Necessary and Sufficient Conditions of Oscillation in First Order Neutral Delay Differential Equations

Abstract

We are concerned with oscillation of the first order neutral delay differential equation [superscript] [ x ( t ) - p x ( t - τ ) ] [variant prime] [/superscript] + q x ( t - σ ) = 0 with constant coefficients, and we obtain some necessary and sufficient conditions of oscillation for all the solutions in respective cases 0 < p < 1 and p > 1 .

(ProQuest: ... denotes non-US-ASCII text omitted.)

Songbai Guo 1, 2 and Youjian Shen 2 and Binbin Shi 3

Academic Editor:Chuangxia Huang

1, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
2, School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China
3, Department of Mathematics and Information Science, Shandong Agricultural University, Tai'an 271018, China

Received 31 December 2013; Accepted 5 April 2014; 27 April 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

Delay differential equations (DDEs) arose widely in many fields, like oscillation theory [1-9], stability theory [10-12], dynamical behavior of delayed network systems [13-15], and so on. Theoretical studies on oscillation of solutions for DDEs have fundamental significance (see [16, 17]). For this reason, DDEs have been attracting great interest of many mathematicians during the last few decades.

In this paper, we consider a class of neutral DDEs [figure omitted; refer to PDF] where _{t 0} is a positive number and p , q , τ , and σ are positive constants. Generally, a solution of (1) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is nonoscillatory. It can be seen in the literature that the oscillation theory regarding solutions of (1) has been extensively developed in the recent years.

In [18], Zhang came to the following conclusion.

Theorem I.

Assume that p ∈ ( 0,1 ) and q σ e > 1 - p ; then all solutions of (1) are oscillatory.

This result in Theorem I improves the corresponding result in [19]. Afterward, many authors have been devoted to studying this problem and have obtained many better results. For details, Gopalsamy and Zhang [20] obtained the improved result shown in Theorem II.

Theorem II.

If p ∈ ( 0,1 ) and q σ e > 1 - p [ 1 + q τ / ( 1 - p ) ] , then all solutions of (1) are oscillatory.

Further, Zhou and Yu [21] proved the following theorem.

Theorem III.

Suppose that p ∈ ( 0,1 ) and q σ e > 1 - p [ 1 + q τ / ( 1 - p ) + ( q τ ^{) 2} / 2 ( 1 - p ^{) 2} ] ; then all solutions of (1) are oscillatory.

Continuing to improve the research work, Xiao and Li [22] obtained the following.

Theorem IV.

Let p ∈ ( 0,1 ) and q σ e > 1 - p ^{e q τ / ( 1 - p )} ; then all solutions of (1) are oscillatory.

Finally, Lin [23] obtained the result shown in Theorem V.

Theorem V.

Assume that p ∈ ( 0,1 ) and q σ e > 1 - p ^{e q τ / ( 1 - p - q σ )} ; then all solutions of (1) are oscillatory.

However, all the conclusions mentioned above are limited to sufficient conditions in the case 0 < p < 1 . The aim of this paper is to establish systematically the necessary and sufficient conditions of oscillation for all solutions of (1) for the cases 0 < p < 1 and p > 1 .

2. Main Results

It is well known [24] that all solutions of (1) are oscillatory if and only if the characteristic equation of (1) [figure omitted; refer to PDF] has no real roots.

Theorem 1.

Assume that p ∈ ( 0,1 ) and let [figure omitted; refer to PDF] Then all solutions of (1) are oscillatory if and only if [figure omitted; refer to PDF] where θ is a unique zero of [straight phi] ( μ ) in ( 0 , 1 / σ ) .

Proof.

It is easy to see that, for λ [= or >, slanted] 0 , we have [figure omitted; refer to PDF] Thus any real root of (2) must be negative.

Next, let [figure omitted; refer to PDF] We consider the monotonicity of the function g ( μ ) ... = f ( - μ ) / μ . Differentiation yields [figure omitted; refer to PDF] where [straight phi] ( μ ) satisfies the following properties:

(1) [straight phi] ( μ ) > 0 for μ ∈ ( 1 / σ , + ∞ ) ;

(2) [straight phi] ( μ ) is strictly increasing on ( 0 , 1 / σ ) since the function ^{μ 2 e ( τ - σ ) μ} is strictly increasing on ( 0 , 1 / σ ) .

In addition, [figure omitted; refer to PDF] Thus, we get that function [straight phi] ( μ ) has a unique zero θ in ( 0 , 1 / σ ) . Hence ^{g [variant prime]} ( μ ) < 0 for μ ∈ ( 0 , θ ) and ^{g [variant prime]} ( μ ) > 0 for μ ∈ ( θ , + ∞ ) , which imply that g ( μ ) is decreasing on ( 0 , θ ) and increasing on ( θ , + ∞ ) . Therefore, g ( μ ) > 0 for μ ∈ ( 0 , + ∞ ) if and only if (7) has no real roots in μ ∈ ( 0 , 1 / σ ) . It is easy to see that g ( θ ) is the minimum value of g ( μ ) in ( 0 , 1 / σ ) . Consequently, g ( μ ) = 0 has no real roots in ( 0 , 1 / σ ) if and only if g ( θ ) > 0 . Since [figure omitted; refer to PDF] we obtain the result immediately.

From Theorem 1, we obtain immediately the following.

Corollary 2.

If p ∈ ( 0,1 ) and τ = σ , then all solutions of (1) are oscillatory if and only if q ^{e θ σ} > σ ^{θ 2} holds, where θ = ( q σ ( q σ + 4 p ) - q σ ) / 2 p σ .

Theorem 3.

Suppose that p ∈ ( 0,1 ) ; then all solutions of (1) are oscillatory if and only if one of the following conditions holds:

( _{H 1} ): q σ e [= or >, slanted] 1 ;

( _{H 2} ): θ ¯ > θ ,

where θ and θ ¯ are the unique zeros of [straight phi] ( μ ) and h ( μ ) (see (3) and (4)) in ( 0 , 1 / σ ) , respectively.

Proof.

Let y ( μ ) = h ( μ ) / ^{μ 2} = q ^{e μ σ} ( ( τ - σ ) / μ + 1 / ^{μ 2} ) - τ ; then [figure omitted; refer to PDF] where z ( μ ) = ( τ - σ ) σ ^{μ 2} + ( 2 σ - τ ) μ - 2 , which satisfies [figure omitted; refer to PDF] If τ [= or >, slanted] σ , we get obviously that z ( μ ) < 0 for all μ ∈ ( 0 , 1 / σ ] . If τ < σ , we also get z ( μ ) < 0 for all μ ∈ ( 0 , 1 / σ ] since ^{z [variant prime]} ( 1 / σ ) = τ > 0 . Thus, z ( μ ) < 0 for all μ ∈ ( 0 , 1 / σ ] . From this and (11) we get that ^{y [variant prime]} ( μ ) < 0 for all μ ∈ ( 0 , 1 / σ ] . Consequently, y ( μ ) is strictly decreasing on ( 0 , 1 / σ ] . Further, [figure omitted; refer to PDF] Therefore, if q σ e [= or >, slanted] 1 , we have y ( θ ) > 0 . Hence h ( θ ) > 0 . If q σ e < 1 , we have y ( 1 / σ ) < 0 . Hence, it is easy to find that both functions y ( μ ) and h ( μ ) have an equal and unique zero θ ¯ ∈ ( 0 , 1 / σ ) . Consequently, h ( θ ) > 0 is equivalent to θ ¯ > θ .

From Theorem 1, all solutions of (1) are oscillatory if and only if one of (H₁ ) or (H₂ ) holds.

Theorem 4.

Assume that p ∈ ( 0,1 ) ; then all solutions of (1) are oscillatory if one of the following conditions holds:

( _{H 1} ): q / θ + q σ [= or >, slanted] 1 - p ;

( _{H 2} ): q σ e [= or >, slanted] 1 - p ^{e q τ / ( 1 - p - q σ )} ,

where θ is a unique zero of [straight phi] ( μ ) in ( 0 , 1 / σ ) .

Proof.

If q / θ + q σ [= or >, slanted] 1 - p , we have that [figure omitted; refer to PDF] From the proof of Theorem 1, all solutions of (1) are oscillatory.

If q σ e [= or >, slanted] 1 - p ^{e q τ / ( 1 - p - q σ )} , we suppose furthermore that q / θ + q σ < 1 - p (otherwise, all solutions of (1) are oscillatory by the above conclusion); that is, θ > q / ( 1 - p - q σ ) . Since q σ e is a minimum value of the function ( q / μ ) ^{e μ σ} at μ = 1 / σ , we have that [figure omitted; refer to PDF] and the result follows.

So far, for p ∈ ( 0,1 ) we have discussed the necessary and sufficient conditions of oscillation for all solutions of (1). Our results have perfected the results in [23] (see Theorem 4). Next, we will discuss the behavior of oscillation of solutions of (1) in the case p > 1 .

Lemma 5.

Let p > 1 ; then all solutions of (1) are oscillatory if and only if the equation [figure omitted; refer to PDF] has no real roots in ( - ln ... p / τ , 0 ) .

Proof.

By (14), we know that g ( μ ) > 0 for μ ∈ ( 0 , ∞ ) . It is not difficult to see that ^{e μ σ} / μ is strictly decreasing on ( - ∞ , 0 ) while ^{e μ τ} is strictly increasing on ( - ∞ , 0 ) . Notice that p ^{e μ τ} - 1 = 0 at μ = - ln ... p / τ ; we find that [figure omitted; refer to PDF] Hence, f ( λ ) has no real roots which is equivalent to g ( μ ) that has no real roots in ( - ln ... p / τ , 0 ) .

Theorem 6.

Suppose that p > 1 and τ = σ ; then all solutions of (1) are oscillatory if and only if [figure omitted; refer to PDF] where θ = ( - q σ ( q σ + 4 p ) - q σ ) / 2 p σ .

Proof.

It is similar to the proof of Theorem 1; g ( θ ) is the maximum value of g ( μ ) for μ ∈ ( - ∞ , 0 ) . This and Lemma 5 imply the result.

Theorem 7.

Assume that p > 1 and τ < σ ; then all solutions of (1) are oscillatory if and only if [figure omitted; refer to PDF] where θ is a unique zero of (3) in ( - ∞ , 0 ) .

Proof.

Firstly, we prove that [straight phi] ( μ ) has a unique zero θ in ( - ∞ , 0 ) . In fact, [figure omitted; refer to PDF] It is easy to verify that ^{[straight phi] [variant prime]} ( μ ) is strictly increasing on ( - ∞ , 0 ) . In addition, [figure omitted; refer to PDF] Therefore, ^{[straight phi] [variant prime]} ( μ ) has a unique zero _{ω 0} in ( - ∞ , 0 ) . Hence, [straight phi] ( μ ) is strictly decreasing on ( - ∞ , _{ω 0} ) and strictly increasing on ( _{ω 0} , 0 ) , so that [straight phi] ( μ ) has a unique zero θ in ( - ∞ , 0 ) as [straight phi] ( 0 ) = - q < 0 and [straight phi] ( μ ) [arrow right] + ∞ ( μ [arrow right] - ∞ ) .

Now, from (8), it follows that g ( θ ) is the maximum value of g ( μ ) in ( - ∞ , 0 ) . By (10), we know that (19) is equivalent to g ( μ ) < 0 for μ ∈ ( - ∞ , 0 ) .

From Theorem 7, we obtain the following corollary that extends Theorem 1 in [25] for τ < σ .

Corollary 8.

If p > 1 , τ < σ , and τ q ^{e - ( σ / τ ) ln ... p} [= or >, slanted] τ ^{ln ... 2} p / ( σ ln ... p + τ ) , then all solutions of (1) are oscillatory.

Proof.

The inequality τ q ^{e - ( σ / τ ) ln ... p} [= or >, slanted] τ ^{ln ... 2} p / ( σ ln ... p + τ ) is equivalent to [straight phi] ( - ln ... p / τ ) [= or <, slanted] 0 . From the proof of Theorem 7, we get that θ [= or <, slanted] - ln ... p / τ . This and (17) imply g ( θ ) < 0 ; therefore, h ( θ ) < 0 .

Theorem 9.

Suppose that p > 1 and τ > σ ; then all solutions of (1) are oscillatory if and only if one of the following conditions holds:

( _{H 1} ): q σ ^{e 2 - 2} [= or >, slanted] ( 2 2 - 2 ) p τ / ( τ - σ ) ;

( _{H 2} ): σ ( τ - σ ) ^{ω 1 2} + ( 2 σ - τ ) _{ω 1} [= or <, slanted] 2 ;

( _{H 3} ): h ( _{θ 2} ) = q ^{e _{θ 2} σ} [ ( τ - σ ) _{θ 2} + 1 ] - τ ^{θ 2 2} < 0 ,

where _{ω 1} is a unique zero of ^{[straight phi] [variant prime]} ( μ ) in ( - 2 / ( τ - σ ) , ( 2 - 2 ) / ( τ - σ ) ) and _{θ 2} is the maximum negative zero of [straight phi] ( μ ) .

Proof.

By Lemma 5, all solutions of (1) are oscillatory if and only if [figure omitted; refer to PDF] From (20), we have that [figure omitted; refer to PDF] and ^{[straight phi] [variant prime]} ( μ ) is strictly decreasing on ( - 2 / ( τ - σ ) , ( 2 - 2 ) / ( τ - σ ) ) and strictly increasing on ( ( 2 - 2 ) / ( τ - σ ) , 0 ) . Thus, ^{[straight phi] [variant prime]} ( ( 2 - 2 ) / ( τ - σ ) ) is the minimum value of ^{[straight phi] [variant prime]} ( μ ) in ( - 2 / ( τ - σ ) , 0 ) .

( 1 ) If ^{[straight phi] [variant prime]} ( ( 2 - 2 ) / ( τ - σ ) ) [= or >, slanted] 0 , which is the case of (H₁ ), we have that [figure omitted; refer to PDF] Combining (23) and (24), we obtain that [figure omitted; refer to PDF] This means that g ( μ ) is strictly decreasing on ( - ∞ , 0 ) and, consequently, [figure omitted; refer to PDF]

( 2 ) If ^{[straight phi] [variant prime]} ( ( 2 - 2 ) / ( τ - σ ) ) < 0 , ^{[straight phi] [variant prime]} ( μ ) has a unique zero _{ω 1} in ( - 2 / ( τ - σ ) , ( 2 - 2 ) / ( τ - σ ) ) and a unique zero _{ω 2} in ( ( 2 - 2 ) / ( τ - σ ) , 0 ) since ^{[straight phi] [variant prime]} ( - 2 / ( τ - σ ) ) = q σ > 0 . Hence [straight phi] ( μ ) is strictly increasing on ( - ∞ , _{ω 1} ) , strictly decreasing on ( _{ω 1} , _{ω 2} ) , and strictly increasing on ( _{ω 2} , 0 ) . Consequently, [straight phi] ( _{ω 1} ) is the maximum value of [straight phi] ( μ ) in ( - ∞ , _{ω 2} ) . Now, it is easy to find that (22) holds if [straight phi] ( _{ω 1} ) [= or <, slanted] 0 .

On the other hand, applying ^{[straight phi] [variant prime]} ( _{ω 1} ) = 0 , we can get [figure omitted; refer to PDF] So [straight phi] ( _{ω 1} ) [= or <, slanted] 0 is equivalent to σ ( τ - σ ) ^{ω 1 2} + ( 2 σ - τ ) _{ω 1} [= or <, slanted] 2 . This is the case of (H₂ ).

If [straight phi] ( _{ω 1} ) > 0 , we obtain that [straight phi] ( μ ) has a unique zero _{θ 1} in ( - ∞ , _{ω 1} ) and a unique zero _{θ 2} in ( _{ω 1} , _{ω 2} ) . Therefore, g ( μ ) is strictly decreasing on ( - ∞ , _{θ 1} ) , strictly increasing on ( _{θ 1} , _{θ 2} ) , and strictly decreasing on ( _{θ 2} , 0 ) . Therefore, it is not difficult to find that (22) holds if and only if g ( _{θ 2} ) < 0 and it is the case of (H₃ ).

From Theorem 9, we obtain the following corollary immediately.

Corollary 10.

If p > 1 , τ > σ , and q σ [= or >, slanted] p τ / ( τ - σ ) , then all solutions of (1) are oscillatory.

Example 11.

Consider the following neutral delay differential equation: [figure omitted; refer to PDF] It is not difficult to see that p = 20 , q = 10.5 , τ = 12 , and σ = 2 . Consequently, τ > σ , and [figure omitted; refer to PDF] so that all the solutions of (28) are oscillatory from Theorem 9.

Acknowledgments

The authors wish to thank the editor and anonymous reviewers for their helpful and valuable comments. This work was supported in part by NSF of Hainan Province under Grant 111004.

Conflict of Interests

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

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