Generalized Pexider Equation on an Open Domain

Chudziak, Małgorzata; Sobek, Barbara

doi:10.1007/s00025-016-0573-4

Cited by 4 publications

(4 citation statements)

References 10 publications

(13 reference statements)

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“…Then, the claim follows by [4,Lemma 2.4]. Putting together Claims 1, 2, and 3, we conclude the proof.…”

Section: Introductionsupporting

confidence: 54%

“…Conversely, let us assume that A : X → Y is a group homomorphism which satisfies (3) and f : K → Y is a function defined by (4). If i≤n β i = 1 and b = 0 then f (x) = A(x) so that…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…also [1,Theorem 4,p.80]. Moreover, see [5,Theorem 5] and [2, 3,4] for related results. The proof of Theorem 1.6 follows in Section 2.…”

Section: Introductionmentioning

confidence: 99%

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The general linear equation on open connected sets

Leonetti

Schwaiger

2019

Acta Math. Hungar.

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Fix non-zero reals α 1 , . . . , α n with n ≥ 2 and let K be a non-empty open connected set in a topological vector space such that i≤n α i K ⊆ K (which holds, in particular, if K is an open convex cone and α 1 , . . . , α n > 0). Let also Y be a vector space over F := Q(α 1 , . . . , α n ). We show, among others, that a function f : K → Y satisfies the general linear equationThe main tool of the proof is a general version of a result Radó and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.

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“…Then, the claim follows by [4,Lemma 2.4]. Putting together Claims 1, 2, and 3, we conclude the proof.…”

Section: Introductionsupporting

confidence: 54%

“…Conversely, let us assume that A : X → Y is a group homomorphism which satisfies (3) and f : K → Y is a function defined by (4). If i≤n β i = 1 and b = 0 then f (x) = A(x) so that…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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The general linear equation on open connected sets

Leonetti

Schwaiger

2019

Acta Math. Hungar.

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“…Let z x ∈ U x and z y ∈ U y . According to[6, Lemma 2.4], there exist n ∈ N andx 0 , x 1 , ..., x n ∈ I such that z x ∈ U x0 , z y ∈ U xn and U xi−1 ∩ U xi = ∅ for i ∈ {1, ..., n}.Thus, in view of (50), we get…”

mentioning

confidence: 94%

Extension problem for principles of equivalent utility

Chudziak

2019

Aequat. Math.

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On the functional equation $$\hbox {f}(\hbox {x}+\hbox {y})=\hbox {g}(\hbox {xy})$$

Erdei,

Glavosits,

Házy

2023

Aequat. Math.

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The functional equation $$f(x+y)=g(xy)$$ f ( x + y ) = g ( x y ) is investigated with unknown functions $$f:A+A\rightarrow Y$$ f : A + A → Y , $$g:A\cdot A\rightarrow Y$$ g : A · A → Y in the following cases: $$A:=\left]\alpha ,\beta \right[\subseteq \mathbb {F}_{+}$$ A : = α , β ⊆ F + where $$\mathbb {F}$$ F is an Archimedean ordered field; A is the set of all positive integers; A is the set of all positive dyadic rational numbers. The set Y is an arbitrarily fixed (infinite) set. The main result of the paper shows that there exists a set $$A\subseteq \mathbb {R}_{+}$$ A ⊆ R + that is closed under addition and multiplication and there exist functions f, $$g:A\rightarrow Y$$ g : A → Y which satisfy the equation $$f(x+y)=g(xy)$$ f ( x + y ) = g ( x y ) for all $$x,y\in A$$ x , y ∈ A such that the range of the function f is infinite. Finally, some application of the above results is also given.

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Generalized Pexider Equation on an Open Domain

Cited by 4 publications

References 10 publications

The general linear equation on open connected sets

The general linear equation on open connected sets

Extension problem for principles of equivalent utility

On the functional equation $$\hbox {f}(\hbox {x}+\hbox {y})=\hbox {g}(\hbox {xy})$$

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