On the quasiuniqueness of solutions of degenerate equations in Hilbert space

Abstract

In this paper, we study the quasiuniqueness (i.e., f1≐f2 if f1−f2 is flat, the function f(t) being called flat if, for any K>0, t−kf(t)→0 as t→0) for ordinary differential equations in Hilbert space. The case of inequalities is studied, too.The most important result of this paper is this:THEOREM 3. Let B(t) be a linear operator with domain DB and B(t)=B1(t)+B2(t) where (B1(t)x,x) is real and Re(B2(t)x,x)=0 for any x∈DB. Let for any x∈DB the following estimate hold:‖B1x−(B1x,x)(x,x)x‖2+Re(B1x,B2x)+t(B1(t)x,x)≥−Ct[|(B˙1(t)x,x)|+(x,x)]                                 with   C≥0. If u(t) is a smooth flat solution of the following inequality in the interval t∈I=(0,1].‖tdudt−B(t)u‖≤tϕ(t)‖u(t)‖with non-negative continuous function ϕ(t), then u(t)≡0 in I. One example with self-adjoint B(t) is given, too.