A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities

The author proves that the abstract differential inequality ‖ u ′ ( t ) − A ( t ) u ( t ) ‖ 2 ≤ γ [ ω ( t ) + ∫ 0 t ω ( η ) d η ] in which the linear operator A ( t ) = M ( t ) + N ( t ) , M symmetric and N antisymmetric, is in general unbounded, ω ( t ) = t − 2 ψ ( t ) ‖ u ( t ) ‖ 2 + ‖ M ( t ) u ( t ) ‖ ‖ u ( t ) ‖ and γ is a positive constant has a nontrivial solution near t = 0 which vanishes at t = 0 if and only if ∫ 0 1 t − 1 ψ ( t ) d t = ∞ . The author also shows that the second order differential inequality ‖ u ″ ( t ) − A ( t ) u ( t ) ‖ 2 ≤ γ [ μ ( t ) + ∫ 0 t μ ( η ) d η ] in which μ ( t ) = t − 4 ψ 0 ( t ) ‖ u ( t ) ‖ 2 + t − 2 ψ 1 ( t ) ‖ u ′ ( t ) ‖ 2 has a nontrivial solution near t = 0 such that u ( 0 ) = u ′ ( 0 ) = 0 if and only if either ∫ 0 1 t − 1 ψ 0 ( t ) d t = ∞ or ∫ 0 1 t − 1 ψ 1 ( t ) d t = ∞ . Some mild restrictions are placed on the operators M and N . These results extend earlier uniqueness theorems of Hile and Protter.