Graph-Theoretic Fixed Point Theorems with Applications to Fractional and Delayed Operator Equations

In this presentation, we investigate the theoretical and practical aspects of fixed point theory on metric spaces equipped with directed graphs. Aiming to relax global contraction conditions, graph related fixed point theorems provide a powerful setting for solving non-linear differential and integral equations. We establish a fixed point theorem for graph preserving self-mappings on such spaces. In addition to presenting a non-trivial example validating our main results, we demonstrate two concrete applications: the existence and uniqueness of solutions for non-linear Caputo fractional differential equations, and the existence of unique periodic trajectories in non-autonomous dynamical systems with time delay.