Abstract: In this paper, parabolic ape differential inclusions with time dependent are considered and this problem is related to the study of the nonlinear distributed parameter control systems. An existence theorem of mild-solutions is proved, and a propety of the solution set is given. The directions and the results by J. P. Aubin et al. are generalized and improved.

Key words: evolution operator, set-valued map, differential inclusion, mildsolution, fixed-point theorem