On Fixed Point Results for Mixed Power-Type Contractions on Complete Metric% Spaces

Abstract

This paper advances fixed-point theory by introducing and investigating two new classes of contraction mappings in complete metric spaces with bounded diameters. We first present the concept of a one-power-type contraction, defined by the inequality $d(Tx,Ty)<c\,d(x,y)^{\lambda }$, which serves as a fundamental and unifying framework for our study. From this simple yet powerful condition, we derive a streamlined proof for the existence and uniqueness of fixed points, and demonstrate that the previously introduced mixed power-type contraction due to Bataihah and Shatnawi \cite{ab} emerges naturally as a direct corollary.

Moreover, we introduce a distinct and more general class called the Generalized one Kannan-type contraction, governed by$%

d(Tx,Ty)<c\,d(x,y)^{\lambda }+a[d(x,Tx)^{1-\lambda }+d(y,Ty)^{1-\lambda }]$.%

We rigorously establish that this class is not equivalent to the Generalized Mixed Kannan-type contraction studied earlier—a fact illustrated by an explicit counterexample.

Our main theorems guarantee the existence and uniqueness of fixed points under these new contractive conditions, thereby extending and generalizing several classical results in the literature. Finally, the practical relevance of our work is confirmed through an application to a class of nonlinear integral equations, demonstrating the utility of the proposed contractions in analysis.