Existence and uniqueness theorems for functional equations

Glavosits, Tamás; Karácsony, Zsolt

doi:10.46298/cm.10830

Cited by 2 publications

(5 citation statements)

References 6 publications

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“…Let us assume that the functions f , g, h satisfy the functional equation (RestPexAdd). By Theorem 3.1 we obtain that they are in the form of (5.1), and by Uniqueness Theorem [10] properties 1., 2., and 3. are fulfilled.…”

Section: The Generalization Of Rimán's Extension Theoremmentioning

confidence: 76%

“…• An ordered field F(+, •, ⩽) is said to be Archimedean ordered if F(+, ⩽) is an Archimedean ordered group. Now, we review some properties of open intervals ( [10,12]). The open intervals are…”

Section: Some Necessary Concepts and Resultsmentioning

confidence: 99%

“…whence by the Extension Theorem [10] we obtain that there exists an additive function a : G → Y such that…”

Section: Extension Theorem For Pexider Additive Functional Equationmentioning

confidence: 99%

“…Take the substitutions: We shall use the Existence Theorem for logarithmic functions in [10] according to which if F(+, •, ⩽) is an Archimedean ordered field, Y (+) is a group, ε ∈ F such that ε > 1, and the function…”

Section: Extension Theorem For Pexider Additive Functional Equationmentioning

confidence: 99%

“…We shall use the Uniqueness Theorem for additive functions [10], according to which if G(+, ⩽) is an Archimedean ordered, Abelian group, a : G → Y is an additive function, C ∈ Y , and ]α, β[ ⊆ Y is a nonempty interval such that a(x) = C for all…”

Section: The Generalization Of Rimán's Extension Theoremmentioning

confidence: 99%

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Short remark to the Rimán’s Theorem

Glavosits

2023

AMI

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In this paper the general solution of the functional equation fan ordered, dense, Abelian group, the topology on G is generated by the open intervals of G, the sets Dx, Dy, Dx+y are defined by Dx := {u ∈ G | ∃v ∈ G : (u, v) ∈ D}, Dy := {v ∈ G | ∃u ∈ G : (u, v) ∈ D}, Dx+y := {z ∈ G | ∃(u, v) ∈ D : z = u + v}, and Y (+) is an Abelian group.The main result of the article is a common generalization of similar results by L. Székelyhidi and J. Rimán. Analogous theorem concerning logarithmic functions is also shown.

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Section: The Generalization Of Rimán's Extension Theoremmentioning

confidence: 76%

Section: Some Necessary Concepts and Resultsmentioning

confidence: 99%

“…whence by the Extension Theorem [10] we obtain that there exists an additive function a : G → Y such that…”

Section: Extension Theorem For Pexider Additive Functional Equationmentioning

confidence: 99%

Section: Extension Theorem For Pexider Additive Functional Equationmentioning

confidence: 99%

Section: The Generalization Of Rimán's Extension Theoremmentioning

confidence: 99%

See 3 more Smart Citations

Short remark to the Rimán’s Theorem

Glavosits

2023

AMI

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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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Existence and uniqueness theorems for functional equations

Abstract: In this paper we give simple extension and uniqueness theorems for restricted additive and logarithmic functional equations.

Cited by 2 publications

References 6 publications

Short remark to the Rimán’s Theorem

Short remark to the Rimán’s Theorem

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

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