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

Abstract

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

Keywords

• Integral inequalities
• integro-differential inequalities
• functional differential equations
• variational problems.