A Note on Fixed-Point Theory

Abstract

In this paper, we narrate in brief the origin and step-by-step development of the fixed-point theory which finds its primary applications in functional analysis. It is a sub-branch of the functional analytic theory in which geometric conditions on the mappings and/or underlying spaces play a crucial role. It is also a major branch of nonlinear functional analysis with close ties to Banach space geometry. In this paper, our main focus is on the single-valued functions. First, we define and describe the fixed-point theorem with the help of definitions and examples. Later, we divide the development of fixed-point theory in four major areas. We explained the history of fixed-point theorem in our divided areas with examples and well-proved theorems.