Graph-based coincidence point theory in complete S_b-metric spaces with applications to integral equations and homotopy analysis

Abstract

In this work, we develop a comprehensive coincidence point theory for a hybrid pair of mappings consisting of a single valued operator and a multivalued operator within the frame work of complete $S_b$ metric spaces endowed with a directed graph structure. A novel contractive condition, governed simultaneously by the graph structure and the Hausdorff  $S_b$ distance, is introduced. 

Under this setting, we establish sufficient conditions ensuring the existence as well as uniqueness of coincidence points for edge-preserving mappings. Furthermore, under a weak compatibility assumption, the coincidence point is shown to reduce to a unique common fixed point. 

The presented results extend and unify several well known fixed point and coincidence point theorems in metric, $b$-metric and graph-based settings. To illustrate the applicability of the theory, appropriate examples are constructed. In addition, the obtained results are applied to demonstrate the existence and uniqueness of solutions for a class of nonlinear Volterra type integral equations. A homotopy invariance result is also derived ensuring the persistence of solutions under continuous deformations