Differential inclusions with unbounded right-hand side. Existence and relaxation theorems

A differential inclusion in which the values of the right-hand side are nonconvex closed possibly unbounded sets is considered in a finite-dimensional space. Existence theorems for solutions and a relaxation theorem are proved. Relaxation theorems for a differential inclusion with bounded right-hand side, as a rule, are proved under the Lipschitz condition. In our paper, in the proof of the relaxation theorem for the differential inclusion, we use the notion of $\rho-H$ Lipschitzness instead of the Lipschitzness of a multivalued mapping.