### Abstract:

As this thesis is concerned with fixed point theorems for non-expansive mappings, the thesis will be confined to functional analysis, where the interaction will mainly be with related areas of mathematics such as General Topology, the theory of Locally Convex Spaces and Measure Theory. Non-expansive mapping refers to a mapping which maps a metric space into itself so that it does not increase distances. Some authors use the term contraction mapping instead of the term non-expensive mapping. The classical Banach fixed point theorem for contraction mappings gives rise to applications in finding solutions to differential equations and integral equations. The necessary definitions and some standard theorems and examples are given. The study examines obtaining fixed points for single non-expansive mappings. The special class of metric spaces known as hyperconvex spaces are considered. An example of a weakly compact convex set which lacks the fixed point property for non-expansive mappings is given. Markov and Kakutani have given a weak extension to Schauder's fixed point theorem by showing the existence of common fixed points for families of commuting continuous mappings which are affine. A theorem of De Marr gives a proper extension to Schauder's fixed point theorem for non-expansive mappings, by assuming the domain to be compact and convex. On the other hand, it is demonstrated that assuming weak compactness instead of compactness may not yield a common fixed point for a family of non-expansive mappings in a Banach space.