A study of Caputo fractional differential equations of variable order via Darbo's fixed point theorem and Kuratowski measure of noncompactness

This paper investigated the existence and stability of solutions for boundary value problems involving Caputo fractional differential equations of variable order. Unlike constant-order models, variable-order equations allow the fractional order to change over time, enabling more flexible and accurate modeling of complex systems with evolving dynamics and memory. Using Darbo's fixed point theorem and the Kuratowski measure of noncompactness, we established new existence results for solutions within a Banach space of continuous functions. Our approach treated the variable order as piecewise constant, transforming the problem into a sequence of more manageable constant-order subproblems. Furthermore, we demonstrated Ulam-Hyers stability of the solutions, ensuring that small perturbations in the system did not lead to significant deviations in the results. To validate the theoretical findings, we provided a detailed example supported by numerical simulations. These results offered a solid foundation for future applications in science and engineering where system dynamics evolve over time.