Fixed Point Theorems for Orbital Contractions on Directed Graphs in JS-Metric Spaces with Applications

This paper introduces a novel fixed-point framework for orbital contractions within the setting of JS-metric spaces endowed with directed graphs. By relaxing conventional metric conditions and incorporating graph-theoretical structures, we define a new class of operators known as orbitally O-transitive edge-preserving mappings. Utilizing an adaptive control function class H(X), we establish sufficient conditions for the existence of fixed points that are specifically tailored to the iterative paths generated by the mappings. The theoretical results are supported by concrete examples demonstrating that our proposed framework strictly generalizes traditional edge-preserving properties. Furthermore, we illustrate the broad applicability and practical significance of our abstract theorems by establishing existence criteria for two major problems: nonlinear integral equations and nonlinear fractional differential equations involving Riemann–Liouville fractional derivatives with nonlocal integral boundary conditions. By reformulating these boundary value problems into equivalent Volterra-type integral equations, we show that our graph-based contraction approach is highly effective in both classical and fractional analytical settings.