Iterative approximations of certain nonlinear optimization, generalized eqilibrium and fixed point problems in Hilbert and Banach spaces.
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Date
2023
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Abstract
In this thesis, in the framework of Banach spaces, we study several iterative methods
for finding the solutions of many important problems in fixed point theory and optimization.
Some of these include the equilibrium problem, monotone variational inclusion
problem, variational inequality problem, split common fixed point problem and split
minimization problem. In addition, we study fixed point problem for some important
and interesting classes of mappings such as nonexpansive mappings, pseudocontractive
mappings, asymptotically demicontractive mappings, quasi-pseudocontractive mappings,
demimetric mappings and multivalued demicontractive mappings in real Hilbert spaces.
Furthermore, we study some other classes of mappings which include the class of Bregman
quasi-nonexpansive mappings and Bregman relatively nonexpansive mappings in real
p-uniformly convex Banach spaces which are also uniformly smooth. Another important
problem considered is the split equality problem. The split equality problem has gained
attention from authors because of its vast applications to real life problems. This problem
is known to contain several other optimization problems as special cases. Based
on its numerous applications, we study a multiple set split equality equilibrium problem
consisting of pseudomonotone bifunctions together with fixed point problem of certain
nonlinear mappings in p-uniformly convex and uniformly smooth Banach spaces. In each
case, we propose and study iterative algorithms for approximating the solutions of these
problems and prove strong convergence theorems under suitable conditions on the control
parameters. In most cases, we incorporate the inertial term which is known to speed up
the convergence rate of iterative schemes. In addition, we employ several efficient iterative
techniques which include the projection and contraction method, alternative regularization
method, modified Halpern’s method, inertial Tseng’s extragradient method and viscosity
approximation method. In all the cases, we design our algorithms in such a way that the
step size does not depend on the knowledge of the Lipschitz constants of the cost operator
or the norm of the bounded linear operators. We present some applications of our results
to solve convex minimization problems, multiple set split variational inequality problem,
image restoration problem, oligopolistic market equilibrium problem, among others. Also,
we present several numerical experiments to demonstrate the efficiency, applicability and
usefulness of our iterative schemes in comparison with several of the existing methods
in the literature. The results obtained in this thesis extend and improve many existing
results in the literature in a unified way.
Description
Doctoral Degree. University of KwaZulu-Natal, Durban.