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# Zeitschrift für Analysis und ihre Anwendungen

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Volume 34, Issue 1, 2015, pp. 17–26
DOI: 10.4171/ZAA/1526

Published online: 2015-01-08

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

Gheorghe Moroșanu

(1) Central European University, Budapest, Hungary

Consider in a real Hilbert space $H$ the differential equation (inclusion) (E): $p(t)u''(t)+q(t)u'(t) \in Au(t) + f(t)$ a.e. in $(0, \infty)$, with the condition (B): $u(0) = x \in \overline{D(A)}$, where $A :D(A)\subset H\rightarrow H$ 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 $\mathrm{ess} \inf \, \big[{(\frac{q}{p})}^2 + 2(\frac{q}{p})^{\prime}\big] >0$. 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 $a(t)=\exp{\big( \int_0^t \frac{q}{p}\, d\tau \big) }$, showing in particular the behavior of $u$ as $t\rightarrow \infty$.

Keywords: Strong solution, weak solution, existence, uniqueness, asymptotic behavior

Moroșanu Gheorghe: On a Class of Second-Order Differential Inclusions on the Positive Half-Line. Z. Anal. Anwend. 34 (2015), 17-26. doi: 10.4171/ZAA/1526