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Wirtinger's Inequalities on Time Scales

Published online by Cambridge University Press:  20 November 2018

Ravi P. Agarwal ,
Victoria Otero-Espinar ,
Kanishka Perera  and
Dolores R. Vivero
Ravi P. Agarwal
Affiliation:
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, U.S.A. e-mail: agarwal@fit.edu
Victoria Otero-Espinar
Affiliation:
Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Galicia, Spain e-mail: lolirv@usc.es e-mail: kperera@fit.edu e-mail: vivioe@usc.es
Kanishka Perera
Affiliation:
Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Galicia, Spain e-mail: lolirv@usc.es e-mail: kperera@fit.edu e-mail: vivioe@usc.es
Dolores R. Vivero
Affiliation:
Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, Galicia, Spain e-mail: lolirv@usc.es e-mail: kperera@fit.edu e-mail: vivioe@usc.es
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Abstract

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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 $\mathbb{T}$. We prove a general inequality for a class of absolutely continuous functions on closed subintervals of an adequate subset of $\mathbb{T}$. By using this expression and by assuming that $\mathbb{T}$ is bounded, we deduce that a general inequality is valid for every absolutely continuous function on $\mathbb{T}$ such that its $\Delta$-derivative belongs to $L_{\Delta }^{2}\,([a,\,b)\,\cap \,\mathbb{T})$ and at most it vanishes on the boundary of $\mathbb{T}$.

Keywords

39A10time scales calculusΔ-integralWirtinger's inequality
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 2 , 01 June 2008 , pp. 161 - 171
Copyright
Copyright © Canadian Mathematical Society 2008

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