Some Fixed-Point Results via Integral Type α-ψ-Contractions with Fixed-Circle Application

Abstract

One of the key areas of mathematics research is metric fixed-point theory. One type of contractive conditions used in this theory is integral type contractive conditions. In this study, new fixed-point results are obtained by defining an integral type α-ψΩ-contractive condition with
the help of the α-admissible concept. A central advantage of the integral formulation is that the Lebesgue-integrable weight function φ enforces contraction in an averaged sense over distance intervals, making the framework strictly more general than its pointwise counterpart: we construct an explicit example of a mapping satisfying our integral condition that fails the non-integral version of [1]. An application to the fixed-circle problem is also obtained; the integral contraction is shown to be geometrically well-suited to characterising stable circular orbits in a metric space, and the radius μ of the fixed circle Cu0,μ is computable directly from the infimum μ = inf{d(Su, u) : Su ̸=u, u ∈ X}.