A non-standard class of variational problems of Herglotz type

In this paper, we extend the variational problem of Herglotz considering the case where the Lagrangian depends not only on the independent variable, an unknown function \begin{document}$ x $\end{document} and its derivative and an unknown functional \begin{document}$ z $\end{document} , but also on the end points conditions and a real parameter. Herglotz's problems of calculus of variations of this type cannot be solved using the standard theory. Main results of this paper are necessary optimality condition of Euler-Lagrange type, natural boundary conditions and the Dubois-Reymond condition for our non-standard variational problem of Herglotz type. We also prove a necessary optimality condition that arises as a consequence of the Lagrangian dependence of the parameter. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases. In addition, two examples are given in order to illustrate our results.