In this paper, we study a new nonlinear differential problem with nonlocal integral conditions and convergent series. The problem involves three fractional order operators: Riemann–Liouville integral, Caputo, and Riemann–Liouville derivatives. The introduced Caputo derivatives in the problem have neither the commutativity property nor the semi-group one. The considered problem can be seen as a more general case for the problem considered in the recent paper: Existence and Mittag–Leffler–Ulam Stability Results for Duffing Type Problem Involving Sequential Fractional Derivatives published in the International Journal of Applied and Computational Mathematics (2022).

We begin by proving a first auxiliary integral result. Then, we demonstrate an existence and uniqueness result by applying the Banach contraction principle. Also, we establish a new existence result using the Leray–Schauder fixed point theorem. We end our paper by presenting some illustrative examples.