On a Class of Measures of Noncompactness in Banach Algebras and Their Application to Nonlinear Integral Equations

Abstract

We introduce a class of measures of noncompactness in Banach algebras satisfying certain condition and we prove a fixed point theorem for the product of two operators being contractions with respect to such measures of noncompactess. We also indicate measures of noncompactness in some Banach function algebras which satisfy the mentioned condition. The obtained results are applied to prove a few theorems on the existence of solutions of nonlinear integral equations in Banach algebras. Some characterizations of solutions of considered integral equations are also derived.