Iterative algorithms for approximating solutions of variational inequality problems and monotone inclusion problems.

Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban, 2017.In this work, we introduce and study an iterative algorithm independent of the operator norm for approximating a common solution of split equality variational inequality prob- lem and split equality xed point problem. Using our algorithm, we state and prove a strong convergence theorem for approximating an element in the intersection of the set of solutions of a split equality variational inequality problem and the set of solutions of a split equality xed point problem for demicontractive mappings in real Hilbert spaces. We then considered nite families of split equality variational inequality problems and proposed an iterative algorithm for approximating a common solution of this problem and the multiple-sets split equality xed point problem for countable families of multivalued type-one demicontractive-type mappings in real Hilbert spaces. A strong convergence re- sult of the sequence generated by our proposed algorithm to a solution of this problem was also established. We further extend our study from the frame work of real Hilbert spaces to more general p-uniformly convex Banach spaces which are also uniformly smooth. In this space, we introduce an iterative algorithm and prove a strong convergence theorem for approximating a common solution of split equality monotone inclusion problem and split equality xed point problem for right Bregman strongly nonexpansive mappings. Finally, we presented numerical examples of our theorems and applied our results to study the convex minimization problems and equilibrium problems

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

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

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

Theory and Application of Fixed Point

In the past few decades, several interesting problems have been solved using fixed point theory. In addition to classical ordinary differential equations and integral equation, researchers also focus on fractional differential equations (FDE) and fractional integral equations (FIE). Indeed, FDE and FIE lead to a better understanding of several physical phenomena, which is why such differential equations have been highly appreciated and explored. We also note the importance of distinct abstract spaces, such as quasi-metric, b-metric, symmetric, partial metric, and dislocated metric. Sometimes, one of these spaces is more suitable for a particular application. Fixed point theory techniques in partial metric spaces have been used to solve classical problems of the semantic and domain theory of computer science. This book contains some very recent theoretical results related to some new types of contraction mappings defined in various types of spaces. There are also studies related to applications of the theoretical findings to mathematical models of specific problems, and their approximate computations. In this sense, this book will contribute to the area and provide directions for further developments in fixed point theory and its applications

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

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

Iterative algorithms for approximating solutions of some optimization problems in Hadamard spaces.

Masters Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF.Some text in red

A hybrid iterative scheme for equilibrium problems and fixed point problems of asymptotically k-strict pseudo-contractions

AbstractIn this paper, we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of asymptotically k-strict pseudo-contractions in the setting of real Hilbert spaces. By using our proposed scheme, we get a weak convergence theorem for a finite family of asymptotically k-strict pseudo-contractions and then we modify these algorithm to have strong convergence theorem by using the two hybrid methods in the mathematical programming. Our results improve and extend the recent ones announced by Ceng, et al.’s result [L.C. Ceng, Al-Homidan, Q.H. Ansari and J.C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math. 223 (2009) 967–974] Qin, Cho, Kang, and Shang, [X. Qin, Y. J. Cho, S. M. Kang, and M. Shang, A hybrid iterative scheme for asymptotically k-strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 70 (2009) 1902–1911] and other authors