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A note on the theorem on differential inequalities. (English) Zbl 1072.34072

Summary: It is proved that if a linear operator \(\ell:C([a,b];\mathbb R)\rightarrow L([a,b];\mathbb R)\) is nonpositive and for the initial value problem \[ u''(t)=\ell(u)(t)+q(t),\quad u(a)=c_1,\quad u'(a)=c_2, \] the theorem on differential inequalities is valid, then \(\ell\) is an \(a\)-Volterra operator.

MSC:

34K12 Growth, boundedness, comparison of solutions to functional-differential equations
34K06 Linear functional-differential equations
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