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

H. Swiatak

Displaying similar documents to “On the functional equation $f (x + y - x y) + f (x y) = f (x) + f (y)$ ”

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

Z. Kominek (1974)

Annales Polonici Mathematici

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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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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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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)

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.

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.

Second order quasilinear functional evolution equations

László Simon (2015)

Mathematica Bohemica

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We consider second order quasilinear evolution equations where also the main part contains functional dependence on the unknown function. First, existence of solutions in $(0, T)$ is proved and examples satisfying the assumptions of the existence theorem are formulated. Then a uniqueness theorem is proved. Finally, existence and some qualitative properties of the solutions in $(0, \infty)$ (boundedness and stabilization as $t \to \infty$ ) are shown.

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 Functional Equation $f (λ) + f (ω λ) f (ω^{- 1} λ) = 1$ , $(ω^{5} = 1)$

Yasutaka Sibuya (1984)

Recherche Coopérative sur Programme n°25

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On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec, Ludwig Reich (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, ${φ |}_{(- \infty, 0]} = 0$ , ${φ |}_{[1, \infty)} = 1$ , = φ: ℝ → ℝ|φ is continuous, ${φ |}_{(- \infty, 0]} = 0$ , ${φ |}_{[1, \infty)} = 1$ . We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the...

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.

On d-characteristic and $d_{Ξ}$ -characteristic of linear operators

D. Przeworska-Rolewicz, S. Rolewicz (1967)

Annales Polonici Mathematici

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The existence of Carathéodory solutions of hyperbolic functional differential equations

Adrian Karpowicz (2010)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We consider the following Darboux problem for the functional differential equation $\partial ² u / \partial x \partial y (x, y) = f (x, y, u_{(x, y)}, \partial u / \partial x (x, y), \partial u / \partial y (x, y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b] $0, a] \times (0, b],$ where the function $u_{(x, y)} : [- a ₀, 0] \times [- b ₀, 0] \to ℝ^{k}$ is defined by $u_{(x, y)} (s, t) = u (s + x, t + y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.

Displaying similar documents to “On the functional equation f ( x + y - x y ) + f ( x y ) = f ( x ) + f ( y ) ”

Displaying similar documents to “On the functional equation $f (x + y - x y) + f (x y) = f (x) + f (y)$ ”