Abstract and Applied AnalysisVolume 2012 (2012), Article ID 546302, 33 pagesdoi:10.1155/2012/546302

Research Article

Bounded Oscillation of a Forced Nonlinear Neutral Differential Equation

Zeqing Liu,¹ Yuguang Xu,² Shin Min Kang,³ and Young Chel Kwun⁴

¹Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China²Department of Mathematics, Kunming University, Kunming, Yunnan 650214, China³Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea⁴Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

Received 20 December 2011; Accepted 5 March 2012

Academic Editor: Miroslava Růžičková

Copyright © 2012 Zeqing Liu et al. 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.

Abstract.

This paper is concerned with the th-order forced nonlinear neutral differential equation . Some necessary and sufficient conditions for the oscillation of bounded solutions and several sufficient conditions for the existence of uncountably many bounded positive and negative solutions of the above equation are established. The results obtained in this paper improve and extend essentially some known results in the literature. Five interesting examples that point out the importance of our results are also included.

1. Introduction

Consider the following th-order forced nonlinear neutral differential equation: where and are constants for . In what follows, we assume that   and satisfy that and there exists such that is positive eventually:   is strictly increasing and in ;   satisfies that for .

During the last decades, the oscillation criteria and the existence results of nonoscillatory solutions for various linear and nonlinear differential equations have been studied extensively, for example, see [1–28] and the references cited therein. In particular, Zhang and Yan [25] obtained some sufficient conditions for the oscillation of the first-order linear neutral delay differential equation with positive and negative coefficients: where , and . Das and Misra [7] studied the nonhomogeneous neutral delay differential equation: where , for , is nondecreasing, Lipschitzian, and satisfies for every , and they obtained a necessary and sufficient condition for the solutions of (1.5) to be oscillatory or tend to zero asymptotically. Parhi and Rath [18] extended Das and Misra’s result to the following forced first-order neutral differential equation with variable coefficients: where , and they got necessary and sufficient conditions which ensures every solution of (1.6) is oscillatory or tends to zero or to as . By using Banach’s fixed point theorem, Zhang et al. [24] proved the existence of a nonoscillatory solution for the first-order linear neutral delay differential equation: where , and for . Çakmak and Tiryaki [6] showed several sufficient conditions for the oscillation of the forced second-order nonlinear differential equations with delayed argument in the form: where , , and . Travis [20] investigated the oscillatory behavior of the second-order differential equation with functional argument: where and satisfies that has the same sign of and when they have the same sign. Lin [12] got some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral differential equation: where with eventually, , is nondecreasing and for . Kulenović and Hadžiomerspahić [9] deduced the existence of a nonoscillatory solution for the neutral delay differential equation of second order with positive and negative coefficients: where , , and for . Utilizing the fixed point theorems due to Banach, Schauder and Krasnoselskii, and Zhou and Zhang [27], and Zhou et al. [28] established some sufficient conditions for the existence of a nonoscillatory solution of the following higher-order neutral functional differential equations: where , and for . Li et al. [11] investigated the existence of an unbounded positive solution, bounded oscillation, and nonoscillation criteria for the following even-order neutral delay differential equation with unstable type: where , and . Zhang and Yan [22] obtained some sufficient conditions for oscillation of all solutions of the even-order neutral differential equation with variable coefficients and delays: where is even, and for , and . Yilmaz and Zafer [21] discussed sufficient conditions for the existence of positive solutions and the oscillation of bounded solutions of the th-order neutral type differential equations: where , , and . Bolat and Akin [4, 5] got sufficient criteria for oscillatory behaviour of solutions for the higher-order neutral type nonlinear forced differential equations with oscillating coefficients: where , is nondecreasing and for for , for , and and are oscillating functions. Zhou and Yu [26] attempted to extend the result of Bolat and Akin [4] and established a necessary and sufficient condition for the oscillation of bounded solutions of the higher-order nonlinear neutral forced differential equation of the form: where , and   for and ;   and are oscillating functions;   and for ;   is nondecreasing function, for and .

That is, they claimed the following result.

Theorem 1.1 (see [26, Theorem 2.1]). Assume that there is an oscillating function such that and ;   is an oscillating function and ;  , .Then, every bounded solution of (1.17) either oscillates or tends to zero if and only if

We, unfortunately, point out that the necessary part in Theorem 1.1 is false, see Remark 4.2 and Example 4.7 below. It is clear that (1.1) includes (1.4)–(1.17) as special cases. To the best of our knowledge, there is no literature referred to the oscillation and existence of uncountably many bounded nonoscillatory solutions of (1.1). The aim of this paper is to establish the bounded oscillation and the existence of uncountably many bounded positive and negative solutions for (1.1) without the monotonicity of the nonlinear term . Our results extend and improve substantially some known results in [4, 5, 9, 10, 20, 24, 26–28] and correct Theorem  2.1 in [26].

The paper is organized as follows. In Section 2, a few notation and lemmas are introduced and proved, respectively. In Section 3, by employing Krasnoselskii’s fixed point theorem and some techniques, the existence of uncountably many bounded positive and negative solutions for (1.1) are given, and some necessary and sufficient conditions for all bounded solutions of (1.1) to be oscillatory or tend to zero as are provided. In Section 4, a number of examples which clarify advantages of our results are constructed.

2. Preliminaries

It is assumed throughout this paper that and By a solution of (1.1), we mean a function for some , such that is times continuously differentiable in and such that (1.1) is satisfied for . As is customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise, it is nonoscillatory, that is, if it is eventually positive or eventually negative. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

Let stand for the Banach space of all bounded continuous functions in with the norm for each and It is easy to see that is a bounded closed and convex subset of the Banach space .

Lemma 2.1. Let and be bounded. If eventually, then (a) exists and for ; furthermore, there exists for odd and for even such that(b) eventually for ;(c) is nonincreasing eventually for .

Proof. Now, we consider two possible cases below.Case 1. Assume that . Let . Note that eventually. It follows that there exists a constant satisfying , for all , which yields that is nonincreasing in . Since is bounded in , it follows that exists.Case 2. Assume that . Notice that is odd. It follows that eventually, which implies that there exists a constant satisfying which means that Suppose that there exists a constant satisfying , which together with (2.4) gives that which guarantees that is increasing in and that is, which means that which contradicts the boundedness of . Consequently, we have Combining (2.4) and (2.9), we conclude easily that there exists a constant with Next, we claim that . Otherwise, there exists a constant satisfying which yields that which gives that which means that which contradicts the boundedness of in . Hence, , that is, Repeating the proof of (2.3)–(2.15), we deduce similarly that which together with the boundedness of implies that is nonincreasing in and exists.Thus, (2.3) and (2.16) yield (a)–(c). This completes the proof.

Lemma 2.2. Let satisfy and where is a fixed constant. Then, .

Proof. Since is a strictly increasing continuous function, in and , it follows that the inverse function of is also strictly increasing continuous, in and , where for all . Equation (2.18) implies that there exists a constant with Using (2.18) and (2.19), we deduce that, for any , there exist sufficiently large numbers and satisfying In view of (2.17), (2.20), and (2.21), we infer that for all which gives that . This completes the proof.

Lemma 2.3. Let , and be in satisfying , (2.17), (2.18), and where , and are constants. Then, there exists such that eventually.

Proof. Obviously, (2.20) holds. It follows from (2.18), (2.23), and (2.24) that for , there exist and satisfying Put . In light of (2.17), we conclude that for each which together with (2.20) and (2.25) yields that for any This completes the proof.

Similar to the proof of Lemma  3.2 in [26], we have the following two lemmas.

Lemma 2.4. Let , and be in satisfying , (2.17), (2.18), and where is a constant. Then, .

Lemma 2.5. Let , , , , and be in satisfying , (2.17), (2.18), (2.23), and (2.29). Then, there exists such that eventually.

Lemma 2.6 (Krasnoselskii’s fixed point theorem). Let be a Banach space, let be a nonempty bounded closed convex subset of , and let , be mappings of into such that for every pair . If is a contraction mapping and is completely continuous, then the mapping has a fixed point in .

3. Main Results

First, we use the Krasnoselskii’s fixed point theorem to show the existence and multiplicity of bounded positive and negative solutions of (1.1).

Theorem 3.1. Let , and hold. Assume that there exist , and satisfying Then, the following hold: (a)for arbitrarily positive constants and with equation (1.1) has uncountably many bounded positive solutions with (b)for arbitrarily positive constants and with equation (1.1) has uncountably many bounded negative solutions with

Proof. It follows from (3.1) and (3.2) that there exists an enough large constant with satisfying (a) Assume that and are arbitrary positive constants satisfying (3.4). Let . First of all, we prove that there exist two mappings and a constant such that has a fixed point , which is also a bounded positive solution of (1.1) with . Put In light of (3.3), (3.9), and , we infer that there exists a sufficiently large number satisfying Define two mappings by for each . In view of (3.1), (3.8), and (3.10)–(3.12), we conclude that for any and which ensures that It follows from (3.11), (3.12), (3.15), and (3.16) that and map into , respectively.Now, we show that is continuous in . Let and with , given . It follows from the uniform continuity of in for and that there exist and satisfying In view of (3.8), (3.12), (3.17), we arrive at which means that is continuous in .Next, we show that is equicontinuous in . Let . Taking into account (3.3) and , we know that there exists satisfying Put It follows from the uniform continuity of and in that there exists satisfying Let and with . We consider three possible cases.Case 1. Let . In view of (3.8), (3.9), (3.12), and (3.19), we conclude that Case 2. Let . In terms of (3.8), (3.9), (3.12), (3.21), we arrive at Case 3. Let . By (3.12), we have Thus, is equicontinuous in . Consequently, is relatively compact by (3.16) and the continuity of . By means of (3.14), (3.15), and Lemma 2.6, we infer that possesses a fixed point , that is, which gives that which mean that is a bounded positive solution of (1.1) with Let and be two arbitrarily different numbers in . Similarly, we conclude that for each there exist two mappings and a sufficiently large number satisfying (3.8)–(3.12), where , and are replaced by , and , respectively, and has a fixed point , which is also a bounded positive solution with , that is, It follows from (3.3) that there exists satisfying Combining (3.8), (3.28), and (3.29), we conclude easily that which guarantees that that is, . Hence, (1.1) has uncountably many bounded positive solutions with .(b) Assume that and are arbitrary positive constants satisfying (3.6) and put Let . It follows from (3.3), (3.8), (3.32), and that there exists satisfying Let the mappings be defined by (3.11) and (3.12), respectively.Using (3.1), (3.8), (3.11), (3.12), and (3.33), we deduce that for any and which give that The rest of the proof is similar to the proof of (a) and is omitted. This completes the proof.

Theorem 3.2. Let , and , hold. Assume that there exist , and satisfying (3.2), (3.3), and Then, the following hold: (a)for arbitrarily positive constants and with equation (1.1) has uncountably many bounded positive solutions with (b)for arbitrarily positive constants and with equation (1.1) has uncountably many bounded negative solutions with

Proof. It follows from (3.2) and (3.36) that there exists a constant with satisfying (a) Assume that and are arbitrary positive constants satisfying (3.37). Let and be defined by (3.9). In light of (3.3), (3.9), and , there exists a sufficiently large number satisfying Define two mappings by (3.12) and for each . In view of (3.12), (3.36), and (3.41)–(3.43), we conclude that for any and which imply (3.15). The rest of the proof is similar to that of Theorem 3.1 and is omitted.(b) Assume that and are arbitrary positive constants satisfying (3.39). Let and be defined by (3.32). Note that (3.3), (3.32), and yield that there exists a sufficiently large number satisfying Let the mappings be defined by (3.12) and (3.43), respectively.Using (3.12), (3.36), (3.41), and (3.45), we infer that for any and which give (3.15). The rest of the proof is similar to the proof of Theorem 3.1 and is omitted. This completes the proof.

Theorem 3.3. Let and hold. Assume that there exist , , and satisfying (3.2), (3.3), and Then, the following hold: (a) for arbitrarily positive constants and with equation (1.1) has uncountably many bounded positive solutions with for arbitrarily positive constants and with equation (1.1) has uncountably many bounded negative solutions with

Proof. It follows from (3.2) and (3.47) that there exists a constant satisfying (a) Assume that and are arbitrary positive constants satisfying (3.48). Let and be defined by (3.9). In light of (3.3), (3.9), and , we infer that there exists a sufficiently large number satisfying Define two mappings by for each . In view of (3.47) and (3.52)–(3.55), we conclude that for any and which yield (3.15). The rest of the proof is similar to that of Theorem 3.1 and is omitted.(b) Assume that and are arbitrary positive constants satisfying (3.50). Let and be defined by (3.32). In light of (3.3), (3.32), and , we infer that there exists a sufficiently large number satisfying Define two mappings by (3.54) and (3.55). In view of (3.47), (3.52), (3.54), (3.55), and (3.57), we conclude that (3.56) holds and Thus, (3.15) follows from (3.58). The rest of the proof is similar to that of Theorem 3.1 and is omitted. This completes the proof.

Second, we provide necessary and sufficient conditions for the oscillation of bounded solutions of (1.1).

Theorem 3.4. Let , and hold. Assume that there exist and satisfying (2.24) and Then, each bounded solution of (1.1) either oscillates or tends to 0 as if and only if

Proof. Sufficiency. Suppose, without loss of generality, that (1.1) possesses a bounded eventually positive solution with , which together with ,, (2.17), (2.24), and (3.60), yields that there exist constants and satisfying Obviously (2.17), (2.24), (3.59), and the boundedness of imply that is bounded. It follows from (2.17), (3.62), Lemmas 2.1 and 2.2 that there exists a constant satisfying Thus, , (3.61), (3.63), and Lemma 2.3 imply that there exist constants and satisfying Put Clearly, guarantees that . Integrating (3.62) from to , by (3.63) and (3.64), we have repeating this procedure, we obtain that which together with (3.64) and means that which gives that which contradicts (3.60).Necessity. Suppose that (3.60) does not hold. Observe that implies that there exist two positive constants and satisfying It follows from Theorem 3.1 or Theorem 3.2 that, for any positive constants and satisfying (3.4) or (3.37), (1.1) possesses uncountably many bounded positive solutions with . This is a contradiction. This completes the proof.

As in the proof of Theorem 3.4, by means of Lemmas 2.1, 2.4, and 2.5, we have

Theorem 3.5. Let and hold. Assume that there exist and satisfying (2.29) and (3.59). Then, each bounded solution of (1.1) either oscillates or tends to 0 as if and only if (3.60) holds.

4. Remarks and Examples

Now, we compare the results in Section 3 with some known results in the literature. In order to illustrate the advantage and applications of our results, five nontrivial examples are constructed.

Remark 4.1. Theorems 3.1–3.3 extend and improve the Theorem in [9], Theorem  8.4.2 in [10], Theorem  1 in [21], Theorems  1–3 in [24], Theorem  2.2 in [26], and Theorems  1–4 in [27, 28].

Remark 4.2. The sufficient part of Theorem 3.5 is a generalization of Theorem  3.1 in [4, 5]. Theorem 3.5 corrects and perfects Theorem  2.1 in [26].

The examples below show that our results extend indeed the corresponding results in [4, 5, 9, 10, 21, 24, 26–28]. Notice that none of the known results can be applied to these examples.

Example 4.3. Consider the th-order forced nonlinear neutral differential equation: where , and . Put , Clearly , , , and (3.1)–(3.3) hold.Let and be arbitrarily positive constants satisfying . It is easy to verify that (3.4) holds. It follows from Theorem 3.1 that (4.1) has uncountably many bounded positive solutions with .Let and be arbitrarily positive constants satisfying . It is easy to verify that (3.6) holds. It follows from Theorem 3.1 that (4.1) has uncountably many bounded negative solutions with .

Example 4.4. Consider the th-order forced nonlinear neutral differential equation: where , and . Put , Clearly , , , (3.2), (3.3), and (3.36) hold.Let and be arbitrarily positive constants satisfying . It is easy to verify that (3.37) holds. It follows from Theorem 3.2 that (4.3) has uncountably many bounded positive solutions with .Let and be arbitrarily positive constants satisfying . It is easy to verify that (3.39) holds. It follows from Theorem 3.2 that (4.3) has uncountably many bounded negative solutions with .

Example 4.5. Consider the th-order forced nonlinear neutral differential equation: where , and . Put , Clearly , , (3.2), (3.3), and (3.47) hold.Let and be arbitrarily positive constants satisfying . It is easy to verify that (3.48) holds. It follows from Theorem 3.3 that (4.5) has uncountably many bounded positive solutions with .Let and be arbitrarily positive constants satisfying . It is easy to verify that (3.50) holds. It follows from Theorem 3.3 that (4.5) has uncountably many bounded negative solutions with .

Example 4.6. Consider the th-order forced nonlinear neutral differential equation: where , and . Put , Clearly , , , (2.24), (3.59), and (3.60) hold. It follows from Theorem 3.4 that each bounded solution of (4.7) either oscillates or tends to 0 as .

Example 4.7. Consider the th-order forced nonlinear neutral differential equation: where , and . Put , Clearly , , (2.29), (3.59), and (3.60) hold. It follows from Theorem 3.5 that each bounded solution of (4.9) either oscillates or tends to 0 as .

Next, we prove that the necessary part of Theorem  2.1 in [26] does not hold by means of (4.9). It is easy to verify that the conditions of Theorem  2.1 in [26] are fulfilled. Suppose that the necessary part of Theorem  2.1 in [26] is true. Because each bounded solution of (4.9) either oscillates or tends to 0 as , it follows that the necessary part of Theorem  2.1 in [26] gives that which yields that which is a contradiction.

Acknowledgments

The authors would like to thank the referees for useful comments and suggestions. This study was supported by research funds from Dong-A University.

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