Existence of Fixed Points for Various Mappings in Abstract Spaces

Abstract

Fixed point theory is an important branch of non-linear analysis. Many problems, occurring in different branches of mathematics, such as differential equations, optimization theory and variational analysis, can be converted into the equation T x = x, where T is some non-linear operator defined on a certain space X. Solutions of this equation are called fixed point of T. Fixed point theory can be classified into three major areas: Metric fixed point theory, Topological fixed point theory and Discrete fixed point theory. The principal findings in these areas are Banach s fixed point theorem, Brouwer s fixed point theorem, and Tarski s fixed point theorem respectively. Abstract space is a set of elements satisfying certain axioms. In 1906, the French mathematician Fr´echet introduced the first abstract space, called metric space. In 1922, Polish mathematician Stefan Banach gave the first fixed point theorem for contraction mappings in metric spaces, and this theorem is famous as the Banach contraction principle. This principle states that every contraction self-mapping defined on a complete metric space has a unique fixed point. This result has become one of the most popular and effective tools in solving existence problem in many branches of mathematics. Banach contraction principle has been generalized in several directions. There are two ways to extend or improve this principle. One way is to extend/improve the condition of contraction mappings, and the second approach is to replace complete metric space with a more general abstract space. In the first direction, there are numerous results in the literature proved by Kannan, Chatterjea, Reich, Hardy and Rogers, C´iri´c, Wang et al., Alber and Delabriae, Samet et al., Shahi et al., Wardowski and many more.