Abstract and Applied Analysis

Volume 2014 (2014), Article ID 238036, 29 pages

http://dx.doi.org/10.1155/2014/238036

Research Article

On Positive Solutions of a Fourth Order Nonlinear Neutral Delay Difference Equation

Zeqing Liu,¹ Xiaoping Zhang,¹ Shin Min Kang,² and Young Chel Kwun³

¹Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China

²Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

³Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

Received 19 August 2013; Accepted 17 December 2013; Published 4 March 2014

Academic Editor: Adem Kılıçman

Copyright © 2013 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.

The existence results of uncountably many bounded positive solutions for a fourth order nonlinear neutral delay difference equation are proved by means of the Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem. A few examples are included.

1. Introduction

In the past few decades, the researchers [1–31] and others studied oscillation, asymptotic behavior, and solvability for a lot of second and third order nonlinear difference equations, some of which are as follows:

By employing a few famous tools in nonlinear analysis including the nonlinear alternative of Leray-Schauder type, Banach's fixed point theorem, Schauder's fixed point theorem, Krasnoselskii's fixed point theorem, coincidence degree theory and critical point theory, the authors [3, 4, 9, 13–16, 18, 25, 27, 28] and others proved the existence of nonoscillatory solutions, uncountably many bounded nonoscillatory solutions and periodic solutions for the difference equations above, where Lipschitz conditions were used in [14, 16]. Recently, the authors [32] used the Krasnoselskii's fixed point theorem to obtain -asymptotic stability results about the zero solution for a very general first order nonlinear neutral differential equation with functional delay.

However, to our knowledge, no one studied the following fourth order nonlinear neutral delay difference equation:

where , , and are real sequences with for and with

The purpose of this paper is to fill this gap in the literature and to study solvability of (2). Under certain conditions, we prove the existence of uncountably many bounded positive solutions of (2) by means of the Krasnoselskii's fixed point theorem and Schauder's fixed point theorem, respectively. Nine examples are included.

This paper is organized as follows. In Section 2 we present some notations, definitions, and lemmas. In Section 3 we establish nine sufficient conditions which guarantee the existence of uncountably many bounded positive solutions of (2) by using fixed point theorems and new techniques. In Section 4 we give nine examples to illustrate the effectiveness and applications of the results presented in Section 3.

2. Preliminaries

Throughout this paper, we assume that and stand for the sets of all integers, positive integers, and nonnegative integers, respectively:

denotes the forward different operator defined by and for . Let denote the Banach space of all bounded sequences on with norm:

Obviously, , , and , are closed bounded and convex subsets of for any .

By a solution of (2), we mean a real sequence with a positive integer such that (2) is satisfied for all .

Definition 1 (see [5]). A subset of is said to be uniformly Cauchy (or equi-Cauchy) if for every there exists a positive integer such that

whenever for any .

Lemma 2 (see [5]). Each bounded and uniformly Cauchy subset of is relatively compact.

Lemma 3 (Krasnoselskii's fixed point theorem). Let be a Banach space, let be a bounded closed convex subset of , and let be mappings from into such that for every pair . If is a contraction and is completely continuous, then the equation

has a solution in .

Lemma 4 (Schauder's fixed point theorem). Let be a nonempty closed convex subset of a Banach space , continuous, and relatively compact. Then has at least one fixed point in .

Lemma 5. Let , , , , and be nonnegative sequences. If

then

where denotes the integer part of number .

Proof. Notice that

That is, (9) holds. This completes the proof.

3. Existence of Uncountably Many Bounded Positive Solutions

Now we study the existence of uncountably many bounded positive solutions for (2) by using the Krasnoselskii's fixed point theorem and Schauder's fixed point theorem, respectively.

Theorem 6. Assume that there exist constants , , , and with and and nonnegative sequences , , , and satisfying

Then (2) possesses uncountably many bounded positive solutions in .

Proof. Set . It follows from (13) that there exists sufficiently large such that

Define two mappings and : by

for each .

Now we prove that

In view of (11), (12), and (14)–(16), we conclude that for any , , and ,

which yield that (17) hold

In order to prove that is completely continuous in , we have to show that is continuous in and is relatively compact. Suppose that is an arbitrary sequence in and with , where for each and . With the help of (12), (13), , and the continuity of , and , we know that for given , there exist , and with satisfying

where

By virtue of (16), (19), and (20), we get that

which means that is continuous in .

Next we prove that is uniformly Cauchy. It follows from (13) that for given , there exists satisfying

Using (16) and (22), we know that for any and ,

which yields that is uniformly Cauchy. It follows from Lemma 2 that is relatively compact. Consequently Lemma 3 guarantees that there exists satisfying , which together with (15) and (16) gives that

which yields that

which means that

that is, is a bounded positive solution of (2) in .

Finally we prove that (2) has uncountably many bounded positive solutions in . Let and . For each , we conclude similarly that there exist a positive integer and two mappings and satisfying (14)–(16), where and are replaced by and , respectively, and has a fixed point , which is a bounded positive solution (2) in ; that is,

Equation (13) ensures that there exists some satisfying

Combining (11), (12), (28), and (29), we deduce that for any ,

which implies that

That is, . Thus (2) has uncountably many bounded positive solutions in . This completes the proof.

Theorem 7. Assume that there exist constants , , , and with and and nonnegative sequences , , , and satisfying (12), (13), and

Then (2) possesses uncountably many bounded positive solutions in .

Proof. Let . It follows from (13) that there exists sufficiently large such that

Let the mappings and be defined by (15) and (16), respectively. By means of (12), (15), (16), (32), and (33), we infer that for any , , and ,

that is, for any . The rest of the proof is similar to that of Theorem 6 and is omitted. This completes the proof.

Theorem 8. Assume that there exist constants , , , and with and and four nonnegative sequences , , , and satisfying (12), (13), and

Then (2) possesses uncountably many bounded positive solutions in .

Proof. Let . It follows from (13) that there exists sufficiently large such that

Let the mappings and be defined by (15) and (16), respectively. Making use of (12), (15), (16), (35), and (36), we derive that for any , and ,

that is, for any . The rest of the proof is similar to that of Theorem 6 and is omitted. This completes the proof.

Theorem 9. Assume that there exist constants , , , and with and and nonnegative sequences , , , and satisfying (13):

Then (2) possesses uncountably many bounded positive solutions in .

Proof. Let . It follows from (13) that there exists an integer satisfying

Define two mappings and by

for each . By means of (13) and (39)–(42), we get that for any , , and ,

which yield that for any . By virtue of (38) and (41), we infer that

that is, is a contraction in because .

In order to prove that is completely continuous in , we have to show that is continuous in and is relatively compact. Suppose that is an arbitrary sequence in and with , where for each and . On account of (13), (39), , and the continuity of , and , we obtain that for given , there exist , and with satisfying

where

It follows from (42)–(46) that

which means that is continuous in . It follows from (13) that for given , there exists satisfying

Next we prove that is uniformly Cauchy. In view of (42) and (48), we infer that for any and ,

Note that (40) and (42) yield that

which gives that is bounded. Thus Lemma 3 means that there exists such that , which is a bounded positive solution of (2) in .

Let and . For any , we deduce similarly that there exist a positive integer , a closed bounded and convex subset of , and two mappings and satisfying (40)–(42), where and are replaced by and , respectively, and possesses a fixed point , which is a bounded positive solution of (2); that is,

Observe that (13) implies that there exists with satisfying

which together with (51) yields that for each

which implies that

that is, . Consequently, (2) has uncountably many bounded positive solutions in . This completes the proof.

Theorem 10. Assume that there exist constants , , , , and with and and nonnegative sequences , , , and satisfying (13):

Then (2) possesses uncountably many bounded positive solutions in .

Proof. Let . It follows from (13) that there exists an integer satisfying

Let the mappings and be defined by (41) and (42), respectively. Using (13) and (41), (42) and (57), we obtain that for any , and ,

which imply that for any . The rest of the proof is similar to that of Theorem 9 and is omitted. This completes the proof.

Theorem 11. Assume that there exist constants , , , and with and and nonnegative sequences , , , and satisfying (13),(39), and

Then (2) possesses uncountably many bounded positive solutions in .

Proof. Let . It follows from (13) that there exists an integer satisfying

Define two mappings and by (42) and

for each . It follows from (42) and (59)–(61) that for any , , and ,

which yield that . The rest of the proof is similar to that of Theorem 9 and hence is omitted. This completes the proof.

Theorem 12. Assume that there exist constants , , , , and with and , , and nonnegative sequences , , , and satisfying (13), (56), and

Then (2) possesses uncountably many bounded positive solutions in .

Proof. Let . It follows from (13) that there exists an integer satisfying

Let the mappings and be defined by (42) and (61), respectively. By means of (42), (56), (61), and (64), we deduce that for any , , and ,

which mean that . The rest of the proof is similar to that of Theorem 9 and hence is omitted. This completes the proof.

Theorem 13. Assume that there exist constants , , and with and nonnegative sequences , , , and satisfying (12), (13), and

Then (2) possesses uncountably many bounded positive solutions in

Proof. Let . Equation (13) ensures that there exists sufficiently large such that

Define a mapping by

for each . In view of (12), (67), and (68), we deduce that for every and ,

which means that and for all . It follows from (13) that for each , there exists satisfying (22). Using (22) and (68), we obtain that for any and ,

which yields that is uniformly Cauchy.

Now we prove that is continuous in . Suppose that is an arbitrary sequence in and with , where for each and . Using (12), (13), , and the continuity of , and , we conclude that for given , there exist , and with satisfying (19) and (20). It follows from (19), (20), and (68) that

which implies that is continuous in . Thus Lemma 4 means that possesses a fixed point ; that is,

which imply that

which means that

which yields that is bounded positive solution of (2). The rest of the proof is similar to that of Theorem 6 and is omitted. This completes the proof.

Theorem 14. Assume that there exist constants , , and with and nonnegative sequences , , , and satisfying (12) and

Then (2) possesses uncountably many bounded positive solution in .

Proof. Let . It follows from (76) that there exists sufficiently large such that

Define a mapping by

for all . By virtue of (12), (77), and (78), we know that for every and ,

which means that and for each .

Next we prove that is uniformly Cauchy. It follows from (76) that for any given there exists with

It follows from (12), (78), and (80) that for any and ,

which yields that is uniformly Cauchy.

Now we prove that is continuous in . Suppose that is an arbitrary sequence in and with , where for each and . By (12), (76), , and the continuity of , and , we get that for given , there exist , and with satisfying

where

In terms of (12), (78), and (83), we have

which yields that is continuous in . It follows from Lemmas 2 and 4 that has a fixed point ; that is,

which imply that

which yields that

that is, is a bounded positive solution of (2). The rest of the proof is similar to that of Theorem 6 and is omitted. This is completes proof.

4. Applications

Now we display nine examples as applications of the results presented in Section 3.

Example 1. Consider the fourth order nonlinear neutral delay difference equation:

It follows from Theorem 6 that (88) possesses uncountably many bounded positive solutions in .

Example 2. Consider the fourth order nonlinear neutral delay difference equation:

It follows from Theorem 7 that (89) possesses uncountably many bounded positive solutions in .

Example 3. Consider the fourth order nonlinear neutral delay difference equation:

Theorem 8 implies that (90) possesses uncountably many bounded positive solutions in .

Example 4. Consider the fourth order nonlinear neutral delay difference equation:

Theorem 9 yields that (91) has uncountably many bounded positive solutions in .

Example 5. Consider the fourth order nonlinear neutral delay difference equation:

Theorem 10 guarantees that (92) possesses uncountably many bounded positive solutions in .

Example 6. Consider the fourth order nonlinear neutral delay difference equation:

Theorem 11 ensures that (93) possesses uncountably many bounded positive solutions in .

Example 7. Consider the fourth order nonlinear neutral delay difference equation:

Theorem 12 guarantees that (94) possesses uncountably many bounded positive solutions in .

Example 8. Consider the fourth order nonlinear neutral delay difference equation:

It follows from Theorem 13 that (95) possesses uncountably many bounded positive solutions in .

Example 9. Consider the fourth order nonlinear neutral delay difference equation:

Theorem 14 means that (96) possesses uncountably many bounded positive solutions in .

Conflict of Interests

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

Acknowledgments

The authors would like to express their thanks to the anonymous referee for her/his valuable suggestions and comments. This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380).

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