Introduction to Metric Fixed Point Theory

Abstract

“The theory of fixed points is one of the most powerful tools of modern mathematics” quoted by Felix Browder, gave a new impetus to the modern fixed point theory via the development of nonlinear functional analysis as an active and vital branch of mathematics. The flourishing field of fixed point theory started in the early days of topology (the work of Poincare, Lefschetz–Hopf, and Leray–Schauder). This theory is applied to many areas of current interest in analysis, with topological considerations playing a crucial role, including the relationship with degree theory. For example, the existence problems are usually translated into a fixed point problem like the existence of solutions to elliptic partial differential equations, or the existence of closed periodic orbits in dynamical systems, and more recently the existence of answer sets in logic programming. Fixed point theory of certain important mappings is very interesting in its own right due to their results having constructive proofs and applications in industrial fields such as image processing engineering, physics, computer science, economics and telecommunication.