# A study of optimization problems and fixed point iterations in Banach Spaces.

## Date

2019

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## Abstract

Abstract
The study of optimization and xed point problems has remained as an attractive area
of research due to its paramount importance in several areas of mathematics and other
sciences. It constitutes a beautiful mixture of pure and applied analysis, topology, geometry,
statistics and mechanics. It has also found several applications in solving nonlinear
phenomena arising in diverse elds such as engineering, economics, biology, management
science, transportation, game theory, physics, computer tomography, etc.
In this thesis, we present some inertial iterative schemes with strong convergence theorems
for approximating solutions of certain optimization problems in real Hilbert spaces.
We further analyze a parallel combination extragradient method for nite family of pseudomonotone
equilibrium problem and xed point of demi-contractive mappings in real Hilbert
spaces. By combining Mann and Krasnolselskii methods with inertial extrapolation term,
we propose a new iterative method which converges strongly to a common solution of split
variational inclusion problem and equilibrium problem with para-monotone equilibria.
More so, we introduce a projection-contraction method for approximating solution of split
generalized equilibrium problem in real Hilbert space. We show that our projectioncontraction
method converges at a linear rate of convergence. Moreover, we extend the
study of projection methods for solving variational inequality problem to re
exive Banach
spaces. We introduce a projection algorithm and prove a strong convergence theorem
for approximating solution of variational inequality problem in re
exive Banach spaces
and give an application of our result to approximating solution of equilibrium problem in
re
exive Banach space without prior knowledge of operator norms.
Furthermore, we introduce a totally relaxed subgradient extragradient method for approximating
a common solution of variational inequality and xed point of quasi-nonexpansive
mapping in a 2-uniformly convex and uniformly smooth Banach space. We also study the
approximation of solution of variational inequality problem using projection-contraction
algorithm in real Hilbert space. Then, we extend the study of split equality monotone
inclusion problem to p-uniformly convex and uniformly smooth real Banach spaces.
Ultimately, we consider the approximation of common xed points of k-strictly pseudocontractive
mappings in a 2-uniformly smooth real Banach space. We introduce a class
of N-generalized Bregman nonspreading mappings and propose an iterative method for
approximating the common xed points of this kind of mappings which is also a solution
of equilibrium problem in a re
exive Banach space. Numerical experiments are presented
to demonstrate the e ciency and performance of our algorithms in comparison with other
existing algorithms in literature. We also achieve strong convergence results using our
algorithms for approximating solutions of the underlying problems in each case.

## Description

Doctoral Degree. University of KwaZulu-Natal, Durban.