International Journal of Mathematics and Mathematical SciencesVolume 11, Issue 1, Pages 129-142http://dx.doi.org/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 Hindawi Publishing Corporation. 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., f1f2 if f1f2 is flat, the function f(t) being called flat if, for any K>0, tkf(t)0 as t0) 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 xDB. Let for any xDB the following estimate hold:B1x(B1x,x)(x,x)x2+Re(B1x,B2x)+t(B1(t)x,x)Ct[|(B˙1(t)x,x)|+(x,x)]withC0. If u(t) is a smooth flat solution of the following inequality in the interval tI=(0,1].tdudtB(t)utϕ(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.