ON Ψα,β-CONTRACTIVE MAPPINGS AND FIXED POINTS
By
Ranjana Maravi, Manoj Ughade and S. S. Shrivastava
Department of Mathematics, Institute for Excellence in Higher Education (IEHE), Bhopal, Madhya Pradesh, India-462016
Email: ranjanamaravi026@gmail.com, manojhelpyou@gmail.com, sss.math@yahoo.co.in
(Received: February 08, 2026; In format: March 24, 2026; Received: June 20, 2026; Accepted: June 29, 2026)
DOI: https://doi.org/10.58250/jnanabha.2026.56117
Abstract
In this paper we introduce a new displacement–distance control function Ψα,β that nonlinearly couples the interpoint distance d(ξ, η) with the self–displacements d(ξ, T ξ) and d(η, T η). This rational gauge generates a novel contractive mechanism that is simultaneously sensitive to the relative geometry of points and to their deviation from fixedness.Under a natural domination condition linking self–displacements to interpoint distances, we establish a new fixed point theorem for Ψα,β–contractive mappings in complete metric spaces.The results guarantee existence, uniqueness, and global convergence of Picard iterates.Several refined corollaries show that the classical Banach, Reich, and Kannan contraction principles are recovered as special cases of the present framework.An orbit–dominated version of the main theorem is also derived, which extends the applicability of the method to nonuniform and state–dependent contractions that are not covered by standard Lipschitz conditions.As an application, the theory is applied to a nonlinear integral equationnon a space of continuous functions, where the associated integral operator is shown to satisfy the new Ψα,β–contractive condition. The proposed displacement–distance paradigm provides a flexible and unifying approach to fixed point theory and opens new directions for generalizations in nonlinear analysis and operator equations.
2020 Mathematical Sciences Classification: 47H10; 54H25; 54E50.
Keywords and Phrases: metric space; fixed point; control function; displacement; rational contraction; Picard iteration; nonlinear integral equation