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A non-autonomous evolution semi-linear differential system undernon-instantaneous impulses, delays, and perturbed by non-local conditions isstudied. Its piece-wise continuous solutions belong to a finite-dimensionalBanach space. The existence and uniqueness of solutions on the interval$[-r,\tau]$ are obtained by applying Karakostas fixed-point theorem. Furtherresults concerning solution prolongation are developed. An example ispresented, and several remarks on the infinite-dimensional case are included.

On the existence and uniqueness of solutions for non-autonomous semi-linear systems with non-instantaneous impulses, delay, and non-local conditions