Some problems concerning different types of vector valued almost periodic functions

Bolis Basit

Displaying similar documents to “Some problems concerning different types of vector valued almost periodic functions”

New spectral criteria for almost periodic solutions of evolution equations

Toshiki Naito, Nguyen Van Minh, Jong Son Shin (2001)

Studia Mathematica

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We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form ẋ = A(t)x + f(t) (*), with f having precompact range, which is then applied to find new spectral criteria for the existence of almost periodic solutions with specific spectral properties in the resonant case where $\bar{e^{i s p (f)}}$ may intersect the spectrum of the monodromy operator P of (*) (here sp(f) denotes the Carleman spectrum of f). We show that if (*)...

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...

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 Hartman almost periodic functions

Guy Cohen, Viktor Losert (2006)

Studia Mathematica

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We consider multi-dimensional Hartman almost periodic functions and sequences, defined with respect to different averaging sequences of subsets in $ℝ^{d}$ or $ℤ^{d}$ . We consider the behavior of their Fourier-Bohr coefficients and their spectrum, depending on the particular averaging sequence, and we demonstrate this dependence by several examples. Extensions to compactly generated, locally compact, abelian groups are considered. We define generalized Marcinkiewicz spaces based upon arbitrary measure...

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...

Partially irregular almost periodic solutions of ordinary differential systems

Alexandr Demenchuk (2001)

Mathematica Bohemica

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Let $f (t, x)$ be a vector valued function almost periodic in $t$ uniformly for $x$ , and let $m o d (f) = L_{1} \oplus L_{2}$ be its frequency module. We say that an almost periodic solution $x (t)$ of the system $\dot{x} = f (t, x), t \in ℝ, x \in D \subset ℝ^{n}$ is irregular with respect to $L_{2}$ (or partially irregular) if $(m o d (x) + L_{1}) \cap L_{2} = {0}$ . Suppose that $f (t, x) = A (t) x + X (t, x),$ where $A (t)$ is an almost periodic $(n \times n)$ -matrix and $m o d (A) \cap m o d (X) = {0} .$ We consider the existence problem for almost periodic irregular with respect to $m o d (A)$ solutions of such system. This problem is reduced to a similar problem for a system of smaller dimension, and sufficient...

A class of $I_{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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Approximative sequences and almost homoclinic solutions for a class of second order perturbed Hamiltonian systems

Marek Izydorek, Joanna Janczewska (2014)

Banach Center Publications

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In this work we will consider a class of second order perturbed Hamiltonian systems of the form $q ̈ + V_{q} (t, q) = f (t)$ , where t ∈ ℝ, q ∈ ℝⁿ, with a superquadratic growth condition on a time periodic potential V: ℝ × ℝⁿ → ℝ and a small aperiodic forcing term f: ℝ → ℝⁿ. To get an almost homoclinic solution we approximate the original system by time periodic ones with larger and larger time periods. These approximative systems admit periodic solutions, and an almost homoclinic solution for the original system...

Functionals on Banach Algebras with Scattered Spectra

H. S. Mustafayev (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let A be a complex, commutative Banach algebra and let $M_{A}$ be the structure space of A. Assume that there exists a continuous homomorphism h:L¹(G) → A with dense range, where L¹(G) is a group algebra of the locally compact abelian group G. The main results of this note can be summarized as follows: (a) If every weakly almost periodic functional on A with compact spectra is almost periodic, then the space $M_{A}$ is scattered (i.e., $M_{A}$ has no nonempty perfect subset). (b) Weakly almost periodic...

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.

A note on an approximative scheme of finding almost homoclinic solutions for Newtonian systems

Robert Krawczyk (2014)

Banach Center Publications

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In this work we will be concerned with the existence of almost homoclinic solutions for a Newtonian system $q ̈ + \nabla_{q} V (t, q) = f (t)$ , where t ∈ ℝ, q ∈ ℝⁿ. It is assumed that a potential V: ℝ × ℝⁿ → ℝ is C¹-smooth and its gradient map $\nabla_{q} V : ℝ \times ℝ ⁿ \to ℝ ⁿ$ is bounded with respect to t. Moreover, a forcing term f: ℝ → ℝⁿ is continuous, bounded and square integrable. We will show that the approximative scheme due to J. Janczewska (see [J2]) for a time periodic potential extends to our case.

Almost homoclinic solutions for a certain class of mixed type functional differential equations

Joanna Janczewska (2011)

Annales Polonici Mathematici

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We shall be concerned with the existence of almost homoclinic solutions for a class of second order functional differential equations of mixed type: $q ̈ (t) + V_{q} (t, q (t)) + u (t, q (t), q (t - T), q (t + T)) = f (t)$ , where t ∈ ℝ, q ∈ ℝⁿ and T>0 is a fixed positive number. By an almost homoclinic solution (to 0) we mean one that joins 0 to itself and q ≡ 0 may not be a stationary point. We assume that V and u are T-periodic with respect to the time variable, V is C¹-smooth and u is continuous. Moreover, f is non-zero, bounded, continuous and square-integrable....

Almost periodic solutions to second order neutral differential equations.

Corduneanu, C. (2005)

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

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Differences of functions in locally convex spaces and applications to almost periodic and almost automorphic functions

Bolis Basit, Magdi Emam (1983)

Annales Polonici Mathematici

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Periodic solutions to a non-linear differential equation of the order $2n+1$

Monika Kubicova (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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A criterion for the existance of periodic solutions of an ordinary differential equation of order k proved by J. Andres and J. Vorâcek for k = 3 is extended to an arbitrary odd k.

A characterization of almost continuity and weak continuity

Chrisostomos Petalas, Theodoros Vidalis (2004)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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It is well known that a function $f$ from a space $X$ into a space $Y$ is continuous if and only if, for every set $K$ in $X$ the image of the closure of $K$ under $f$ is a subset of the closure of the image of it. In this paper we characterize almost continuity and weak continuity by proving similar relations for the subsets $K$ of $X$ .