Limit periodic homogeneous linear difference systems

doi:10.1016/j.amc.2015.06.008

Applied Mathematics and Computation

Volume 265, 15 August 2015, Pages 958-972

https://doi.org/10.1016/j.amc.2015.06.008 Get rights and content

Abstract

We study limit periodic homogeneous linear difference systems, where the coefficient matrices belong to a bounded group. We find groups of matrices with the property that the systems, which do not possess any non-zero asymptotically almost periodic solution, form a dense subset in the space of all considered systems. Analogously, we analyse almost periodic systems as well.

Introduction

Let us consider the limit periodic (and almost periodic) homogeneous linear difference systems $x_{k + 1} = A_{k} \cdot x_{k}, k \in Z .$ Limit periodic systems (1) form the smallest class of systems which generalize pure periodic systems (1) and which can have at least one non-almost periodic solution (for complex matrices A_k from a bounded group). The basic motivation of our research comes from papers [20], [41], where we study non-almost periodic solutions of almost periodic systems. In this paper, we improve the main results of [20], [41], because we analyse non-almost periodic solutions in the limit periodic case. In addition, we obtain results about non-asymptotically almost periodic solutions. Since the method used in this paper is substantially different from the processes applied in [20], [41], we also obtain new results for almost periodic systems. We remark that the treated systems are viewed in the general setting, when the elements of considered sequences are from a field with an absolute value (although we get new results even for the field of complex numbers).

For example, let us point out coefficient matrices A_k from the unitary group. It is proved in [35] (see also [33]) that, in any neighbourhood of an almost periodic unitary system (1), there exist almost periodic unitary systems (1) which possess non-almost periodic solutions. In this paper, we prove direct generalisations of this property of the almost periodic unitary systems. Note that the corresponding result about orthogonal difference, skew-Hermitian differential, and skew-symmetric differential systems is proved in [38], [40] (see also [34]), and in [42], respectively. The reason for the study of non-almost periodic solutions lies in the fact that the considered systems have only almost periodic solutions in many cases (for unitary, orthogonal, skew-Hermitian, and skew-symmetric systems, see [23], [24], [36], [38]).

The necessity of generalisations of periodic mathematical models is implied by various oscillatory phenomena in natural sciences. The models induce the research of limit periodic, almost periodic, and asymptotically almost periodic sequences in connection with difference equations. There are many significant books dealing with (asymptotically) almost periodic solutions of difference and differential equations, e.g., [6], [9], [12], [15], [44]. We also refer to the references given in these books.

Now we mention a short literature overview concerning almost periodic difference equations. The basics of the theory of almost periodicity (limit periodicity, asymptotic almost periodicity) can be found in classical books [4], [12], [15], [31], [45]. Complex almost periodic systems (1) and their properties (namely exponential dichotomy) are studied in [3], [22], [30]. Concerning the almost periodicity of solutions of almost periodic linear difference equations, see papers [7], [8], [11], [18], [21], [46]. Criteria ensuring the existence of almost periodic solutions of general difference systems are proved, e.g., in [19], [47], [48]. For linear almost periodic equations with $k \in N,$ we refer to [1], [32].

In this paper, we use a method based on a construction of limit periodic sequences. Similar method was firstly applied in [16], where non-almost periodic solutions of homogeneous linear difference equations are found as classes of constructible sequences. A method of constructions of minimal cocycles, which are obtained as solutions of homogeneous linear differential systems, is described in [28] (see also [29]). Special constructions of homogeneous linear differential systems with almost periodic coefficients are used in [25], [26], [40], [42] as well.

This paper is organised as follows. We begin with the notation. In Section 3, we recall the concepts of limit periodicity, almost periodicity, and asymptotic almost periodicity and we mention some properties of the considered generalisations of periodicity. In Section 4, we state the notion of (weakly) transformable groups of matrices and we state the strongest known result concerning non-almost periodic solutions of (1), when the coefficient matrices belong to a (weakly) transformable group. Then we introduce properties (denoted by P and P^*) which allow us to improve this result. Our results are formulated and proved in Section 5. We also give several examples.

Section snippets

Preliminaries

In this section, we mention the used notation which corresponds to the standard one. Let (F, ⊕, ⊙) be an infinite field with a zero e₀ and a unit e₁. Let $| \cdot | : F \to R$ be an absolute value on F; i.e., let

(a)
| f | ≥ 0 and $| f | = 0$ if and only if $f = e_{0},$
(b)
$| f ⊙ g | = | f | \cdot | g |,$
(c)
$| f \oplus g | \leq | f | + | g |$

for all f, g ∈ F. As $F,$ we will understand each one of the fields $C, R, Q$ with the usual absolute value. We remark that $i \in C$ will stand for the imaginary unit.

Let $m \in N$ be arbitrarily given (as the dimension of systems under

Limit periodic, almost periodic, and asymptotically almost periodic sequences

Since we will consider various metric spaces, we recall the notion of limit periodicity, almost periodicity, and asymptotic almost periodicity for a general metric space (Y, τ). Note that, we mention only the properties of considered sequences which are necessary to prove our results.

Definition 1

A sequence ${φ_{k}}_{k \in Z} \subseteq Y$ is called limit periodic if there exists a sequence of periodic sequences ${φ_{k}^{n}}_{k \in Z} \subseteq Y$ for $n \in N$ such that $\lim_{n \to \infty} φ_{k}^{n} = φ_{k}$ uniformly with respect to $k \in Z$ .

Remark 1

The periods of sequences ${φ_{k}^{n}}$ in Definition 1

Homogeneous linear difference systems

At first we mention a well-known result with its corollary concerning periodic systems in the most important case when $F = F$ .

Theorem 5

Let a periodic sequence ${A_{k}}_{k \in Z} \subset M a t (F, m)$ of non-singular matrices be given. Then a solution of(1)is almost periodic if and only if it is bounded.

Proof

See, e.g., [41, Corollary 3.9]. □

Corollary 1

Let $F = F$ . All solutions of any system ${A_{k}} \in P (X)$ are almost periodic.

Theorem 5 motivates our requirement on the boundedness of the considered matrix group $X$ . We repeat that we analyse limit periodic

Results

Now we can formulate and prove our main result.

Theorem 7

Let $X$ have property P. For any ${A_{k}} \in LP (X)$ and ε > 0, there exists a system ${B_{k}} \in O_{ɛ}^{σ} ({A_{k}}) \cap LP (X)$ which does not have any non-zero asymptotically almost periodic solution.

Proof

Let ${u_{n}}_{n \in N} \subset F^{m}$ be a sequence of non-zero vectors such that $\bar{{u_{n} : n \in N}} = F^{m}$ . We know that there exists K > 1 satisfying ‖ M ‖ < K for all $M \in X$ . In addition, for any u_n, there exists L_n > 0 satisfying $∥ M u_{n} ∥ > L_{n}, M \in X .$ Indeed, $\lim_{j \to \infty} ∥ M_{j} u_{n} ∥ = 0$ for a sequence of $M_{j} \in X$ implies the following

Acknowledgement

The corresponding author would like to thank to Prof. Werner Kratz for his support and hospitality at the University of Ulm (Universität Ulm) during the period in which this paper was partially written.

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¹: Supported by the Czech Science Foundation under grant P201/10/1032.

²: Supported by the project Employment of Best Young Scientists for International Cooperation Empowerment (CZ.1.07/2.3.00/30.0037) co-financed from European Social Fund and the state budget of the Czech Republic.

View full text

Limit periodic homogeneous linear difference systems

Abstract

Introduction

Section snippets

Preliminaries

Limit periodic, almost periodic, and asymptotically almost periodic sequences

Homogeneous linear difference systems

Results

Acknowledgement

J. Diff. Equ.

Appl. Math. Comput.

J. Math. Anal. Appl.

Appl. Math. Comput.

Linear almost periodic difference equations

J. Comput. Math. Optim.

Semi-periodic solutions of difference and differential equations

Bound. Value Probl.

The exponential dichotomy of solutions to systems of linear difference equations with almost periodic coefficients

Russ. Math.

Almost Periodic Functions

Periodic, almost-periodic, and semiperiodic sequences

Michigan Math. J.

Method of Averaging for Differential Equations on an Infinite Interval. Theory and Applications

Almost periodic and almost automorphic solutions of linear differential equations

Discrete Contin. Dyn. Syst.

Asymptotically Almost Periodic Solutions of Differential Equations

Almost Periodic Functions

Almost periodic discrete processes

Libertas Math.

Almost Periodic Oscillations and Waves

Limit-periodic arithmetical functions and the ring of finite integral Adeles

Lith. Math. J.

Les fonctions asymptotiquement presque-périodiques d’une variable entière et leur application à l’étude de l’itération des transformations continues

Math. Z.

Almost Periodic Differential Equations

Strict ergodicity and transformation of the torus

Am. J. Math.

The Theory of Matrices

Almost periodic solutions of nonautonomous linear difference equations

Appl. Anal.

Existence of an almost periodic solution in a difference equation by Lyapunov functions

Nonlinear Stud.

The existence of almost periodic solutions for a class of differential equations with piecewise constant argument

Nonlin. Anal.

The method of Lyapunov functions for systems of linear difference equations with almost periodic coefficients

Siberian Math. J.