Optimal Solution for the System of Differential Inclusion in Hilbert Space

In this paper, we study the existence of the following optimal solution for the system of differential inclusion y′ ∈ Φ(t,y(t)) a.e. t ∈ I = [t0,b] and y(t0) = u2, y′ ∈ Ψ(t,y(t)) a.e. t ∈ I = [t0,b] and y(t0) = u1. in a Hilbert space, where Φ and Ψ are multivalued maps. Our existence result is obtained via selection technique and the best proximity point methods reducing the problem to a differential inclusion.