Fixed point theorems for nonexpansive mappings in Banach spaces

In this Thesis study, in the first chapter, a preface and some former studies arerelated to main subject. Chapter two contains the necessary definetions andsome standard theorems and examples. Chapter three is focused on obtainingfixed points for single nonexpansive mappings. It is thus interesting to ask awhat ways one can weaken the assumptions on the domain and/or space, and stilobtain a fixed point. We show that a nonexpansive mapping T :C ? C has afixed point if any one of the following is true: Let C be a bounded closed convexset in a normed space X . Let S ? C be compact and let T :C ? C benonexpansive such that S is repeatedly approached by all orbits of T . Thenthere exists at least one fixed point for T in S . Let C be a nonempty boundedclosed convex set in a Banach space X . Suppose that C has normal structureand let S be a weakly compact subset C . Let T :C ? C be nonexpansive forwhich all orbits of T repeatedly approach S . Then there exists at least onefixed point for T in S . Let C be a nonempty bounded closed convex set in areflexive Banach space X . Suppose that C has normal structure. If T :C ? Cis nonexpansive than T has a fixed point. Let C be a nonempty closed convex( not necessarily bounded ) set with normal structre in a reflexive Banach spaceX and let T :C ? C be nonexpansive. Let the {T np} be bounded for somep ?C . Than T has a fixed point. Let C be a nonempty, bounded closed andconvex set in a uniformly convex Banach space X . If T :C ? C is nonexpansivethan T has a fixed point. Let X be a reflexive Banach space. Let C be anonempty bounded closed convex set in X . Suppose that C has a simptoticnormal structure. If T :C ? C be nonexpansive mapping then T has a fixedpoint in C .Since Key Words : Fixed point, Banach space, nonexpansive mapping, weaklycompact, normal structure, asimptotic normed structure,unformly convex