A useful algebra for functional calculus

Mohammed Hemdaoui

Displaying similar documents to “A useful algebra for functional calculus”

$C^{r}$ -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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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...

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.

On the Schröder equation

M. Kuczma

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CONTENTSPART IIntroduction............................................................................................... 31. General solution.................................................................................. 42. Preliminaries and notation................................................................ 53. $C^{p}$ solutions in $ℭ$ *................................................ 74. Change of variables..............................................................................

A symmetry problem in the calculus of variations

Graziano Crasta (2006)

Journal of the European Mathematical Society

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We consider the integral functional $J (u) = \int_{Ω} [f (| D u |) - u] d x$ , $u \in W_{0}^{1, 1} (Ω)$ , where $Ω \subset ℝ^{n}$ , $n \geq 2$ , is a nonempty bounded connected open subset of $ℝ^{n}$ with smooth boundary, and $ℝ ∋ s \mapsto f (| s |)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W_{0}^{1, 1} (Ω)$ depending only on the distance from the boundary of $Ω$ , then $Ω$ must be a ball.

A note on functional tightness and minitightness of space of the $G$ -permutation degree

Dimitrios N. Georgiou, Nodirbek K. Mamadaliev, Rustam M. Zhuraev (2023)

Commentationes Mathematicae Universitatis Carolinae

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We study the behavior of the minimal tightness and functional tightness of topological spaces under the influence of the functor of the permutation degree. Analytically: a) We introduce the notion of $τ$ -open sets and investigate some basic properties of them. b) We prove that if the map $f : X \to Y$ is $τ$ -continuous, then the map $S P^{n} f : S P^{n} X \to S P^{n} Y$ is also $τ$ -continuous. c) We show that the functor $S P^{n}$ preserves the functional tightness and the minimal tightness of compacts. d) Finally, we give some facts and properties...

$H^{\infty}$ calculus and dilatations

Andreas M. Fröhlich, Lutz Weis (2006)

Bulletin de la Société Mathématique de France

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We characterise the boundedness of the $H^{\infty}$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $- A$ generates a bounded analytic $C_{0}$ semigroup $(T_{t})$ on a UMD space, then the $H^{\infty}$ calculus of $A$ is bounded if and only if $(T_{t})$ has a dilation to a bounded group on $L^{2} ([0, 1], X)$ . This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^{p}$ space we can choose another $L^{p}$ space in place of $L^{2} ([0, 1], X)$ .

On square functions associated to sectorial operators

Christian Le Merdy (2004)

Bulletin de la Société Mathématique de France

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We give new results on square functions ${∥ x ∥}_{F} = ∥ {(\int_{0}^{\infty} {|F (t A) x|}^{2} \frac{d t}{t})}^{1 / 2} ∥_{p}$ associated to a sectorial operator $A$ on $L^{p}$ for $1 < p < \infty$ . Under the assumption that $A$ is actually $R$ -sectorial, we prove equivalences of the form $K^{- 1} {∥ x ∥}_{G} \leq {∥ x ∥}_{F} \leq K {∥ x ∥}_{G}$ for suitable functions $F, G$ . We also show that $A$ has a bounded $H^{\infty}$ functional calculus with respect to ${∥ . ∥}_{F}$ . Then we apply our results to the study of conditions under which we have an estimate $∥ {(\int_{0}^{\infty} | C e^{- t A} {(x) |}^{2} d t)^{1 / 2} ∥}_{q} \leq M {∥ x ∥}_{p}$ , when $- A$ generates a bounded semigroup $e^{- t A}$ on $L^{p}$ and $C : D (A) \to L^{q}$ is a linear mapping.

The topology of the space of $ℋ𝒦$ integrable functions in $ℝ^{n}$

Varayu Boonpogkrong (2025)

Czechoslovak Mathematical Journal

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It is known that there is no natural Banach norm on the space $ℋ𝒦$ of $n$ -dimensional Henstock-Kurzweil integrable functions on $[a, b]$ . We show that the $ℋ𝒦$ space is the uncountable union of Fréchet spaces $ℋ𝒦 (X)$ . On each $ℋ𝒦 (X)$ space, an $F$ -norm ${∥ \cdot ∥}^{X}$ is defined. A ${∥ \cdot ∥}^{X}$ -convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$ -norm is also defined for a ${∥ \cdot ∥}^{X}$ -continuous linear operator. Hence, many important results in functional analysis hold for the $ℋ𝒦 (X)$ space. It is well-known that every...

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.