Electron. J. Diff. Eqns., Vol. 2004(2004), No. 04, pp. 1-24.

On Oleck-Opial-Beesack-Troy integro-differential inequalities

Evgeniy I. Bravyi & Sergey S. Gusarenko

Abstract:
We obtain necessary and sufficient conditions for the integro-differential inequality
$$ \int_a^b\dot x^2(t)\,dt\geq\gamma\int_a^bq(t)\,|\dot x(t)x(t)|\,dt $$
to be valid with one of the three boundary conditions: $x(a)=0$, or $x(b)=0$, or $x(a)=x(b)=0$. For a power functions $q$, the best constants $\gamma$ are found.

Submitted October 27, 2003. Published January 2, 2004.
Math Subject Classifications: 34K10, 34B30, 34B05, 41A44, 49J40, 58E35.
Key Words: Integral inequalities, integro-differential inequalities, functional differential equations, variational problems.

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Evgeniy I. Bravyi
Research Centre "Functional Differential Equations"
Perm State Technical University, Perm 614099, Russia
email: bravyi@pi.ccl.ru
Sergey S. Gusarenko
Department of Mathematical Analysis
Perm State University, Perm 614099, Russia
email: gusarenko@psu.ru

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