Wirtinger's Inequalities on Time Scales
Printed: Jun 2008
Ravi P. Agarwal
Dolores R. Vivero
This paper is devoted to the study of Wirtinger-type
inequalities for the Lebesgue $\Delta$-integral on an arbitrary time scale $\T$.
We prove a general inequality for a class of absolutely continuous
functions on closed subintervals of an adequate subset of $\T$.
By using this expression and by assuming that $\T$ is bounded,
we deduce that
a general inequality is valid for every absolutely continuous function on $\T$
such that its $\Delta$-derivative belongs to $L_\Delta^2([a,b)\cap\T)$ and at most it vanishes
on the boundary of $\T$.
time scales calculus, $\Delta$-integral, Wirtinger's inequality
39A10 - Difference equations, additive