Fixed Point Theorems, Coincidence Point Theorems and Their Applications

Abstract

This study aims to illuminate a general framework for fixed point and coincidence point theorems. Our theorems work with functions defined on ball spaces (X,\cB). This notion provides the minimal structure that is needed to express the basic assumptions which are used in the proofs of such theorems when they are concerned with functions that are contractive in some way. We present a general fixed point theorem which can be seen as the underlying principle of proof for fixed point theorems of Banach and of Prie{\ss}-Crampe and Ribenboim. Also we study two types of general coincidence point theorems and their applications to metric spaces (Theorem due to K.~Goebel) and ultametric spaces (Theorem due to Prie{\ss}-Crampe and Ribenboim). Further, we find an alternative approach to coincidence point theorems. We introduce a general B_x theorem which does not deal with obtaining a coincidence point for two functions f,g directly, but allows a variety of applications. Then we present two coincidence point theorems as its applications. Finally, we introduce three different coincidence point theorems for ultrametric spaces. These theorems are: a special case of one of the general B_x theorem's applications, a coincidence point theorem due to Prie{\ss}-Crampe and Ribenboim, and an ultrametric version of Goebel's theorem. We study the logical relation between these theorems.