Study Of Lipschitzian and Non Lipschitzian Mappings With Application In Fractals

Abstract

The achievements in the field of functional analysis in the last few decades are mainly the developments of new imaginative ways to use the fundamental tools of functional analysis, either in theory where they had not been applied before or in the construction of more powerful methods to handle functional analysis. This thesis can be looked at as a study in a part of pure mathematics and application of the same in the Fractal theory. The goal of the first chapter is to introduce the reader to the history and noteworthy contributions to the field of work discussed in this thesis. Other than the first Chapter each chapter is divided into three parts. The first part gives an insight for the problem discussed in the Chapter and its relevance. Second part gives some useful definitions and Lemmas required for understanding the main result of the Chapter and the third part comprises the main result. Every Chapter ends with the conclusion that can be drawn from the main part of the Chapter. The purpose of Chapter-3 is to establish the strong convergence of modified generalized Ishikawa iterative sequence for a pair of quasi-nonexpansive and nearly asymptotically nonexpansive mappings in a uniformly convex Banach space. newlineII newlineIn Chapter-4, the strong convergence theorem for the sequence of iterates for a finite family of nearly asymptotically nonexpansive mapping has been discussed using an implicit iteration process. In Chapter-5, the equivalence of convergence of G-iteration scheme with that of the convergence of Mann-Ishikawa iteration schemes for Lipschitzian and non-Lipschitzian mapping has been shown. Chapter-6 deals with the theory behind fractal dimensions. Also some examples of fractal dimension and application of calculation of fractal dimension in the real world has been discussed. Iterated function system has been studied in Chapter-7 and attempt has been to find ways for improving and generalizing it. These results can then be used to solve inverse problem. Thus Kannan Iterated Function System has been introduce