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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 191254, 21 pages
http://dx.doi.org/10.1155/2012/191254
Research Article

Existence of Bounded Positive Solutions for Partial Difference Equations with Delays

1Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
2Department of Mathematics, Dong-A University, Busan 614-714, Republic of Korea
3Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 3 February 2012; Accepted 14 March 2012

Academic Editor: Agacik Zafer

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 deals with solvability of the third-order nonlinear partial difference equation with delays . With the help of the Banach fixed-point theorem, the existence results of uncountably many bounded positive solutions for the partial difference equation are given; some Mann iterative schemes with errors are suggested, and the error estimates between the iterative schemes and the bounded positive solutions are discussed. Three nontrivial examples illustrating the results presented in this paper are also provided.

1. Introduction and Preliminaries

In the past twenty years many authors studied the oscillation, nonoscillation, asymptotic behavior, and solvability for various neutral delay difference and partial difference equations; see, for example, [114] and the references cited therein.

By using the Banach fixed-point theorem, Cheng [2] investigated the existence of a nonoscillatory solution for the second-order neutral delay difference equation with positive and negative coefficients under the condition . Applying a nonlinear alternative of Leray-Schauder type for condensing operators, Agarwal et al. [1] discussed the existence of a bounded nonoscillatory solution for the discrete equation: Liu et al. [6] introduced the second-order nonlinear neutral delay difference equation with respect to all and gave the existence of uncountably many bounded nonoscillatory solutions for (1.3) by utilizing the Banach fixed-point theorem. Kong et al. [3] investigated a class of BVPs for the third-order functional difference equation and established the existence of positive solutions for (1.4) under certain conditions. Using the Schauder fixed-point theorem, Yan and Liu [12] studied the existence of a bounded nonoscillatory solution for third order nonlinear delay difference equation and provided also a necessary and sufficient condition for the existence of a bounded nonoscillatory solution of (1.5).

Karpuz and Öcalan [4] discussed the first-order linear partial difference equation: where is a nonnegative sequence and and obtained sufficient conditions under which every solution of (1.6) is oscillatory. Yang and Zhang [14] considered oscillations of the partial difference equation with several nonlinear terms of the form and established some new oscillatory criteria by making use of frequency measures. Wong and Agarwal [10] considered the partial difference equations and offered sufficient conditions for the oscillation of all solutions for (1.8) and (1.9), respectively. Wong [9] established the existence of eventually positive and monotone decreasing solutions for the partial difference inequalities where and are some deviating arguments for .

However, to the best of our knowledge, there is no literature referred to the following third order nonlinear partial difference equation with delays: where , are real sequences with , for , and with

The aim of this paper is to establish three sufficient conditions of the existence of uncountably many bounded positive solutions for (1.11) by using the Banach fixed-point theorem, to suggest some Mann iterative methods with errors for these bounded positive solutions and to compute the error estimates between the bounded positive solutions and the sequences generated by the Mann iterative methods with errors. In order to explain the results presented in this paper, three nontrivial examples are constructed.

Throughout this paper, the forward partial difference operators and are defined by and , respectively the second and third-order partial difference operators are defined by and , respectively. Let , and denote the sets of all positive integers and integers, respectively,

represents the Banach space of all bounded sequences on with the norm

It is not difficult to see that is a bounded closed and convex subset of the Banach space . By a solution of (1.11), we mean a sequence with positive integers and such that (1.11) is satisfied for all and .

Lemma 1.1 (see [15]). Let be nonnegative sequences satisfying the inequality where with , and . Then .

2. Existence of Uncountably Many Bounded Positive Solutions and Mann Iterative Schemes with Errors

Utilizing the Banach fixed-point theorem, we now investigate the existence of uncountably many bounded positive solutions for (1.11), suggest the Mann type iterative schemes with errors and discuss the error estimates between the bounded positive solutions and the sequences generated by the Mann iterative schemes.

Theorem 2.1. Assume that there exists positive constants and , nonnegative constants and , and nonnegative sequences and satisfying Then
(a)for any , there exist , and such that for any , the Mann iterative sequence with errors generated by the scheme: converges to a bounded positive solution of (1.11) and has the following error estimate: where is an arbitrary sequence in , and are any sequences in such that
(b) (1.11) possesses uncountably many bounded positive solutions in .

Proof. First of all we show that (a) holds. Set . It follows from (2.1), (2.2), and (2.5) that there exist , and such that Define a mapping by for each . By employing (2.1)–(2.4) and (2.10)–(2.13), we infer that for and which lead to Consequently, (2.15) means that is a contraction mapping in and it has a unique fixed-point , which together with (2.13) gives that for which yields that for that is, is a bounded positive solution of (1.11) in .
Using (2.6), (2.13), and (2.15), we infer that for any and which yields that That is, (2.7) holds. Consequently, Lemma 1.1 and (2.7)–(2.9) imply that .
Next we show that (b) holds. Let and let . As in the proof of (a), we infer that for each , there exist and satisfying (2.10)–(2.13), where , , , and are replaced by and , respectively, and the mapping has a fixed-point , which is a bounded positive solution of (1.11), that is, In order to show that the set of bounded positive solutions of (1.11) is uncountable, it is sufficient to prove that . It follows from (2.3), (2.10), (2.11), (2.20) that for which implies that that is, . This completes the proof.

Theorem 2.2. Assume that there exist positive constants and , negative constants and and nonnegative sequences and satisfying (2.3)–(2.5) and Then
(a) for any , there exist , and such that for each , the Mann iterative sequence with errors generated by the scheme: converges to a bounded positive solution of (1.11) and has the error estimate (2.7), where is an arbitrary sequence in , and are any sequences in satisfying (2.8) and (2.9);
(b) (1.11) possesses uncountably many bounded positive solutions in .

Proof. First of all we show (a). Taking , from (2.5), (2.23), and (2.24) we infer that there exist , and such that Define a mapping by for each . It follows from (2.3), (2.4), (2.23), (2.24), and (2.26)–(2.29) that for , and which imply that (2.15) holds. Consequently, (2.15) ensures that is a contraction mapping in and it has a unique fixed-point , which together with (2.29) gives that which yields that for which implies that is a bounded positive solution of (1.11) in .
It follows from (2.15), (2.25) and (2.29) that for any and which yields (2.7). Thus Lemma 1.1 and (2.7)–(2.9) ensure that .
Next we show that (b) holds. Let let and . As in the proof of (a), we infer that for each , there exist and satisfying (2.26)–(2.29), where and are replaced by and , respectively, and the mapping has a fixed-point , which is a bounded positive solution of (1.11), that is: In order to show that the set of bounded positive solutions of (1.11) is uncountable, it is sufficient to prove that . It follows from (2.3), (2.26), (2.27), (2.34), and (2.35) that for which implies that that is, . This completes the proof.

Theorem 2.3. Assume that there exist positive constants and , nonnegative constants and , and nonnegative sequences and satisfying (2.3)–(2.5), (2.24) and Then
(a) for any , there exist , and such that for each , the Mann iterative sequence with errors generated by (2.25) converges to a bounded positive solution of (1.11) and has the error estimate (2.7), where