Asymptotic Properties of Solutions to Third-Order Nonlinear Neutral Differential Equations

Abstract

The aim of this work is to discuss asymptotic properties of a class of third-order nonlinear neutral functional differential equations. The results obtained extend and improve some related known results. Two examples are given to illustrate the main results.

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Qi Li 1 and Rui Wang 1 and Fanwei Meng 2 and Jianxin Han 3

Academic Editor:Tongxing Li

1, Qingdao Technological University, Feixian, Shandong 273400, China
2, Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China
3, College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, China

Received 26 March 2014; Accepted 12 May 2014; 25 May 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 this work, we study the asymptotic behavior of solutions of the third-order neutral differential equation [figure omitted; refer to PDF] We always assume that the following conditions hold:

(H₁ ): a(t),b(t),p(t),q(t)∈C([_t0 ,∞),[0,∞)),0...4;p(t)...4;_p0 <1 ;

(H₂ ): σ(t),τ(t)∈C([_t0 ,∞),[0,∞)) , σ(t)...4;t , τ(t)...4;t , _{lim...t[arrow right]∞} σ(t)=_{lim...t[arrow right]∞} τ(t)=∞ ;

(H₃ ): f∈C(R,R) , f(x)/x...5;K>0 , for all x...0;0 ;

(H₄ ): g∈C(R,[L,∞)) , L>0 ;

(H₅ ): ^{∫_t0 ∞} ...(1/a(t))dt=^{∫_t0 ∞} ...(1/b(t))dt=∞ .

Set z(t)=x(t)+p(t)x(σ(t)) . By a solution of (1), we mean a nontrivial function x(t)∈C([_Tx ,∞),R) , _Tx ...5;_t0 , which has the properties z(t)∈^C1 ([_Tx ,∞),R) , b(t)^{z[variant prime]} (t)∈^C1 ([_Tx ,∞)) , and a(t)^{(b(t)z[variant prime] (t))[variant prime]} ∈^C1 ([_Tx ,∞)) and satisfies (1) on [_Tx ,∞) . We consider only those solutions x(t) of (1) which satisfies sup...{|x(t)|:t...5;T}>0 for all T...5;_Tx . We assume that (1) possesses such a solution. A solution of (1) is called oscillatory if it has arbitrarily large zeros on [_Tx ,∞) ; otherwise, it is called nonoscillatory.

Recently, great attention has been devoted to the oscillation of various classes of differential equations. See, for example, [1-19]. Hartman and Wintner [1] and Erbe et al. [3] studied the third-order differential equation [figure omitted; refer to PDF]

Paper [5] studied the oscillation of third-order trinomial delay differential equation [figure omitted; refer to PDF]

Li et al. [7] discussed (1) with f(x(τ(t)))=x(τ(t)) and g(^{x[variant prime]} (t))=1 . Han [8] examined the oscillation of (1) with b(t)=1 .

In this work, we establish some oscillation criteria for (1) which extend and improve the results in [7, 8].

2. Main Results

In the following, all functional inequalities considered are assumed to hold eventually for all t large enough. Without loss of generality, we deal only with the positive solutions of (1).

Theorem 1.

Suppose that [figure omitted; refer to PDF] If for some function ρ∈^C1 ([_t0 ,∞),(0,∞)) , for all sufficiently large _t2 >_t1 >_t0 , one has [figure omitted; refer to PDF] where [figure omitted; refer to PDF] then all solutions of (1) are oscillatory or convergent to zero asymptotically.

Proof.

Assume that x is a positive solution of (1). Based on condition (_H5 ) , there are two possible cases:

(1) z(t)>0 , ^{z[variant prime]} (t)>0 , ^{(b(t)z[variant prime] (t))[variant prime]} >0 , ^{[a(t)(b(t)z[variant prime] (t))[variant prime] ][variant prime]} <0 ;

(2) z(t)>0 , ^{z[variant prime]} (t)<0 , ^{(b(t)z[variant prime] (t))[variant prime]} >0 , ^{[a(t)(b(t)z[variant prime] (t))[variant prime] ][variant prime]} <0 .

First, consider that z(t) satisfies (1). We have [figure omitted; refer to PDF] From (1), (_H3 ) , and (_H4 ) , we get [figure omitted; refer to PDF] Define a function ω by [figure omitted; refer to PDF] we obtain ω(t)>0 . Then [figure omitted; refer to PDF] By the proof of [7, Theorem 2.1], we have [figure omitted; refer to PDF] where Q(t) is defined as in (6). We obtain [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] which contradicts (5). Assume that case (2) holds. Using the similar proof of [8, Lemma 4], we can get _{lim...t[arrow right]∞} x(t)=0 . This completes the proof.

Based on Theorem 1, we present a Kamenev-type criterion for (1).

Theorem 2.

Assume that (4) holds. If for some function ρ∈^C1 ([_t0 ,∞),(0,∞)) , for all sufficiently large _t1 >_t0 , one has [figure omitted; refer to PDF] then all solutions of (1) are oscillatory or convergent to zero asymptotically.

Proof.

Assume that x(t) is a positive solution of (1). Then by the proof of Theorem 1, we have cases (1) and (2). Let case (1) hold. Proceeding as in the proof of Theorem 1, we have (11). Then we have [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] which contradicts (14). Assume that case (2) holds. We can get _{lim...t[arrow right]∞} x(t)=0 . The proof is completed.

Next, we present a Philos-type criterion for (1). Let [figure omitted; refer to PDF] We say that a function H∈C(D,R) belongs to a function class P , if it satisfies

(i) H(t,t)=0 , t...5;_t0 ; H(t,s)>0 , (t,s)∈_D0 ;

(ii) H has a continuous and nonpositive partial derivative on _D0 with respect to the second variable, and such that [figure omitted; refer to PDF]

Theorem 3.

Assume that (4) holds. If for some function ρ∈^C1 ([_t0 ,∞),(0,∞)) , for all sufficiently large _t1 >_t0 , one has [figure omitted; refer to PDF] where Q(t) is defined as in (6), B(t)=1/ρ(t)a(t) , and [figure omitted; refer to PDF] then all solutions of (1) are oscillatory or convergent to zero asymptotically.

Proof.

Assume that x(t) is a positive solution of (1), and z(t) has the case of (1); ω(t) is defined as in (9). Then [figure omitted; refer to PDF] Let B(t)=1/ρ(t)a(t) , we have [figure omitted; refer to PDF] We obtain [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] which contradicts (19). Assume that (2) holds. We can get _{lim...t[arrow right]∞} x(t)=0 . The proof is completed.

3. Examples

In this section, we will present two examples to illustrate the main results.

Example 1.

Consider the third-order nonlinear neutral differential equation: [figure omitted; refer to PDF] where _p1 ∈[0,1) , and K=L=1 . Let ρ(t)=t . It follows from Theorem 1 that every solution x(t) of (25) is oscillatory or convergent to zero asymptotically.

Example 2.

Consider the third-order nonlinear neutral differential equation: [figure omitted; refer to PDF] where λ>0 , t...5;1 . We have [figure omitted; refer to PDF] we see that (4) and (H₁ )-(H₅ ) hold. Let H(t,s)=(t-s⁾² , ρ(t)=1 . Then _h1 (t,s)=2 . It follows, from Theorem 3, that the solutions of (26) are oscillatory or convergent to zero asymptotically.

Acknowledgment

This research was partially supported by the NNSF of China (11171178).

Conflict of Interests

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

References

[1] P. Hartman, A. Wintner, "Linear differential and difference equations with monotone solutions," American Journal of Mathematics , vol. 75, pp. 731-743, 1953.

[2] L. Erbe, "Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations," Pacific Journal of Mathematics , vol. 64, no. 2, pp. 369-385, 1976.

[3] L. H. Erbe, Q. Kong, B. G. Zhang Oscillation Theory for Functional-Differential Equations , vol. 190, of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1995.

[4] R. P. Agarwal, S. R. Grace, D. O'Regan, "Oscillation criteria for certain n th order differential equations with deviating arguments," Journal of Mathematical Analysis and Applications , vol. 262, no. 2, pp. 601-622, 2001.

[5] J. Dzurina, "Asymptotic properties of the third order delay differential equations," Nonlinear Analysis. Theory, Methods & Applications , vol. 26, no. 1, pp. 33-39, 1996.

[6] T. Candan, R. S. Dahiya, "Functional differential equations of third order," in Proceedings of the Conference on Differential Equations and Applications in Mathematical Biology, vol. 12, pp. 47-56, 2005.

[7] T. Li, C. Zhang, G. Xing, "Oscillation of third-order neutral delay differential equations," Abstract and Applied Analysis , vol. 2012, 2012.

[8] Z.-Y. Han, "Oscillations of a class of third order nonlinear neutral functional differential equations," Journal of East China Normal University , no. 1, pp. 113-120, 2012.

[9] B. Baculíková, J. Dzurina, "Oscillation of third-order functional differential equations," Electronic Journal of Qualitative Theory of Differential Equations , vol. 43, pp. 1-10, 2010.

[10] M. F. Aktas, A. Tiryaki, A. Zafer, "Oscillation criteria for third-order nonlinear functional differential equations," Applied Mathematics Letters , vol. 23, no. 7, pp. 756-762, 2010.

[11] S. R. Grace, R. P. Agarwal, R. Pavani, E. Thandapani, "On the oscillation of certain third order nonlinear functional differential equations," Applied Mathematics and Computation , vol. 202, no. 1, pp. 102-112, 2008.

[12] R. P. Agarwal, S. R. Grace, D. O'Regan, "The oscillation of certain higher-order functional differential equations," Mathematical and Computer Modelling , vol. 37, no. 7-8, pp. 705-728, 2003.

[13] S. H. Saker, "Oscillation criteria of third-order nonlinear delay differential equations," Mathematica Slovaca , vol. 56, no. 4, pp. 433-450, 2006.

[14] S. H. Saker, J. Dzurina, "On the oscillation of certain class of third-order nonlinear delay differential equations," Mathematica Bohemica , vol. 135, no. 3, pp. 225-237, 2010.

[15] R. P. Agarwal, M. F. Aktas, A. Tiryaki, "On oscillation criteria for third order nonlinear delay differential equations," Archivum Mathematicum , vol. 45, no. 1, pp. 1-18, 2009.

[16] T. S. Hassan, "Oscillation of third order nonlinear delay dynamic equations on time scales," Mathematical and Computer Modelling , vol. 49, no. 7-8, pp. 1573-1586, 2009.

[17] T. Li, Z. Han, S. Sun, Y. Zhao, "Oscillation results for third order nonlinear delay dynamic equations on time scales," Bulletin of the Malaysian Mathematical Sciences Society , vol. 34, no. 3, pp. 639-648, 2011.

[18] E. Thandapani, T. Li, "On the oscillation of third-order quasi-linear neutral functional differential equations," Archivum Mathematicum , vol. 47, no. 3, pp. 181-199, 2011.

[19] R. P. Agarwal, B. Baculikova, J. Dzurina, T. Li, "Oscillation of third-order nonlinear functional differential equations with mixed arguments," Acta Mathematica Hungarica , vol. 134, no. 1-2, pp. 54-67, 2012.