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

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.