Dynamical systems method (DSM) for unbounded operators

Abstract:

Let $L$ be an unbounded linear operator in a real Hilbert space $H$, a generator of a $C_0$ semigroup, and let $g:H\to H$ be a $C^2_{loc}$ nonlinear map. The DSM (dynamical systems method) for solving equation $F(v):=Lv+g(v)=0$ consists of solving the Cauchy problem $\dot {u}=\Phi (t,u)$, $u(0)=u_0$, where $\Phi$ is a suitable operator, and proving that i) $\exists u(t) \quad \forall t>0$, ii) $\exists u(\infty )$, and iii) $F(u(\infty ))=0$. Conditions on $L$ and $g$ are given which allow one to choose $\Phi$ such that i), ii), and iii) hold.