Dual Connected-Image Contractions for a Pair of Mappings in Directed Graph Metric Spaces with Applications

In this paper, we introduce a new class of contractions, called dual connected-image contractions, for a pair of self-mappings on a metric space endowed with a directed graph. This concept extends the notion of connected-image contractions by incorporating the interaction between two mappings through the graph structure. By employing a class of auxiliary functions, we establish existence and uniqueness results for common fixed points under this generalized contractive framework. The proposed approach not only unifies and extends several known results in the literature but also provides greater flexibility in handling nonlinear problems. As an application, the theoretical results are applied to a class of nonlinear fractional differential equations with nonlocal integral boundary conditions, where the existence of solutions is established by reformulating the problems as equivalent integral equations and applying the developed common fixed point framework.