Existence of solutions for nonconvex second-order differential inclusions in the infinite dimensional space

Abstract

We prove the existence of solutions to the differential inclusion ẍ(t) ∈ F(x(t), ẋ(t)) + ƒ(t, x(t), ẋ(t)), x(0) = x0, ẋ(0) = y0, where ƒ is a Carathéodory function and F with nonconvex values in a Hilbert space such that F(x, y) ⊂ γ(∂g(y)), with g a regular locally Lipschitz function and γ a linear operator.