Study Of Fixed Point Theorems in Topological Spaces With Application To Fractal

Abstract

Topology which is almost entirely the discovery of present century, was formerly regarded as the study of situation, but it is now generally accepted that it is the study of continuity. It is supposed to be a branch of geometry, dealing with the properties which are unaffected by change in shape and size. And so it is sometimes regarded as volumes written on a branch of geometry. But the most important thing which is set out as a problem in this present research work is that suppose we have a very elastic or approximately perfect elastic sheet of rubber and draw some geometrical figures on it. Without cutting or tearing the sheet if it is allowed to stretch and bend, then it is remarkable that distance between any two points on unstretched form, may be made large at will , but the path between the two points remains such that it does not cross itself whatever mode of stretching or bending is adopted. newlineThe problems associated in the present research work which are taken care of are related to the study of fixed point theorems in semi Hausdorff space, uniform space and Banach space along with studying the applications of fixed point theorem in economics and analyzing the theory of fractals and application of fixed point theory in it and lastly the study of the convergence of Ishikawa iterates as well. newlineFor studying the above said problems, books, journals and monograph related to fixed point, fractal and economics have been studied along with consideration of mathematical reviews, note abstract of recent results, collecting reprints of papers for the research followed by publishing results developed during the tenure of research in suitable journals. newlineIn this subsequent parts of research attempts has been made to introduce the application of weaker condition of continuity in Hausdorff space, thereby generalizing and improving several known results of Semi Hausdorff space. The non-expansive type mappings in quasi uniform space are discussed and a fixed point theorem in quasi uniform space was determined. newline