On contractions involving an auxiliary mapping and fixed point results

Abstract

Two classes of mappings $F: (M,d)\to (M,d)$ satisfying contractions involving an auxiliary mapping $S: M\times M\to M$, are introduced, where $(M,d)$ is a metric space. The first one includes the class of contractions, and the second includes the class of Kannan contractions. For each class, we study the existence and uniqueness of fixed points. Iterative algorithms converging to the fixed points, as well as the size of the convergence errors, are also provided. For particular choices of the auxiliary mapping $S$, we recover the Banach and Kannan fixed point theorems. Some examples illustrating the obtained results and an application to cyclic contractions are given.