International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 1, Pages 129-142
doi:10.1155/S0161171288000183
On the quasiuniqueness of solutions of degenerate equations in Hilbert space
Departamento de Mathematicas del Centro de Investigacion y de Estudios Avanzados del I.P.N. Apartado, Postal 14-740, Mexico, D.F. CP 07000, Mexico
Received 22 December 1981; Revised 15 April 1982
Copyright © 1988 Vladimir Schuchman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.