Geraghty–Pata–Suzuki-Type Proximal Contractions and Related Coincidence Best Proximity Point Results

Abstract

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an α-θ-Geraghty–Pata–Suzuki-type proximal contraction; an α-θ-generalized Geraghty–Pata–Suzuki-type proximal contraction; and an α-θ-modified Geraghty–Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches.

Keywords

• <i>α</i>-<i>θ</i>-proximal admissible Picard sequence
• <i>α</i>-<i>θ</i>-Geraghty–Pata–Suzuki-type proximal contraction
• <i>α</i>-<i>θ</i>-generalized Geraghty–Pata–Suzuki-type proximal contraction
• <i>α</i>-<i>θ</i>-modified Geraghty–Pata–Suzuki-type proximal contraction
• best proximity point
• the coincidence of best proximity point