Nonlinear Analysis 83 (2013) 69–81 Contents lists available at SciVerse ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Second-order differential equations on R + governed by monotone operators Gheorghe Moroşanu ∗ Department of Mathematics, Central European University, Nador u. 9, 1051 Budapest, Hungary article info Article history: Received 31 December 2012 Accepted 10 January 2013 Communicated by Enzo Mitidieri Dedicated to Professor Jean Mawhin on his 70th anniversary MSC: 47J25 47H05 47H09 Keywords: Monotone operator Existence Uniqueness Nonlinear semigroup Bounded solutions abstract 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 (possibly set-valued) 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. Recall that equation (E) in the case p ≡ 1, q ≡ 0, f ≡ 0, subject to u(0) = x and sup t ≥0 ∥u(t )∥ < ∞, was investigated in the early 1970s by V. Barbu, who derived in particular from his results a definition for the square root of the nonlinear operator A. Subsequently H. Brézis, N.H. Pavel, L. Véron and others have paid 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 = L 2 b (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. This is a classic solution if 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 −  (c 2 /4)I + A. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction Throughout this paper H will be a real Hilbert space with respect to an inner product (·, ·) and the induced norm ∥x∥= (x, x) 1/2 . Consider the nonlinear second-order equation (inclusion) p(t )u ′′ (t ) + q(t )u ′ (t ) ∈ Au(t ) + f (t ), for a.a. t ∈ R + =[0, ∞), (E) with the condition u(0) = x ∈ D(A), (B) where (H1) A: D(A) ⊂ H → H is a maximal monotone operator, with 0 ∈ D(A) and 0 ∈ A0; (H2) p, q ∈ L ∞ (R + ) := L ∞ (R + ; 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 which will be described later. In other words, (H2) says that both p and q are measurable and satisfy 0 < p 0 ≤ p(t ) ≤ p 1 < ∞ for a.a. t ∈ R + , ∗ Tel.: +36 1 3273000x2430. E-mail address: morosanug@ceu.hu. 0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2013.01.007