Application to fixed point theory of "Equation missing" -Geraghty Pata proximal contractions

Abstract In this paper, we introduce the concepts of "Equation missing"-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 $\alpha -\theta $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> <mml:mo>−</mml:mo> <mml:mi>θ</mml:mi> </mml:math> -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.