Regularization for evolution equations in Hilbert spaces involving monotone operators via the semi-flows method
Konstantina G. Palaska, George L. Karakostas · DOAJ (DOAJ: Directory of Open Access Journals) · 2007
In a Hilbert space $H$ consider the equation $$ frac{d}{dt}x(t)+T(t)x(t)+alpha(t)x(t)=f(t),quad tgeq 0, $$ where the family of operators $T(t)$, $tgeq 0$ 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 $xin H$ such that $langle Sx-h, x-z angle=0$, for all $zin S^{-1}(h)$. 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].