  Consider in a real Hilbert space $H$ the differential equation (inclusion)
$(E)$: $p(t)u^{\prime \prime}(t)+q(t)u^{\prime}(t)\in Au(t)+f(t)$ for a.a.
$t>0$, with the condition $(B)$: $u(0)=x \in \overline{D(A)}$, where $A\colon
D(A)\subset H\rightarrow H$ is a (possibly set-valued) maximal monotone
operator whose range contains $0$; $p,q\in L^{\infty}(0,\infty)$, with
$\mathrm{ess} \inf \ p>0$ and $q^+ \in L^1(0, \infty)$. Existence in the
non-homogeneous case has received less attention. On the other hand, much
attention has been paid by several authors to the asymptotic behavior of
bounded solutions (if they exist) as $t\rightarrow \infty $, both in the
homogeneous and nonhomogeneous case. Recently, I established jointly with H.
Khatibzadeh [Set-Valued Var. Anal. DOI 10.1007/s11228-013-0270-3] the existence
of (weak and strong) bounded solutions to $(E)$, $(B)$, in the case $p \equiv
1$, $q\equiv 0$, under the optimal condition $tf(t) \in L^1(0,\infty ; H)$. In
this paper, this result is extended to the general case of non-constant
functions $p, \, q$ satisfying the mild conditions above, thus compensating for
the lack of existence theory for such kind of second order problems. Note that
our results open up the possibility to apply Lions' method of artificial
viscosity towards approximating the solutions of some nonlinear parabolic and
hyperbolic problems, as shown in the last section of the paper.
