Analytical approach for coupled Atangana-Baleanu Caputo fractional differential equations with time-dependent delays and integral boundary conditions

Abstract This study explores the existence, uniqueness, and Ulam-Hyers stability of solutions for coupled Atangana-Baleanu-Caputo fractional differential equations with time-dependent delays and integral boundary conditions. These equations, characterized by nonlocal and nonsingular kernels, effectively capture memory and hereditary properties essential for modelling complex systems in science and engineering. To establish the existence of at least one solution, Schaefer’s fixed point theorem is employed, while uniqueness is demonstrated using Banach’s fixed point theorem. A key contribution of this research is the formulation of novel Ulam-Hyers stability criteria, ensuring the small deviation in initial conditions yields bounded variations in the solution, thereby confirming the robustness of the system. An illustrated example is provided to validate the theoretical findings. This work expands the Ulam-Hyers stability framework of the complex group of fractional integro-differential equations, filling in gaps in existing research and improving the use of mathematical models that involve fractional-order dynamics with delays and integral limits.