Types Of Generalized Equilibrium Problem Fixed Point Iteration Method

In recent years, variational inequality theory has been becomevery effective and powerful tools for studying a wide class of nonlinearproblems arising in many diverse fields of pure mathematics andapplied sciences. As a generalization of variational inequality problem,the equilibrium problem becomes one of the most important and mostuseful problems at present. It had a huge impact on many branches ofpure mathematics and applied mathematics. At the same time, theequilibrium problem has been extensively and intensively applied tomechanics, physics, modern control, nonlinear programming,economical equilibrium and many other areas. It is well known that oneof the most important and difficult problems in variational inequalitytheory is to develop effective and implementable iterative algorithmsfor computing approximate solutions and to analyse the convergenceof the algorithms. Therefore, among the tasks of studying the equilibriumproblem, one important task is the construction of iterative methods forthe common solution of equilibrium problems, variational inequalityproblems and fixed point problems. The aim of this thesis is to investigateiterative methods for the common solution of these problems withapplications in concrete examples. The thesis consists of six chapters.In Chapter1, we first state the research background and give thesummary of the work in this paper.In Chapter2, we are devoted to investigating a new iterativealgorithm for finding a common solution of a generalized equilibriumproblem with perturbed mapping and the fixed point problem of a strictpseudocontractive mapping. Then the strong convergence of thesequence generated by this iterative algorithm is proven.In Chapter3, we are devoted to investigating a new iterativescheme for finding a common solution of two generalized mixed equilibrium problems with perturbed mappings and a common fixedpoint problem of a finite family of strict pseudocontractive mappings.Strong convergence theorem is established in the framework of Hilbertspaces. Applying this result, we also get some new and interestingresults.In Chapter4, we are devoted to investigating a new iterativescheme for finding a common solution of two generalized equilibriumproblems with perturbed mappings and of a common fixed pointproblem of an infinite family of nonexpansive mappings. Strongconvergence of the sequence generated by this iterative scheme isproven.In Chapter5, we are devoted to investigating a new iterativescheme for finding a common solution of two equilibrium problems fortwo infinite families of inverse-strongly monotone mappings and acommon fixed point problems if an infinite family of nonexpansivemappings. Then we prove some strong convergence theorems.In Chapter6, we are devoted to investigating a modification ofthe relaxed extragradient-like method for finding a common solution ofa generalized mixed equilibrium problem with perturbed mapping, ageneral system of generalized nonlinear mixed composite-typeequilibria and a fixed point problem of a strict pseudocontractivemapping, and then obtain a strong convergence theorem. Utilizing thistheorem, we establish some new strong convergence results for fixedpoint problems, variational inequalities, mixed equilibrium problems andsystems of generalized nonlinear mixed composite-type equilibria.