Iterative schemes for approximating common solutions of certain optimization and fixed point problems in Hilbert spaces.
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Date
2021
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Abstract
In this dissertation, we introduce a shrinking projection method of an inertial type with
self-adaptive step size for finding a common element of the set of solutions of Split Gen-
eralized Equilibrium Problem (SGEP) and the set of common fixed points of a countable
family of nonexpansive multivalued mappings in real Hilbert spaces. The self-adaptive step
size incorporated helps to overcome the difficulty of having to compute the operator norm
while the inertial term accelerates the rate of convergence of the propose algorithm. Under
standard and mild conditions, we prove a strong convergence theorem for the sequence
generated by the proposed algorithm and obtain some consequent results. We apply our
result to solve Split Mixed Variational Inequality Problem (SMVIP) and Split Minimiza-
tion Problem (SMP), and present numerical examples to illustrate the performance of
our algorithm in comparison with other existing algorithms. Moreover, we investigate the
problem of finding common solutions of Equilibrium Problem (EP), Variational Inclusion
Problem (VIP)and Fixed Point Problem (FPP) for an infinite family of strict pseudo-
contractive mappings. We propose an iterative scheme which combines inertial technique
with viscosity method for approximating common solutions of these problems in Hilbert
spaces. Under mild conditions, we prove a strong theorem for the proposed algorithm and
apply our results to approximate the solutions of other optimization problems. Finally,
we present a numerical example to demonstrate the efficiency of our algorithm in comparison with other existing methods in the literature. Our results improve and complement
contemporary results in the literature in this direction.
Description
Masters Degree. University of KwaZulu-Natal, Durban.