Functionals on Banach Algebras with Scattered Spectra

H. S. Mustafayev

Displaying similar documents to “Functionals on Banach Algebras with Scattered Spectra”

Characterising weakly almost periodic functionals on the measure algebra

Matthew Daws (2011)

Studia Mathematica

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Let G be a locally compact group, and consider the weakly almost periodic functionals on M(G), the measure algebra of G, denoted by WAP(M(G)). This is a C*-subalgebra of the commutative C*-algebra M(G)*, and so has character space, say $K_{W A P}$ . In this paper, we investigate properties of $K_{W A P}$ . We present a short proof that $K_{W A P}$ can naturally be turned into a semigroup whose product is separately continuous; at the Banach algebra level, this product is simply the natural one induced by the Arens...

A class of $I_{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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Remotely $c$ -almost periodic type functions in $ℝ^{n}$

Marco Kostić, Vipin Kumar (2022)

Archivum Mathematicum

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In this paper, we relate the notions of remote almost periodicity and quasi-asymptotical almost periodicity; in actual fact, we observe that a remotely almost periodic function is nothing else but a bounded, uniformly continuous quasi-asymptotically almost periodic function. We introduce and analyze several new classes of remotely $c$ -almost periodic functions in $ℝ^{n},$ slowly oscillating functions in $ℝ^{n},$ and further analyze the recently introduced class of quasi-asymptotically $c$ -almost periodic...

On the uniqueness of periodic decomposition

Viktor Harangi (2011)

Fundamenta Mathematicae

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Let $a ₁, . . ., a_{k}$ be arbitrary nonzero real numbers. An $(a ₁, . . ., a_{k})$ -decomposition of a function f:ℝ → ℝ is a sum $f ₁ + \dots + f_{k} = f$ where $f_{i} : ℝ \to ℝ$ is an $a_{i}$ -periodic function. Such a decomposition is not unique because there are several solutions of the equation $h ₁ + \dots + h_{k} = 0$ with $h_{i} : ℝ \to ℝ a_{i}$ -periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the $(a ₁, . . ., a_{k})$ -decomposition is essentially unique. We characterize those periods for which essential...

Three periodic solutions for a class of higher-dimensional functional differential equations with impulses

Yongkun Li, Changzhao Li, Juan Zhang (2010)

Annales Polonici Mathematici

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By using the well-known Leggett–Williams multiple fixed point theorem for cones, some new criteria are established for the existence of three positive periodic solutions for a class of n-dimensional functional differential equations with impulses of the form ⎧y’(t) = A(t)y(t) + g(t,yt), $t \neq t_{j}$ , j ∈ ℤ, ⎨ ⎩ $y (t ⁺_{j}) = y (t ¯_{j}) + I_{j} (y (t_{j}))$ , where $A (t) = {(a_{i j} (t))}_{n \times n}$ is a nonsingular matrix with continuous real-valued entries.

Generalized $c$ -almost periodic type functions in $ℝ^{n}$

M. Kostić (2021)

Archivum Mathematicum

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In this paper, we analyze multi-dimensional quasi-asymptotically $c$ -almost periodic functions and their Stepanov generalizations as well as multi-dimensional Weyl $c$ -almost periodic type functions. We also analyze several important subclasses of the class of multi-dimensional quasi-asymptotically $c$ -almost periodic functions and reconsider the notion of semi- $c$ -periodicity in the multi-dimensional setting, working in the general framework of Lebesgue spaces with variable exponent. We provide...

Existence and uniqueness of periodic solutions for odd-order ordinary differential equations

Yongxiang Li, He Yang (2011)

Annales Polonici Mathematici

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The paper deals with the existence and uniqueness of 2π-periodic solutions for the odd-order ordinary differential equation $u^{(2 n + 1)} = f (t, u, u^{'}, . . ., u^{(2 n)})$ , where $f : ℝ \times ℝ^{2 n + 1} \to ℝ$ is continuous and 2π-periodic with respect to t. Some new conditions on the nonlinearity $f (t, x ₀, x ₁, . . ., x_{2 n})$ to guarantee the existence and uniqueness are presented. These conditions extend and improve the ones presented by Cong [Appl. Math. Lett. 17 (2004), 727-732].

Probabilistic properties of a Markov-switching periodic $G A R C H$ process

Billel Aliat, Fayçal Hamdi (2019)

Kybernetika

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In this paper, we propose an extension of a periodic $G A R C H$ ( $P G A R C H$ ) model to a Markov-switching periodic $G A R C H$ ( $M S$ - $P G A$ $R C H$ ), and provide some probabilistic properties of this class of models. In particular, we address the question of strictly periodically and of weakly periodically stationary solutions. We establish necessary and sufficient conditions ensuring the existence of higher order moments. We further provide closed-form expressions for calculating the even-order moments as well...

Existence of nonnegative periodic solutions in neutral integro-differential equations with functional delay

Imene Soulahia, Abdelouaheb Ardjouni, Ahcene Djoudi (2015)

Commentationes Mathematicae Universitatis Carolinae

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The fixed point theorem of Krasnoselskii and the concept of large contractions are employed to show the existence of a periodic solution of a nonlinear integro-differential equation with variable delay $x^{'} (t) = - \int_{t - τ (t)}^{t} a (t, s) g (x (s)) d s + \frac{d}{d t} Q (t, x (t - τ (t))) + G (t, x (t), x (t - τ (t))) .$ We transform this equation and then invert it to obtain a sum of two mappings one of which is completely continuous and the other is a large contraction. We choose suitable conditions for $τ$ , $g$ , $a$ , $Q$ and $G$ to show that this sum of mappings fits into the framework of a modification of...

A version of the law of large numbers

Katusi Fukuyama (2001)

Colloquium Mathematicae

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By the method of Rio [10], for a locally square integrable periodic function f, we prove $(f (μ ₁^{t} x) + . . . + f (μ ₙ^{t} x)) / n \to \int_{0}^{1} f$ for almost every x and t > 0.

Positive periodic solutions of a neutral functional differential equation with multiple delays

Yongxiang Li, Ailan Liu (2018)

Mathematica Bohemica

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This paper deals with the existence of positive $ω$ -periodic solutions for the neutral functional differential equation with multiple delays ${(u (t) - c u (t - δ))}^{'} + a (t) u (t) = f (t, u (t - τ_{1}), \dots, u (t - τ_{n})) .$ The essential inequality conditions on the existence of positive periodic solutions are obtained. These inequality conditions concern with the relations of $c$ and the coefficient function $a (t)$ , and the nonlinearity $f (t, x_{1}, \dots, x_{n})$ . Our discussion is based on the perturbation method of positive operator and fixed point index theory in cones.

Existence and global attractivity of periodic solutions in a higher order difference equation

Chuanxi Qian, Justin Smith (2018)

Archivum Mathematicum

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Consider the following higher order difference equation $x (n + 1) = f (n, x (n)) + g (n, x (n - k)), n = 0, 1, \dots$ where $f (n, x)$ and $g (n, x) : {0, 1, \dots} \times [0, \infty) \to [0, \infty)$ are continuous functions in $x$ and periodic functions in $n$ with period $p$ , and $k$ is a nonnegative integer. We show the existence of a periodic solution ${\tilde{x} (n)}$ under certain conditions, and then establish a sufficient condition for ${\tilde{x} (n)}$ to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also...

Special sets of reals and weak forms of normality on Isbell--Mrówka spaces

Vinicius de Oliveira Rodrigues, Victor dos Santos Ronchim, Paul J. Szeptycki (2023)

Commentationes Mathematicae Universitatis Carolinae

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We recall some classical results relating normality and some natural weakenings of normality in $Ψ$ -spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$ -sets, $λ$ -sets and $σ$ -sets. We introduce a new class of special sets of reals which corresponds to the corresponding almost disjoint family of branches being $ℵ_{0}$ -separated. This new class fits between $λ$ -sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $𝒜$ being...

C*-algebras have a quantitative version of Pełczyński's property (V)

Hana Krulišová (2017)

Czechoslovak Mathematical Journal

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A Banach space $X$ has Pełczyński’s property (V) if for every Banach space $Y$ every unconditionally converging operator $T : X \to Y$ is weakly compact. H. Pfitzner proved that $C^{*}$ -algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C (K)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover,...

Periodic solutions for a class of non-autonomous Hamiltonian systems with $p (t)$ -Laplacian

Zhiyong Wang, Zhengya Qian (2024)

Mathematica Bohemica

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We investigate the existence of infinitely many periodic solutions for the $p (t)$ -Laplacian Hamiltonian systems. By virtue of several auxiliary functions, we obtain a series of new super- $p^{+}$ growth and asymptotic- $p^{+}$ growth conditions. Using the minimax methods in critical point theory, some multiplicity theorems are established, which unify and generalize some known results in the literature. Meanwhile, we also present an example to illustrate our main results are new even in the case $p (t) \equiv p = 2$ . ...

Almost demi Dunford--Pettis operators on Banach lattices

Hedi Benkhaled (2023)

Commentationes Mathematicae Universitatis Carolinae

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We introduce new concept of almost demi Dunford–Pettis operators. Let $E$ be a Banach lattice. An operator $T$ from $E$ into $E$ is said to be almost demi Dunford–Pettis if, for every sequence ${x_{n}}$ in $E_{+}$ such that $x_{n} \to 0$ in $σ (E, E^{'})$ and $∥ x_{n} - T x_{n} ∥ \to 0$ as $n \to \infty$ , we have $∥ x_{n} ∥ \to 0$ as $n \to \infty$ . In addition, we study some properties of this class of operators and its relationships with others known operators.

The Lebesgue constant for the periodic Franklin system

Markus Passenbrunner (2011)

Studia Mathematica

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We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_{j}$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n, ν}$ be the space of piecewise linear continuous functions on the torus with knots $t_{j} : 0 \leq j \leq N - 1$ . Finally, let $P_{n, ν}$ be the orthogonal projection operator from L²([0,1)) onto $V_{n, ν}$ . The main result is $l i m_{n \to \infty, ν = 1} | | P_{n, ν} : L^{\infty} \to L^{\infty} | | = s u p_{n \in ℕ, 0 \leq ν \leq n} | | P_{n, ν} : L^{\infty} \to L^{\infty} | | = 2 + (33 - 18 \sqrt 3) / 13$ . This shows in particular that the Lebesgue constant of the classical...