Coincidence Point and Common Fixed Point in C^*-algebra Valued R-metric Space using Picard-Jungck Iteration Process with Application in Graph Theory

This paper focuses on establishing the existence and uniqueness of coincidence points and common fixed points in C^*-algebra valued R-metric spaces, utilizing the Picard-Jungck iteration technique. By introducing a novel approach and employing weaker contraction conditions, the study proves results for a pair of self-mappings in C^*-algebra valued R-metric spaces. These results are sufficiently generalized to encompass coincidence and common fixed point results in C^*-algebra valued ordered metric spaces, C^*-algebra valued metric spaces, and general metric spaces. Furthermore, the paper demonstrates the coincidence and common fixed point results within a C^*-algebra valued metric space equipped with a directed graph. The utility of the proved results has been substantiated by ascertaining the existence of the solution of a Caputo fractional differential equation.