Generalization of Proinov contraction in non-triangular metric space

Abstract

The main purpose of this paper is to find the conditions under which it will be sufficient to establish the existence of a unique fixed point in the non-triangular metric space for the auxiliary functions ψ and ϕ satisfying the contractive condition ψ(d(Sy, Sz)) ≤ ϕ(d(y, z)). In 2020, Proinov [24] has proved some fixed point results using his contractive type conditions in metric
space, and recently in 2022, Erdal Karpanar et. al. [16] introduced the extended Proinov contractions by avoiding the monotone condition on auxiliary function ψ in the metric space. We have generalized these in non-triangular metric space. Further, as an application, we find the existence and uniqueness of a solution of the homogeneous Fredholm integral equation in non-triangular metric space using Proinov contraction. Illustrative examples and numerical calculations are given to support the obtained results which extend some of the theorems in the recent literatures.