Boundary value problem for differential inclusions in fréchet spaces with multiple solutions of the homogeneous problem

The paper deals with the multivalued boundary value problemx' Є A(t, x)x + F(t, x) for a.a. t Є [a, b], Mx(a)+Nx(b) = 0 in a separable, reflexive Banach space E. The nonlinearity F is weakly upper semicontinuous in x. We prove the existenceof global solutions in the Sobolev space W1,p([a, b], E) with 1 < p < ∞ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneouslinearized problem. An example completes the discussion.