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Wirtinger's Inequalities on Time Scales

Open Access article
 Printed: Jun 2008
  • Ravi P. Agarwal
  • Victoria Otero-Espinar
  • Kanishka Perera
  • Dolores R. Vivero
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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 $\T$.
Keywords: time scales calculus, $\Delta$-integral, Wirtinger's inequality time scales calculus, $\Delta$-integral, Wirtinger's inequality
MSC Classifications: 39A10 show english descriptions Difference equations, additive 39A10 - Difference equations, additive

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