Abstract

Consider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u"(t) + q(t)u'(t) ∈ Au(t) + f (t) for a.a. t ∈ R+ = [0,∞), subject to the condition u(0) = x ∈ Cl {D(A)}, where A: D(A) ⊂ H → H is a (possibly set-valued) maximal monotone operator whose range contains 0; p, q ∈ L∞(R+), with ess inf p > 0 and either ess inf q > 0 or ess sup q < 0. Eq. (E) has been investigated under various assumptions by Barbu [1]-[2], Brezis [4], V´eron [9]-[10] and others. Recently (see Morosanu [8]), we proved existence and uniqueness of a strong solution to Eq. (E) subject to u(0) = x ∈ D(A) in the weighted space X = L^2_b(R+; H), where b(t) = a(t)/p(t), a(t) = exp(\int_0^t q(s)/p(s) ds), under our weak conditions on p and q (see above). Here we extend this result to the general case x ∈ Cl {D(A)}, while keeping the other assumptions unchanged, thus solving a long standing problem. In addition, the proof of our recent result, which is the starting point of the present paper, is considerably simplified. Furthermore, some qualitative properties of solutions are pointed out, an application to a minimization problem is given, and some open problems are formulated.