General quantum variational calculus

Cruz, Artur M. C. Brito da; Martins, Natália

doi:10.19139/soic.v6i1.467

Cited by 8 publications

(4 citation statements)

References 19 publications

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“…Theory of quantum difference equations helps us to avoid proving results twice, once for Jackson qdifference equations and once for Hahn difference equations (see [8]). For related results and applications to quantum difference operators, see [7]. We denote by…”

Section: Introductionmentioning

confidence: 99%

Quantum Lp-spaces and inequalities of Hardy's type

Hamza,

Alghamdi,

Alasmi

2023

J. Math. Computer Sci.

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Section: Introductionmentioning

confidence: 99%

Quantum Lp-spaces and inequalities of Hardy's type

Hamza,

Alghamdi,

Alasmi

2023

J. Math. Computer Sci.

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“…Let p, q : I ⟶ ℂ be continuous functions at s 0 and satisfy the condition 1 + ðβðtÞ − tÞpðtÞ ≠ 0 for all t ∈ I, then the following properties hold [24]: Journal of Function Spaces Recently, Cardoso [25] investigated the β-Lagrange's identity for the β-Sturm-Liouville eigenvalue problem and proved that it is self-adjoint in L 2 β ð½a, bÞ. For more results in β-calculus, we refer the readers to see [26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

A $β$ -Convolution Theorem Associated with the General Quantum Difference Operator

Shehata

Zafarani

2022

Journal of Function Spaces

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In this paper, we prove some properties of the β -partial derivative. We define the β -convolution of two functions associated with the general quantum difference operator, D β f t = f β t − f t / β t − t ; β is a strictly increasing continuous function. Moreover, we study the shift, the associative law, and the β -differentiability of the β -convolution. Furthermore, we prove the β -convolution theorem and give some applications.

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“…The calculus of variations is a field of mathematical analysis that uses, as the name indicates, variations, which are small changes in functions, to find maxima and minima of the considered functionals: mappings from a set of functions to the real numbers. In the non-Newtonian framework, instead of the classical variations of the form y(⋅) + h(⋅), proposed by Lagrange (1736-1813) and still used nowadays in all recent formulations of the calculus of variations [16][17][18], for example, in the fractional calculus of variations [19,20], quantum variational calculus [21,22] and the calculus of variations on time scales [23,24], we propose here to use "multiplicative variations". More precisely, in contrast with the calculi of variations found in the literature, we show here, for the first time, how to consider variations of the form y(⋅) ⋅ ln h(⋅) .…”

Section: Introductionmentioning

confidence: 99%

On a Non-Newtonian Calculus of Variations

Torres

2021

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The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler–Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler–Lagrange type. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given.

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General quantum variational calculus

Cited by 8 publications

References 19 publications

Quantum Lp-spaces and inequalities of Hardy's type

Quantum Lp-spaces and inequalities of Hardy's type

A $β$ -Convolution Theorem Associated with the General Quantum Difference Operator

On a Non-Newtonian Calculus of Variations

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General quantum variational calculus

Cited by 8 publications

References 19 publications

Quantum Lp-spaces and inequalities of Hardy's type

Quantum Lp-spaces and inequalities of Hardy's type

A β -Convolution Theorem Associated with the General Quantum Difference Operator

On a Non-Newtonian Calculus of Variations

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A $β$ -Convolution Theorem Associated with the General Quantum Difference Operator