Total asymptotically nonexpansive mappings in uniformly convex metric spaces

We approximate common fixed point of a pair of total asymptotically nonexpansive mappings in the setting of a uniformly convex metric space. The proposed algorithm is computationally simpler than the existing ones in the literature of metric fixed point theory. Our results are new and are valid in Hilbert spaces, CAT(0) spaces and uniformly convex Banach spaces satisfying Opial's property, simultaneously. - 2019, Politechnica University of Bucharest. All rights reserved.Acknowledgments. The authors wish to thank the anonymous reviewer(s) and handling editor for careful reading and valuable suggestions to improve the quality of the paper. The first author would like to acknowledge the support provided by the Deanship of Scientific Research(DSR) at King Fahd University of Petroleum & Minerals (KFUPM) for funding this work through project No. IN151014.Scopu

Demiclosedness Principle for Total Asymptotically Nonexpansive Mappings in C

We prove the demiclosedness principle for a class of mappings which is a generalization of all the forms of nonexpansive, asymptotically nonexpansive, and nearly asymptotically nonexpansive mappings. Moreover, we establish the existence theorem and convergence theorems for modified Ishikawa iterative process in the framework of CAT(0) spaces. Our results generalize, extend, and unify the corresponding results on the topic in the literature

An Implicit Algorithm for a Family of Total Asymptotically Nonexpansive Mappings in CAT(0) Spaces

In this paper, we establish some strong convergence theorems of an implicit algorithm for a finite family of of total asymptotically nonexpansive mappings in the setting of CAT(0) spaces. Our results extend and generalize several recent results from the current existing literatures (see, e.g., [2, 9, 14, 16, 17, 25, 29])

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

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

Quantitative Analysis of Iterative Algorithms in Fixed Point Theory and Convex Optimization

The ongoing program of `proof mining' aims to extract new, quantitative information in the form of bounds and rates from prima facie noneffective proofs in mathematics. In doing so, proof mining and the thesis at hand draws a bridge between mathematics and logic, prompting a lively interaction between mathematical practice and logical theory. This thesis applies proof mining paradigms to several convergence results in fixed point theory and nonlinear optimization, resulting in new complexity information in the form of rates of convergence or rates of metastability. A first -- yet from the proof mining perspective trivial -- example is Banach's famous fixed point theorem; The theorem itself asserts that any contraction mapping on a complete metric space possesses a unique fixed point, and that this fixed point may be approximated by means of the Picard iteration. The proof, on the other hand, exhibits the well known rate of convergence, which can be read off immediately from it. Moreover, the rate is uniform in the mapping, the underlying metric space and the starting point in that it only depends on a Lipschitz constant for the mapping in question and an upper bound on its initial displacement. In other words, for given contraction constant $q$ and positive real number $b$, the rate of convergence is valid for the class of all metric spaces, all mappings on this metric space with contraction constant $q$ and all starting points that are displaced by the operator by a distance of at most $b$. The reason that Banach's fixed point theorem admits such a uniform rate of convergence and that its proof divulges the rate so easily has two reasons: It is constructive and convergence to the limit point is monotone. In general, however, neither is the case. It is precisely in these cases that proof mining, or rather the general logical theorems and tools behind the name, come into play. Under vastly general conditions on the proof that admit non-constructive reasoning in the form of ideal principles and classical logic, so-called metatheorems guarantee the existence of uniform complexity information. For instance, a large part of classical analysis is covered by those metatheorems. Furthermore, the complexity information is not only guaranteed to exist, but can be extracted from the proof at hand. The original proof is moreover transformed into a new proof of the new statement which exhibits the additional complexity information. The new proof then exhibits no trace of its proof-theoretic manipulation and is carried out without reference to any mathematical logic. This has the further advantage that the complexity bound is not only valid, but its proof is easily accessible to the experts of the respective mathematical field and can be published in the corresponding journals

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