Dynamical systems method (DSM) for unbounded operators

A. G. Ramm

paper

Dynamical systems method (DSM) for unbounded operators

arXiv:math/0404436

Abstract

Let $L$ be an unbounded linear operator in a real Hilbert space $H$ , a generator of $C_{0}$ semigroup, and $g : H \to H$ be a $C_{l oc}^{2}$ nonlinear map. The DSM (dynamical systems method) for solving equF(v):=Lv+gv=0 $co n s i s t so f so l v in g t h e C a u c h y p r o b l e m$ \dot {u}=Φ(t,u) $,$ u(0)=u_0 $, w h er e$ Φ $i s a s u i t ab l eo p er a t or, an d p r o v in g t ha t i)$ \exists u(t) \quad \forall t>0 $, ii)$ \exists u(\infty) $, an d iii)$ F(u(\infty))=0$.