L'Hopital-Type Rules for Monotonicity with Application to Quantum Calculus

Martins, Natália; Torres, Delfim F. M.

doi:10.48550/arxiv.1011.4880

Cited by 3 publications

(4 citation statements)

References 27 publications

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“…Let X(q, p) T = (X q (q, p), X p (q, p)) be a vector field. The differential operator O ,X given by (33) If such properties of an underlying calculus exist, then the Helmholtz conditions will always be the same up to some conditions on the working space of functions.…”

Section: Nondifferentiable Helmholtz Problemmentioning

confidence: 99%

“…Several types of quantum calculus are available in the literature, including Jackson's quantum calculus [28,34], Hahn's quantum calculus [10,29,30], the time-scale q-calculus [8,33], the power quantum calculus [2], and the symmetric quantum calculus [11,12,13]. Cresson introduced in 2005 his quantum calculus on a set of Hölder functions [15].…”

Section: Introductionmentioning

confidence: 99%

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

Pierret¹,

Torres

2016

Discrete Dynamics in Nature and Society

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We derive the Helmholtz theorem for nondifferentiable Hamiltonian systems in the framework of Cresson's quantum calculus. Precisely, we give a theorem characterizing nondifferentiable equations, admitting a Hamiltonian formulation. Moreover, in the affirmative case, we give the associated Hamiltonian. 2010 Mathematics Subject Classification. 49N45, 70S05. Key words and phrases. Cresson's quantum calculus, nondifferentiable calculus of variations, nondifferentiable Hamiltonian systems, inverse problem of the calculus of variations.

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Section: Nondifferentiable Helmholtz Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Pierret¹,

Torres

2016

Discrete Dynamics in Nature and Society

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“…Quantum calculus, sometimes called "calculus without limits", is analogous to traditional infinitesimal calculus without the notion of limits [17]. Several dialects of quantum calculus are available in the literature, including Jackson's quantum calculus [17,22], Hahn's quantum calculus [6,18,19], the time-scale q-calculus [5,21], the power quantum calculus [1], and the symmetric quantum calculi [7][8][9]. Here we consider the recent quantum calculus of Cresson. Motivated by Nottale's theory of scale relativity without the hypothesis of spacetime differentiability [23,24], Cresson introduced in 2005 his quantum calculus on a set of Hölder functions [11].…”

Section: Introductionmentioning

confidence: 99%

Noether's Theorem with Momentum and Energy Terms for Cresson's Quantum Variational Problems

Frederico,

Torres

2014

Preprint

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“…Introduction. Due to its many applications, quantum operators are recently subject to an increase number of investigations [24][25][26]. The use of quantum differential operators, instead of classical derivatives, is useful because they allow to deal with sets of nondifferentiable functions [4,10].…”

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confidence: 99%

Hahn's symmetric quantum variational calculus

Cruz¹,

Martins²,

Torres³

2013

Numerical Algebra, Control &Amp; Optimization

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Abstract. 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. 1.Introduction. Due to its many applications, quantum operators are recently subject to an increase number of investigations [24][25][26]. The use of quantum differential operators, instead of classical derivatives, is useful because they allow to deal with sets of nondifferentiable functions [4,10]. Applications include several fields of physics, such as cosmic strings and black holes [27], quantum mechanics [12,29], nuclear and high energy physics [18], just to mention a few. In particular, the q-symmetric quantum calculus has applications in quantum mechanics [17].In 1949, Hahn introduced his quantum difference operator [13], which is a generalization of the quantum q-difference operator defined by Jackson [14]. However, only in 2009, Aldwoah [1] defined the inverse of Hahn's difference operator, and short after, Malinowska and Torres [24] introduced and investigated the Hahn quantum variational calculus. For a deep understanding of quantum calculus, we refer the reader to [2,5,6,11,15,16] and references therein.For a fixed q ∈ ]0, 1[ and an ω ≥ 0, we introduce here the Hahn symmetric difference operator of function f at point t = ω 1 − q bỹ

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L'Hopital-Type Rules for Monotonicity with Application to Quantum Calculus

Abstract: We prove new l'Hôpital rules for monotonicity valid on an arbitrary time scale. Both delta and nabla monotonic l'Hôpital rules are obtained. As an example of application, we give some new upper and lower bounds for the exponential function of quantum calculus restricted to the q-scale.

Cited by 3 publications

References 27 publications

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

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

Noether's Theorem with Momentum and Energy Terms for Cresson's Quantum Variational Problems

Hahn's symmetric quantum variational calculus

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