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2013 | Buch

Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces

verfasst von: Toka Diagana

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book presents a comprehensive introduction to the concepts of almost periodicity, asymptotic almost periodicity, almost automorphy, asymptotic almost automorphy, pseudo-almost periodicity, and pseudo-almost automorphy as well as their recent generalizations. Some of the results presented are either new or else cannot be easily found in the mathematical literature. Despite the noticeable and rapid progress made on these important topics, the only standard references that currently exist on those new classes of functions and their applications are still scattered research articles. One of the main objectives of this book is to close that gap. The prerequisites for the book is the basic introductory course in real analysis. Depending on the background of the student, the book may be suitable for a beginning graduate and/or advanced undergraduate student. Moreover, it will be of a great interest to researchers in mathematics as well as in engineering, in physics, and related areas. Further, some parts of the book may be used for various graduate and undergraduate courses.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Metric, Banach, and Hilbert Spaces

Abstract

In this introductory chapter, we collect the basic background on metric, Banach, and Hilbert spaces needed in the sequel. The exposition is self-contained, with a wealth of illustrative examples. Proofs of some of the most important results are given including that of the Banach fixed-point theorem.

Toka Diagana

Chapter 2. Linear Operators on Banach Spaces

Abstract

Let $(\mathcal{X},\|\cdot \|)$ and $(\mathcal{Y},\|\cdot \|_{1})$ be two Banach spaces over the same field $\mathbb{F}$. A mapping $A: D(A) \subset \mathcal{X} \rightarrow \mathcal{Y}$ satisfying

$$\displaystyle{A(\alpha x +\beta y) =\alpha Ax +\beta Ay}$$

for all x,y∈D(A) and $\alpha,\beta \in \mathbb{F}$, is called a linear operator or a linear transformation.

Toka Diagana

Chapter 3. Almost Periodic Functions

Abstract

The theory of almost periodic functions was introduced in the literature around 1924–1926 with the pioneering work of the Danish mathematician Bohr [25]. A decade later, various significant contributions were then made to that theory mainly by Bochner [24], von Neumann [159], and van Kampen [155]. The notion of almost periodicity, which generalizes the concept of periodicity, plays a crucial role in various fields including harmonic analysis, physics, dynamical systems, etc.

Toka Diagana

Chapter 4. Almost Automorphic Functions

Abstract

The concept of almost automorphy was introduced in the literature by S. Bochner in 1955 in the context of differential geometry [21] (see also Bochner [22, 23]). Since then, this concept has been extended in various directions. Veech [156] extended this concept to groups and then obtained various properties of these functions including the existence of their corresponding Fourier series (see also [157, 158]).

Toka Diagana

Chapter 5. Pseudo-Almost Periodic Functions

Abstract

Toka Diagana

Chapter 6. Pseudo-Almost Automorphic Functions

Abstract

The concept of pseudo-almost automorphy, which is a generalization of the notions of almost periodicity, almost automorphy, and that of the pseudo-almost periodicity, was introduced in the literature a few years ago by Xiao et al. [164].

Toka Diagana

Chapter 7. Existence Results for Some Second-Order Differential Equations

Abstract

Let α ∈ (0,1). Fix once and for all a separable infinite dimensional complex Hilbert space $(\mathcal{H},\langle \cdot,\cdot \rangle,\|\cdot \|)$.

Toka Diagana

Chapter 8. Existence Results to Some Integrodifferential Equations

Abstract

In this chapter we study the existence of asymptotically almost automorphic mild solutions to the abstract partial neutral integrodifferential equation

$$\displaystyle\begin{array}{rcl} \frac{d} {dt}D(t,u_{t})& =& AD(t,u_{t}) +\int _{ 0}^{t}B(t - s)D(s,u_{ s})ds + g(t,u_{t}),\ t \in [\sigma,\sigma +a),{}\end{array}$$

(8.1)

$$\displaystyle\begin{array}{rcl} u_{\sigma }& =& \varphi \in \mathcal{B},{}\end{array}$$

(8.2)

where $A,B(t): D(A) \subset \mathcal{X} \rightarrow \mathcal{X}$ are densely defined closed linear operators with a common domain D(A), which is independent of t; the history

$$\displaystyle{u_{t}: (-\infty,0] \rightarrow \mathcal{X},\ \ \mbox{ defined by}\ \ u_{t}(\theta ):= u(t+\theta )}$$

belongs to an abstract phase space $\mathcal{B}$ defined axiomatically, f, g are functions subject to some additional conditions, and

$$\displaystyle{D(t,\varphi ) =\varphi (0) + f(t,\varphi ).}$$

For that, we will make extensive use of the concept of compact asymptotically almost automorphy and the so-called resolvent of operators.

Toka Diagana

Chapter 9. Existence of C (m)-Pseudo-Almost Automorphic Solutions

Abstract

In this chapter we study and obtain the existence of C ^(m)-pseudo-almost automorphic solutions to some classes of first-order, second-order, and higher-order differential equations with operator coefficients whose forcing term is C ^(m)-pseudo-almost automorphic. For that, we will make extensive use of various tools including analytic semigroup and exponential dichotomy techniques.

Toka Diagana

Chapter 10. Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations

Abstract

Motivated by the recent work by Diagana [49, 52, 61], in this chapter using the Schauder fixed-point theorem (Theorem 1.98), the Banach fixed-point principle (Theorem 1.96), and the dichotomy techniques, we study the problem which consists of the existence of pseudo-almost periodic (respectively, weighted pseudo-almost periodic) solutions to the nonautonomous third-order differential equations

$$\displaystyle\begin{array}{rcl} \frac{d} {dt}\Big[{u}^{{\prime\prime}} + g(t,Bu(t))\Big] = w(t)Au(t) + f(t,Cu(t)),\ \ t \in \mathbb{R}& &{}\end{array}$$

(10.1)

Toka Diagana

Chapter 11. Pseudo-Almost Automorphic Solutions to Some Sobolev-Type Equations

Abstract

Toka Diagana

Chapter 12. Stability Results for Some Higher-Order Difference Equations

Abstract

The main motivation of this chapter comes from Diagana et al. [58], in which not only a basic theory for almost periodic sequences on $\mathbb{Z}_{+}$ was introduced and studied but also discrete dichotomy techniques were utilized to find various sufficient conditions for the existence of globally attracting almost periodic solutions to some first-order nonautonomous system of difference equations. Furthermore, Diagana et al. [58] subsequently applied their abstract results to study discretely reproducing populations with and without overlapping generations.

Toka Diagana

Backmatter

Titel: Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces
verfasst von: Toka Diagana
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-00849-3
Print ISBN: 978-3-319-00848-6
DOI: https://doi.org/10.1007/978-3-319-00849-3

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Metric, Banach, and Hilbert Spaces

Chapter 2. Linear Operators on Banach Spaces

Chapter 3. Almost Periodic Functions

Chapter 4. Almost Automorphic Functions

Chapter 5. Pseudo-Almost Periodic Functions

Chapter 6. Pseudo-Almost Automorphic Functions

Chapter 7. Existence Results for Some Second-Order Differential Equations

Chapter 8. Existence Results to Some Integrodifferential Equations

Chapter 9. Existence of C (m)-Pseudo-Almost Automorphic Solutions

Chapter 10. Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations

Chapter 11. Pseudo-Almost Automorphic Solutions to Some Sobolev-Type Equations

Chapter 12. Stability Results for Some Higher-Order Difference Equations

Backmatter

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