On the quasiuniqueness of solutions of degenerate equations in Hilbert space

In this paper, we study the quasiuniqueness (i.e., f 1 ≐ f 2 if f 1 − f 2 is flat, the function f ( t ) being called flat if, for any K > 0 , t − k f ( 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 D B and B ( t ) = B 1 ( t ) + B 2 ( t ) where ( B 1 ( t ) x , x ) is real and Re ( B 2 ( t ) x , x ) = 0 for any x ∈ D B . Let for any x ∈ D B the following estimate hold: ‖ B 1 x − ( B 1 x , x ) ( x , x ) x ‖ 2 + Re ( B 1 x , B 2 x ) + t ( B 1 ( t ) x , x ) ≥ − C t [ | ( 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 ] . ‖ t d u d t − B ( t ) u ‖ ≤ t ϕ ( 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.