Existence results for second-order monotone differential inclusions on the positive half-line

Abstract

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.