Alien Functional Equations: A Selective Survey of Results

Ger, Roman; Sablik, Maciej

doi:10.1007/978-3-319-61732-9_6

Cited by 17 publications

(6 citation statements)

References 29 publications

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“…Then the problem of alienation was studied by many authors, we cite here just some of them [2][3][4]6,9,11,13,17], for the more detailed study of the history and present state of this problem see [7].…”

Section: Introductionmentioning

confidence: 99%

Alienation of two general linear functional equations

Szostok

2019

Aequat. Math.

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We study the alienation problem for two general linear equations i.e. we compare the solutions of the system of equations n i=1 α i f (p i x + q i y) = 0 m j=1 β j g(s j x + t j y) = 0 with the solutions of the single equation n i=1 α i f (p i x + q i y) = m j=1 β j g(s j x + t j y). To this end we introduce the notion of l-alienation-alienation in the class of monomial functions of order l. We use our results among others to study the alienation properties of two monomial functional equations.

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Section: Introductionmentioning

confidence: 99%

Alienation of two general linear functional equations

Szostok

2019

Aequat. Math.

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“…We end the paper with a remark connecting the results obtained here with the topic called alienation of functional equations (for some details concerning the problem of alienation of functional equations see the survey paper of R. Ger and the third author [12]). which usually has some solutions (depending on n and the constants involved).…”

Section: Examplementioning

confidence: 74%

On a new class of functional equations satisfied by polynomial functions

et al. 2021

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The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi’s result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation $$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$ F ( x + y ) - F ( x ) - F ( y ) = y f ( x ) + x f ( y ) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.

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“…This concept was developed by J. Dhombres in the paper [3]. However, the interested reader should also consult the survey paper Ger-Sablik [5].…”

Section: Further Interpretations and Open Questionsmentioning

confidence: 99%

Remarks on the notion of homo-derivations

Gselmann,

Kiss

2020

Preprint

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The purpose of this paper is to study the (different) notions of homo-derivations. These are additive mappings f of a ring R that also fulfill the identityor (in case of the other notion) the system of equationsOur primary aim is to investigate the above equations without additivity as well as the following Pexiderized equationThe obtained results show that under rather mild assumptions homo-derivations can be fully characterized, even without the additivity assumption.Dedicated to the 70 th birthday of Professor Antal Járai.Remark. Let Q be a ring and let P be a subring of Q. Functions f : P → Q fulfilling the Leibniz rule only, will be termed Leibniz functions.Among derivations one can single out so-called inner derivations, similarly as in the case of automorphisms.Definition 3. Let R be a ring and b ∈ R, then the mapping ad b : R → R defined byis a derivation. A derivation f : R → R is termed to be an inner derivation if there is a b ∈ R so that f = ad b . We say that a derivation is an outer derivation if it is not inner.An another fundamental example for derivations is the following.Remark. Let F be a field, and let in the above definitionbe the ring of polynomials with coefficients from F. For a polynomial pwhere p ′ (x) = n k=1 ka k x k−1 is the derivative of the polynomial p. Then the function f clearly fulfillsClearly, commutative rings admit only trivial inner derivations. At the same time, it not so evident whether commutative rings (or fields) do or do not admit nontrivial outer derivations. To answer this problem partially, here we recall Theorem 14.2.1 from Kuczma [7].Theorem 1. Let K be a field of characteristic zero, let F be a subfield of K, let S be an algebraic base of K over F, if it exists, and let S = ∅ otherwise. Let f : F → K be a derivation. Then, for every function u : S → K, there exists a unique derivation g : K → K such that g| F = f and g| S = u.

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Alien Functional Equations: A Selective Survey of Results

Cited by 17 publications

References 29 publications

Alienation of two general linear functional equations

Alienation of two general linear functional equations

On a new class of functional equations satisfied by polynomial functions

Remarks on the notion of homo-derivations

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