Abstract and Applied Analysis

Volume 2012 (2012), Article ID 237036, 12 pages

doi:10.1155/2012/237036

Research Article

Bounded Positive Solutions for a Third Order Discrete Equation

Zeqing Liu,¹ Ming Jia,² Shin Min Kang,³ and Young Chel Kwun⁴

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

²Dalian Hongwen Middle School, Dalian, Liaoning 116029, China

³Gyeongsang National University, Department of Mathematics and RINS, Jinju 660-701, Republic of Korea

⁴Dong-A University, Department of Mathematics, Pusan 614-714, Republic of Korea

Received 21 February 2012; Accepted 12 April 2012

Academic Editor: Josef Diblík

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 studies the following third order neutral delay discrete equation , where , , , , are real sequences with for , with for and . By using a nonlinear alternative theorem of Leray-Schauder type, we get sufficient conditions which ensure the existence of bounded positive solutions for the equation. Three examples are given to illustrate the results obtained in this paper.

1. Introduction and Preliminaries

The oscillatory, nonoscillatory and asymptotic behaviors and existence of solutions for various difference equations have received more and more attentions in recent years. For details, we refer the reader to [1–11] and the references therein.

In 2005, M. Migda and J. Migda [10] studied the asymptotic behavior of solutions for the second order neutral difference equation

where , is a nonnegative integer and . In 2008, Cheng and Chu [7] established sufficient and necessary conditions of oscillation for the second order difference equation

where is the quotient of two odd positive integers and for . In 2000, Li et al. [9] gave several necessary and/or sufficient conditions of the existence of unbounded positive solution for the nonlinear difference equation

where is a fixed nonnegative integer, and . In 2003, using the Leray-Schauder's nonlinear alternative theorem, Agarwal et al. [1] presented the existence of nonoscillatory solutions for the discrete equation

where are fixed nonnegative integers, and is continuous. In 1995, Yan and Liu [11] proved the existence of a bounded nonoscillatory solution for the third order difference equation

by utilizing the Schauder's fixed point theorem. In 2005, Andruch-Sobio and Migda [2] studied the third order linear difference equations of neutral type

and obtained sufficient conditions under which all solutions of (1.6) are oscillatory.

The aim of this paper is to study the following third order neutral delay discrete equation

where are real sequences with for , with for and . By making use of the Leray-Schauder's nonlinear alternative theorem, we establish the existence results of bounded positive solutions for (1.7), which extend substantially Theorem 2 in [11]. Three nontrivial examples are given to illustrate the superiority and applications of the results presented in this paper.

Let us recall and introduce the below concepts, signs and lemmas. Let and denote the sets of all real numbers, integers and positive integers, respectively,

and stand for the Banach space of all bounded sequences on with norm

For any constants , put

It is easy to verify that is a nonempty closed convex subset of and is a nonempty open subset of .

By a solution of (1.7), we mean a sequence with a positive integer such that (1.7) holds for all .

For any subset of a Banach space , let and denote the closure and boundary of in , respectively.

Lemma 1.1 (see [8]). A bounded, uniformly Cauchy subset of is relatively compact.

Lemma 1.2 (Leray-Schauder's Nonlinear Alternative Theorem [1]). Let be a nonempty closed convex subset of a Banach space and be an open subset of with . Also is a continuous, condensing mapping with bounded. Then either

has a fixed point in ; or there are and with .

2. Existence of Bounded Positive Solutions

Now we investigate sufficient conditions of the existence of bounded positive solutions for (1.7) by using the Leray-Schauder's Nonlinear Alternative Theorem.

Theorem 2.1. Assume that there exist constants and and satisfying

Then (1.7) possesses a bounded positive solution in .

Proof. Let . It follows from (2.1)–(2.3) that there exists a positive integer sufficiently large satisfying

Choose with and

Note that

which implies that . Define two mappings by

for all .

We now show that

For each , by (2.4)–(2.9), we have

We next assert that

Let be an arbitrary sequence and with

Since is closed, it follows that . Given . Using (2.1), (2.13) and the continuity of , we infer that there exists with satisfying

Combining (2.9) and (2.14)–(2.17), we conclude that

which means that is continuous in . On the other hand, in light of (2.6) and (2.9), we get that for each

which yields that is a bounded subset of . By virtue of (2.1) and (2.2), we deduce that for any , there exists satisfying

which together with (2.9) gives that for any

which means that is uniformly Cauchy. Thus Lemma 1.1 ensures that is a relatively compact subset of .

Let . In view of (2.3), (2.4) and (2.8), we know that

which implies that

which together with (2.10) and (2.12) guarantees that is a continuous, condensing mapping.

In order to show the existence of a fixed point of , we need to prove that in Lemma 1.2 does not hold. Otherwise there exist and such that . Let

It is easy to verify that . Now we have to discuss two possible cases as follows:

Case  1. Let . It follows from (2.3), (2.4), (2.8) and (2.9) that

which yield that

which is a contradiction;

Case  2. Let . If , by (2.3), (2.4), (2.8) and (2.9), we deduce that

which is impossible; if , by (2.3), (2.4), (2.8) and (2.9), we arrive at

which is absurd.

Consequently Lemma 1.2 ensures that there is such that , which is a bounded positive solution of (1.7). This completes the proof.

Remark 2.2. Under the conditions of Theorem 2.1 we prove also that (1.7) has uncountably many bounded positive solutions in .

In fact, as in the proof of Theorem 2.1, for any different we conclude that for each , there exist a constant and two mappings satisfying (2.6)–(2.9) and

where and are replaced by and , respectively, and has a fixed point , which is a bounded positive solution of (1.7). In order to prove that (1.7) possesses uncountably many bounded positive solutions in , we need only to prove that . It follows from (2.8), (2.9) and (2.30) that for

which implies that

which yields that .

Remark 2.3. If either or , then Theorem 2.1 reduces to the below results, respectively.

Theorem 2.4. Assume that there exist constants and and satisfying (2.1), (2.2) and

Then (1.7) possesses a bounded positive solution in .

Theorem 2.5. Assume that there exist constants and and satisfying (2.1), (2.2) and

Then (1.7) possesses a bounded positive solution in .

Remark 2.6. Theorems 2.1–2.5 include Theorem 2 in [11] as special cases. Examples 3.1–3.3 in Section 3 explain that Theorems 2.1–2.5 are genuine generalizations of Theorem 2 in [11].

3. Examples and Applications

Now we construct three nontrivial examples to explain the superiority and applications of Theorems 2.1–2.5, respectively.

Example 3.1. Consider the third order neutral delay discrete equation

where is fixed. Let ,

It is easy to verify that (2.1)–(2.4) hold. It follows from Theorem 2.1 that (3.1) has a bounded positive solution in . However Theorem 2 in [11] is useless for (3.1).

Example 3.2. Consider the third order neutral delay discrete equation

where is fixed. Let ,

It is clear that (2.1), (2.2) and (2.33) hold. Consequently Theorem 2.4 guarantees that (3.3) has a bounded positive solution in . But Theorem 2 in [11] is inapplicable for (3.3).

Example 3.3. Consider the third order neutral delay discrete equation

where is fixed. Let ,

Obviously, (2.1), (2.2) and (2.34) hold. Thus Theorem 2.5 ensures that (3.5) has a bounded positive solution in . While Theorem 2 in [11] is unfit for (3.5)

Acknowledgment

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

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