Convergence Of Equilibrium Problems And Split Variational Inequalities

The fundamental task of nonlinear functional analysis is to solve the solutions of equations generated by the operators in infinite dimension spaces with analysis structures. But the accurate solution is usually very di?cult to obtain.Therefore, finding approximate solutions has been became an important content in the research of nonlinear functional analysis. Constructing e?ective iteration algorithms to investigate the solutions of nonlinear problems has been one kind of main tools. In recent years, scholars constructed some iterative algorithms to approximate the solutions of various nonlinear functional problem. But there exists the shortcomings that they are not easy to compute.So, in this thesis§we use the research experience of predecessors and establish some iterative algorithms which may easily compute to find a common element of the set of solutions of equilibrium problems and the set of fixed points of two di?erent nonlinear mappings in Hilbert space and Banach space. Then, using the idea of split feasibility problem, we study the convergence of the solutions of generalized variational inequality problems and mixed equilibrium problems in two di?erent Hilbert spaces. The results presented in this thesis improves and extends some recent corresponding results. This thesis discuss the following aspects.Firstly, the background and current research state of equilibrium problems and variational problem are introduced in briefly, and the research interests are illustrated?Secondly, an iterative scheme for finding a common element of the set of solutions of classical equilibrium problems and the set of fixed points of k-asymptotical strictly pseudo-nonspreading mappings is introduced in Hilbert spaces, and strong convergence theorems are obtained.Thirdly, by using sunny generalized nonexpansive retraction, an iterative scheme for finding a common element of the set of solutions of mixed equilibrium problem and the set of fixed points of total quasi-φ-asymptotically nonexpansive mapping is introduced in Banach spaces, some weak and strong convergence theorems are obtained.Finally, an iterative scheme is introduced to approximate the solutions of split generalized variational inequality problems and split mixed equilibrium problems in two di?erent Hilbert spaces, some strong convergence theorems are obtained.