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Mixed Type Of Additive And Quintic Functional Equations

100%
Bodaghi A., Narasimman P., Ravi K., Shojaee B.
Annales Mathematicae Silesianae
|
2015
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tom 29
|
nr 1
35-50
EN
In this paper, we investigate the general solution and Hyers–Ulam–Rassias stability of a new mixed type of additive and quintic functional equation of the form [...] f(3x+y)−5f(2x+y)+f(2x−y)+10f(x+y)−5f(x−y)=10f(y)+4f(2x)−8f(x) $$f\left( {3x + y} \right) - 5f\left( {2x + y} \right) + f\left( {2x - y} \right) + 10f\left( {x + y} \right) - 5f\left( {x - y} \right) = 10f\left( y \right) + 4f\left( {2x} \right) - 8f\left( x \right)$$ in the set of real numbers.
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On the superstability of the cosine and sine type functional equations

100%
Lehlou F., Moussa M., Roukbi A., Kabbaj S.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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2016
|
tom 15
EN
In this paper, we study the superstablity problem of the cosine and sine type functional equations: f(xσ(y)a)+f(xya)=2f(x)f(y) $$f(x\sigma (y)a) + f(xya) = 2f(x)f(y)$$ and f(xσ(y)a)−f(xya)=2f(x)f(y), $$f(x\sigma (y)a) - f(xya) = 2f(x)f(y),$$ where f : S → ℂ is a complex valued function; S is a semigroup; σ is an involution of S and a is a fixed element in the center of S.
3
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Alienation of the Jensen, Cauchy and d’Alembert Equations

100%
Sobek B.
Annales Mathematicae Silesianae
|
2016
|
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.
4
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Weighted entropies

64%
Ebanks B.
Open Mathematics
|
2010
|
tom 8
|
nr 3
602-615
EN
We present an axiomatic characterization of entropies with properties of branching, continuity, and weighted additivity. We deliberately do not assume that the entropies are symmetric. The resulting entropies are generalizations of the entropies of degree α, including the Shannon entropy as the case α = 1. Such “weighted” entropies have potential applications to the “utility of gambling” problem.
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