Fixed point, game and selection theory: from the hairy ball theorem to a non hair-pulling conversation

Abstract

This PHD dissertation develops fixed point theory issues. Fixed point theory is at the heart of the nonlinear analysis. One amongst the first and famous results is Brouwer's fixed point theorem (1910). Two years after, Brouwer proved another topological result. The first chapter of this thesis is dedicated to this result. Namely, the hairy ball theorem. We introduced an equivalent version of the hairy ball theorem in the form of a `fixed point theorem'. In this way, ensuring the proof of this equivalent theorem gives a new insight on the hairy ball theorem. On the other hand, the theorem of Brouwer was extended in many ways. The first extension was made by kakutani who generalised the result for multifunctions. More precisely, he ensured the exsitence of fixed point theorem for upper semicontinuous correspondence. Yet, since the upper semicontinuity and the lower semincontinuity are two notions generalizing the classical continuity for single valued functions, then one can ask the following question: What about the class of lower semicontinuous correspondences? This class plays a major role in selection theory and chapter 2 of this thesis is related to Michael's selection theorem. We proved a selection theorem related to one result of Michael. One of the most important tools in order to prove our result is the introduction of a new concept of convex analysis: The peeling concept. Finally, it is known that Kakutani's fixed point theorem has many applications. Namely, in game theory. Indeed, it has a vital connection with the Nash equilibrium theorem. The last chapter of this thesis presents an unusual application of Brouwer's fixed point theorem and the Nash equilibrium theorem in order to modelise a non-cooperative game. We have two agents, endowed with their own individual convex categorisations, negotiate over the construction of a common convex categorisation. The common convex categorisation emerges as the (unique) equilibrium of the game