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 Co semigroup, and let g : H → H be a C̀ 2loc nonlinear map. The DSM (dynamical systems method) for solving equation F(v) := Lv + g(v) = 0 consists of solving the Cauchy problem u̇= Φ(t,u), u(0) = u 0, where Φ is a suitable operator, and proving that i) ∃u(t) ∀ > 0, ii) ∃u(∞), and iii) F(u(∞)) = 0. Conditions on L and g are given which allow one to choose Φ such that i), ii), and iii) hold. ©2005 American Mathematical Society.