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

Bravyi, Evgeniy I.; Gusarenko, Sergey S.

Electron. J. Differ. Equ. 2004, Paper No. 04, 23 p. (2004).

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.

Reviewer: Cristian Vladimirescu (Craiova)

MSC:

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

Keywords:

integral inequalities; integro-differential inequalities; functional differential equations; variational problems