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

This study explores both theoretical and applied dimensions 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. Here, we derive a novel fixed point theorem concerning graph preserving self-mappings in these spaces. In addition to presenting a non-trivial example validating our main results, we demonstrate two concrete applications: establishing solution uniqueness for non-linear Caputo fractional differential equations, and proving the presence of distinct periodic trajectories in time-dependent non autonomous dynamical systems with time delay.