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

Abstract

We obtain necessary and sufficient conditions for the integro-differential inequality ∫b a ẋ2(t) dt ≥ γ ∫b a q(t) |ẋ(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 γ are found.