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 inclusiony′ ∈ 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.