A Banach-Type Fixed Point Theorem via Discrepancy Functionals

Abstract

In this paper we establish a Banach-type fixed point theorem for mappings that contract a general nonnegative functional rather than a metric. The principal assumption is that the functional dominates the underlying metric through a simple lower bound, together with a compatibility condition at limits. Our result provides a common framework that recovers recent fixed point theorems in perturbed metric spaces and for polynomial contractions as particular cases, while showing that neither symmetry nor the triangle inequality is essential for the contraction argument. Consequently, the contractive quantity may be interpreted as a general discrepancy or divergence measure rather than a metric. The results highlight that the essential ingredient behind Banach's contraction principle is not the metric structure itself, but the ability of the contracted functional to control the geometry of the underlying space. This perspective provides a simple and flexible framework for obtaining fixed point results for broad classes of non-metrical contractive quantities.