On the quasiuniqueness of solutions of degenerate equations in Hilbert space

[摘要] In this paper, we study the quasiuniqueness (i.e.,f1≐f2iff1−f2is flat, the functionf(t)being called flat if, for anyK>0,t−kf(t)→0ast→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. LetB(t)be a linear operator with domainDBandB(t)=B1(t)+B2(t)where(B1(t)x,x)is real andRe(B2(t)x,x)=0for anyx∈DB. Let for anyx∈DBthe 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. Ifu(t)is a smooth flat solution of the following inequality in the intervalt∈I=(0,1].‖tdudt−B(t)u‖≤tϕ(t)‖u(t)‖with non-negative continuous functionϕ(t), thenu(t)≡0inI. One example with self-adjointB(t)is given, too.

[学科分类] 数学（综合）