Existence Theorems of Solutions for a System of Nonlinear Inclusions with an Application
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Research Article Existence Theorems of Solutions for a System of Nonlinear Inclusions with an Application Ke-Qing Wu, Nan-Jing Huang, and Jen-Chih Yao Received 7 June 2006; Revised 3 November 2006; Accepted 18 December 2006 Recommended by H. Bevan Thompson
By using the iterative technique and Nadler’s theorem, we construct a new iterative algorithm for solving a system of nonlinear inclusions in Banach spaces. We prove some new existence results of solutions for the system of nonlinear inclusions and discuss the convergence of the sequences generated by the algorithm. As an application, we show the existence of solution for a system of functional equations arising in dynamic programming of multistage decision processes. Copyright © 2007 Ke-Qing Wu 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. 1. Introduction It is well known that the iterative technique is a very important method for dealing with many nonlinear problems (see, e.g., [1–4]). Let E be a real Banach space, let X be a nonempty subset of E, and let A,B : X × X → E be two nonlinear mappings. Chang and Guo [5] introduced and studied the following nonlinear problem in Banach spaces: A(u,u) = u,
B(u,u) = u,
(1.1)
which has been used to study many kinds of differential and integral equations in Banach spaces. If A = B, then problem (1.1) reduces to the problem considered by Guo and Lakshmikantham [1]. On the other hand, Huang et al. [6] introduced and studied the problem of finding u ∈ X, x ∈ Su, and y ∈ Tu such that A(y,x) = u,
(1.2)
2
Journal of Inequalities and Applications
where A : X × X → X is a nonlinear mapping and S,T : X → 2X are two set-valued mappings. They constructed an iterative algorithm for solving this problem and gave an application to the problem of the general Bellman functional equation arising in dynamic programming. Let A,B : X × X → E be two nonlinear mappings, let g : X → E be a nonlinear mapping, and let S,T : X → 2X be two set-valued mappings. Motivated by above works, in this paper, we study the following system of nonlinear inclusions problem of finding u ∈ X, x ∈ Su, and y ∈ Tu such that A(y,x) = gu,
B(x, y) = gu.
(1.3)
It is easy to see that the problem (1.3) is equivalent to the following problem: find u ∈ X such that
gu ∈ A Tu,Su ,
gu ∈ B Su,Tu ,
(1.4)
which was considered by Huang and Fang [7] when g is an identity mapping. It is well known that problem (1.3) includes a number of variational inequalities (inclusions) and equilibrium problems as special cases (see, e.g, [8–10] and the references therein). By using the iterative technique and Nadler’s theorem [11], we construct a new algorithm for solving the system of nonlinear inclusions problem (1.3) in Banach spaces. We prove the existence of solution for the system of nonlinear inclusions problem (1.3) and the convergence of the sequences generated by the algorithm. As an a
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