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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 470181, 16 pages
http://dx.doi.org/10.1155/2014/470181
Research Article

On Positive Solutions and Mann Iterative Schemes of a Third Order Difference Equation

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 14 October 2013; Accepted 16 December 2013; Published 28 January 2014

Academic Editor: Zhi-Bo Huang

Copyright © 2014 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 of uncountably many positive solutions and convergence of the Mann iterative schemes for a third order nonlinear neutral delay difference equation are proved. Six examples are given to illustrate the results presented in this paper.

1. Introduction and Preliminaries

Recently, many researchers studied the oscillation, nonoscillation, and existence of solutions for linear and nonlinear second and third order difference equations and systems see, for example, [123] and the references cited therein. By means of the Reccati transformation techniques, Saker [18] discussed the third order difference equation and presented some sufficient conditions which ensure that all solutions are to be oscillatory or tend to zero. Utilizing the Schauder fixed point theorem, Yan and Liu [22] proved the existence of a bounded nonoscillatory solution for the third order difference equation Agarwal [2] established the oscillatory and asymptotic properties for the third order nonlinear difference equation Andruch-Sobiło and Migda [4] studied the third order linear difference equation of neutral type and obtained sufficient conditions which ensure that all solutions of the equation are oscillatory. Grace and Hamedani [6] discussed the difference equation and gave some new criteria for the oscillation of all solutions and all bounded solutions.

Our goal is to discuss solvability and convergence of the Mann iterative schemes for the following third order nonlinear neutral delay difference equation: where, , , , , , , , , and By employing the Banach fixed point theorem and some new techniques, we establish the existence of uncountably many positive solutions of (6), conceive a few Mann iterative schemes for approximating these positive solutions, and prove their convergence and the error estimates. Six nontrivial examples are included.

Throughout this paper, we assume thatis the forward difference operator defined by, ,and denote the sets of nonnegative integers and positive integers, respectively, and represents the Banach space of all real sequences onwith norm It is easy to see thatis a closed and convex subset of. By a solution of (6), we mean a sequencewith a positive integersuch that (6) holds for all.

Lemma 1. Letbe a nonnegative sequence and.(i)If then.(ii)If then.

Proof. Note that that is, As in the proof of (10), we infer that which implies that This completes the proof.

2. Uncountably Many Positive Solutions and Mann Iterative Schemes

In this section, using the Banach fixed point theorem and Mann iterative schemes, we establish the existence of uncountably many positive solutions of (6), prove convergence of the Mann iterative schemes relative to these positive solutions, and compute the error estimates between the Mann iterative schemes and the positive solutions.

Theorem 2. Assume that there exist two constantsandwithand four nonnegative sequences, , and satisfying Then one has the following.(a)For any, there existandsuch that, for each, the Mann iterative sequence generated by the scheme converges to a positive solutionof (6) with and has the following error estimate: where is an arbitrary sequence insuch that (b)Equation (6) possesses uncountably many positive solutions in.

Proof. Firstly, we show that (a) holds. Put. It follows from (16)~(18) that there existandsatisfying Define a mapping by for each. In light of (14), (15), (22), (23), and (25), we obtain that for each which yield that which implies thatis a contraction in. The Banach fixed point theorem and (27) ensure thathas a unique fixed point; that is, which mean that which yields that which gives that which together with (24) implies thatis a positive solution of (6) in. Note that which guarantees that. It follows from (19), (22), (24), (25), and (27) that for anyand which implies that That is, (20) holds. Thus Lemma 1, (20), and (21) guarantee that .
Next we show that (b) holds. Let, and. As in the proof of (a), we deduce similarly that, for each, there exist constantsandand a mappingsatisfying (22)(27), where, andare replaced by, , and, respectively, and the mappinghas a fixed point, which is a positive solution of (6) inwith. It follows that which together with (14) and (20) means that for which yields that that is,This completes the proof.

Theorem 3. Assume that there exist two constantsandwithand four nonnegative sequences, , , andsatisfying (14), (15), and Then one has the following.(a)For any, there existandsuch that, for each, the Mann iterative sequencegenerated by the scheme converges to a positive solutionof (6) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).(b)Equation (6) possesses uncountably many positive solutions in.

Proof. Let. It follows from (38)(40) that there existandsatisfying Define a mappingby for each. Using (14), (15), (42), (43), and (45), we get that for eachand which imply (27). The rest of the proof is similar to the proof of Theorem 2 and is omitted. This completes the proof.

Theorem 4. Assume that there exist three constants, , andwithand four nonnegative sequences, , andsatisfying (14), (15), (38), (39) and Then one has the following.(a)For any, there existandsuch that, for any, the Mann iterative sequence generated by the scheme converges to a positive solutionof (6) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).(b)Equation (6) possesses uncountably many positive solutions in.

Proof. Put. It follows from (38), (39), and (47) that there existandsatisfying Define a mappingby for each. In view of (14), (15), and (49) and (50), we obtain that for eachand which imply (27). The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 5. Assume that there exist constants, , andwithand four nonnegative sequences, , , and satisfying (14), (15), (38), (39), and Then one has the following.(a)For any, there existandsuch that, for any, the Mann iterative sequencegenerated by (48) converges to a positive solutionof (6) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).(b)Equation (6) possesses uncountably many positive solutions in.

Proof. Put. It follows from (38), (39), and (52) that there existandsatisfying Define a mappingby (50). By virtue of (15), (50), (53), and (55), we infer that for all, and That is, (27) holds. The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 6. Assume that there exist constants, , and withand four nonnegative sequences, , , and satisfying (14), (15), (38), (39), and Then one has the following.(a)For any, there existandsuch that, for any , the Mann iterative sequence generated by the scheme converges to a positive solutionof (6) withand has the error estimate (20), whereis an arbitrary sequence insatisfying (21).(b)Equation (6) possesses uncountably many positive solutions in.

Proof. Put. It follows from (38), (39), and (57) that there existandsatisfying Define a mappingby for each. In view of (14), (15), and (59)(62), we obtain that for each, and which imply (27). The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 7. Assume that there exist constants, , andwithand four nonnegative sequences, , , and satisfying (14), (15), (38), (39) and Then one has the following.(a)For any, there existandsuch that, for any, the Mann iterative sequencegenerated by the scheme converges to a positive solution of (6) with and has the error estimate (20), where is an arbitrary sequence insatisfying (21).(b)Equation (6) possesses uncountably many positive solutions in.

Proof. Put. It follows from (38), (39), and (64) that there existandsatisfying