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

Abstract

<p>We obtain necessary and sufficient conditions for the integro-differential inequality</p> <pre>∫^b_α ẋ²(t) dt ≥ γ ∫^b_α q(t) |ẋ(t)x(t) | dt</pre> <p>to be valid with one of the three boundary conditions: x(α) = 0, or x(b) = 0, or x(α) = x(b) = 0. For a power functions q, the best constants γ are found.</p>