On a functional equation with derivative and symmetrization

Adam Bobrowski; Małgorzata Kubalińska

Displaying similar documents to “On a functional equation with derivative and symmetrization”

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

Z. Kominek (1974)

Annales Polonici Mathematici

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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...

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 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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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.

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

M. Malenica (1982)

Matematički Vesnik

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Cauchy-like functional equation based on a class of uninorms

Feng Qin (2015)

Kybernetika

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Commuting is an important property in any two-step information merging procedure where the results should not depend on the order in which the single steps are performed. In the case of bisymmetric aggregation operators with the neutral elements, Saminger, Mesiar and Dubois, already reduced characterization of commuting $n$ -ary operators to resolving the unary distributive functional equations. And then the full characterizations of these equations are obtained under the assumption that...

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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Semigroups generated by convex combinations of several Feller generators in models of mathematical biology

Adam Bobrowski, Radosław Bogucki (2008)

Studia Mathematica

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Let be a locally compact Hausdorff space. Let $A_{i}$ , i = 0,1,...,N, be generators of Feller semigroups in C₀() with related Feller processes $X_{i} = X_{i} (t), t \geq 0$ and let $α_{i}$ , i = 0,...,N, be non-negative continuous functions on with $\sum_{i = 0}^{N} α_{i} = 1$ . Assume that the closure A of $\sum_{k = 0}^{N} α_{k} A_{k}$ defined on $⋂_{i = 0}^{N} (A_{i})$ generates a Feller semigroup T(t), t ≥ 0 in C₀(). A natural interpretation of a related Feller process X = X(t), t ≥ 0 is that it evolves according to the following heuristic rules: conditional on being at a point p ∈ , with probability...

On the Cauchy problem for convolution equations

(2013)

Colloquium Mathematicae

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We consider one-parameter (C₀)-semigroups of operators in the space $^{'} (ℝ ⁿ; ℂ^{m})$ with infinitesimal generator of the form ${(G *) |}_{^{'} (ℝ ⁿ; ℂ^{m})}$ where G is an $M_{m \times m}$ -valued rapidly decreasing distribution on ℝⁿ. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces $(ℝ ⁿ; ℂ^{m})$ , $_{L^{p}} (ℝ ⁿ; ℂ^{m})$ , p ∈ [1,∞], $(_{a}) (ℝ ⁿ; ℂ^{m})$ , a ∈ ]0,∞[, or the spaces $_{L^{q}}^{'} (ℝ ⁿ; ℂ^{m})$ , q ∈ ]1,∞], of bounded distributions.

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.

Positive solutions for a class of non-autonomous second order difference equations via a new functional fixed point theorem

Lydia Bouchal, Karima Mebarki, Svetlin Georgiev Georgiev b (2022)

Archivum Mathematicum

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In this paper, by using recent results on fixed point index, we develop a new fixed point theorem of functional type for the sum of two operators $T + S$ where $I - T$ is Lipschitz invertible and $S$ a $k$ -set contraction. This fixed point theorem is then used to establish a new result on the existence of positive solutions to a non-autonomous second order difference equation.

Surjectivity of partial differential operators on ultradistributions of Beurling type in two dimensions

Thomas Kalmes (2010)

Studia Mathematica

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We show that if Ω is an open subset of ℝ², then the surjectivity of a partial differential operator P(D) on the space of ultradistributions $_{(ω)}^{'} (Ω)$ of Beurling type is equivalent to the surjectivity of P(D) on $C^{\infty} (Ω)$ .

Asymptotic behavior of a stochastic combustion growth process

Alejandro Ramírez, Vladas Sidoravicius (2004)

Journal of the European Mathematical Society

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We study a continuous time growth process on the $d$ -dimensional hypercubic lattice $𝒵^{d}$ , which admits a phenomenological interpretation as the combustion reaction $A + B \to 2 A$ , where $A$ represents heat particles and $B$ inert particles. This process can be described as an interacting particle system in the following way: at time 0 a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hypercubic lattice; then, as soon as any random walk visits a site...

Analytic semigroups on vector valued noncommutative $L^{p}$ -spaces

Cédric Arhancet (2013)

Studia Mathematica

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We give sufficient conditions on an operator space E and on a semigroup of operators on a von Neumann algebra M to obtain a bounded analytic or R-analytic semigroup ( ${(T \otimes I d_{E})}_{t \geq 0}$ on the vector valued noncommutative $L^{p}$ -space $L^{p} (M, E)$ . Moreover, we give applications to the $H^{\infty} (Σ_{θ})$ functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu.