Electron. J. Diff. Eqns., Vol. 2007(2007), No. 145, pp. 1-18.

Regularization for evolution equations in Hilbert spaces involving monotone operators via the semi-flows method

George L. Karakostas, Konstantina G. Palaska

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.

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George L. Karakostas
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: gkarako@uoi.gr
Konstantina G. Palaska
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece

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