Fixed Point Theorems for Reich-Type Contractions in Perturbed Metric Spaces

The Banach contraction principle [2], a cornerstone of nonlinear analysis, provides a robust framework for establishing the existence and uniqueness of solutions to various mathematical problems, ranging from differential equations to dynamic programming. However, the standard metric structure often fails to adequately model experimental environments where distance measurements are inherently subject to non-zero errors or perturbations. Addressing this limitation, Jleli and Samet [1] recently introduced the topological structure of perturbed metric spaces, where the distance function is modified by a perturbation mapping to account for such inaccuracies. While their work successfully established a Banach-type fixed point theorem within this framework, the contractive condition imposed remains restrictive, excluding a large class of discontinuous or non-linear mappings. In this paper, we significantly extend the scope of perturbed metric fixed point theory by introducing the concept of Reich-type perturbed contractions. This new class of mappings generalizes the classical contraction conditions proposed by Reich [4], which historically unified the independent results of Banach [2] and Kannan [3]. We establish sufficient conditions for the existence and uniqueness of fixed points for such mappings in the setting of complete perturbed metric spaces. Our main results demonstrate that the fixed point theorems obtained by Jleli and Samet [1] are specific corollaries of our work. Furthermore, we explore the topological relationship between perturbed metrics and partial metric spaces as discussed by Matthews [7], providing a broader context for our findings. To validate the applicability of our theoretical results, we provide illustrative examples and a concrete application to the solvability of nonlinear Fredholm integral equations.