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A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities. (English) Zbl 0699.34014
Summary: The author proves that the abstract differential inequality $\| u'(t)-A(t)u(t)\|^ 2\leq \gamma [\omega (t)+\int^{t}_{0}\omega (\eta)d\eta]$ in which the linear operator $$A(t)=M(t)+N(t)$$, M symmetric and N antisymmetric, is in general unbounded, $$\omega (t)=t^{-2}\psi (t)\| u(t)\|^ 2+\| M(t)u(t)\| \quad \| u(t)\|$$ and $$\gamma$$ is a positive constant has a nontrivial solution near $$t=0$$ which vanishes at $$t=0$$ if and only if $$\int^{1}_{0}t^{- 1}\psi (t)dt=\infty$$. The author also shows that the second order differential inequality $\| u''(t)-A(t)u(t)\|^ 2\leq \gamma [\mu (t)+\int^{t}_{0}\mu (\eta)d\eta]$ in which $$\mu (t)=t^{-4}\psi_ 0(t)\| u(t)\|^ 2+t^{-2}\psi_ 1(t)\| u'(t)\|^ 2$$ has a nontrivial solution near $$t=0$$ such that $$u(0)=u'(0)=0$$ if and only if either $$\int^{1}_{0}t^{-1}\psi_ 0(t)dt=\infty$$ or $$\int^{1}_{0}t^{-1}\psi_ 1(t)dt=\infty$$. Some mild restrictions are placed on the operators M and N. These results extend earlier uniqueness theorems of Hile and Protter.

##### MSC:
 34A40 Differential inequalities involving functions of a single real variable 34G10 Linear differential equations in abstract spaces 34G20 Nonlinear differential equations in abstract spaces
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