We examine the solution to problem fractional with respect derivatives of Caputo. Our approach is mainly based on fractional calculus, particularly fractional derivatives and integrals, together with fixed point theory. First, we apply the Banach fixed point theorem to establish the existence and uniqueness of the solution. Then, we use the Schauder fixed point theorem to prove the existence of at least one solution. We also provide examples with optimization graphs to further demonstrate the validity of our results.