D'Alembert's functional equation on groups

Henrik Stetkær

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

On the functional equation $f (x + y - x y) + f (x y) = f (x) + f (y)$

H. Swiatak (1968)

Matematički Vesnik

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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 $f : G \to H$ 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 $x, y, z \in G$ . Let $Q^{*} (G, H)$ be the set of solutions $f : G \to H$ 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 $x, y, z \in G$ . In this paper we determine solutions of the Whitehead equation on groups. We show that every solution $f : G \to H$ of the Whitehead equation is of the form $4 f = 2 ϕ + 2 ψ$ , where $2 ϕ \in J^{*} (G, H)$ and $2 ψ \in Q^{*} (G, H)$ . Moreover, if $H$ has the additional property that $2 h = 0$ implies $h = 0$ for all $h \in H$ ,...

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.

The functional equation $f^{2} (x) = g (x)$

James C. Lillo (1967)

Annales Polonici Mathematici

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Continuous solutions of the functional equation $φ (f (x)) = G (x, φ (x))$ for vector-valued functions φ

Z. Krzeszowiak (1969)

Annales Polonici Mathematici

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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 ⎧ $ẋ (t) \in F (t, x_{t}, y_{t})$ (0) ⎨ ⎩ $ẋ (t) \in G (t, x_{t}, y_{t})$ (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) $g (x) = \sum_{i = 1}^{r} p_{i} g (c_{i} x)$ , where g:(0,∞) → (0,∞) is unknown, $(p ₁, . . ., p_{r})$ 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.

On the functional equation $f (x) + \sum_{i = 1}^{n} g_{i} (y_{i}) = h (T (x, y_{1}, y_{2}, . . ., y_{n}))$

C. T. Ng (1973)

Annales Polonici Mathematici

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$C^{r}$ -solutions of a system of functional equations

Z. Kominek (1974)

Annales Polonici Mathematici

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Functional equations of delay type in $L^{1}$ spaces

Rosanna Villella-Bressan (1985)

Annales Polonici Mathematici

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On the solutions of the functional equation $ϕ (x) + ϕ^{2} (x) = F (x)$

M. Malenica (1982)

Matematički Vesnik

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On some symmetrical equations of the form $f_{1} (x_{1} + . . . + x_{n}) = \sum_{i_{1}, . . ., i_{n}}) f_{1} (x_{i} 1) . . . f_{n} (x_{i} n)$

H. Światak (1967)

Annales Polonici Mathematici

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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 $𝒞^{(\infty)}$ $ℕ$ -tempered functions on $ℝ^{+}$ (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.