On existence of extremal solutions of nonlinear functional integral equations in Banach algebras

Abstract

An algebraic fixed point theorem involving the three operators in a Banach algebra is proved using the properties of cones and they are further applied to a certain nonlinear integral equations of mixed type x(t)=k(t,x(μ(t)))+[f(t,x(θ(t)))](q(t)+∫0σ(t)v(t,s)g(s,x(η(s)))ds) for proving the existence of maximal and minimal solutions. Our results include the earlier fixed point theorems of Dhage (1992 and 1999) as special cases with a different but simple method.