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 (possibly set-valued) maximal monotone operator, with [0,0]∈A (or, more generally, 0∈R(A)); p,q∈L∞(R+), with essinfp>0 and either essinfq>0 or esssupq<0. Recall that equation (E) in the case p≡1,q≡0,f≡0, subject to u(0)=x and supt≥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=Lb2(R+;H), where b(t)=a(t)/p(t), a(t)=exp(∫0tq(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−(c2/4)I+A.