On the d-Fenchel–Young distance function with fixed point results

Abstract

The aim of this work is to introduce a d-Fenchel–Young distance function associated with the 5-tuple (α, β, γ, ξ, f), where d is a metric on a nonempty set X,  α : [0, +∞) → [0, +∞) is a monotonically increasing and invertible function from (0, +∞) onto (0, +∞), ξ : [0, +∞) → [0, +∞) is an auxiliary function related to α through the structural condition c := inf_{t > 0} ξ(t) / α^-1(t) > 0,  β,γ : X × X → [0, +∞) satisfy β(x,y) ⩽ γ(x,y) for all distinct x,y ∈ X, and γ is related to a function f : [0, +∞) → [0, +∞). Within this framework, we establish several fundamental properties of the proposed distance and derive fixed point results for self-mappings on metric spaces satisfying μ-comparison contractive conditions with respect to this distance. For suitable choices of (α, β, γ, ξ, f), the proposed framework reduces to the classical metric and the d-Young distance. Consequently, the Banach fixed point theorem and related fixed point results in the d-Young setting are recovered as special cases. Finally, we provide an explicit example for which neither the Banach fixed point theorem nor the existing fixed point results based on the d-Young distance are applicable, whereas our generalized framework guarantees the existence and uniqueness of a fixed point.