Positivity and stability with decaying rate for delay nonlocal evolution systems involving multi-kernel

This paper explores the qualitative properties of solutions to nonlocal delay systems with time-dependent coefficients, focusing on nonlocal derivatives associated with a sequence of distinct kernels. Both uni-kernel and multi-kernel cases are considered. We first introduce a natural framework for mild solutions and examine their solvability and positivity using fixed-point theory. For the uni-kernel case, we analyze the long-term behavior of mild solutions, providing explicit convergence rates by utilizing the properties of relaxation functions and novel comparison techniques. Extended results for nonlinear Lipschitz cases are also presented. In the multi-kernel setting, we establish the stability of mild solutions with decay rates under sufficient conditions that capture interactions among singular kernels. Several examples illustrate and substantiate the theoretical findings, significantly broadening and refining previous results.