Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 145, pp. 1-18.
Title: Regularization for evolution equations in Hilbert spaces
involving monotone operators via the semi-flows method
Authors: George L. Karakostas (Univ. of Ioannina, Greece)
Konstantina G. Palaska (Univ. of Ioannina, Greece)
Abstract:
In a Hilbert space $H$ consider the equation
$$
\frac{d}{dt}x(t)+T(t)x(t)+\alpha(t)x(t)=f(t),\quad t\geq 0,
$$
where the family of operators $T(t)$, $t\geq 0$ converges in a
certain sense to a monotone operator $S$, the function $\alpha$
vanishes at infinity and the function $f$ converges to a point
$h$. In this paper we provide sufficient conditions that
guarantee the fact that full limiting functions of any solution of
the equation are points of the orthogonality set
$\mathcal{O}(h;S)$ of $S$ at $h$, namely the set of all $x\in H$
such that $\langle Sx-h, x-z\rangle=0$, for all $z\in S^{-1}(h)$.
If the set $\mathcal{O}(h;S)$ is a singleton, then the original
solution converges to a solution of the algebraic equation $Sz=h$.
Our problem is faced by using the semi-flow theory and it extends
to various directions the works [1,12].
Submitted September 10, 2007. Published October 30, 2007.
Math Subject Classifications: 39B42, 34D45, 47H05, 47A52, 70G60.
Key Words: Hilbert spaces; monotone operators; regularization; evolutions;
semi-flows; limiting equations; full limiting functions;
convergence.
An addendum was posted on September 15, 2008. The authors correct some misprints and clarify a uniform boundedness. Please see the last page of this manuscript.