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Displaying similar documents to “D'Alembert's functional equation on groups”

Solution of Whitehead equation on groups

Valeriĭ A. Faĭziev, Prasanna K. Sahoo (2013)

Mathematica Bohemica

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Let G be a group and H an abelian group. Let J * ( G , H ) be the set of solutions f : G H of the Jensen functional equation f ( x y ) + f ( x y - 1 ) = 2 f ( x ) satisfying the condition f ( x y z ) - f ( x z y ) = f ( y z ) - f ( z y ) for all x , y , z G . Let Q * ( G , H ) be the set of solutions f : G H of the quadratic equation f ( x y ) + f ( x y - 1 ) = 2 f ( x ) + 2 f ( y ) satisfying the Kannappan condition f ( x y z ) = f ( x z y ) for all x , y , z G . In this paper we determine solutions of the Whitehead equation on groups. We show that every solution f : G H of the Whitehead equation is of the form 4 f = 2 ϕ + 2 ψ , where 2 ϕ J * ( G , H ) and 2 ψ Q * ( G , H ) . Moreover, if H has the additional property that 2 h = 0 implies h = 0 for all h H ,...

On the superstability of the cosine and sine type functional equations

Fouad Lehlou, Mohammed Moussa, Ahmed Roukbi, Samir Kabbaj (2016)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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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 σ ( y ) a ) + f ( x y a ) = 2 f ( x ) f ( y ) and f(xσ(y)a)−f(xya)=2f(x)f(y), f ( x σ ( y ) a ) - f ( x y a ) = 2 f ( 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.

A pair of linear functional inequalities and a characterization of L p -norm

Dorota Krassowska, Janusz Matkowski (2005)

Annales Polonici Mathematici

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It is shown that, under some general algebraic conditions on fixed real numbers a,b,α,β, every solution f:ℝ → ℝ of the system of functional inequalities f(x+a) ≤ f(x)+α, f(x+b) ≤ f(x)+β that is continuous at some point must be a linear function (up to an additive constant). Analogous results for three other similar simultaneous systems are presented. An application to a characterization of L p -norm is given.

Method of averaging for the system of functional-differential inclusions

Teresa Janiak, Elżbieta Łuczak-Kumorek (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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The basic idea of this paper is to give the existence theorem and the method of averaging for the system of functional-differential inclusions of the form ⎧ ( t ) F ( t , x t , y t ) (0) ⎨ ⎩ ( t ) G ( t , x t , y t ) (1)

The law of large numbers and a functional equation

Maciej Sablik (1998)

Annales Polonici Mathematici

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We deal with the linear functional equation (E) g ( x ) = i = 1 r p i g ( c i x ) , where g:(0,∞) → (0,∞) is unknown, ( p , . . . , p r ) is a probability distribution, and c i ’s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli’s Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.

On the superstability of generalized d’Alembert harmonic functions

Iz-iddine EL-Fassi (2016)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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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 + σ ( z ) ) + f ( x + σ ( y ) + z ) + f ( σ ( x ) + y + z ) = 4 f ( 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.

A useful algebra for functional calculus

Mohammed Hemdaoui (2019)

Mathematica Bohemica

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We show that some unital complex commutative LF-algebra of 𝒞 ( ) -tempered functions on + (M. Hemdaoui, 2017) equipped with its natural convex vector bornology is useful for functional calculus.

Solutions for the p-order Feigenbaum’s functional equation h ( g ( x ) ) = g p ( h ( x ) )

Min Zhang, Jianguo Si (2014)

Annales Polonici Mathematici

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This work deals with Feigenbaum’s functional equation ⎧ h ( g ( x ) ) = g p ( h ( x ) ) , ⎨ ⎩ g(0) = 1, -1 ≤ g(x) ≤ 1, x∈[-1,1] where p ≥ 2 is an integer, g p is the p-fold iteration of g, and h is a strictly monotone odd continuous function on [-1,1] with h(0) = 0 and |h(x)| < |x| (x ∈ [-1,1], x ≠ 0). Using a constructive method, we discuss the existence of continuous unimodal even solutions of the above equation.