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

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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.


Doctoral Degree. University of KwaZulu-Natal, Durban.