Perturbed Iterative Approximation of Solutions for Nonlinear General -Monotone Operator Equations in Banach Spaces
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Research Article Perturbed Iterative Approximation of Solutions for Nonlinear General A-Monotone Operator Equations in Banach Spaces Xing Wei,1 Heng-you Lan,2 and Xian-jun Zhang2 1
College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066, China 2 Department of Mathematics, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, China Correspondence should be addressed to Heng-you Lan, [email protected] Received 2 January 2009; Accepted 19 March 2009 Recommended by Ram U. Verma We introduce and study a new class of nonlinear general A-monotone operator equations with multivalued operator. By using Alber’s inequalities, Nalder’s results, and the new proximal mapping technique, we construct some new perturbed iterative algorithms with mixed errors for solving the nonlinear general A-monotone operator equations and study the approximationsolvability of the nonlinear operator equations in Banach spaces. The results presented in this paper improve and generalize the corresponding results on strongly monotone quasivariational inclusions and nonlinear implicit quasivariational inclusions. Copyright q 2009 Xing Wei 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 Let X be a real Banach space with the topological dual space of X∗ , let x, y be the pairing ∗ between x ∈ X∗ and y ∈ X, let 2X denote the family of all subsets of X∗ , and let CBX denote the family of all nonempty closed bounded subsets of X. We denote by the z, x zx for ∗ all x ∈ X and z ∈ X∗ . Let f : X → X∗ , T : X → 2X , g : X → X, and A : X → X∗ ∗ be nonlinear operators, and let M : X → 2X be a general A-monotone operator such that gX ∩ dom M· / ∅. We will consider the following nonlinear general A-monotone operator equation with multivalued operator. Find x ∈ X such that u ∈ T x and A A gx − ρ fx u , gx PM
1.1
2
Journal of Inequalities and Applications
A A ρM−1 is the proximal mapping associated with where ∈ 0, 1 is a constant and PM the general A-monotone operator M due to Cui et al. 1. It is easy to see that the problem 1.1 is equivalent to the problem of finding x ∈ X such that
A A gx − ρ fx T x . gx ∈ PM
1.2
Example 1.1. If ≡ 1, then the problem 1.1 is equivalent to finding x ∈ X such that u ∈ T x and A A gx − ρ fx u . gx PM
1.3
A , 1.3 can be written as Based on the definition of the proximal mapping PM
0 ∈ fx u M gx .
1.4
Example 1.2. If T : X → X∗ is a single-valued operator, then a special case of the problem 1.3 is to determine element x ∈ X such that A A gx − ρQx 0, gx − PM
1.5
where Q : X → X∗ is defined by Qx fx T x for all x ∈ X. The problem 1.5 was studied by Xia and Huang 2 when M is a general H-monotone mapping. Further, the problem
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