Fixed Point Theory in Modular Function Spaces

Abstract

Fixed point theory has been a flourishing area of mathematical research for decades, because of its many diverse applications. It is a combination of geometry, topology and analysis. This theory has been discovered as a very influential and essential mechanism in learning of nonlinear phenomena. It has a lot of applications in almost all branches of mathematical sciences, for example, proving the existence of solutions of ODE’S, PDE’S, integral equations, system of linear equations, closed orbit of dynamical systems and of equilibria in economics.

In particular fixed point techniques have been applied in such different fields as economics, engineering biology, physics and chemistry. It has very fruitful applications in control theory, game theory, category theory, mathematical economics, mathematical physics, functional equations, integral equations, mathematical chemistry, mathematical biology, W* algebra, functional analysis and many other areas.

The concept of fixed point plays a key role in analysis. Also, fixed point theorems are mainly used in existence theory of random differential equations, numerical methods like Newton-Rapshon method and Picard’s existence theorem and in other related areas. Fixed point theorems based on the consideration of order have importance in algebra, the theory of automata, mathematical linguistics, linear functional analysis, approximation theory and theory of critical points. Fixed point theorems play a key role in applications of variational inequalities, linear inequalities, optimization techniques and approximation theory. Thus the theory of fixed point has been studied by many researchers extensively.

From the perspective of different settings, methods and applications, the fixed point theory is typically separated into three main branches:

(i) Metric fixed point theory. (ii) Topological fixed point theory. (iii) Discrete fixed point theory.

In history the boundary lines between the aforesaid three branches was defined by the creation of the following three main theorems:

(i) Banach’s Fixed Point Theorem (1922). (ii) Brouwer’s Fixed Point Theorem (1912). (iii) Tarski’s Fixed Point Theorem (1955).

Fixed point theory in modular function spaces is closely related to the metric theory, in that it provides modular equivalents of norm and metric concepts. Modular spaces are extensions of the classical Lebesgue and Orlicz spaces, and in many instances conditions cast in this framework are more natural and more easily verified than their metric analogs.