Fixed-points of $(\dot{\alpha},\dot{\beta})$-$(\dot{\psi},\dot{\phi})$-$\mathcal{Z}_{\mathcal{C_G}}$-Geraghty type contraction maps in strong $b$-metric spaces with applications

Fixed-points of $(\dot{\alpha},\dot{\beta})$-$(\dot{\psi},\dot{\phi})$-$\mathcal{Z}_{\mathcal{C_G}}$-Geraghty type contraction maps ...

Abstract

This paper introduces the novel notion of an $(\dot{\alpha},\dot{\beta})$-$(\dot{\psi},\dot{\phi})$-$\mathcal{Z}_{\mathcal{C_G}}$-Geraghty type contraction applicable to both single-valued and multi-valued mappings within a strong $b$-metric space. We derive multiple fixed-point theorems for this contraction class. These outcomes generalize several known fixed-point results in the literature, notably including Geraghty’s theorem. Supporting examples and applications for integral equations and functional equations that are arise in dynamic programming are provided to demonstrate the efficacy of the results.