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We extend the DuBois-Reymond necessary optimality condition and Noether'sfirst theorem to variational problems of Herglotz type with time delay. Ourresults provide, as corollaries, the DuBois-Reymond necessary optimalitycondition and the first Noether theorem for variational problems with timedelay recently proved in [Numer. Algebra Control Optim. 2 (2012), no. 3,619-630]. Our main result is also a generalization of the first Noether-typetheorem for the generalized variational principle of Herglotz proved in [Topol.Methods Nonlinear Anal. 20 (2002), no. 2, 261-273].

Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem