A new method with regularization for solving split variational inequality problem in real Hilbert spaces.

Abstract

The concept of the optimization problem, fixed point theory and its application constitute the nucleus of nonlinear analysis, which is a major branch of mathematics. Optimization theory, fixed point theory and its applications have a wide range of application in practically every field of science, particularly mathematical sciences. The theory of optimization and fixed point have received great attention from authors around the world and these areas will continue to receive such great attention. The theory has been well developed by well-known researchers in these areas. However, there are still a lot of work to be done. The goal of this thesis is to advance the theory of optimization and fixed point in the framework of Hilbert and Banach spaces. The substance of this thesis is separated into two parts. The research efforts of the first part of this thesis (Chapter 3 to Chapter 6) has to do with introducing some new iterative methods for approximating the solution of a variational inequality problems, split variational inequality problems, equilibrium problems, split monotone variational inclusion problem, split generalized mixed equilibrium problem and fixed point problems in the framework of a Hilbert and Banach spaces. In addition, we introduce a new class of bilevel problem in the framework of real Hilbert spaces and a new regularization technique, and inertial terms for approximating solutions of split bilevel variational inequality problems. Furthermore, we establish that the proposed iterative methods converges strongly to the solution of the aforementioned problems as the case may be. Then, we present some numerical experiments to show the efficiency and applicability of our proposed methods in comparison with other state-of-the-art iterative methods in the literature. The second part (Chapter 7) of this thesis deals with developing iterative algorithms and introducing some nonlinear mappings in the framework of the Hilbert and Banach spaces. First, we present a modified (improved) generalized Miteration with the inertial technique for three quasi-nonexpansive multivalued mappings in a real Hilbert space. In addition, we present some fixed point results for a general class of nonexpansive mappings in the framework of the Banach space and also proposed a new iterative scheme for approximating the fixed point of this class of mappings in the framework of uniformly convex Banach spaces. Finally, we apply our convergence results to certain optimization problems, integral equations, and we present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature.