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1
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Some remarks on the stability of the multi-Jensen equation

100%
Schwaiger J.
Open Mathematics
|
2013
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tom 11
|
nr 5
966-971
EN
First a stability result of Prager-Schwaiger [Prager W., Schwaiger J., Stability of the multi-Jensen equation, Bull. Korean Math. Soc., 2008, 45(1), 133–142] is generalized by admitting more general domains of the involved function and by allowing the bound to be not constant. Next a result by Cieplinski [Cieplinski K., On multi-Jensen functions and Jensen difference, Bull. Korean Math. Soc., 2008, 45(4), 729–737] is discussed. Finally a characterization of the completeness of a normed space in terms of stability requirements for multi-Jensen functions is presented.
2
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On Baire measurable solutions of some functional equations

81%
Baron K.
Open Mathematics
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2009
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tom 7
|
nr 4
804-808
EN
We establish conditions under which Baire measurable solutions f of $$ \Gamma (x,y,|f(x) - f(y)|) = \Phi (x,y,f(x + \phi _1 (y)),...,f(x + \phi _N (y))) $$ defined on a metrizable topological group are continuous at zero.
3
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On the superstability of generalized d’Alembert harmonic functions

81%
EL-Fassi I.-i.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2016
|
tom 15
EN
The aim of this paper is to study the superstability problem of the d’Alembert type functional equation f(x+y+z)+f(x+y+σ(z))+f(x+σ(y)+z)+f(σ(x)+y+z)=4f(x)f(y)f(z) $$f(x + y + z) + f(x + y + \sigma (z)) + f(x + \sigma (y) + z) + f(\sigma (x) + y + z) = 4f(x)f(y)f(z)$$ for all x, y, z ∈ G, where G is an abelian group and σ : G → G is an endomorphism such that σ(σ(x)) = x for an unknown function f from G into ℂ or into a commutative semisimple Banach algebra.
4
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Alienation of the Jensen, Cauchy and d’Alembert Equations

81%
Sobek B.
Annales Mathematicae Silesianae
|
2016
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tom 30
|
nr 1
181-191
EN
Let (S, +) be a commutative semigroup, σ : S → S be an endomorphism with σ2 = id and let K be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions f, g : S → K of the functional equation f(x+y)+f(x+σ(y))+g(x+y)=2f(x)+g(x)g(y)     for  x,y∈S. $$f(x + y) + f(x + \sigma (y)) + g(x + y) = 2f(x) + g(x)g(y)\;\;\;\;{\rm for}\;\;x,y \in S.$$ We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy equations.
5
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Commutativity of set-valued cosine families

62%
Smajdor A., Smajdor W.
Open Mathematics
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2014
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tom 12
|
nr 12
1871-1881
EN
Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. If {F t: t ≥ 0} is a regular cosine family of continuous additive set-valued functions F t: K → cc(K) such that x ∈ F t(x) for t ≥ 0 and x ∈ K, then $F_t \circ F_s (x) = F_s \circ F_t (x)fors,t \geqslant 0andx \in K$.
6
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On The Continuous Dependence Of Solutions To Orthogonal Additivity Problem On Given Functions

62%
Baron K.
Annales Mathematicae Silesianae
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2015
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tom 29
|
nr 1
19-23
EN
We show that the solution to the orthogonal additivity problem in real inner product spaces depends continuously on the given function and provide an application of this fact.
7
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Superstability of functional equations related to spherical functions

62%
Székelyhidi L.
Open Mathematics
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2017
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tom 15
|
nr 1
427-432
EN
In this paper we prove stability-type theorems for functional equations related to spherical functions. Our proofs are based on superstability-type methods and on the method of invariant means.
8
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On Popoviciu-Ionescu Functional Equation

62%
Almira J. M.
Annales Mathematicae Silesianae
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2016
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tom 30
|
nr 1
5-15
EN
We study a functional equation first proposed by T. Popoviciu [15] in 1955. It was solved for the easiest case by Ionescu [9] in 1956 and, for the general case, by Ghiorcoiasiu and Roscau [7] and Radó [17] in 1962. Our solution is based on a generalization of Radó’s theorem to distributions in a higher dimensional setting and, as far as we know, is different than existing solutions. Finally, we propose several related open problems.
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