A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities - Alan V. Lair (International Journal of Mathematics and Mathematical Sciences)
The author proves that the abstract differential inequality ‖u′(t)−A(t)u(t)‖2≤γ[ω(t)+∫0tω(η)dη] in which the linear operator A(t)=M(t)+N(t), M symmetric and N antisymmetric, is in general unbounded, ω(t)=t−2ψ(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 ∫01t−1ψ(t)dt=∞. The author also shows that the second order differential inequality ‖u″(t)−A(t)u(t)‖2≤γ[μ(t)+∫0tμ(η)dη] in which μ(t)=t−4ψ0(t)‖u(t)‖2+t−2ψ1(t)‖u′(t)‖2 has a nontrivial solution near t=0 such that u(0)=u′(0)=0 if and only if either ∫01t−1ψ0(t)dt=∞ or ∫01t−1ψ1(t)dt=∞. Some mild restrictions are placed on the operators M and N. These results extend earlier uniqueness theorems of Hile and Protter.