Second-order differential equations on R+ governed by monotone operators

Consider in a real Hilbert space H the differential equation (E) : p(t)u′′(t) + q(t)u′(t) ∈ Au(t) + f(t), for a.a. t ∈ R+ = [0,∞), with the condition u(0) = x ∈ D(A), where A : D(A) ⊂ H → H is a maximal monotone operator, with [0, 0] ∈ A (or, more generally, 0 ∈ R(A)); p, q ∈ L∞(R+), with ess inf p > 0 and either ess inf q > 0 or ess sup q < 0; and f : R+ → H is a given function. Recall that equation (E) in the case p ≡ 1, q ≡ 0, f ≡ 0, subject to u(0) = x and the boundedness condition supt≥0 ‖u(t)‖ < ∞, was investigated in early 1970’s by V. Barbu, who derived in particular from his results a definition for the square root of the nonlinear operator A. Subsequently H. Brezis, N. H. Pavel, L. Véron and others have paid much attention to equation (E). In this paper we prove the existence and uniqueness of the solution to equation (E) subject to u(0) = x ∈ D(A) in the weighted space X = Lb(R+;H), where b(t) = a(t)/p(t), a(t) = exp( ∫ t 0 q(s)/p(s) ds), under our weak assumptions on p and q (see above) and f ∈ X. For x ∈ D(A) we prove the existence of a generalized solution in the case of general variable coefficients p, q and a classic solution in the case p ≡ 1, q ≡ c ∈ R \ {0}. If p ≡ 1, q(t) ≡ c ∈ R \ {0}, f ≡ 0 the solutions give rise to a nonlinear semigroup of contractions. If A is linear its infinitesimal generator G is given by G = − c 2 I − √ c2 4 I + A.