Some recent applications of functional equations

Dhombres, Jean

doi:10.1007/978-94-009-6320-7_8

Cited by 31 publications

(47 citation statements)

References 10 publications

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“…Let (G,+) be a set with a binary operation and let f : G → G be a solution of Eq. (2). Then for every n ∈ N and x, y ∈ G the following condition holds:…”

Section: Case G = H = Idmentioning

confidence: 99%

On a modification of some conditional Cauchy equation

Balcerowski¹

2018

Aequat. Math.

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“…Let (G,+) be a set with a binary operation and let f : G → G be a solution of Eq. (2). Then for every n ∈ N and x, y ∈ G the following condition holds:…”

Section: Case G = H = Idmentioning

confidence: 99%

On a modification of some conditional Cauchy equation

Balcerowski¹

2018

Aequat. Math.

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“…His main result reads as follows. Studies were continued by Gudder and Strawther [94][95][96] and Dhombres [55]. The manuscript [94] from 1974 contains a collection of properties and results without proofs, but we can find there the first axiomatization of the orthogonality relation containing the former situations as special cases.…”

Section: Approximate Orthogonalitiesmentioning

confidence: 99%

“…In his book [55], Dhombres states the open problem whether the regularity assumption of a considered function may be omitted in order to derive that in a normed space of dimension not less than 2 the existence of an even nonzero orthogonally additive mapping characterizes inner product spaces.…”

Section: Approximate Orthogonalitiesmentioning

confidence: 99%

Orthogonalities and functional equations

Sikorska

2014

Aequat. Math.

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Abstract. In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations.Mathematics Subject Classification. 39B52, 39B82, 46B20, 47B49, 15A86.

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“…In [2] Dhombres investigated solutions of the equation f (xf (x)) = f (x) 2 in the real domain. In the complex domain the generalized Dhombres functional equation is given by f (zf (z)) = ϕ(f (z)), (1) where the function ϕ is known.…”

Section: Introductionmentioning

confidence: 99%

Local solutions of the generalized Dhombres functional equation in a neighbourhood of infinity

Tomaschek

Reich

2014

ESAIM: Proc.

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Abstract. We study the generalized Dhombres functional equation in the complex domain, we investigate solutions in a neighbourhood of infinity. After some transformations a characterization of those solutions which have a complex number different from zero or infinity as value is given. Moreover the convergence of the solutions and solutions represented as infinite product are figured out.Résumé. Nous considérons leséquations fonctionnelles de Dhombres généralisées dans le plan complexe. Nousétudions les solutions dans un voisinage de l'infini. Après quelques transformations, nous caractérisons les solutions qui ont comme valeur un nombre complexe non nul ou l'infini. De plus, nous considérons le problème de la convergence des solutions et celui de la caractérisation des solutions qui admettent une représentation sous forme de produit infini. Mathematics Subject Classification (2013). 30D05, 39B12, 39B32

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Some recent applications of functional equations

Cited by 31 publications

References 10 publications

On a modification of some conditional Cauchy equation

On a modification of some conditional Cauchy equation

Orthogonalities and functional equations

Local solutions of the generalized Dhombres functional equation in a neighbourhood of infinity

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