Fixed point theorems for general contractive mappings in cone Banach algebras with applications

Abstract

In this paper, we develop a series of fixed point theorems for contractive mappings defined on complete cone metric spaces over Banach algebras. These theorems not only broaden the scope of existing literature but also provide a unified framework for tackling fixed point problems. The theoretical exposition is complemented by several illustrative examples. Furthermore, we demonstrate the practical utility of our established results by proving the existence and uniqueness of solutions for Urysohn integral equations and Caputo fractional differential equations, leveraging the well-known Banach contraction principle. Keywords: Fixed point technique; Banach algebra; Urysohn integral equation; Fractional derivative.