Generalizing the variational theory on time scales to include the delta indefinite integral

TL;DR

This paper extends the calculus of variations on time scales by incorporating delta indefinite integrals into the Lagrangian, providing generalized necessary optimality conditions that encompass previous results and new special cases.

Contribution

It generalizes the variational theory on time scales to include delta indefinite integrals, broadening the scope of optimality conditions.

Findings

Provides generalized Euler-Lagrange type conditions for new variational problems

Includes previous results as special cases

Offers additional interesting optimality conditions

Abstract

We prove necessary optimality conditions of Euler-Lagrange type for generalized problems of the calculus of variations on time scales with a Lagrangian depending not only on the independent variable, an unknown function and its delta derivative, but also on a delta indefinite integral that depends on the unknown function. Such kind of variational problems were considered by Euler himself and have been recently investigated in [Methods Appl. Anal. 15 (2008), no. 4, 427-435]. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases.