Some fixed point theorems in analysis

Abstract

The main object of this thesis is to study the Contraction Mapping Principle given by Banach. The principle states: -- Theorem. Let f be a self mapping of a complete metric space X. -- If there exists a real number λ ε (0, 1) such that the condition -- d(f(x), f(y)) < λd(x, y) -- holds for every pair of points x, y ε X, then f has a unique fixed point. -- This theorem has been used extensively in proving existence and uniqueness theorems of differential and integral equations. Some examples have been given to illustrate its applications. -- Several generalizations of Banach's contraction principle have been given in recent years. We have tried to give some further generalizations in Chapter II. -- We have also studied Contractive mappings and Eventually contractive mappings. A few new results have been investigated related to these mappings. -- The converse statements of Banach's contraction principle have been given by a few mathematicians. We have also obtained a few new results on the converse of the Banach contraction principle. -- A few simple but interesting results related to commuting functions and common fixed points have been given. Some new results on commuting polynomials and common fixed points have been obtained.