# American Institute of Mathematical Sciences

March  2022, 15(3): 573-586. doi: 10.3934/dcdss.2021152

## A non-standard class of variational problems of Herglotz type

 Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

* Corresponding author: natalia@ua.pt

Received  February 2020 Revised  February 2021 Published  March 2022 Early access  November 2021

Fund Project: This work is supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020

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 $x$ and its derivative and an unknown functional $z$, 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.

Citation: Natália Martins. A non-standard class of variational problems of Herglotz type. Discrete & Continuous Dynamical Systems - S, 2022, 15 (3) : 573-586. doi: 10.3934/dcdss.2021152
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