# Contribution to iterative algorithms for certain optimization problems and fixed-point problems in Banach spaces.

## Date

2018

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## Abstract

We study the convergence analysis of the xed points set of common solution of a one-
parameter nonexpansive semigroup, the set of solution of constrained convex minimization
problem and the set of solutions of generalized equilibrium problem in a real Hilbert space
using the idea of regularized gradient-projection algorithm. Also, we look at the strong
convergence of a modi ed gradient projection algorithm and forward-backward algorithm
in Hilbert spaces with numerical computations. We also introduce an iterative algorithm
for approximating a common solution of generalized mixed equilibrium problem and xed
point problem in a real re
exive Banach space. Using our algorithm, a strong convergence
theorem is proved concerning an element in the intersection of set of solutions of general-
ized mixed equilibrium problem and the set of solutions of xed point for a nite family
of Bregman strongly nonexpansive mappings.
Moreover, we study and analyze an iterative method for nding a common element of
the xed points set of an in nite family of k-demicontractive mappings which is also a
solution to a zero of the sum of two monotone operators, with one operator being maximal
monotone and the other inverse-strongly monotone. We further extend our study from the
frame work of real Hilbert spaces to more general real smooth and uniformly convex Banach
spaces. In this space, we introduce an iterative algorithm with Meir-Keeler contractions
for nding zeros of the sum of nite families of m-accretive operators and nite family of
inverse strongly accretive operators. We apply our result to the approximation of solution
of certain integro-di erential equation with generalized p-Laplacian operators.
Furthermore, we study the convergence theorem for a new class of split variational inequal-
ity and variational inclusion problem in Hilbert space. We further considered split equality
for minimization problem and xed point sets, split xed point problem and monotone
inclusion problems, split equilibrium problem and xed point set for multivalued map-
pings. All these of our algorithms involve a step-size selected in such a way that their
implementation does not require the computation or an estimate of the spectral radius.
Again, an iterative algorithm that does not require any knowledge of the operator norm
for approximating a solution of split equality equilibrium and xed point problems in
the frame work of p-uniformly convex Banach spaces which are also uniformly smooth is
introduced of which we studied the approximation of solution of split equality generalized
mixed equilibrium problem and xed point problem for right Bregman strongly quasi-
nonexpansive mappings in q-uniformly convex Banach spaces which are also uniformly
smooth. We also study and analyze an iterative algorithm for nding a common element
of the set of the split equality for monotone inclusion problem and xed point of a right
Bregman strongly nonexpansive mapping T in the setting of p-uniformly convex uniformly
smooth Banach spaces. Finally, we present numerical examples of our theorems and apply
our results to study the convex minimization problems and equilibrium problems.

## Description

Doctoral Degree. University of KwaZulu-Natal, Durban.