JournalszaaVol. 34, No. 1pp. 17–26

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

  • Gheorghe Moroșanu

    Central European University, Budapest, Hungary
On a Class of Second-Order Differential Inclusions on the Positive Half-Line cover

Abstract

Consider in a real Hilbert space HH the differential equation (inclusion) (E): p(t)u(t)+q(t)u(t)Au(t)+f(t)p(t)u''(t)+q(t)u'(t) \in Au(t) + f(t) a.e. in (0,)(0, \infty), with the condition (B): u(0)=xD(A)u(0) = x \in \overline{D(A)}, where A:D(A)HHA :D(A)\subset H\rightarrow H is a (possibly set-valued) maximal monotone operator whose range contains 00; p,qL(0,)p, q\in L^{\infty}(0,\infty ), such that essinf p>0\mathrm{ess} \inf \ p>0, qp\frac{q}{p} is differentiable a.e., and essinf[(qp)2+2(qp)]>0\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 uu to (E), (B), satisfying a12uL(0,;H)a^{\frac{1}{2}}u \in L^{\infty}(0,\infty ;H) and t12a12uL2(0,;H)t^{\frac{1}{2}}a^{\frac{1}{2}}u^{\prime} \in L^2(0,\infty ;H), where a(t)=exp(0tqpdτ)a(t)=\exp{\big( \int_0^t \frac{q}{p}\, d\tau \big) }, showing in particular the behavior of uu as tt\rightarrow \infty.

Cite this article

Gheorghe Moroșanu, On a Class of Second-Order Differential Inclusions on the Positive Half-Line. Z. Anal. Anwend. 34 (2015), no. 1, pp. 17–26

DOI 10.4171/ZAA/1526