Banach Contraction Principle and Its Generalizations

Abstract

In 1922, the Polish mathematician Stefan Banach established a remarkable fixed point theorem known as the “Banach Contraction Principle” (BCP) which is one of the most important results of analysis and considered as the main source of metric fixed point theory. It is the most widely applied fixed point result in many branches of mathematics because it requires the structure of complete metric space with contractive condition on the map which is easy to test in this setting. The BCP has been generalized in many different directions. In fact, there is vast amount of literature dealing with extensions/generalizations of this remarkable theorem. In this chapter, it is impossible to cover all the known extensions / generalizations of the BCP. However, an attempt is made to present some extensions of the BCP in which the conclusion is obtained under mild modified conditions and which play important role in the development of metric fixed point theory.