Hahn's Symmetric Quantum Variational Calculus
  We introduce and develop the Hahn symmetric quantum calculus with applications to the calculus of variations. Namely, we obtain a necessary optimality condition of Euler-Lagrange type and a sufficient optimality condition for variational problems within the context of Hahn's symmetric calculus. Moreover, we show the effectiveness of Leitmann's direct method when applied to Hahn's symmetric variational calculus. Illustrative examples are provided. 
Artur M. C. Brito da Cruz, Natalia Martins, Delfim F. M. Torres
math.OC