Perimetric Hardy--Rogers and Interpolative Contractions with a Hyers--Ulam Estimate

Abstract

We study self-maps of a metric space that contract the perimeter of atriangle, in the sense of Petrov. We introduce a Hardy--Rogers typecondition that combines, in a single linear inequality, the perimeter ofthe image triangle, the displacements of the three vertices, and thecross-distances between vertices and images. The perimetric contractionsof Petrov and the generalized Kannan, Chatterjea, and Ćirić--Reich--Rus types are recovered as special cases, with theirknown parameter ranges. We prove that a map satisfying the new conditionis weakly Picard with at most two fixed points, in two ways: by a graphiccontraction in the original metric, and by a graphic contraction in theauxiliary metric of Cvetković. We then consider two interpolativeconditions, in which the linear combination is replaced by a geometricmean of two or of three functionals; the two thresholdsthat bound the admissible range are not nested, and for the interpolationparameter above an explicit crossing point the auxiliary-metric routeadmits a strictly larger range of parameters than the linear one, whilebelow it the two are incomparable. A Hyers--Ulam stability estimate isobtained in both a classical two-point and a perimetricthree-point form, whose constant grows without bound as the parametersapproach the boundary of the admissible region. We alsoexhibit a discontinuous mapping that satisfies the generalized Kannancondition despite admitting no two-point Lipschitz constant at all, sothe perimetric route does not simply reduce to a Banach contraction indisguise. Finally, the integral operator attached to a class ofnonlinear measure differential equations studied by Shah and Xia is shownto fall within this setting: uniqueness follows from the underlyingLipschitz estimate, and the perimetric reading recovers the convergence ofthe successive approximations together with the Hyers--Ulam stability ofsolutions.