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: urn:x-wiley:01611712:media:ijmm512093:ijmm512093-math-0001 urn:x-wiley:01611712:media:ijmm512093:ijmm512093-math-0002 . If u ( t ) is a smooth flat solution of the following inequality in the interval t ∈ I = (0, 1]. urn:x-wiley:01611712:media:ijmm512093:ijmm512093-math-0003 with non‐negative continuous function ϕ ( t ), then u ( t ) ≡ 0 in I . One example with self‐adjoint B ( t ) is given, too.