The DuBois–Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler–Lagrange Equation Involving Only Derivatives of Caputo

  • Matheus J. Lazo
  • Delfim F. M. TorresEmail author


Derivatives and integrals of noninteger order were introduced more than three centuries ago but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is currently in fast development. However, all current formulations for the fractional variational calculus fail to give an Euler–Lagrange equation with only Caputo derivatives. In this work, we propose a new approach to the fractional calculus of variations by generalizing the DuBois–Reymond lemma and showing how Euler–Lagrange equations involving only Caputo derivatives can be obtained.


Fractional calculus Fractional calculus of variations DuBois–Reymond lemma Euler–Lagrange equations in integral and differential forms 



The authors are grateful to two referees for their valuable comments and helpful suggestions.


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© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Instituto de Matemática, Estatística e FísicaFURGRio GrandeBrazil
  2. 2.Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal

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