Application to fixed point theory of -Geraghty Pata proximal contractions

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 2025; no. 1; pp. 19 - 24
• Main Authors: Alam, Khairul Habib, Rohen, Yumnam, Albargi, Amer Hassan, Hussain, Aftab
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2025; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Approximation; Differential equations; Differential Geometry; Fixed points (mathematics); Fractional calculus; Geraghty Pata proximal contraction; Geraghty Pata type coupled fixed point; Mapping; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Metric space; Theorems; Topology; Weak -Geraghty Pata proximal contraction; α − θ $\alpha -\theta $ -proximal admissibility; Weak; 47H10; 54E50; 46T99; Geraghty Pata proximal contraction; Geraghty Pata type coupled fixed point; 54H10; proximal admissibility
• ISSN: 2730-5422
• DOI: 10.1186/s13663-025-00799-0

In this paper, we introduce the concepts of -Geraghty Pata proximal contractions and their weak ϕ -variants, which generalize and unify several well-known contraction conditions in fixed point theory. By integrating the ideas of Geraghty and Pata contractions with the framework of α − θ -proximal admissibility, we establish best proximity point theorems for both single-valued and multivalued non-self mappings in metric spaces. Furthermore, we extend our results to the context of coupled fixed points and partially ordered metric spaces, thus broadening their applicability. The presented theorems generalize a range of existing results and offer a more flexible setting for analyzing nonlinear problems. We also include a concrete application to fractional differential equations and provide illustrative examples to demonstrate the effectiveness of our results.