Strong convergence theorems for nonlinear operator equations with total quasi-ϕ-asymptotically nonexpansive mappings and applications

• Published in: Fixed point theory and applications (Hindawi Publishing Corporation) Vol. 2012; no. 1; pp. 1 - 16
• Main Authors: Wang, Xiong-rui, Chang, Shih-sen, Wang, Lin, Tang, Yong-Kun, Xu, Yu Guang
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2012; BioMed Central Ltd
• Subjects: Algorithms; Analysis; Applications of Mathematics; Banach space; Classification; Convergence; Differential Geometry; Mapping; Mathematical analysis; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Metal oxide semiconductors; Nonlinearity; Topology; asymptotically nonexpansive mappings; Hammerstian type equation; fixed point; quasi; relatively nonexpansive mapping; variational inequalities; total quasi; generalized mixed equilibrium problems
• ISSN: 1687-1812; 1687-1820; 1687-1812
• DOI: 10.1186/1687-1812-2012-34

The purpose of this article is first to introduce the concept of total quasi-ϕ - asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases, and then by using the hybrid algorithm to introduce a new iterative scheme for finding a common element of set of solutions for a system of generalized mixed equilibrium problems, the set of common fixed points of a countable family of total quasi- ϕ -asymptotically nonexpansive mappings and null spaces of finite family of γ-inverse strongly monotone mappings in a 2-uniformly convex and uniformly smooth real Banach space. As an application, we shall utilize our results to study the iterative solutions of the nonlinear Hammerstian type equation. The results presented in the article improve and extend the corresponding results announced by some authors. AMS (MOS) subject classification : 47H06; 47H09; 47J05; 47J25.