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On Oleck-Opial-Beesack-Troy integro-differential inequalities. (English) Zbl 1061.26013

In this interesting paper, the authors find necessary and sufficient conditions for the following integro-differential inequality \[ \int_a^b\dot x^2(t) \,dt\geq\gamma\int_a^bq(t) | \dot x(t)x(t)| \,dt \] to hold with respect to one of the following conditions on the boundary: \(x(a)=0\), or \(x(b)=0\), or \(x(a)=x(b)=0\). The proof ideas are based on the reduction of the problems above to minimization problems for adequate functionals.
For various types of power functions \(q\), the best constants \(\gamma\) are determined.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
34K10 Boundary value problems for functional-differential equations
45J05 Integro-ordinary differential equations
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