On a Nonlocal Cauchy Problem for Differential Inclusions

where F : J ×Rn → (Rn) is a multivalued map, (Rn) is the family of all subsets of Rn, y0 ∈ Rn, and 0 ≤ t1 < t2 < ··· < tp ≤ b, p ∈ N, ck = 0, k = 1,2, . . . , p. The single-valued case of problem (1.1) was studied by Byszewski [5], in which a new definition of mild solution was introduced. In the same paper, it was remarked that the constants ck, k = 1, . . . , p, from condition (1.1b) can satisfy the inequalities |ck| ≥ 1, k = 1, . . . , p. As pointed out by Byszewski [4], the study of initial value problems with nonlocal conditions is of significance since they have applications in problems in physics and other areas of applied mathematics. The initial value problem (1.1) was studied by Benchohra and Ntouyas [1] in the semilinear case where the right-hand side is assumed to be convex-valued. Here, we drop this restriction and consider problem (1.1) with a nonconvex-valued right-hand side. By using the fixed-point theorem for contraction multivalued maps due to Covitz and Nadler [7] and the Schaefer’s theorem combined with a selection theorem of Bressan