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Positive Solutions and Mann Iterative Algorithms for a Nonlinear Three-Dimensional Difference System

DOI: 10.1155/2014/954182

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The existence of uncountably many positive solutions and Mann iterative approximations for a nonlinear three-dimensional difference system are proved by using the Banach fixed point theorem. Four illustrative examples are also provided. 1. Introduction In recent years, the oscillation, nonoscillation, asymptotic behavior, existence and multiplicity of solutions, bounded solutions, unbounded solutions, positive solutions, and nonoscillatory solutions for some two- and three-dimensional difference systems have been studied by many authors and a significant number of important results have been found; see [1–12] and the references therein. In order to solve the problem that the Picard iteration fails to converge under some conditions, Mann [13] introduced a modified iteration, which is now called Mann iterative scheme. It is well known that the Mann iterative schemes are often used in the fields of nonlinear differential equations, nonlinear equations, nonlinear mappings, optimization, variational inequalities, nonlinear analysis, and so forth. Note that the difference systems in [1–12] are as follows: where ,?? ,?? and are nonnegative sequences, and is a real sequence with : where , ,?? and are real sequences with , and ?? is a nonnegative sequence: where ,?? , ?? , , , for , and ,?? ,?? ,?? ?? with However, to the best of our knowledge, there exists no result in the literature dealing with the following nonlinear three-dimensional difference system: which is abbreviated as, for convenience, where ,?? ,?? ,?? , , and ,? with The main purpose of this paper is to study solvability and convergence of the Mann iterative schemes for the system (6). Sufficient conditions for the existence of uncountably many positive solutions of the system (6) and convergence of the Mann iterative schemes relative to these positive solutions are provided by utilizing the Banach fixed point theorem. Four illustrative examples are given. 2. Preliminaries Throughout this paper, we assume that is the forward difference operator defined by , , , , , and ,?? , and stand for the sets of all integers, positive integers, and nonnegative integers, respectively, as Let denote the Banach space of all sequences in with norm and put It is easy to see that for each , is a nonempty closed convex subset of the Banach space with norm for each . By a solution of the system (6), we mean a three-dimensional sequence with a positive integer such that (6) holds for all . Let be a nonempty convex subset of a Banach space and let be a mapping. For any given , the sequence defined by where is a


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