We consider the well-posedness of the abstract CAUCHY problem for the doubly nonlinear evolution inclusion equation of second order given by { u ''(t) + partial derivative Psi (u '(t)) + B(t, u(t)) is not an element of f (t), t is an element of (0, T), T > 0, { u(0) = u(0), u '(0) = v(0) where the function u takes values in a real separable HILBERT space, denoted by H. Here, u(0) lies in H, nu(0) is in the intersection dom(partial derivative Psi)<overline>boolean AND dom(Psi), and f belongs to L-2(0, T; H). The functional Psi: H -> (-infinity, +infinity] is assumed to be proper, lower semicontinuous, and convex. Additionally, the nonlinear operator B :[0, T] x H -> H is assumed to satisfy either a global or a local LIPSCHITZ condition. In the case where B satisfies a global Lipschitz condition, we can establish the existence and uniqueness of strong solutions u belonging to H-2(0, T-& lowast; ; H). Furthermore, these solutions continuously depend on the data. We derive these results using the theory of nonlinear semigroups combined with the BANACH fixed-point theorem. On the other hand, when B satisfies a local LIPSCHITZ condition, we can guarantee the existence of strong local solutions.
