Dynamics of Third-Order Nonlinear Neutral Equations

Abstract

The aim of this paper is to study oscillatory and asymptotic properties of the third-order nonlinear neutral equation with continuously distributed delays of the form (r(t)([xt+[superscript]∫ab[/superscript] pt,μxτt,μdμ[superscript]][variant prime][variant prime][/superscript] [superscript])α[/superscript] [superscript])[variant prime][/superscript] +[superscript]∫cd[/superscript] qt,ξ[superscript]fα[/superscript] xgt,ξdξ=0 . Applying suitable generalized Riccati transformation and integral averaging technique, we present new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of this equation. Obtained results essentially improve and complement earlier ones.

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Hua Wang 1 and Li Liu 1 and Yanxiang Tan 2

Academic Editor:Taishan Yi

1, School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410114, China
2, College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China

Received 12 March 2014; Accepted 18 June 2014; 14 July 2014

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.

1. Introduction

In recent years, the dynamics theory such as oscillation theory and asymptotic behavior of differential equations and their applications have been and still are receiving intensive attention [1-4]. In fact, in the last few years several monographs and hundreds of research papers have been written; see, for example, the monograph [5]. Determining oscillation criteria for particular second-order differential equations has received a great deal of attention in the last few years [6-8]. For example, [9] considered [figure omitted; refer to PDF] and obtained oscillatory criteria of Philos type. In [10], by means of Riccati transformation technique, Han et al. established some new oscillation criteria for the second-order Emden Fowler delay dynamic equations on a time scale T : [figure omitted; refer to PDF] However, compared to second-order differential equations, the study of oscillation and asymptotic behavior of third-order differential equations has received considerably less attention in the literature [11-15]. In [16], Qiu investigated the oscillation criteria for the third-order neutral differential equations taking the following form: [figure omitted; refer to PDF] By using a generalized Riccati transformation and integral averaging technique, Zhang et al. [17] established some new sufficient conditions which ensure that every solution of the following equation oscillates or converges to zero: [figure omitted; refer to PDF] As we know, the dynamics theory such as oscillation theory and asymptotic behavior of the following equation have not been investigated up to now: [figure omitted; refer to PDF] With the help of a generalized Riccati transformation and integral averaging technique, this paper aims to establish some new sufficient conditions of Philos type which ensure that every solution of (5) oscillates or converges to zero. Our results improve and complement the corresponding results in [6, 11-17]. We should point out that, in this paper, α is any quotient of odd positive integers and α...4;1 ; it is more general than that reported in [17] where α=1 .

We are interested in (5) in the case of t...5;_t0 . Throughout this paper, we assume that the following hypotheses hold:

(_H1 ) : r(t)∈^C1 ([_t0 ,∞),(0,∞)),^{∫_t0 ∞} ...(1/r(t))dt=∞ ;

(_H2 ) : P(t,μ)∈C([_t0 ,∞)×[a,b],R), 0...4;p(t)...1;^∫ab ...p(t,μ)dμ...4;p<1 ;

(_H3 ) : τ(t,μ)∈C([_t0 ,∞)×[a,b],R) is not a decreasing function for ξ , and τ(t,μ)...4;t, _{lim...t[arrow right]∞} mi_nξ∈[a,b] τ(t,μ)=∞ ;

(_H4 ) : q(t,ξ)∈C([_t0 ,∞)×[c,d],(0,∞)) ;

(_H5 ) : g(t,ξ)∈C([_t0 ,∞)×[c,d],R) is not a decreasing function for ξ , such that g(t,ξ)...4;t,_{lim...t[arrow right]∞} mi_nξ∈[c,d] g(t,ξ)=∞ ;

(_H6 ) : f(x)∈C(R,R),(f(x)/^xα )...5;δ>, x...0;0 .

We also define the following function: [figure omitted; refer to PDF] As far as a solution of (5) is concerned, we mean a nontrivial function x(t)∈^{C[variant prime]} ([_Tx ,∞),R) , _Tx ...5;_t0 , which has the property r(t)^{z[variant prime][variant prime]} (t)∈^{C[variant prime]} ([_Tx ,∞)) and satisfies (5) on [_Tx ,∞) .

We restrict our attention to those solutions of (5) which satisfy sup...{|x(t)|:t...5;T}>0 for all T>_Tx . A solution of (5) is said to be oscillatory on [_Tx ,∞) if it is neither eventually positive nor eventually negative. Otherwise it is called nonoscillatory.

The rest of this paper is organized as follows. In Section 2, we will present some lemmas which are useful for the proof of our main results. In Section 3, we present new criteria of Philos type for oscillation or certain asymptotic behavior of nonoscillatory solutions of (5).

2. Several Lemmas

Lemma 1.

Let x(t) be a positive solution of (5), and ^{r[variant prime]} (t)...5;0, ^{z[variant prime][variant prime][variant prime]} (t)<0 . Then z(t) which is defined as in (6) has only one of the following two properties:

(I ): z(t)>0, ^{z[variant prime]} (t)>, ^{z[variant prime][variant prime]} (t)>0 ;

(II ): z(t)>0, ^{z[variant prime]} (t)<0, ^{z[variant prime][variant prime]} (t)>0 .

Proof.

Letting x(t) be a positive solution of (5) on [_t0 ,∞) , from (6), we have z(t)>x(t)>0 and (r(t)(^{z[variant prime][variant prime]} (t)^{)α)[variant prime]} =-^∫cd ...q(t,ξ)^fα (x[g(t,ξ)])dξ<0 . Then r(t)(^{z[variant prime][variant prime]} (t)^)α is a decreasing function and of one sign, and following α∈(0,1) and α=p/q where p and q are odd positive integers, we have that (^{z[variant prime][variant prime]} (t)^)α and ^{z[variant prime][variant prime]} (t) have the same sign, so ^{z[variant prime][variant prime]} (t) is either eventually positive or eventually negative on t...5;_t1 ...5;_t0 ; that is, ^{z[variant prime][variant prime]} (t)<0 or ^{z[variant prime][variant prime]} (t)>0 . If ^{z[variant prime][variant prime]} (t)<0 , then there exists a constant M>0 , such that r(t)^{z[variant prime][variant prime]} ...4;-M<0 . By integrating from _t1 to t , we get [figure omitted; refer to PDF] Letting t[arrow right]∞ and using (_H1 ) , we have ^{z[variant prime]} (t)[arrow right]-∞ . Thus ^{z[variant prime]} (t)<0 eventually; since ^{z[variant prime][variant prime]} (t)<0 and ^{z[variant prime]} (t)<0 , we have z(t)<0 , which contradicts assumption z(t)>0 , so ^{z[variant prime][variant prime]} (t)>0 . Therefore, z(t) has only one of the two properties (I) and (II).

Lemma 2.

Let x(t) be a positive solution of (5), and correspondingly z(t) has property (II) . Assume that [figure omitted; refer to PDF] Then [figure omitted; refer to PDF]

Proof.

Let x(t) be a positive solution of (5). Since z(t) has property (II), then there exists finite limit _{lim...t[arrow right]∞} z(t)=l . We assert that l=0 . Assuming that l>0 , then we have l<z(t)<l+... , for all ...>0 . Choosing ...∈(0,l(1-p)/p) , we obtain [figure omitted; refer to PDF] where k=(l-p(l+...))/(l+...)>0 . Using (_H6 ) and x(t)>kz(t) , from (5), we find that [figure omitted; refer to PDF] Note that z(t) has property (II) and (_H5 ) ; we have [figure omitted; refer to PDF] where _q1 (t)=kδ^∫cd ...q(t,ξ)dξ,_g1 (t)=g(t,d) . Integrating inequality (13) from t to ∞ , we get [figure omitted; refer to PDF] Using z[_g1 (t)]...5;l , then we have [figure omitted; refer to PDF] Integrating inequality (15) from t to ∞ , we have [figure omitted; refer to PDF] Integrating the last inequality from _t1 to ∞ , we obtain [figure omitted; refer to PDF] we have a contradiction with (8) and so it follows that _{lim...t[arrow right]∞} x(t)=0 .

Lemma 3 (see [18]).

Let z(t)>0,^{z[variant prime]} (t)>0,^{z[variant prime][variant prime]} (t)...4;0,t>_t0 . Then, for each β∈(0,1) , there exists _Tβ ...5;_t0 such that [figure omitted; refer to PDF]

Lemma 4 (see [19]).

Letting z(t)>0, ^{z[variant prime]} (t)>0, ^{z[variant prime][variant prime]} ...5;0, ^{r[variant prime]} >0, ^{z[variant prime][variant prime][variant prime]} (t)...4;0, t...5;_Tβ , then there exist γ∈(0,1) and _Tγ ...5;_Tβ such that [figure omitted; refer to PDF]

Lemma 5.

For all α>0 , then for all A>0,B>0 , one has [figure omitted; refer to PDF]

Proof.

Let u...5;0, α>0 . We investigate the maximal value and minimal value of the function f(u)=Bu-A^u(α+1)/α .

At first, for all A>0, B>0 , the derivative of function f(u)=Bu-A^u(α+1)/α is ^{f[variant prime]} (u)=B-A((α+1)/α)^u1/α . It is clear that when u>(B/A^)α ·(α/(α+1)^)α , we have ^{f[variant prime]} (u)<0 , and when u<(B/A^)α ·(α/(α+1)^)α , we have ^{f[variant prime]} (u)>0 . Hence the function f(u)=Bu-A^u(α+1)/α attains its maximum value (^αα /(α+1^)α+1 )·(^Bα+1 /^Aα ) at u=(B/A^)α ·(α/(α+1)^)α . This completes the proof.

3. Main Result

Theorem 6.

Assume that the condition of Lemma 2 holds, and there exists ρ∈^C1 ([_t0 ,∞),(0,∞)) , such that ^{ρ[variant prime]} >0 and [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then every solution x(t) of (5) either is oscillatory or converges to zero.

Proof.

Assume that (5) has a nonoscillatory solution x(t) . Without loss of generality we may assume that x(t)>0,t...5;_t1 , x[τ(t,μ)]>0, (t,μ)∈[_t1 ,∞)×[a,b]; x[g(t,ξ)]>0, (t,ξ)∈[_t1 ,∞)×[c,d] , and z(t) is defined as in (6). By Lemma 1, we have that z(t) has property (I) or property (II). At first, when z(t) has property (I), we obtain [figure omitted; refer to PDF] Using (_H5 ) and (_H6 ) , we get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] so [figure omitted; refer to PDF] Letting u(t)=^{z[variant prime]} (t) , from Lemma 3, we obtain [figure omitted; refer to PDF] Using Lemma 4, we get [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] where Q(t) is defined as (21). Letting A(t)=^{ρ[variant prime]} (t)/ρ(t), B(t)=α/(ρ(t)r(t)^)1/α , we have that [figure omitted; refer to PDF] and, from Lemma 5, we obtain [figure omitted; refer to PDF] Integrating inequality (33) from T to t , [figure omitted; refer to PDF] we obtain [figure omitted; refer to PDF] which contradicts (21). If z(t) has property (II), since (8) holds, then the conditions in Lemma 2 are satisfied. Hence _{lim...t[arrow right]∞} x(t)=0 .

This completes the proof.

Acknowledgments

This work is supported by the Scientific Research Funds of Hunan Provincial Science and Technology Department of China (no. 12FJ4252 and no. 2013SK3143), National Natural Science Foundation of China (nos. 11101053, 11326116).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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