On a Class of Second-Order Differential Inclusions on the Positive Half-Line

Abstract

Consider in a real Hilbert space $H$ the differential equation (inclusion) (E): $p(t)u^{\prime \prime }(t)+q(t)u^{\prime }(t)\in Au(t)+f(t)$ a.e. in $(0,\infty )$, with the condition (B): $u(0)=x\in \overline{D(A)}$, where $A:D(A)\subset H\to H$ is a (possibly set-valued) maximal monotone operator whose range contains $0$; $p,q\in L^{\infty }(0,\infty )$, such that $\mathrm{ess}\inf p>0$, $\frac{q}{p}$ is differentiable a.e., and $\mathrm{ess}\inf [{(\frac{q}{p})}^2+2(\frac{q}{p}{)}^{\prime }]>0$. We prove existence of a unique (weak or strong) solution $u$ to (E), (B), satisfying $a^{\frac{1}{2}}u\in L^{\infty }(0,\infty ;H)$ and $t^{\frac{1}{2}}{a}^{\frac{1}{2}}{u}^{\prime }\in L^2(0,\infty ;H)$, where $a(t)=\exp \left({\int }_0^t\frac{q}{p}d\tau \right)$, showing in particular the behavior of $u$ as $t\to \infty$.