The contingent epiderivative and the calculus of variations on time scales

Abstract

The calculus of variations on time scales is considered. We propose a new approach to the subject that consists in 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.

Submitted 06/March/2010; revised 12/May/2010; accepted 03/July/2010; for publication in "Optimization---A Journal of Mathematical Programming and Operations Research"

References in corpus (5)

• Noether's Theorem on Time Scales
• Calculus of Variations on Time Scales with Nabla Derivatives
• Higher-Order Calculus of Variations on Time Scales
• Isoperimetric problems on time scales with nabla derivatives
• Necessary and sufficient conditions for local Pareto optimality on time scales

Cited by in corpus (5)

• Generalizing the variational theory on time scales to include the delta indefinite integral
• Optimality conditions for the calculus of variations with higher-order delta derivatives
• The Variational Calculus on Time Scales
• Direct and Inverse Variational Problems on Time Scales: A Survey
• Existence of Three Positive Solutions to Some $p$-Laplacian Boundary Value Problems