A study of optimization and fixed point problems in certain geodesic metric spaces.

Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF

On common fixed points approximation of countable families of certain multi-valued maps in hilbert spaces.

Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2017.Fixed point theory and its applications have been widely studied by many researchers. Di erent iterative algorithms have been used extensively to approximate solutions of xed point problems and other related problems such as equilibrium problems, variational in- equality problems, optimization problems and so on. In this dissertation, we rst introduce an iterative algorithm for nding a common solution of multiple-set split equality mixed equilibrium problem and xed point problem for in nite families of generalized ki-strictly pseudo-contractive multi-valued mappings in real Hilbert spaces. Using our iterative algo- rithm, we obtain weak and strong convergence results for approximating a common solution of multiple-set split equality mixed equilibrium problem and xed point problem. As ap- plication, we utilize our result to study the split equality mixed variational inequality and split equality convex minimization problems . Also, we present another iterative algorithm that does not require the knowledge of the oper- ator norm for approximating a common solution of split equilibrium problem and xed point problem for in nite family of multi-valued quasi-nonexpansive mappings in real Hilbert spaces. Using our iterative algorithm, we state and prove a strong convergence result for approximating a common solution of split equilibrium problem and xed point problem for in nite family of multi-valued quasi-nonexpansive mappings in real Hilbert spaces. We apply our result to convex minimization problem and also present a numerical example

Alternative iterative methods for nonexpansive mappings, rates of convergence and application

Alternative iterative methods for a nonexpansive mapping in a Banach space are proposed and proved to be convergent to a common solution to a fixed point problem and a variational inequality. We give rates of asymptotic regularity for such iterations using proof-theoretic techniques. Some applications of the convergence results are presented

Investigations in Hadamard spaces

Kjo tezë e doktoratës hulumton ndërveprimin midis gjeometrisë dhe analizës konvekse në hapësirat Hadamard. E motivuar nga aplikime të shumta të gjeometrisë CAT(0), puna jonë bazohet në rezultatet e shumë autorëve të mëparshëmnë mbi analizën konvekse dhe gjeometrinë në sensin e Alexandrovit. Hetimet tona u përgjigjen disa pyetjeve në teorinë e hapësirave CAT(0) prej të cilave disa janë parashtruar si probleme të hapura në literaturën e fundit. Teza jonë e doktoratës zhvillohet sipas linjave të mëposhtme: 1. Topologjitë e dobëta në hapësirat Hadamard, 2. Konveksifikimi i bashkësive kompakte, 3. Problemi i pemës mesatare në hapësirat e pemëve filogjenetike, 4. Konvergjenca Mosko në hapësirat Hadamard, 5. Operatorët (plotësisht) jo-ekspansivë dhe aplikimet e tyre në hapësirat Hadamard.Diese Doktorarbeit untersucht das Zusammenspiel zwischen Geometrie und konvexer Analyse in Hadamardräumen. Motiviert durch zahlreiche Anwendungen der CAT(0)-Geometrie baut unsere Arbeit auf den Ergebnissen vieler früherer Autoren in der konvexen Analysis und der Alexandrov-Geometrie auf. Unsere Untersuchungen beantworten mehrere Fragen in der Theorie von CAT(0)-Räumen, von denen einige in der neueren Literatur als offene Probleme gestellt wurden. Zusammengefasst entwickelt sich unsere Dissertation in folgende Richtungen: 1. Schwache Topologien in Hadamard-Räumen, 2. Konvexe Hüllen kompakter Mengen, 3. Mittleres Baumproblem in phylogenetischen Baumräumen, 4. Mosco-Konvergenz in Hadamard-Räumen, 5. Fest nichtexpansive Operatoren und ihre Anwendungen in Hadamard-Räumen.This thesis investigates the interplay between geometry and convex analysis in Hadamard spaces. Motivated by numerous applications of CAT(0) geometry, our work builds upon the results in convex analysis and Alexandrov geometry of many previous authors. Our investigations answer several questions in the theory of CAT(0) spaces some of which were posed as open problems in recent literature. In a nutshell our thesis develops along the following lines: 1. Weak topologies in Hadamard spaces, 2. Convex hulls of compact sets, 3. Mean tree problem in phylogenetic tree spaces, 4. Mosco convergence in Hadamard spaces, 5. Firmly nonexpansive operators and their applications in Hadamard spaces

Alternative iterative methods for nonexpansive mappings, rates of convergence and applications

Alternative iterative methods for a nonexpansive mapping in a Banach space are proposed and proved to be convergent to a common solution to a fixed point problem and a variational inequality. We give rates of asymptotic regularity for such iterations using proof-theoretic techniques. Some applications of the convergence results are presented

Approximation methods for solutions of some nonlinear problems in Banach spaces.

Doctor of Philosophy in Mathematics. University of KwaZulu-Natal, Durban 2016.Abstract available in PDF file

Self-adaptive inertial algorithms for approximating solutions of split feasilbility, monotone inclusion, variational inequality and fixed point problems.

Masters Degree. University of KwaZulu-Natal, Durban.In this dissertation, we introduce a self-adaptive hybrid inertial algorithm for approximating a solution of split feasibility problem which also solves a monotone inclusion problem and a fixed point problem in p-uniformly convex and uniformly smooth Banach spaces. We prove a strong convergence theorem for the sequence generated by our algorithm which does not require a prior knowledge of the norm of the bounded linear operator. Numerical examples are given to compare the computational performance of our algorithm with other existing algorithms. Moreover, we present a new iterative algorithm of inertial form for solving Monotone Inclusion Problem (MIP) and common Fixed Point Problem (FPP) of a finite family of demimetric mappings in a real Hilbert space. Motivated by the Armijo line search technique, we incorporate the inertial technique to accelerate the convergence of the proposed method. Under standard and mild assumptions of monotonicity and Lipschitz continuity of the MIP associated mappings, we establish the strong convergence of the iterative algorithm. Some numerical examples are presented to illustrate the performance of our method as well as comparing it with the non-inertial version and some related methods in the literature. Furthermore, we propose a new modified self-adaptive inertial subgradient extragradient algorithm in which the two projections are made onto some half spaces. Moreover, under mild conditions, we obtain a strong convergence of the sequence generated by our proposed algorithm for approximating a common solution of variational inequality problems and common fixed points of a finite family of demicontractive mappings in a real Hilbert space. The main advantages of our algorithm are: strong convergence result obtained without prior knowledge of the Lipschitz constant of the the related monotone operator, the two projections made onto some half-spaces and the inertial technique which speeds up rate of convergence. Finally, we present an application and a numerical example to illustrate the usefulness and applicability of our algorithm