Some Basic Topological Properties on Non-Newtonian Real Line

Duyar, Cenap; Sagr, Birsen; Oğur, Oğuz

doi:10.9734/bjmcs/2015/17941

Cited by 16 publications

(8 citation statements)

References 11 publications

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“…Corollary 2 already shows the importance of DuBois-Reymond condition to establish constants of motion, that is, quantities, like the one given by the left-hand side of equality (7), that are conserved along the solutions of a problem of the calculus of variations. The constant of motion given by the Erdmann condition is obtained under the assumption that the Lagrangian is autonomous, that is, time-invariant (i.e., there exists a symmetry of the problem under time translations).…”

Section: The Dubois-reymond Conditionmentioning

confidence: 98%

“…Here we follow the notations and the results published open access in [31]. We just recall: the basic four operations, x ⊕ y = x ⋅ y; x ⊖ y = x y ; x ⊙ y = x ln(y) ; x ⊘ y = x 1 ln(y) , y ≠ 1; the fact that (R + , ⊕, ⊙) forms a field; and that with such arithmetic a real analysis is available, together with all fundamental topological properties for the non-Newtonian metric space and a full calculus, including non-Newtonian differential and integral equations [20,25,7,3,13,21,24]. We also refer the reader to the recent book [4].…”

Section: Introductionmentioning

confidence: 99%

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A Non-Newtonian Noether's Symmetry Theorem

Torres

2021

Preprint

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The universal principle obtained by Emmy Noether in 1918, asserts that the invariance of a variational problem with respect to a one-parameter family of symmetry transformations implies the existence of a conserved quantity along the Euler-Lagrange extremals. Here we prove Noether's theorem for the recent non-Newtonian calculus of variations. The proof is based on a new necessary optimality condition of DuBois-Reymond type.

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Section: The Dubois-reymond Conditionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

A Non-Newtonian Noether's Symmetry Theorem

Torres

2021

Preprint

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“…For more on the α-arithmetic, its generalized real analysis, its fundamental topological properties related to non-Newtonian metric spaces and its calculus, including non-Newtonian differential equations and its applications, see [7,33,[44][45][46][47][48]. For gentle, thorough and modern introduction to the subject of non-Newtonian calculi, we also refer the reader to the recent book [49].…”

Section: Integralsmentioning

confidence: 99%

On a Non-Newtonian Calculus of Variations

Torres

2021

Axioms

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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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“…After, Ç akmak and Başar [5] obtained some properties of continuous functions in non-Newtonian calculus. Also, Duyar, Sagır and Ogur [8] studied on some properties of non-Newtonian real line.…”

Section: Introductionmentioning

confidence: 99%

On characterization of non-Newtonian superposition operators in some sequence spaces

Sağır

Erdogan

2019

Filomat

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In this paper, we define a non-Newtonian superposition operator N P f where f : N × R(N) α → R(N) β by N P f (x) = f (k, x k) ∞ k=1 for every non-Newtonian real sequence x = (x k). Chew and Lee [4] have characterized P f : p → 1 and P f : c 0 → 1 for 1 ≤ p < ∞. The purpose of this paper is to generalize these works respect to the non-Newtonian calculus. We characterize N P f : ∞ (N) → 1 (N) , N P f : c 0 (N) → 1 (N) , N P f : c (N) → 1 (N) and N P f : p (N) → 1 (N), respectively. Then we show that such N P f : ∞ (N) → 1 (N) is *-continuous if and only if f (k, .) is *-continuous for every k ∈ N.

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Some Basic Topological Properties on Non-Newtonian Real Line

Cited by 16 publications

References 11 publications

A Non-Newtonian Noether's Symmetry Theorem

A Non-Newtonian Noether's Symmetry Theorem

On a Non-Newtonian Calculus of Variations

On characterization of non-Newtonian superposition operators in some sequence spaces

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