This paper proposes and studies a new class of nonlinear nonlocal problem driven by a tempered Caputo-type fractional derivative with respect to an arbitrary smooth kernel. The novelty lies in treating a single nonlocal integro-delay setting that simultaneously couples an arbitrary kernel, exponential tempering, a delayed state, a lower-order distributed fractional memory term, and multipoint nonlocal initial data, rather than introducing a new fractional operator. The resulting problem can be viewed as a rigorous well-posedness prototype for hereditary systems with delayed feedback, tempered memory, and nonlocal initialization. First, an equivalent Volterra integral equation is derived. Then, the existence and uniqueness of solutions are obtained by the Banach contraction principle in a suitable Banach space of continuous functions. Next, a Picard successive approximation procedure is introduced and shown to converge uniformly to the unique solution, together with an explicit a priori error estimate. Moreover, a continuous dependence result is proved with respect to perturbations in the initial constants, the multipoint coefficients, and the nonlinear term. Finally, the main results are illustrated with two examples enhanced by graphs of explicit Picard approximations and convergence tables.
Ennaceur et al. (Tue,) studied this question.