Progressive contractions, measures of non-compactness and quadratic integral equations

Abstract

Classical fixed point theorems often begin with the assumption that we have a mapping P of a non-empty, closed, bounded, convex set G in a Banach space into itself. Then a number of conditions are added which will ensure that there is at least one fixed point in the set G . These fixed point theorems have been very effective with many problems in applied mathematics, particularly for integral equations containing a term ∫ t 0 A(t− s)v(t,s,x(s))ds, because such terms frequently map sets of bounded continuous functions into compact sets. But there is a large and important class of integral equations from applied mathematics containing such a term with a coefficient function f (t,x) which destroys all compactness. Investigators have then turned to Darbo’s fixed point theorem and measures of non-compactness to get a (possibly non-unique) fixed point. In this paper: a) We offer an elementary alternative to measures of non-compactness and Darbo’s theorem by using progressive contractions. This method yields a unique fixed point (unlike Darbo’s theorem) which, in turn, by default yields asymptotic stability as introduced in [1]. b) We lift the growth requirements in both x and t seen using Darbo’s theorem. c) We offer a technique for finding the mapping set G .