
0 þ
1 t ¼ 0: ð2 þ x4 ðtÞÞx tþ2 2
ð17Þ
Let
qðtÞ ¼ 8t1 and v ðtÞ ¼ t2 : Then,
t4 16t 16 : t2 ðt þ 1Þ
wðtÞ ¼
An application of Corollary 3 with n ¼ 3 establishes the oscillation of Eq. (17) since, for any b P 1,
lim supt!þ1
Z t s4 16s 16 ðt sÞ2 8b ds ¼ þ1: sðs þ 1Þ 1
1 t2
Theorem 5. Suppose that (3) holds. Assume also that there exist functions H 2 I; q 2 C 1 ð½t 0 ; þ1Þ; RÞ and / 2 Cð½t0 ; þ1Þ; RÞ such that, for all T P t 0 and for some b > 1,
lim supt!þ1 where w and
1 Hðt; TÞ
Z t bv ðsÞrðrðsÞÞ 2 Hðt; sÞwðsÞ ðt; sÞ ds P /ðTÞ; h 4r0 ðsÞ T
v are as in Theorem 2. Suppose further that
lim supt!þ1
Z
t
t0
r0 ðsÞ/2þ ðsÞ ds ¼ þ1; v ðsÞrðrðsÞÞ
ð18Þ
where /þ ðtÞ ¼ maxð/ðtÞ; 0Þ. Then Eq. (1) is oscillatory. Proof. Without loss of generality, assume again that Eq. (1) possesses a solution xðtÞ such that xðtÞ > 0 and xðrðtÞÞ > 0 on ½T 0 ; þ1Þ, for some T 0 P t 0 . Proceeding as in the proof of Theorem 2, we arrive at the inequality (15), which yields, for all t P T 3 and for any b P 1,
Z t
1 Hðt; T 3 Þ
/ðT 3 Þ 6 lim supt!þ1
Hðt; sÞwðsÞ
T3
1 6 uðT 3 Þ lim inf t!þ1 Hðt; T 3 Þ
Z
t
T3
bv ðsÞrðrðsÞÞ 2 ðt; sÞ ds h 4r0 ðsÞ
ðb 1Þr0 ðsÞHðt; sÞ 2 u ðsÞ ds: bv ðsÞrðrðsÞÞ
The latter inequality implies that, for all t P T 3 and for all b P 1,
/ðT 3 Þ þ lim inf t!þ1
1 Hðt; T 3 Þ
Z
t
T3
ðb 1Þr0 ðsÞHðt; sÞ 2 u ðsÞ ds 6 uðT 3 Þ: bv ðsÞrðrðsÞÞ
Consequently,
/ðT 3 Þ 6 uðT 3 Þ
ð19Þ
and
lim inf t!þ1
1 Hðt; T 3 Þ
Z
t
T3
r0 ðsÞHðt; sÞ 2 b ðuðT 3 Þ /ðT 3 ÞÞ < þ1: u ðsÞ ds 6 b1 v ðsÞrðrðsÞÞ
ð20Þ
Assume now that
Z
þ1
T3
r0 ðsÞu2 ðsÞ
v ðsÞrðrðsÞÞ
ds ¼ þ1:
ð21Þ
Condition (3) implies existence of a # > 0 such that
lim inf t!þ1
Hðt; sÞ > # > 0: Hðt; t0 Þ
ð22Þ
It follows from (21) that, for any positive constant g, there exists a T 4 > T 3 such that, for all t P T 4 ,
Z
t
T3
r0 ðsÞu2 ðsÞ
v ðsÞrðrðsÞÞ
ds P
g #
:
Using integration by parts and (23), we have, for all t P T 4 ,
ð23Þ
M. Hasanbulli, Yu.V. Rogovchenko / Applied Mathematics and Computation 215 (2010) 4392–4399
1 Hðt; T 3 Þ
Z
t
Hðt; sÞ T3
r0 ðsÞu2 ðsÞ 1 ds ¼ Hðt; T 3 Þ v ðsÞrðrðsÞÞ
Z
t
Hðt; sÞ d T3
Z t Z
Z
s
T3
r0 ðnÞu2 ðnÞ dn v ðnÞrðrðnÞÞ
s
4397
r0 ðnÞu2 ðnÞ @Hðt; sÞ ds dn @s T3 T 3 v ðnÞrðrðnÞÞ Z t Z s 0 Z t 1 r ðnÞu2 ðnÞ @Hðt; sÞ g 1 @Hðt; sÞ ds P ds dn P Hðt; T 3 Þ T 4 T 3 v ðnÞrðrðnÞÞ @s @s # Hðt; T 3 Þ T 4 g Hðt; T 4 Þ g Hðt; T 4 Þ
1 ¼ Hðt; T 3 Þ
¼
# Hðt; T 3 Þ
P
# Hðt; t0 Þ
:
By virtue of (22), there exists a T 5 P T 4 such that, for all t P T 5 ,
Hðt; T 4 Þ P #; Hðt; t0 Þ which implies that
1 Hðt; T 3 Þ
Z
t
Hðt; sÞ T3
r0 ðsÞu2 ðsÞ ds P g: v ðsÞrðrðsÞÞ
Since g is an arbitrary positive constant,
lim inf t!þ1
Z
1 Hðt; T 3 Þ
t
Hðt; sÞ
T3
r0 ðsÞu2 ðsÞ ds ¼ þ1; v ðsÞrðrðsÞÞ
and the latter contradicts (20). Consequently,
Z
þ1
T3
r0 ðsÞu2 ðsÞ ds < þ1; v ðsÞrðrðsÞÞ
and, by virtue of (19),
Z
þ1
T3
r0 ðsÞ/2þ ðsÞ ds 6 v ðsÞrðrðsÞÞ
Z
þ1
T3
r0 ðsÞu2 ðsÞ
v ðsÞrðrðsÞÞ
ds < þ1;
which contradicts (18). Therefore, Eq. (1) is oscillatory.
h
Choosing H as in Corollary 3, we observe that condition (3) holds because
lim
t!þ1
Hðt; sÞ ðt sÞn1 ¼ 1: ¼ lim Hðt; t0 Þ t!þ1 ðt t0 Þn1
Thus, we derive from Theorem 5 a useful oscillation test for Eq. (1). Corollary 6. Assume that there exist functions q 2 C 1 ð½t 0 ; þ1Þ; RÞ and / 2 Cð½t 0 ; þ1Þ; RÞ such that, for all T P t0 , some integer n > 2 and for some b > 1,
lim supt!þ1
1 t n1
Z T
t
v ðsÞrðrðsÞÞ ðt sÞn3 ðt sÞ2 wðsÞ bðn 1Þ2 ds P /ðTÞ: 4r0 ðsÞ
Suppose also that (18) holds, where w, v and /þ are as in Theorem 5. Then Eq. (1) is oscillatory. Example 7. For t P 1, consider the nonlinear neutral differential equation
t ðrðtÞðxðtÞ þ pðtÞxðt 1ÞÞ0 Þ0 þ qðtÞðx2 ðtÞ þ 2Þx ¼ 0; 2 where
1 1 1 t ð2 þ cos 2tÞ; rðtÞ ¼ ; qðtÞ ¼ t2 þ 3; þ 4 4t 2 3 2 ð4t 2 ð4tðt 1Þ 1Þ 1Þ cos 2t 2 þ t sin 2t : pðtÞ ¼ 1 þ 32t 4 ð4t2 þ 3Þ 96t4 rðtÞ ¼
We apply Corollary 6 with n ¼ 3 and
qðtÞ ¼
8ðt 2 þ 3Þð2 þ cos tÞ : tð4t 2 þ 3Þð2 þ cos 2tÞ
ð24Þ
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M. Hasanbulli, Yu.V. Rogovchenko / Applied Mathematics and Computation 215 (2010) 4392–4399
Correspondingly, v ðtÞ ¼ t2 and
2 wðtÞ ¼ ð2t t 2 þ 1Þ cos t þ : 3 Let b ¼ 2: Then,
lim supt!þ1 ¼
1 t2
2 Z t 2 s def ðt sÞ2 2s s2 þ 1 cos s þ þ 1 ð2 þ cos sÞ ds ¼ /ðTÞ 3 3 T
5 2T þ T 2 sin T þ 2T cos T 3 sin T 2 cos T 2T sin T; 3 3
and it follows from
lim supt!þ1
Z
t
2
1
ðs2 =3
/2þ ðsÞ ds P 2 lim supt!þ1 þ 1Þð2 þ cos sÞ
Z 1
t
/2þ ðsÞ ds ¼ þ1 s2 þ 3
that Eq. (24) is oscillatory. Note that in this example
lim supt!þ1
1 t2
Z t 2 s þ 1 ð2 þ cos sÞ ds ¼ þ1; 3 1
which means that an analogue of the condition (4) in Theorem 1, not requested for our oscillation criterion, fails to hold. Theorem 8. Let (3) be satisfied. Assume also that there exist functions H 2 I; q 2 C 1 ð½t0 ; þ1Þ; RÞ and / 2 Cð½t 0 ; þ1Þ; RÞ such that, for all T P t 0 and for some b > 1,
lim inf t!þ1
1 Hðt; TÞ
Z t T
Hðt; sÞwðsÞ
bv ðsÞrðrðsÞÞ 2 ðt; sÞ ds P /ðTÞ; h 4r0 ðsÞ
where w, v, and /þ are as in Theorem 5. Suppose further that (18) holds. Then Eq. (1) is oscillatory. Proof. The conclusion of the theorem follows immediately from the properties of the limits
Z t 1 bv ðsÞrðrðsÞÞ 2 Hðt; sÞwðsÞ ðt; sÞ ds h Hðt; TÞ T 4r0 ðsÞ Z t 1 bv ðsÞrðrðsÞÞ 2 Hðt; sÞwðsÞ 6 lim supt!þ1 h ðt; sÞ ds Hðt; TÞ T 4r0 ðsÞ
/ðTÞ 6 lim inf t!þ1
and Theorem 5.
h
The following result, analogous to Corollary 6, is derived by choosing again a Kamenev-type function Hðt; sÞ ¼ ðt sÞn1 : Corollary 9. Assume that there exist functions q 2 C 1 ð½t 0 ; þ1Þ; RÞ and / 2 Cð½t0 ; þ1Þ; RÞ such that, for all T P t 0 , for some integer n > 2, and for some b > 1,
lim inf t!þ1
1 t n1
Z T
t
v ðsÞrðrðsÞÞ ðt sÞn3 ðt sÞ2 wðsÞ bðn 1Þ2 ds P /ðTÞ: 4r0 ðsÞ
Suppose also that (18) holds, where w, v, and /þ are as in Theorem 5. Then Eq. (1) is oscillatory. Remark 10. Note that it is much simpler to determine an appropriate function hðt; sÞ coupled with one’s selection of Hðt; sÞ using (5) rather than the condition (iii) in Theorem 1 since the latter involves, in addition to functions H and h, functions k and q. Examples 4 and 7 clearly demonstrate that our approach leads to more flexible and easily verifiable criteria for oscillation. Remark 11. We would like to stress that it is very important that the parameter b in Theorems 5 and 8 is strictly larger than one. This allows us to eliminate in both results condition similar to (4) which has been assumed in most papers on the subject. Furthermore, modifications of the proofs through the refinement of the standard integral averaging method allowed us to shorten significantly the proofs of Theorem 5 and Theorem 8, cf. [18]. If one selects b ¼ 1 in Theorems 5 and 8, all advantages of a new technique are lost, and assumptions similar to (4) should be introduced. We also note that a different approach, which allows one to eliminate assumption (4) or alike using an elementary quadratic inequality, is suggested in [16], cf. also [19]. Remark 12. It is not difficult to extend the results reported in this paper to a neutral differential equation
ðrðtÞwðxðtÞÞðxðtÞ þ pðtÞxðt sÞÞ0 Þ0 þ qðtÞf ðt; xðtÞ; xðrðtÞÞÞ ¼ 0;
M. Hasanbulli, Yu.V. Rogovchenko / Applied Mathematics and Computation 215 (2010) 4392–4399
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where the function f 2 CðR3 ; RÞ satisfies xf ðt; x; yÞ for all xy > 0 and
f ðt; x; yÞ P lðtÞ; cðyÞ l 2 CðR; ½0; 1ÞÞ; xcðxÞ > 0 for all x–0, and c0 ðxÞ P 0, cf. also [13]. We note that one can easily apply our idea of introducing a parameter b and rearranging the terms in the main inequality (15) to other classes of neutral differential equations as well. This, with obvious modifications, immediately leads to elimination of assumption (4) and alike along with simplification of the proofs of oscillation results similar to our Theorems 5 and 8. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
J. Dzˇurina, D. Lacková, Oscillation results for second-order nonlinear differential equations, Stud. Univ. Zˇilina Math. Ser. 17 (2003) 79–86. J. Dzˇurina, B. Mihalı´ková, Oscillation criteria for second-order neutral differential equations, Math. Bohem. 125 (2000) 145–153. J. Dzˇurina, I.P. Stavroulakis, Oscillation criteria for second-order delay differential equations, Appl. Math. Comput. 140 (2003) 445–453. L.H. Erbe, Q. Kong, B.G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker Inc., New York, Basel, Hong Kong, 1995. M. Gai, B. Shi, D. Zhang, Oscillation criteria for second order nonlinear differential equations of neutral type, Appl. Math. J. Chin. Univ. Ser. B 16 (2001) 122–126. J.R. Graef, M.K. Grammatikopoulos, P.W. Spikes, On the asymptotic behavior of solutions of a second order nonlinear neutral delay differential equation, J. Math. Anal. Appl. 156 (1991) 23–39. J.R. Graef, P.W. Spikes, Asymptotic behavior of solutions of a forced second order neutral delay equation, Ann. Differ. Equations 12 (1996) 137–146. M.K. Grammatikopoulos, G. Ladas, A. Meimaridou, Oscillation and asymptotic behavior of second order neutral differential equations, Ann. Mater. Pure Appl. (4 14) (1987) 29–40. M.K. Grammatikopoulos, G. Ladas, A. Meimaridou, Oscillation of second order neutral delay differential equations, Rad. Mat. 1 (1985) 267–274.. I. Györi, G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. M. Hasanbulli, Yu.V. Rogovchenko, Asymptotic behavior of nonoscillatory solutions of second order nonlinear neutral differential equations, Math. Ineq. Appl. 10 (2007) 607–618. M. Hasanbulli, Yu.V. Rogovchenko, Oscillation of nonlinear neutral functional differential equations, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 16 (Suppl. S1) (2009) 227–233. W. Li, Classification and existence of nonoscillatory solutions of second order nonlinear neutral differential equations, Ann. Polon. Math. 67 (1997) 283–302. H.J. Li, W.L. Liu, Oscillation for second order neutral differential equations, Can. J. Math. 48 (1996) 871–886. Yu. V. Rogovchenko, F. Tuncay, Oscillation criteria for second-order nonlinear differential equations with damping, Nonlinear Anal. 69 (2008) 208–221. S. Ruan, Oscillation of second order neutral delay differential equations, Can. Math. Bull. 36 (1993) 485–496. W. Shi, P. Wang, Oscillation criteria of a class of second-order neutral functional differential equations, Appl. Math. Comput. 146 (2003) 211–226. Z. Xu, B. Jia, D. Ma, Oscillation results for a second order neutral delay differential equations, Ann. Differ. Equations 19 (2003) 229–237. A. Zafer, Oscillation criteria for even order neutral differential equations, Appl. Math. Lett. 11 (1998) 21–25.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Oscillation criteria for second order nonlinear neutral differential equations q Mustafa Hasanbulli a, Yuri V. Rogovchenko a,b,* a b
Department of Mathematics, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey School of Pure and Applied Natural Sciences, University of Kalmar, SE-391 82 Kalmar, Sweden
a r t i c l e
i n f o
Keywords: Nonlinear neutral differential equations Second order Asymptotic behavior Oscillatory solutions Integral averaging
a b s t r a c t By refining the standard integral averaging technique, we obtain new oscillation criteria for a class of second order nonlinear neutral differential equations of the form
ðrðtÞðxðtÞ þ pðtÞxðt sÞÞ0 Þ0 þ qðtÞf ðxðtÞ; xðrðtÞÞÞ ¼ 0: Assumptions in our theorems are less restrictive, whereas the proofs are significantly simpler compared to those in the recent paper by Shi and Wang [18] and related contributions to the subject. Examples are provided to illustrate the relevance of new theorems. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction In this paper, we are concerned with the oscillatory behavior of second order nonlinear neutral delay differential equations
ðrðtÞðxðtÞ þ pðtÞxðt sÞÞ0 Þ0 þ qðtÞf ðxðtÞ; xðrðtÞÞÞ ¼ 0; 1
ð1Þ 2
where t P t 0 > 0; s P 0 is a constant, r; r 2 C ð½t 0 ; þ1Þ; ð0; þ1ÞÞ; p; q 2 Cð½t 0 ; þ1Þ; RÞ, and f 2 CðR ; RÞ. By a solution of Eq. (1) we mean a continuous function xðtÞ, defined on ½tx ; þ1Þ, such that rðtÞðxðtÞ þ pðtÞxðt sÞÞ0 is continuously differentiable and xðtÞ satisfies (1) for t P tx : In the sequel, we assume that solutions of Eq. (1) exist and can be continued indefinitely to the right. Recall that a nontrivial solution xðtÞ of Eq. (1) is called oscillatory if there exists a sequence of real numbers ftk g1 k¼1 , diverging to þ1, such that xðt k Þ ¼ 0. Neutral differential Eq. (1) is said to be oscillatory if all its solutions oscillate. Recently, an increasing interest in obtaining sufficient conditions for oscillatory or non-oscillatory behavior of different classes of differential and functional differential equations has been manifested. In particular, investigation of neutral differential equations is important since they are encountered in many applications in science and technology and are used, for instance, to describe distributed networks with lossless transmission lines, in the study of vibrating masses attached to an elastic bar, as well as in some variational problems, see [11]. It is well known [4] that the presence of a neutral term in a differential equation can cause oscillation, but it can also destroy oscillatory nature of a differential equation. In general, investigation of neutral differential equations is more difficult in comparison with ordinary differential equations, although certain similarities in the behavior of solutions of ordinary and neutral differential equations can be observed, see [1–20].
q
Research of the second author has been supported in part through the grant from the Faculty of Sciences and Technology of the University of Kalmar. * Corresponding author. Present address: University of Kalmar, School of Pure and Applied Natural Sciences, Famagusta, 391 82 Kalmar, SE, Sweden. E-mail addresses: [email protected], [email protected] (Yu.V. Rogovchenko).
0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.001
M. Hasanbulli, Yu.V. Rogovchenko / Applied Mathematics and Computation 215 (2010) 4392–4399
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In what follows, we briefly review several important oscillation results obtained for second order neutral differential equations. Grammatikopoulos et al. [9] established that condition
Z
þ1
qðsÞ½1 pðs rÞ ds ¼ þ1;
t0
ensures oscillation of a linear neutral differential equation
ðxðtÞ þ pðtÞxðt sÞÞ00 þ qðtÞxðt rÞ ¼ 0: Sufficient conditions for the oscillation of solutions of a slightly more general neutral differential equation
ðxðtÞ þ pðtÞxðt sÞÞ00 þ qðtÞxðrðtÞÞ ¼ 0; including the case when p ¼ 1, were obtained by Dzˇurina and Mihalı´ková [2]. By using the integral averaging method, Ruan [17] derived a number of general oscillation criteria for a nonlinear neutral differential equation
ðrðtÞðxðtÞ þ pðtÞxðt sÞÞ0 Þ0 þ qðtÞf ðxðt rÞÞ ¼ 0;
ð2Þ
whereas Li [14] provided classification of non-oscillatory solutions of Eq. (2) and established necessary and/or sufficient conditions for the existence of eventually positive solutions. Ruan’s results for Eq. (2) have been further improved by Li and Liu [15] who exploited a generalized Riccati transformation. Interesting applications of the integral averaging technique to oscillation of several classes of nonlinear neutral differential equations can be found in the papers by Dzˇurina and Lacková [1], Gai et al. [5], and Xu et al. [19]. In particular, the latter paper addresses the oscillation of a nonlinear neutral differential equation
ðrðtÞðxðtÞ þ pðtÞxðt sÞÞ0 Þ0 þ f ðt; xðtÞ; xðt rÞ; x0 ðtÞÞ ¼ 0; where
f ðt; xðtÞ; xðt rÞ; x0 ðtÞÞ P qðtÞf1 ðxðtÞÞf2 ðxðt rÞÞgðx0 ðtÞÞ; f1 ðxÞ P k1 > 0; f2 ðxÞ=x P k2 > 0; gðxÞ P k3 > 0. Recently, Shi and Wang [18] proved several oscillation criteria for Eq. (1), one of which we present below for the convenience of the reader. In what follows, we use the following notation:
D0 ¼ fðt; sÞ : t 0 6 s < t < þ1g and D ¼ fðt; sÞ : t 0 6 s 6 t < þ1g:
Theorem 1 [18, Theorem 2]. Let the following conditions hold: (A1) (A2) (A3) (A4)
for all t P t0 ; 0 6 pðtÞ 6 1; qðtÞ P 0, and qðtÞ is not identically zero for large t, R þ1 1 r ðsÞ ds ¼ þ1, 0 t P t0 ; rðtÞ 6 t; r ðtÞ > 0, and limt!þ1 rðtÞ ¼ þ1, for all f ðx;yÞ P K > 0, for y–0, and f ðx; yÞ has the sign of x and y if they have the same sign. y
Suppose further that there exist functions H 2 C 1 ðD; RÞ; h 2 CðD0 ; RÞ, and k; q 2 C 1 ð½t 0 ; þ1Þ; ð0; þ1ÞÞ satisfying
ðiÞ Hðt; tÞ ¼ 0; t P t0 ; Hðt; sÞ > 0; t > s P t0 ; @H ðiiÞ ðt; sÞ 6 0; ðt; sÞ 2 D0 ; @s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ q0 ðsÞ ðiiiÞ ½Hðt; sÞkðsÞ þ Hðt; sÞkðsÞ ¼ hðt; sÞ Hðt; sÞkðsÞ: @s qðsÞ Assume also that
Hðt; sÞ 6 þ1; 0 < inf lim inf t!þ1 sPt 0 Hðt; t 0 Þ
ð3Þ
and
lim supt!þ1
1 Hðt; t 0 Þ
Z
t
t0
rðrðsÞÞqðsÞ 2 h ðt; sÞ ds < þ1: r0 ðsÞ
If there exists a function B 2 Cð½t0 ; þ1Þ; RÞ such that
lim supt!þ1
Z
t
t0
r0 ðsÞB2þ ðsÞ ds ¼ þ1; kðsÞqðsÞrðrðsÞÞ
and
lim supt!þ1
1 Hðt; TÞ
Z t rðrðsÞÞqðsÞ 2 KHðt; sÞkðsÞqðsÞqðsÞð1 pðrðsÞÞÞ ðt; sÞ ds P BðTÞ; h 4r0 ðsÞ T
ð4Þ
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for any T P t0 , where Bþ ðtÞ ¼ maxðBðtÞ; 0Þ, then Eq. (1) is oscillatory. Very recently, Rogovchenko and Tuncay [16] established new oscillation criteria for a second order nonlinear differential equation with a damping term
ðrðtÞx0 ðtÞÞ0 þ pðtÞx0 ðtÞ þ qðtÞf ðxðtÞÞ ¼ 0; without an assumption that has been required in related results reported in the literature over the last two decades. The purpose of this paper is to strengthen oscillation results obtained for Eq. (1) by Shi and Wang [18] using a generalized Riccati transformation and developing ideas exploited by Rogovchenko and Tuncay [16]. In order to illustrate the relevance of our theorems, two examples are provided. 2. Main results We say that a continuous function H : D ! ½0; þ1Þ belongs to the class I if: (i) Hðt; tÞ ¼ 0 and Hðt; sÞ > 0 for ðt; sÞ 2 D0 ; (ii) H has a continuous partial derivative with respect to the second variable satisfying, for some locally integrable continuous function h,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ Hðt; sÞ ¼ hðt; sÞ Hðt; sÞ: @s
ð5Þ
Theorem 2. Let conditions ðA1 Þ ðA3 Þ of Theorem 1 hold with ðA4 Þ ðA4 Þ f ðx;yÞ y P j > 0, for y–0, and xf ðx; yÞ > 0, for xy > 0. Suppose that there exits a function q 2 C 1 ð½t 0 ; þ1Þ; RÞ such that, for some b P 1 and for some H 2 I,
lim supt!þ1
1 Hðt; t 0 Þ
Z t
Hðt; sÞwðsÞ
t0
bv ðsÞrðrðsÞÞ 2 ðt; sÞ ds ¼ þ1; h 4r0 ðsÞ
replaced
by
ð6Þ
where
wðtÞ ¼ v ðtÞ
jqðtÞ½1 pðrðtÞÞ þ r0 ðtÞ
r2 ðtÞq2 ðtÞ ðrðtÞqðtÞÞ0 rðrðtÞÞ
ð7Þ
and
Z
t
v ðtÞ ¼ expð2 r0 ðsÞ
rðsÞqðsÞ dsÞ: rðrðsÞÞ
ð8Þ
Then Eq. (1) is oscillatory. Proof. Let xðtÞ be a non-oscillatory solution of Eq. (1). Then, there exists a T 0 P t 0 such that xðtÞ–0 for all t P T 0 . Without loss of generality, we may assume that xðtÞ > 0 and xðrðtÞÞ > 0, for all t P T 0 P t0 . Define
zðtÞ ¼ xðtÞ þ pðtÞxðt sÞ;
t P T0:
Obviously, for all t P T 0 ; zðtÞ P xðtÞ > 0, and rðtÞz0 ðtÞ is non-increasing because
ðrðtÞz0 ðtÞÞ0 ¼ qðtÞf ðxðtÞ;
xðrðtÞÞÞ 6 0:
0
We claim that z ðtÞ P 0, for all t P T 0 . Otherwise, there should exist a T 1 P T 0 P t 0 such that z0 ðT 1 Þ < 0, which implies that rðT 1 Þz0 ðT 1 Þ < 0. Since rðtÞz0 ðtÞ is non-increasing and qðtÞ does not eventually vanish, there exists a T 2 P T 1 such that rðT 2 Þz0 ðT 2 Þ < 0 and rðtÞz0 ðtÞ 6 rðT 2 Þz0 ðT 2 Þ < 0, for all t P T 2 . Thus,
z0 ðtÞ 6 rðT 2 Þz0 ðT 2 Þ
1 : rðtÞ
Integration of the latter inequality from T 2 to t yields
zðtÞ 6 zðT 2 Þ þ rðT 2 Þz0 ðT 2 Þ
Z
t
T2
1 ds: rðsÞ
Passing in (9) to the limit as t ! þ1 and using ðA2 Þ, we conclude that
lim zðtÞ ¼ 1;
t!þ1
which contradicts the fact that zðtÞ > 0. Note that condition ðA4 Þ implies that
ð9Þ
M. Hasanbulli, Yu.V. Rogovchenko / Applied Mathematics and Computation 215 (2010) 4392–4399
ðrðtÞz0 ðtÞÞ0 þ jqðtÞxðrðtÞÞ 6 0:
4395
ð10Þ
On the other hand,
xðtÞ ¼ zðtÞ pðtÞxðt sÞ P zðtÞ pðtÞzðt sÞ P ð1 pðtÞÞzðtÞ: Since limt!þ1 rðtÞ ¼ þ1, there exists a T 3 P T 2 > 0 such that, for all t P T 3 ,
xðrðtÞÞ P ð1 pðrðtÞÞÞzðrðtÞÞ:
ð11Þ
It follows from (10) and (11) that, for all t P T 3 ,
ðrðtÞz0 ðtÞÞ0 6 jqðtÞð1 pðrðtÞÞÞzðrðtÞÞ:
ð12Þ
Introduce the generalized Riccati transformation by
uðtÞ ¼ v ðtÞrðtÞ
z0 ðtÞ þ qðtÞ ; zðrðtÞÞ
ð13Þ
where q is a C 1 function and v is defined by (8). Differentiating (13) and using (1), ðA3 Þ and ðA4 Þ, after some algebra we conclude that, for all t P T 3 ,
v 0 ðtÞ ðrðtÞz0 ðtÞÞ0 r0 ðtÞz0 ðrðtÞÞ z0 ðtÞ 2 þ v ðtÞðrðtÞqðtÞÞ0 uðtÞ þ v ðtÞ v ðtÞrðtÞ z0 ðtÞ zðrðtÞÞ v ðtÞ zðrðtÞÞ 2 v 0 ðtÞ r 2 ðtÞ uðtÞ 6 uðtÞ jv ðtÞqðtÞð1 pðrðtÞÞÞ v ðtÞr0 ðtÞ qðtÞ þ v ðtÞðrðtÞqðtÞÞ0 : rðrðtÞÞ v ðtÞrðtÞ v ðtÞ
u0 ðtÞ ¼
The latter inequality yields, for all t P T 3 ,
u0 ðtÞ 6 wðtÞ r0 ðtÞ
u2 ðtÞ ; v ðtÞrðrðtÞÞ
ð14Þ
where w is defined by (7). Multiplying (14) by Hðt; sÞ and integrating between T 3 and t, we have, for all b P 1 and for all t P T3,
Z t Z t bv ðsÞrðrðsÞÞ 2 ðb 1Þr0 ðsÞHðt; sÞ 2 Hðt; sÞwðsÞ h ðt; sÞ ds 6 Hðt; T 3 ÞuðT 3 Þ u ðsÞ ds 0 4r ðsÞ bv ðsÞrðrðsÞÞ T3 T3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi #2 Z t" r0 ðsÞHðt; sÞ bv ðsÞrðrðsÞÞ ds: uðsÞ þ hðt; sÞ bv ðsÞrðrðsÞÞ 4r0 ðsÞ T3
ð15Þ
Using monotonicity of H, we conclude that, for all t P T 3 ,
Z t bv ðsÞrðrðsÞÞ 2 Hðt; sÞwðsÞ ðt; sÞ ds 6 Hðt; T 3 ÞjuðT 3 Þj 6 Hðt; t 0 ÞjuðT 3 Þj; h 4r0 ðsÞ T3 and, correspondingly,
Z t
Hðt; sÞwðsÞ
t0
Z T3 bv ðsÞrðrðsÞÞ 2 ðt; sÞ ds 6 Hðt; t Þ juðT Þj þ jwðsÞj ds : h 0 3 4r0 ðsÞ t0
ð16Þ
By virtue of (16),
lim supt!þ1
1 Hðt; t 0 Þ
Z t Z T3 bv ðsÞrðrðsÞÞ 2 Hðt; sÞwðsÞ ðt; sÞ ds 6 juðT Þj þ jwðsÞj ds < þ1; h 3 4r0 ðsÞ t0 t0
which contradicts (6). Therefore, all solutions of Eq. (1) are oscillatory. h Efficient oscillation tests can be derived from Theorem 2 with the appropriate choice of the functions H and h. For instance, for ðt; sÞ 2 D, a Kamenev-type function H defined by Hðt; sÞ ¼ ðt sÞn1 , where n > 2 is an integer, belongs to the class I, and hðt; sÞ ¼ ðn 1Þðt sÞðn3Þ=2 : Thus, a consequence of Theorem 2 is the following oscillation criterion. Corollary 3. Suppose that there exists a function q 2 C 1 ð½t 0 ; þ1Þ; RÞ such that, for some integer n > 2 and for some b P 1,
lim supt!þ1 where w and
1 t n1
v ðsÞrðrðsÞÞ ðt sÞn3 ðt sÞ2 wðsÞ bðn 1Þ2 ds ¼ þ1; 4r0 ðsÞ t0
Z
t
v are as in Theorem 2. Then Eq. (1) is oscillatory.
Example 4. For t P 1, consider the second order neutral differential equation
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1 t ðxðtÞ þ xðt 1ÞÞ0 2t þ 1 t2
0 þ
1 t ¼ 0: ð2 þ x4 ðtÞÞx tþ2 2
ð17Þ
Let
qðtÞ ¼ 8t1 and v ðtÞ ¼ t2 : Then,
t4 16t 16 : t2 ðt þ 1Þ
wðtÞ ¼
An application of Corollary 3 with n ¼ 3 establishes the oscillation of Eq. (17) since, for any b P 1,
lim supt!þ1
Z t s4 16s 16 ðt sÞ2 8b ds ¼ þ1: sðs þ 1Þ 1
1 t2
Theorem 5. Suppose that (3) holds. Assume also that there exist functions H 2 I; q 2 C 1 ð½t 0 ; þ1Þ; RÞ and / 2 Cð½t0 ; þ1Þ; RÞ such that, for all T P t 0 and for some b > 1,
lim supt!þ1 where w and
1 Hðt; TÞ
Z t bv ðsÞrðrðsÞÞ 2 Hðt; sÞwðsÞ ðt; sÞ ds P /ðTÞ; h 4r0 ðsÞ T
v are as in Theorem 2. Suppose further that
lim supt!þ1
Z
t
t0
r0 ðsÞ/2þ ðsÞ ds ¼ þ1; v ðsÞrðrðsÞÞ
ð18Þ
where /þ ðtÞ ¼ maxð/ðtÞ; 0Þ. Then Eq. (1) is oscillatory. Proof. Without loss of generality, assume again that Eq. (1) possesses a solution xðtÞ such that xðtÞ > 0 and xðrðtÞÞ > 0 on ½T 0 ; þ1Þ, for some T 0 P t 0 . Proceeding as in the proof of Theorem 2, we arrive at the inequality (15), which yields, for all t P T 3 and for any b P 1,
Z t
1 Hðt; T 3 Þ
/ðT 3 Þ 6 lim supt!þ1
Hðt; sÞwðsÞ
T3
1 6 uðT 3 Þ lim inf t!þ1 Hðt; T 3 Þ
Z
t
T3
bv ðsÞrðrðsÞÞ 2 ðt; sÞ ds h 4r0 ðsÞ
ðb 1Þr0 ðsÞHðt; sÞ 2 u ðsÞ ds: bv ðsÞrðrðsÞÞ
The latter inequality implies that, for all t P T 3 and for all b P 1,
/ðT 3 Þ þ lim inf t!þ1
1 Hðt; T 3 Þ
Z
t
T3
ðb 1Þr0 ðsÞHðt; sÞ 2 u ðsÞ ds 6 uðT 3 Þ: bv ðsÞrðrðsÞÞ
Consequently,
/ðT 3 Þ 6 uðT 3 Þ
ð19Þ
and
lim inf t!þ1
1 Hðt; T 3 Þ
Z
t
T3
r0 ðsÞHðt; sÞ 2 b ðuðT 3 Þ /ðT 3 ÞÞ < þ1: u ðsÞ ds 6 b1 v ðsÞrðrðsÞÞ
ð20Þ
Assume now that
Z
þ1
T3
r0 ðsÞu2 ðsÞ
v ðsÞrðrðsÞÞ
ds ¼ þ1:
ð21Þ
Condition (3) implies existence of a # > 0 such that
lim inf t!þ1
Hðt; sÞ > # > 0: Hðt; t0 Þ
ð22Þ
It follows from (21) that, for any positive constant g, there exists a T 4 > T 3 such that, for all t P T 4 ,
Z
t
T3
r0 ðsÞu2 ðsÞ
v ðsÞrðrðsÞÞ
ds P
g #
:
Using integration by parts and (23), we have, for all t P T 4 ,
ð23Þ
M. Hasanbulli, Yu.V. Rogovchenko / Applied Mathematics and Computation 215 (2010) 4392–4399
1 Hðt; T 3 Þ
Z
t
Hðt; sÞ T3
r0 ðsÞu2 ðsÞ 1 ds ¼ Hðt; T 3 Þ v ðsÞrðrðsÞÞ
Z
t
Hðt; sÞ d T3
Z t Z
Z
s
T3
r0 ðnÞu2 ðnÞ dn v ðnÞrðrðnÞÞ
s
4397
r0 ðnÞu2 ðnÞ @Hðt; sÞ ds dn @s T3 T 3 v ðnÞrðrðnÞÞ Z t Z s 0 Z t 1 r ðnÞu2 ðnÞ @Hðt; sÞ g 1 @Hðt; sÞ ds P ds dn P Hðt; T 3 Þ T 4 T 3 v ðnÞrðrðnÞÞ @s @s # Hðt; T 3 Þ T 4 g Hðt; T 4 Þ g Hðt; T 4 Þ
1 ¼ Hðt; T 3 Þ
¼
# Hðt; T 3 Þ
P
# Hðt; t0 Þ
:
By virtue of (22), there exists a T 5 P T 4 such that, for all t P T 5 ,
Hðt; T 4 Þ P #; Hðt; t0 Þ which implies that
1 Hðt; T 3 Þ
Z
t
Hðt; sÞ T3
r0 ðsÞu2 ðsÞ ds P g: v ðsÞrðrðsÞÞ
Since g is an arbitrary positive constant,
lim inf t!þ1
Z
1 Hðt; T 3 Þ
t
Hðt; sÞ
T3
r0 ðsÞu2 ðsÞ ds ¼ þ1; v ðsÞrðrðsÞÞ
and the latter contradicts (20). Consequently,
Z
þ1
T3
r0 ðsÞu2 ðsÞ ds < þ1; v ðsÞrðrðsÞÞ
and, by virtue of (19),
Z
þ1
T3
r0 ðsÞ/2þ ðsÞ ds 6 v ðsÞrðrðsÞÞ
Z
þ1
T3
r0 ðsÞu2 ðsÞ
v ðsÞrðrðsÞÞ
ds < þ1;
which contradicts (18). Therefore, Eq. (1) is oscillatory.
h
Choosing H as in Corollary 3, we observe that condition (3) holds because
lim
t!þ1
Hðt; sÞ ðt sÞn1 ¼ 1: ¼ lim Hðt; t0 Þ t!þ1 ðt t0 Þn1
Thus, we derive from Theorem 5 a useful oscillation test for Eq. (1). Corollary 6. Assume that there exist functions q 2 C 1 ð½t 0 ; þ1Þ; RÞ and / 2 Cð½t 0 ; þ1Þ; RÞ such that, for all T P t0 , some integer n > 2 and for some b > 1,
lim supt!þ1
1 t n1
Z T
t
v ðsÞrðrðsÞÞ ðt sÞn3 ðt sÞ2 wðsÞ bðn 1Þ2 ds P /ðTÞ: 4r0 ðsÞ
Suppose also that (18) holds, where w, v and /þ are as in Theorem 5. Then Eq. (1) is oscillatory. Example 7. For t P 1, consider the nonlinear neutral differential equation
t ðrðtÞðxðtÞ þ pðtÞxðt 1ÞÞ0 Þ0 þ qðtÞðx2 ðtÞ þ 2Þx ¼ 0; 2 where
1 1 1 t ð2 þ cos 2tÞ; rðtÞ ¼ ; qðtÞ ¼ t2 þ 3; þ 4 4t 2 3 2 ð4t 2 ð4tðt 1Þ 1Þ 1Þ cos 2t 2 þ t sin 2t : pðtÞ ¼ 1 þ 32t 4 ð4t2 þ 3Þ 96t4 rðtÞ ¼
We apply Corollary 6 with n ¼ 3 and
qðtÞ ¼
8ðt 2 þ 3Þð2 þ cos tÞ : tð4t 2 þ 3Þð2 þ cos 2tÞ
ð24Þ
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M. Hasanbulli, Yu.V. Rogovchenko / Applied Mathematics and Computation 215 (2010) 4392–4399
Correspondingly, v ðtÞ ¼ t2 and
2 wðtÞ ¼ ð2t t 2 þ 1Þ cos t þ : 3 Let b ¼ 2: Then,
lim supt!þ1 ¼
1 t2
2 Z t 2 s def ðt sÞ2 2s s2 þ 1 cos s þ þ 1 ð2 þ cos sÞ ds ¼ /ðTÞ 3 3 T
5 2T þ T 2 sin T þ 2T cos T 3 sin T 2 cos T 2T sin T; 3 3
and it follows from
lim supt!þ1
Z
t
2
1
ðs2 =3
/2þ ðsÞ ds P 2 lim supt!þ1 þ 1Þð2 þ cos sÞ
Z 1
t
/2þ ðsÞ ds ¼ þ1 s2 þ 3
that Eq. (24) is oscillatory. Note that in this example
lim supt!þ1
1 t2
Z t 2 s þ 1 ð2 þ cos sÞ ds ¼ þ1; 3 1
which means that an analogue of the condition (4) in Theorem 1, not requested for our oscillation criterion, fails to hold. Theorem 8. Let (3) be satisfied. Assume also that there exist functions H 2 I; q 2 C 1 ð½t0 ; þ1Þ; RÞ and / 2 Cð½t 0 ; þ1Þ; RÞ such that, for all T P t 0 and for some b > 1,
lim inf t!þ1
1 Hðt; TÞ
Z t T
Hðt; sÞwðsÞ
bv ðsÞrðrðsÞÞ 2 ðt; sÞ ds P /ðTÞ; h 4r0 ðsÞ
where w, v, and /þ are as in Theorem 5. Suppose further that (18) holds. Then Eq. (1) is oscillatory. Proof. The conclusion of the theorem follows immediately from the properties of the limits
Z t 1 bv ðsÞrðrðsÞÞ 2 Hðt; sÞwðsÞ ðt; sÞ ds h Hðt; TÞ T 4r0 ðsÞ Z t 1 bv ðsÞrðrðsÞÞ 2 Hðt; sÞwðsÞ 6 lim supt!þ1 h ðt; sÞ ds Hðt; TÞ T 4r0 ðsÞ
/ðTÞ 6 lim inf t!þ1
and Theorem 5.
h
The following result, analogous to Corollary 6, is derived by choosing again a Kamenev-type function Hðt; sÞ ¼ ðt sÞn1 : Corollary 9. Assume that there exist functions q 2 C 1 ð½t 0 ; þ1Þ; RÞ and / 2 Cð½t0 ; þ1Þ; RÞ such that, for all T P t 0 , for some integer n > 2, and for some b > 1,
lim inf t!þ1
1 t n1
Z T
t
v ðsÞrðrðsÞÞ ðt sÞn3 ðt sÞ2 wðsÞ bðn 1Þ2 ds P /ðTÞ: 4r0 ðsÞ
Suppose also that (18) holds, where w, v, and /þ are as in Theorem 5. Then Eq. (1) is oscillatory. Remark 10. Note that it is much simpler to determine an appropriate function hðt; sÞ coupled with one’s selection of Hðt; sÞ using (5) rather than the condition (iii) in Theorem 1 since the latter involves, in addition to functions H and h, functions k and q. Examples 4 and 7 clearly demonstrate that our approach leads to more flexible and easily verifiable criteria for oscillation. Remark 11. We would like to stress that it is very important that the parameter b in Theorems 5 and 8 is strictly larger than one. This allows us to eliminate in both results condition similar to (4) which has been assumed in most papers on the subject. Furthermore, modifications of the proofs through the refinement of the standard integral averaging method allowed us to shorten significantly the proofs of Theorem 5 and Theorem 8, cf. [18]. If one selects b ¼ 1 in Theorems 5 and 8, all advantages of a new technique are lost, and assumptions similar to (4) should be introduced. We also note that a different approach, which allows one to eliminate assumption (4) or alike using an elementary quadratic inequality, is suggested in [16], cf. also [19]. Remark 12. It is not difficult to extend the results reported in this paper to a neutral differential equation
ðrðtÞwðxðtÞÞðxðtÞ þ pðtÞxðt sÞÞ0 Þ0 þ qðtÞf ðt; xðtÞ; xðrðtÞÞÞ ¼ 0;
M. Hasanbulli, Yu.V. Rogovchenko / Applied Mathematics and Computation 215 (2010) 4392–4399
4399
where the function f 2 CðR3 ; RÞ satisfies xf ðt; x; yÞ for all xy > 0 and
f ðt; x; yÞ P lðtÞ; cðyÞ l 2 CðR; ½0; 1ÞÞ; xcðxÞ > 0 for all x–0, and c0 ðxÞ P 0, cf. also [13]. We note that one can easily apply our idea of introducing a parameter b and rearranging the terms in the main inequality (15) to other classes of neutral differential equations as well. This, with obvious modifications, immediately leads to elimination of assumption (4) and alike along with simplification of the proofs of oscillation results similar to our Theorems 5 and 8. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
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