Coupled Coincidence Point of $\phi$-Contraction Type $T$-Coupling and $(\phi,\psi)$-Contraction Type Coupling in Metric Spaces

TL;DR

This paper introduces new types of couplings and contraction mappings in metric spaces, proving existence and uniqueness of coupled coincidence and fixed points, extending previous results to non-complete spaces and addressing open problems.

Contribution

It generalizes existing contraction couplings to $ ext{T}$-couplings and $( ext{phi}, ext{psi})$-couplings, establishing new fixed point theorems in broader metric space contexts.

Findings

Proved existence of coupled coincidence points in non-complete metric spaces.

Established uniqueness of strong coupled fixed points for $( ext{phi}, ext{psi})$-contraction couplings.

Provided examples illustrating the applicability of the main theorems.

Abstract

In this research article, we discuss two topics. Firstly, we introduce SCC-Map and $\phi$-contraction type $T$-coupling. By using these two definitions, we generalize $\phi$-contraction type coupling given by H. Aydi et al. [3] to $\phi$-contraction type $T$-coupling and proved the existence theorem of coupled coincidence point for metric spaces which are not complete. Secondly, we attempt to give an answer to an open problem presented by choudhury et al. [7] concerning the investigation of fixed point and related properties for couplings satisfying other type of inequalities. In this direction we prove the existence and uniqueness theorem of strong coupled fixed point for $(\phi,\psi)$-contraction type coupling. We give examples to illustrate our main results.