Behavior at infinity of solutions of operator differential equations

Abstract

dy(t)/at + Ag(O =0, tE [0, co), (I) where A is a positive self-adjoint operator in a Hilbert space H, (Af, f) >~ O, Vf E D (A), and (.,.) is the scalar product in H. We call a vector-valued function y(t) with values in H a solution of Eq. (i) if: a) y(t) is strongly continuous on [0, ~) and continuously differentiable in (0, ~); b) y (t) E ~ (A) for tE (0, co) (~ (A) is the domain of the operator A) ; c) y(t) satisfies (I).