Research Article New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive and Negative Coefficients

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1 Abstract and Applied Analysis Volume 2010, Article ID , 11 pages doi: /2010/ Research Article New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive and Negative Coefficients Yuzhen Bai and Lihua Liu School of Mathematical Sciences, Qufu Normal University, Qufu , China Correspondence should be addressed to Yuzhen Bai, Received 23 November 2009; Revised 8 May 2010; Accepted 11 June 2010 Academic Editor: AğacıkZafer Copyright q 2010 Y. Bai and L. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We establish the oscillation and asymptotic criteria for the second-order neutral delay differential equations with positive and negative coefficients having the forms x t i R c i t x α i t r t x t i R c i t x α i t i P p i t x β i t i Q q i t x γ i t 0 and x t i R c i t x α i t r t x t i R c i t x α i t i P p i t x β i t i Q q i t x γ i t f t. The obtained new oscillation criteria extend and improve the recent results given in the paperof B. Karpuz et al Introduction In this paper, we consider the oscillation of all solutions of the second-order neutral delay differential equations with positive and negative coefficients having the forms [ x t [ i t x α i t ] r t x t i Rc ] i t x α i t i Rc p i t x ( β i t ) q i t x ( γ i t ) 0, i P i Q 1.1 [ x t [ i t x α i t ] r t x t i Rc ] i t x α i t i Rc p i t x ( β i t ) q i t x ( γ i t ) f t. i P i Q 1.2 We introduce the following class of functions D t 0, equipped with the functions satisfying the following properties: P 1 f C 1 t 0,, R is strictly increasing and lim t f t holds,

2 2 Abstract and Applied Analysis P 1 f t t holds for all t t 0. In this paper, we make the following assumptions: H 1 R, P, Q are bounded starting segments of positive integers; that is, R {1, 2,...,R 0 }, P {1, 2,...,P 0 }, Q {1, 2,...,Q 0 }, R 0,P 0,Q 0 N; 1.3 H 2 c i C t 0,, R for all i R, p i C t 0,, R for all i P, and q i C t 0,, R for all i Q; H 3 α i D t 0, with lim inf t α i t > 0 for all i R,β i D t 0, for all i P, and γ i D t 0, for all i Q; H 4 r C 1 t 0,, R, and r t 0; H 5 f C t 0,, R and that there exists a function F C 2 t 0,, R which satisfies F f and lim t F t 0. In order to establish our main results, we will assume that there exists a mapping ψ : Q P satisfying the following conditions: A 1 γ i t β ψ i t for all t t 0 and i Q; A 2 h i C t 0,, R for all i P, where p i t h i t : p i t, j Q, ψ j i ρ j t q j( ρj t ), i ψ Q, i/ ψ Q, 1.4 and ρ i t : γ 1 i β ψ i t for all i Q and t t 0 ; A 3 there exists i 0 P such that lim inf t h i0 t > 0 and lim sup t β i 0 t <. A function x is called a solution of 1.1 or 1.2 provided that x satisfies 1.1 or 1.2 identically on t 0,, x i R c i t x α i C 2 t 0,, R and x C t 1,, R, where t 1 : min{α, β, γ}, α : min{α i t 0 : i R}, β : min{β i t 0 : i P}, andγ : min{γ i t 0 : i Q}. We restrict our attention only to the nontrivial solution x, that is, to the solution x such that sup{ x t : t } > 0 for all t 0. A nontrivial solution of 1.1 or 1.2 is called oscillatory if it has arbitrary large zeros, otherwise, it is called nonoscillatory. The oscillation and nonoscillation of solutions of second-order neutral delay differential equations have been studied by many authors; see However, to the best of our knowledge,there seem to be few oscillation results for 1.1 and 1.2. Recently,Manojlović etal. 4 andweng and Sun 10 have studied oscillation and asymptotic behavior of all solutions of the following equations: [ l m n x t c i t x t τ i ] p i t x t δ i q i t x t σ i 0, t t 0, i 1 i 1 i 1 [ l m n x t c i t x t τ i ] p i t x t δ i q i t x t σ i f t, t t 0, i 1 i 1 i and several well-known results have been obtained.

3 Abstract and Applied Analysis 3 By using weaker conditions than in 4, 10, Karpuz et al. 1 have established oscillation criteria for differential equation [ x t ] i t x α i t i Rr p i t x ( β i t ) q i t x ( γ i t ) f t. i P i Q 1.6 In this paper, we shall continue in the direction to study the oscillatory properties of 1.1 and 1.2. We establish new oscillation criteria for 1.1 and 1.2, which extend and improve the corresponding results in 1, 4, 10. We also give two examples to illustrate our main results. 2. Main Results The following properties of the set L 1 t 0, in 1 are needed for our subsequent discussion. Property 1. If f L 1 t 0, and f C t 0,, R, then lim inf t f t 0. Corollary 2.1. Suppose that f L 1 t 0, and lim t f t exists; then lim t f t 0. Property 2. If f C t 0,, R and f L 1,, where t 0, then we have f L 1 t 0,. Property 3. Let be such that g t 0. Suppose g D, with lim sup t g t < and f C t 0,, R. If f g L 1, holds, then f L 1 t 0,. Property 4. Let be such that g t 0. Suppose g D, with lim inf t g t > 0,f g C,, R, and f C t 0,, R. If f L 1 t 0, holds, then f g L 1, holds. For simplicity, we denote the set of bounded functions by B t 0, : { f C t 0,, R : f < }, 2.1 where f : sup { f t,t t0 }. 2.2 For an arbitrary function ψ : Q P, which satisfies A 1 A 3, we denote the function ϕ : t 0, R by ϕ t : i Q ρ i t q i v dv, t t In this section, for convenience, we suppose that q i 0holdsforalli Q on t 1,t 0. We start with the following Theorem.

4 4 Abstract and Applied Analysis Theorem 2.2. Assume that H 1 H 4 hold and there exists a mapping ψ : Q P which satisfies A 1 A 3 and that i R c i B t 0,. If ϕ L 1 t 0,, then every solution x of 1.1 is oscillatory. Proof. Suppose that x is a nonoscillatory solution of 1.1. Without loss of generality, we may assume that x t > 0fort t 0. Therefore, we may assume existence of t 0 such that ϕ u du < 1, t, x α i t > 0, i R, x ( β i t ) > 0, i P. 2.4 Now, we set w t : x t i Rc i t x α i t, t, 2.5 z t, s : w t u r u w u du i Q s s ρ i u q i v x ( γ i v ) dv du, t s. 2.6 By z t, s, we denote differential of functions with respect to the first component. Considering 2.5, we rewrite 1.1 in the form w t r t w t p i t x ( β i t ) i t x i P i Qq ( γ i t ) on,. By Leibnitz s rule, 2.7 and H 4, we have z t, w t r t w t r t w t q i t x ( γ i t ) i Q i Qρ i t q ( i ρi t ) x ( β ψ i t ) w t r t w t q i t x ( γ i t ) i Q i Qρ i t q ( i ρi t ) x ( β ψ i t ) p i t x ( β i t ) i P i Qρ i t q ( i ρi t ) x ( β ψ i t ) p i t x ( β i t ) p i t x ( β i t ) i ψ Q i ψ Q p i t j Q, ψ j i i / ψ Q i P h i t x ( β i t ) 0, t, ρ j t q ( j ρj t ) x ( β i t ) i ψ Q j Q, ψ j i i / ψ Q ρ j t q ( j ρj t ) x ( β i t ) p i t x ( β i t ). 2.8 which implies z, and z, is eventually strictly monotonic on,. Hence there exists t 2 such that either z t, < 0orz t, > 0holdsforallt t 2.

5 Abstract and Applied Analysis 5 We consider the following two possible cases: Case 1 z t, > 0 for all t t 2. Integrating 2.8 from t 2 to, we have >z t 2, z t 2, lim t z t, t 2 z u, du i P t 2 h i u x ( β i u ) du, 2.9 which implies that i P h i t x β i t L 1 t 2,. Therefore, for i 0 P for which A 3 holds, we have x β i0 L 1 t 2,. Then we conclude that x L 1 t 0, by Property 3. Hence i R c i B t 0,. H 3 and Property 4 imply that w L 1,. Since r t is bounded, b t r t w t is also integrable in,. So we obtain that there exists a constant M>0 such that r u w u du M, t Let u t w t r u w u du, t t From 2.6, we have u t z t, i Q ρ i t q i v x ( γ i v ) dv > Then u t is bounded and monotonous and lim t u t exists. We can suppose that lim t u t μ>0sinceu t >x t > 0andu t > 0. So there exists t 3 t 2 such that u t >μ ε/2, t t 3, for arbitrary ε 0,μ M ;by u t w t r u w u du > μ ε t 3 2, 2.13 we have w t > r u w u du μ ε t 3 2 >μ 1 2( μ M ) M ( μ M ). This implies that w t / L 1,, which is a contradiction.

6 6 Abstract and Applied Analysis Case 2 z t, < 0 for all t t 2. Since z, is nonincreasing by 2.8, the inequality z t, z t 2, implies that lim t z t,. Hence z, / B,. We claim that x B t 0,. On contrary, there exists t 3 t 2 such that z t 3, < 0andx t 3 sup{x t : t t 0,t 3 }. We get the following contradiction: 3 0 >z t 3, w t 3 r u w u du 3 u q i v x ( γ i v ) dv du i Q ρ i u 3 x t 3 x t 3 r u du 3 u q i v x ( γ i v ) dv du i Q ρ i u [ 3 3 ] x t 3 1 r u du ϕ u du > 0, 2.15 since 3 ϕ u du < ϕ u du < Thus x <. Accordingly, by 2.4 and 2.6, it follows that z t, w t r u w u du x ϕ u du x ϕ u du > x, t t Therefore, x z, holds and we see that z, B,, which is a contradiction. Therefore, we completed the proof by considering both possible cases. Remark 2.3. When r t 0, and Theorem 2.2 reduces to Theorem 3.1 in 1. So Theorem 2.2 extends and improves the corresponding results in 1, 4, 10. Theorem 2.4. Assume that H 1 H 5 hold and there exists a mapping ψ : Q P which satisfies A 1 A 3. Furthermore, assume that i R c i B t 0,. If ϕ L 1 t 0,, then every solution x of 1.2 is oscillatory or tends to zero asymptotically. Proof. Suppose that x is a nonoscillatory solution of 1.2. Without loss of generality, we assume that x t > 0fort t 0. Therefore, we may assume existence of a constant ε > 0 and t 0 such that 2.4 and F t ε hold for all t. Now, for t s, set Z t, s : W t W t : w t F t, u r u w u du i Q s s ρ i u q i v x ( γ i v ) dv du ε,

7 Abstract and Applied Analysis 7 where w t is defined on the interval, as in 2.5. Then as in 2.8, we have Z t, i P h i t x ( β i t ) 0, t Thus there exists t 2 satisfying either Z t, > 0orZ t, < 0 for all t t 2. We consider the following two possible cases. Case 1 Z t, > 0 for all t t 2. In this case, one can show that w L 1, as shown in above proofs. Since r t is bounded, r t w t is also integrable in,. Let v t W t r u w u du, t t By 2.19, we have v t Z t, s i Q ρ i t q i v x ( γ i v ) dv > 0, t t 2 ; 2.22 then v t is bounded and monotonous and lim t v t exists. By 2.21, we can obtain that lim t w t exists. Letting a t w t r u w u du, 2.23 we can obtain that lim t a t exists. Suppose that lim t a t θ, where θ 0,. We claim θ 0. Suppose that θ 0,. By r t w t L 1,, we see that there exists N>0, such that r u w u du N Because lim t a t θ > 0, there exists t 3 arbitrary ɛ 0,θ N. But by t 2, such that a t >θ 1/2 ɛ, t t 3, for w t r u w u du > θ ɛ t 3 2, 2.25

8 8 Abstract and Applied Analysis we have w t > r u w u du θ ɛ t 3 2 >θ 1 θ N N θ N ; 2 this implies that w t / L 1 t 2,, which is a contradiction. Therefore, lim t a t 0. Since 0 <x t <a t for t t 2, we have that lim t x t 0. Case 2 Z t, < 0 for all t t 2. Then we have that Z, / B, by We claim that x B t 0,. On contrary, there exists t 3 t 2 such that z t 3, < 0and x t 3 sup{x t : t t 0,t 3 } hold and lim t Z t, < 0. Taking 2.4, 2.5, 2.18, and 2.19 into account, we get the following contradiction: 3 0 >Z t 3, W t 3 r u w u du 3 u q i v x ( γ i v ) dv du ɛ i Q ρ i u [ 3 3 ] x t 3 1 r u du ϕ u du > Hence x <. Accordingly, using 2.18, 2.19, and the fact that w t > 0on t 2,, we have that Z t, t 2 : W t r u w u du t 2 i Q u s ρ i u q i v x ( γ i v ) dv du ɛ w t r u w u du x t 2 ϕ u du t x, t t 2. Thus x Z, and Z, B,, which is a contradiction. Therefore, we completed the proof by considering both possible cases. Remark 2.5. If there exists F C 2 t 0,, R such that F f on t 0, and l 0 : lim t F t exists and is finite, then G : F l 0 on t 0, satisfies H 5.

9 Abstract and Applied Analysis 9 Corollary 2.6. If all conditions of Theorem 2.4 hold, then every nonoscillatory solution of 1.2 converges to zero at infinity. Remark 2.7. When r t 0, Theorem 2.4 reduces to Theorem 3.3in 1. So Theorem 2.4 extends and improves the corresponding results in 1, 4, Examples In this section, we provide two examples to illustrate our main results. Example 3.1. Consider the following equation: x t 3x t π 1 1 t x t 3x t 2 π t 1 x t 2π e t 2π x t π 0, t Here, we have c 1 t 3, α 1 t π, β 1 t 2π, γ 1 t π, p 1 t t 1, q 1 t e t 2π, r t t. 2 Set the function ψ with ψ 1 1, then ρ 1 t t π. By simple calculation, we have h 1 t t 1 e t π, 1 ϕ t e t π e t 2π, lim inf t h 1 t, ϕ u du e 1 π e 1 2π. 3.3 Therefore, according to Theorem 2.2, every solution x of 3.1 is oscillatory. Clearly, the known results in 1 10 cannot be applied to 3.1. Example 3.2. Consider the following equation: [ ( t x t e t x 2 ( t e t x 2 3π 2 )] [ 1 x t e t x 1 t 2 ) 3 ( t 4 e t x ( )] t ( ) ( ) t t x 2 2 2π x 2 π ) 2 π e t 1t sin 1t 3.4, t 0. 3

10 10 Abstract and Applied Analysis Here, we have R {1}, P {1, 2}, Q {1, 2}, c 1 t e t, p 1 t p 2 t 1, q 1 t e t, q 2 t 3 4 e t, α 1 t t 2, β 1 t t 2 2π, β 2 t t 2 π, 3.5 γ 1 t t 2 3π 2, γ 2 t t 2 π, f t e t 1 t 3 sin 1 t, r t 1 1 t 2. Set the function ψ : Q P with ψ i 1fori 1, 2; then ρ 1 t t π and ρ 2 t t 2π. By simple calculation, we have F t e t t sin 1 t, lim inf t h 2 t 1, h 1 t 1 e t π 3 4 e t 2π, h 2 t 1, ϕ t 0 t π ϕ u du 3 4 e v du 3 4 t 2π ( ) e 2π 1 e v dv, e π 1, lim F t 1. t 3.6 Therefore, according to Theorem 2.4 and Remark 2.5, every solution x of 3.4 is oscillatory or tends to zero asymptotically. Clearly, the known results in 1 10 cannot be applied to 3.4. Acknowledgment The authors gratefully acknowledge the referee who pointed out the important reference 1 and gave some valuable suggestions which improved the results in this paper. Project supported by the National Natural Science Foundation of China , STPF of University in Shandong Province of China J09LA04 and the Research Foundation of Qufu Normal University XJ0620. References 1 B. Karpuz, J. V. Manojlović, Ö. Öcalan, and Y. Shoukaku, Oscillation criteria for a class of secondorder neutral delay differential equations, Applied Mathematics and Computation, vol. 210, no. 2, pp , X. Lin, Oscillation of second-order nonlinear neutral differential equations, Mathematical Analysis and Applications, vol. 309, no. 2, pp , Z. Liu and S. M. Kang, Infinitely many nonoscillatory solutions for second order nonlinear neutral delay differential equations, Nonlinear Analysis, vol. 70, no. 12, pp , 2009.

11 Abstract and Applied Analysis 11 4 J. V. Manojlović, Y. Shoukaku, T. Tanigawa, and N. Yoshida, Oscillation criteria for second order differential equations with positive and negative coefficients, Applied Mathematics and Computation, vol. 181, no. 2, pp , S. Padhi, Oscillation and asymptotic behaviour of solutions of second order homogeneous neutral differential equations with positive and negative coefficients, Functional Differential Equations, vol. 14, no. 2 4, pp , N. Parhi and S. Chand, Oscillation of second order neutral delay differential equations with positive and negative coefficients, The the Indian Mathematical Society. New Series, vol. 66, no. 1 4, pp , R. N. Rath, N. Misra, and P. P. Mishra, Non-oscillatory criteria for a class of second order non-linear forced neutral-delay differential equations, Mathematica Slovaca, vol. 59, no. 4, pp , Y. Şahiner, On oscillation of second order neutral type delay differential equations, Applied Mathematics and Computation, vol. 150, no. 3, pp , W. Shi and P. Wang, Oscillatory criteria of a class of second-order neutral functional differential equations, Applied Mathematics and Computation, vol. 146, no. 1, pp , A. Weng and J. Sun, Oscillation of second order delay differential equations, Applied Mathematics and Computation, vol. 198, no. 2, pp , 2008.

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