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Abstract
The calculus of variations on time scales is considered. We propose a new approach to the subject that consists of applying a differentiation tool called the contingent epiderivative. It is shown that the contingent epiderivative applied to the calculus of variations on time scales is very useful: it allows to unify the delta and nabla approaches previously considered in the literature. Generalized versions of the Euler–Lagrange necessary optimality conditions are obtained, both for the basic problem of the calculus of variations and isoperimetric problems. As particular cases one gets the recent delta and nabla results.
Acknowledgements
The authors are supported by the R&D unit Center for Research and Development in Mathematics and Applications (CIDMA) via The Portuguese Foundation for Science and Technology (FCT) and the European Community fund FEDER/POCI 2010. Girejko is also supported by the FCT post-doc fellowship SFRH/BPD/48439/2008; Malinowska by Białystok University of Technology, via a project of the Polish Ministry of Science and Higher Education Wsparcie miedzynarodowej mobilnosci naukowcow and Torres by the project UTAustin/MAT/0057/2008.