Fixed Point Theorems For Set-Valued Maps

Preface In this thesis we provide an introduction to fixed point theory for set-valued maps. It is not our goal in the present work to give an outline of the status quo of fixed point theory with all its newest achievements, but rather to give a thorough overview of the basic results in this discipline. It then should be possible for the reader to better and easier understand the newest developments as generalizations and continuations of the results we present here. After a short recollection about classical fixed point theorems for single-valued maps, we will first give an introduction to the theory of set-valued maps. We will generalize the notion of continuity for such maps and also give an example on how one can reduce problems involving set-valued maps to classical maps. We then show how to achieve fixed point theorems by using two principles: First we will generalize the well known concept of contractivity of a map. Thus we will achieve results similar to the famous Banach Fixed Point Theorem. We will also adress the question of convergence of fixed points for sequences of set-valued contraction. Then we will derive fixed point theorems for maps from geometrical properties. There, we will see generalizations of the important fixed point theorems by Schauder and Tychonoff. We will also give a small excursus to the concept of the KKM-principle and give a few examples of results, that can be achieved that way.