Solution Methods of Monotone Inclusion Problems, Equilibrium Problems and Fixed Point Problems in Banach Spaces
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Date
2024-09
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Addis Ababa University
Abstract
In this dissertation, we discuss four fixed point approximation methods that can be applied to solve optimization problems, differential equations, variational inequalities and equilibrium problems. In the first main result of the dissertation, we propose an inertial algorithm for solving split equality of monotone inclusion and fixed point of Bregman relatively f-nonexpansive mapping problems in reflexive real Banach spaces and established strong convergence theorems for the algorithm.
Secondly, we establish a strong convergence theorem for approximating a common element of sets of solutions of a finite family of generalized mixed equilibrium problem, sets of semi-fixed points of a finite family of continuous semi-pseudocontractive mappings and sets of solutions of a finite family of variational inequality for a finite family of monotone and L-Lipschitz mappings in Banach spaces.
Thirdly, we constructed and proved a strong convergence of an algorithm for approximating a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of f-fixed points of a finite family of f-pseudocontractive mappings and the set of solutions of a finite family of variational inequality problems for Lipschitz monotone mappings in real reflexive Banach spaces.
In the fourth main result of the dissertation, we introduce an iterative process which converges strongly to a common point of sets of solutions of a finite family of generalized equilibrium problems, fixed points of a finite family of continuous asymptotically quasi-ϕ-nonexpansive mapping in intermediate sense, and zeros of a finite family of γ-inverse strongly monotone operators in uniformly convex and uniformly smooth real Banach space. We give numerical examples to demonstrate the behavior of the convergence of the algorithms analysed in each of the four main results of the thesis.
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Solution Methods, Monotone Inclusion Problems, Equilibrium Problems, Fixed Point Problems, Banach Spaces