Variational problems for Hölderian functions with free terminal point

Abstract

We develop the new variational calculus introduced in 2011 by J. Cresson and I. Greff, where the classical derivative is substituted by a new complex operator called the scale derivative. In this paper, we consider several nondifferentiable variational problems with free terminal point with and without constraints of first and higher‐order type. Copyright © 2014 John Wiley & Sons, Ltd.

Helmholtz Theorem for Nondifferentiable Hamiltonian Systems in the Framework of Cresson’s Quantum Calculus

“…Cresson introduced in 2005 his quantum calculus on a set of Hölder functions [15]. This calculus attracted attention due to its applications in physics and the calculus of variations and has been further developed by several different authors (see [3,4,14,16] and references therein). Cresson's calculus of 2005 [15] presents, however, some difficulties, and in 2011 Cresson and Greff improved it [17,18].…”

“…For the state of the art on the quantum calculus of variations we refer the reader to the recent book [31]. With respect to Cresson's approach, the quantum calculus of variations is still in its infancy: see [3,5,6,17,18,23]. In [17] nondifferentiable Euler-Lagrange equations are used in the study of PDEs.…”

“…This result is further extended in [24] by considering invariance transformations that also change the time variable, thus obtaining not only the generalized momentum term of [18] but also a new energy term. In [3], nondifferentiable variational problems with a free terminal point, with or without constraints, of first and higher-order, are investigated. Here, we continue to develop Cresson's quantum calculus in obtaining a result for Hamiltonian systems and by considering the so-called inverse problem of the calculus of variations.…”