Fixed Point Theorems for Generalized Contractions in Perturbed 2-Banach Spaces

Abstract - The Banach contraction principle serves as a foundational result in nonlinear analysis and is extensively applied to establish the existence and uniqueness of solutions to mathematical problems, including differential equations and dynamic programming. The classical metric space framework, however, presumes ideal precision in distance measurements. To address the impact of experimental errors, Jleli and Samet recently introduced perturbed metric spaces, in which a perturbation mapping modifies the distance function to account for inherent measurement inaccuracies. Concurrently, the geometric generalization of functional analysis has advanced through the study of 2-Banach spaces, a concept introduced by Gähler and subsequently formalized by White. In these spaces, the traditional notion of distance between two points is replaced by the area determined by three points. This framework provides a multidimensional perspective on fixed-point theory, as recently examined by Ettayb. This paper unifies these research directions by introducing the concept of Perturbed 2-Banach Spaces. This framework facilitates rigorous analysis of two-dimensional geometric structures subject to non-zero perturbation errors. The study extends contraction mapping theory in this context by examining Hardy-Rogers-type contractions, which unify and generalize the contraction conditions of Banach, Kannan, and Reich. Sufficient conditions are established for the existence and uniqueness of fixed points for such mappings in complete perturbed 2-Banach spaces. To illustrate the significance and applicability of these results, examples are provided that differentiate the findings from classical 2-normed space theory, along with a concrete application to the solvability of nonlinear integral equations. Key Words: Perturbed metric space, 2-Banach space, Hardy-Rogers contraction, Fixed point theory, Error analysis, Integral equations.