This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Semi-periodic solutions of difference and differential equations

Jan Andres1* and Denis Pennequin2

Author affiliations

1 Department of Mathematical Analysis, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 771 46, Czech Republic

2 Centre PMF, Laboratoire SAMM, Université Paris I Panthéon-Sorbonne, 90, Rue de Tolbiac, Paris Cedex 13, 75 634, France

For all author emails, please log on.

Citation and License

Boundary Value Problems 2012, 2012:141  doi:10.1186/1687-2770-2012-141


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/141


Received:1 October 2012
Accepted:7 November 2012
Published:28 November 2012

© 2012 Andres and Pennequin; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The spaces of semi-periodic sequences and functions are examined in the relationship to the closely related notions of almost-periodicity, quasi-periodicity and periodicity. Besides the main theorems, several illustrative examples of this type are supplied. As an application, the existence and uniqueness results are formulated for semi-periodic solutions of quasi-linear difference and differential equations.

MSC: 34C15, 34C27, 34K14, 39A10, 42A16, 42A75.

Keywords:
semi-periodic sequences; semi-periodic functions; semi-periodic solutions; difference equations; differential equations

Introduction

In [1], it is observed that although the set of periodic sequences forms a linear space, its uniform closure is not the space of almost-periodic sequences but of semi-periodic sequences. In fact, the space of semi-periodic sequences was shown there to be Banach.

The whole Sections I.6, I.7 in [2] and Sections II.4, II.5 in [3] are devoted to semi-periodic continuous functions, called there limit periodic functions (cf. also [[4], p.129]). This class was shown there to be identical with the one of uniformly almost-periodic functions with one-term ℚ-base and, in case of integral one-term base, it reduces to the one of purely periodic functions. For some more references concerning limit periodic functions, see, e.g., [5,6]. In fact, limit periodic functions were already considered by Bohr in 1925, as pointed out in [[3], p.113].

In the following section, we define analogously to [1] the class of semi-periodic continuous functions (with values in a Banach space) and show that it is the same as the class of limit periodic functions considered in [2,3] (see Theorem 1 below). Let us note that many different notions with the same name (i.e., semi-periodic), like functions satisfying Floquet boundary conditions (see, e.g., [7,8]) or those describing Bloch waves (see, e.g., [7], and the references therein), exist in the literature (cf. also [9,10]).

Hence, after giving a definition of semi-periodic functions, which is analogous to [1], we prove that the uniform closure of the set of periodic functions is again the one of semi-periodic functions. Unlike in the discrete case, the space of semi-periodic functions is, however, not linear and so not Banach. In order to clarify transparently the position of semi-periodic sequences and functions in the hierarchy of closely related spaces, we decided to illustrate it by means of Venn’s diagrams. Thus, the spaces of almost-periodic, semi-periodic, quasi-periodic and periodic functions and sequences and some of their sums (in the continuous case) are compared in this way. For this, the semi-periodicity is considered by means of the Fourier-Bohr coefficients.

There are even more general interesting classes of almost-periodic functions (for their hierarchy, see, e.g., [11,12]), but for our needs here only those which are uniformly (Bohr) a.p. will be taken into account. It is well known that uniformly continuous Stepanov a.p. functions are Bohr a.p. (see, e.g., [4,11]). Another nontraditional characterization of Bohr almost-periodicity was recently done in [13], namely that Stepanov a.p. functions with Stepanov a.p. derivatives are also Bohr a.p.

In order to make applications to difference and differential equations, we still need to define the notion of uniform semi-periodicity and prove that the associated Nemystkii operators map the set of semi-periodic sequences into themselves. This is unfortunately not true in the case of functions. On this basis, we finally give two examples about the existence of semi-periodic solutions in the form of theorems, both in the discrete and in the continuous cases. Although many various sorts of periodic-type solutions were investigated (for their panorama, see [7]), as far as we know, semi-periodic solutions in the sense of definitions below of difference or differential equations have been only considered in [14] and in a certain sense also in [5]. Nevertheless, as pointed out in [14], Johnson [15] and Millionshchikov [16] have already given examples of limit periodic differential equations which admit almost automorphic solutions, but not limit periodic ones.

Before passing to semi-periodic functions in the next section, it will be convenient to mention some facts about semi-periodic sequences.

Hence, denoting as usually by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M1">View MathML</a> ℤ the set of (positive) integers and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M2">View MathML</a> to be a Banach space endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M3">View MathML</a>, let us recall the definition of semi-periodic sequences (cf.[1]).

Definition 1 A sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M4">View MathML</a> is called semi-periodic (s.p.) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M5">View MathML</a>

One can readily check that Definition 1 can be regarded as a discrete version of Definition 2 below for semi-periodic functions. Similarly, the definition of quasi-periodic (q.p.) sequences can be regarded as a discretized (i.e., restricted to ℤ) version of the one for quasi-periodic functions recalled below. A q.p. extending function has the Fourier-Bohr expansion with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M6">View MathML</a> to be finitely generated which is also true for q.p. sequences. For more properties and details concerning q.p. functions, see, e.g., [17].

In this light, since the analogy of Theorem 2 below holds for sequences (see Remark 4) and since the discrete (i.e., restricted to ℤ) analogies of Examples 1-3 below can be constructed, one can illustrate the relationship of these classes by means of Venn’s diagram in Figure 1. For more properties about s.p. sequences, see, e.g., [1,18,19].

thumbnailFigure 1. Venn’s diagram: discrete case.

On the other hand, the situation in Figure 1 is much simpler than in Figure 2 for continuous functions, because under the restriction to ℤ, the sum of (semi-)periodic sequences remains (semi-)periodic while Stepanov almost-periodic (a.p.) sequences were shown in [20] to coincide with Bohr a.p. sequences.

thumbnailFigure 2. Venn’s diagram: continuous case.

Continuous semi-periodic functions

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M7">View MathML</a> be the set of continuous T-periodic functions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M8">View MathML</a>

be the set of periodic functions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M9">View MathML</a> be the set of continuous bounded functions. The last one is a Banach space with the uniform norm (written <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M10">View MathML</a>).

Definition 2 A continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M11">View MathML</a> is said to be semi-periodic (s.p.) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M12">View MathML</a>

Such a T will be called an ε-semi-period of f.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M13">View MathML</a> denote the set of semi-periodic functions.

It is easy to see from the definition that every continuous periodic function is semi-periodic. Moreover, if f is semi-periodic, then f is uniformly (Bohr) almost-periodic (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M14">View MathML</a>), and so it is bounded. Thus, we can rewrite Definition 2 as follows.

Definition 3 A (bounded) continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M11">View MathML</a> is said to be semi-periodic (s.p.) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M16">View MathML</a>

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M17">View MathML</a>

From this, we can consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M13">View MathML</a> as a metric space, when using

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M19">View MathML</a>

As we will see later, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M13">View MathML</a> is not a linear space, but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M13">View MathML</a> is a complete metric space.

Lemma 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M23">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M24">View MathML</a>be anε-semi-period off. Then there exists a continuous<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M24">View MathML</a>-periodic functionφs.t.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M26">View MathML</a>

Proof Consider a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M24">View MathML</a>-periodic function ψ such that its restriction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M28">View MathML</a> is the same as the one of f. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M29">View MathML</a>, we can write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M30">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M32">View MathML</a>. Thus, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M33">View MathML</a>

Since ψ is not necessarily continuous, consider still <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M34">View MathML</a> such that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M36">View MathML</a>. Define a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M24">View MathML</a>-periodic continuous function φ which is equal to ψ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M38">View MathML</a> and which is linear on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M39">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M35">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M41">View MathML</a>

and subsequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M42">View MathML</a>

 □

Remark 1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M43">View MathML</a>, unlike for semi-periodic functions in the sense of Definition 2 or Definition 3, in fact the same lemma was already proved in [[3], pp.114-115], but for limit periodic functions. As already pointed out in the foregoing section, these classes will be shown to coincide by Theorem 1 below, whose proof is just based on Lemma 1.

We are ready to give the first theorem.

Theorem 1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M13">View MathML</a>is the closure of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M45">View MathML</a>in the sup-norm.

Proof Assume firstly that f is s.p. Taking in Lemma 1 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M46">View MathML</a>, we obtain a sequence of periodic functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M47">View MathML</a> s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M48">View MathML</a>.

Reversely, assume that f is in the closure of the set of continuous T-periodic functions. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M23">View MathML</a>, we can find a periodic φ s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M50">View MathML</a>. Let T be its period. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M51">View MathML</a>,

 □

Remark 2 In view of Theorem 1, one can now also define a semi-periodic function, equivalently w.r.t. Definition 2 and Definition 3, as the uniform limit of a uniformly convergent sequence of continuous purely periodic functions. This was so done, e.g., in [2,3,5,6,14].

In the following proposition, we look for the link between s.p. sequences and functions. Given a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M53">View MathML</a>, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M54">View MathML</a>, the function s.t. its restriction to ℤ is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M55">View MathML</a> and which is linear on each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M57">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M58">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M59">View MathML</a> is the fractional part of u, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M61">View MathML</a>.

Proposition 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M4">View MathML</a>. All the following statements are equivalent:

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M63">View MathML</a>is s.p. with a semi-period in ℕ,

2. there exists a s.p. function with a semi-period inwhose restriction tois<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M55">View MathML</a>,

3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M55">View MathML</a>is s.p.

Proof For (1) ⇒ (2), take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M63">View MathML</a> in (2). For (2) ⇒ (3), take T as an ε-semi-period for the function f in (2). Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M67">View MathML</a>

For (3) ⇒ (1), given T as an ε-semi-period of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M55">View MathML</a>, we have for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M69">View MathML</a>

 □

Let us now consider the Fourier expansion of a semi-periodic function. Recall that every a.p. function has the Fourier-Bohr expansion,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M71">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M72">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M73">View MathML</a>

is the mean operator (see, e.g., [3,4,11]). It follows from the above formula that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M74">View MathML</a> is 1-Lipschizian (and so it is continuous) from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M75">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M2">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M77">View MathML</a> and denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M78">View MathML</a> the ℤ-modulus generated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M79">View MathML</a>. Recall that an a.p. function is quasi-periodic (q.p.) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M78">View MathML</a> has a finite ℤ-basis, and that T is a period of f if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M81">View MathML</a> (see, e.g., [4,17]).

Proposition 2 (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M43">View MathML</a>, cf. [[2], p.32])

Set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M83">View MathML</a>and consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M84">View MathML</a>

for a fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M85">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M86">View MathML</a>

Thenfis s.p.

Proof Consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M87">View MathML</a>

Clearly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M88">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M89">View MathML</a> is a period of the nth term. The same is obviously true for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M90">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M91">View MathML</a> is a period of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M92">View MathML</a> which is so periodic. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M93">View MathML</a>

which already proves that f is s.p. □

The following result is also, at least for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M43">View MathML</a>, well known (see, e.g., [14], [[3], pp.118-119], and the references therein).

Lemma 2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M95">View MathML</a>, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M85">View MathML</a>s.t.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M97">View MathML</a>

Proof Let us consider λ and μ s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M99">View MathML</a> and a sequence of periodic functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M100">View MathML</a> s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M101">View MathML</a>, uniformly. It follows from the continuity that, for sufficiently large N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M103">View MathML</a>, but since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M92">View MathML</a> is periodic, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M105">View MathML</a>. □

Remark 3

1. This proof also demonstrates that, for a sufficiently large n, the period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M106">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M107">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M108">View MathML</a>.

2. It indicates that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M109">View MathML</a> is not a linear space. For instance, a simple q.p. function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M110">View MathML</a> is not s.p. although it is a sum of two s.p. functions. On the other hand, the sum of two a.p. functions is trivially a.p.

Example 1 On the basis of Proposition 2 and Lemma 2, we can easily give the following example of a purely s.p. (i.e., not periodic) function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M111">View MathML</a>

Moreover, one can readily check that the function f can be obtained as a uniform limit of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M112">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M92">View MathML</a> is a continuous <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M114">View MathML</a>-periodic function,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M115">View MathML</a>

Theorem 2Every s.p. function which is also q.p. is in fact periodic:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M116">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M117">View MathML</a>. Since f is q.p., we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M118">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M119">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M120">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M121">View MathML</a> is an additive subgroup of ℝ. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M22">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M85">View MathML</a> s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M124">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M125">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M126">View MathML</a> is another additive subgroup of ℝ, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M127">View MathML</a> is a subgroup of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M121">View MathML</a> which contains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M79">View MathML</a>. Since G is a subgroup of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M121">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M131">View MathML</a> and positive ℤ-independent real numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M132">View MathML</a> s.t.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M133">View MathML</a>

Let us show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M134">View MathML</a>. Once we have it, we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M135">View MathML</a> which proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M136">View MathML</a> is a period of f. Since, for each i, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M137">View MathML</a>, we know that, for each i, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M138">View MathML</a> s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M139">View MathML</a>. This proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M140">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M141">View MathML</a>, which is impossible. □

Remark 4 In view of Proposition 1 and its analogy for q.p. sequences mentioned in the foregoing section, a discrete (i.e., restricted to ℤ) analogy of Theorem 2 holds for sequences.

Example 2 As an example of a function which is almost-periodic (a.p.) but neither quasi-periodic nor a sum of semi-periodic functions, consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M142">View MathML</a>

where the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M143">View MathML</a>’s are constructed by induction, say for all k,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M144">View MathML</a>

We will prove that we cannot find a finite set of numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M145">View MathML</a> s.t.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M146">View MathML</a>

Firstly, assume this has already been proved. Then if f is a sum of semi-periodic functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M147">View MathML</a>, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M148">View MathML</a>, we could find, according to Lemma 2, for each j a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M149">View MathML</a> s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M150">View MathML</a>. This implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M151">View MathML</a>

which is not true. If f were quasi-periodic, we could find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M145">View MathML</a> s.t.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M153">View MathML</a>

which is again wrong. Now, we can make the first part of the proof. So, let us assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M154">View MathML</a>

We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M155">View MathML</a>. Thus, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M156">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M157">View MathML</a> s.t.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M158">View MathML</a>

Let us now consider the square matrix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M159">View MathML</a>

If it is invertible, we can express <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M145">View MathML</a> linearly (with rational coefficients) depending on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M161">View MathML</a>. This proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M162">View MathML</a> should be a (rational) linear combination of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M161">View MathML</a>, which is not true.

Assuming that the matrix is singular, its rows are linearly dependent. So, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M164">View MathML</a> s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M165">View MathML</a>, for each j. Multiplying it by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M149">View MathML</a> and then summing over j, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M167">View MathML</a> which is not possible.

Example 3 As an example of a function which is quasi-periodic (q.p.) but not a sum of periodic functions, consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M168">View MathML</a>

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M169">View MathML</a>, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M170">View MathML</a>, i.e., f is q.p. Assume that f is a sum of a finite number of periodic functions. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M171">View MathML</a> be the periods. According to [21], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M172">View MathML</a>

where

An easy calculation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M174">View MathML</a>

by which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M175">View MathML</a>

Since ℕ is infinite, we can find two different integers m, n with the same <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M176">View MathML</a>. Thus, there exist two integers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M178">View MathML</a> s.t.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M179">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M180">View MathML</a>, and we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M181">View MathML</a>

which is not possible.

Remark 5 We know (see, e.g., [4,11]) that every almost-periodic (a.p.) f is a uniform limit of a sequence of a finite sum of periodic functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M100">View MathML</a>. Writing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M183">View MathML</a>

we can see that every a.p. function can be expressed as a series of periodic functions. Reversely, a uniformly convergent series of periodic functions is a.p.

Summing up the above observations, we can present in Figure 2 Venn’s diagram for continuous functions under our investigation. The classes of almost-periodic, semi-periodic and quasi-periodic functions are in circles, while sums of semi-periodic functions are in the ellipse. Sums of periodic functions are in the intersection of the classes of quasi-periodic functions and sums of semi-periodic functions. In fact, one can check by similar arguments as in the proof of Theorem 2 that a sum of periodic functions is exactly the sum of semi-periodic functions which is quasi-periodic. Periodic functions are, according to Theorem 2, at the same time semi-periodic and quasi-periodic. Purely semi-periodic functions are in the grey strip.

Now, consider the primitives of s.p. functions.

Lemma 3Assume thatfis a.p. and consider<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M184">View MathML</a>. Assume that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M185">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M186">View MathML</a>s.t.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M187">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M188">View MathML</a>.

Indeed, φ is necessarily differentiable, and integrating the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M189">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M190">View MathML</a>

because φ is bounded. This already proves Lemma 3. It is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M191">View MathML</a> is a necessary and sufficient condition for F to be periodic, provided f is so. It is, however, not sufficient in the case of a.p. functions. For more details, see, e.g., [22]. Despite the approximation by periodic functions, it is also not sufficient in the case of s.p. functions, as demonstrated by the following example.

Example 4 Let us consider the s.p. function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M192">View MathML</a>

We have a normal convergence, so the series exists and defines a s.p. function for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M191">View MathML</a>. A formal candidate to be its primitive is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M194">View MathML</a>

We have a uniform convergence on each compact set, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M195">View MathML</a>. Thus, this series also exists and defines a primitive of f. If F were s.p., it should be a.p. which is obviously not true, because the Parseval equality does not apply.

Uniformly semi-periodic functions with respect to a parameter

Definition 4 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M196">View MathML</a>, where M is a subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M197">View MathML</a>. We say that f is uniformly semi-periodic (u.s.p.) if for any compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M198">View MathML</a>, we have

Since such a function is u.a.p., we know that given a compact subset K of M, f is bounded and uniformly continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M200">View MathML</a>.

Proposition 3Any u.s.p. function is a uniform limit, on each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M200">View MathML</a>, of a sequence of continuous functions which are periodic w.r.t. their first variables.

Proof Let T be given by the definition and consider a T-periodic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M202">View MathML</a> such that its restriction to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M203">View MathML</a> is the same as the one of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M204">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M29">View MathML</a>, we can write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M206">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M208">View MathML</a>. Thus, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M209">View MathML</a>

uniformly w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M210">View MathML</a>. Since φ is not necessarily continuous, consider still <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M211">View MathML</a> such that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M212">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M210">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M214">View MathML</a>. This is possible, because K is compact. Define a T-periodic continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M215">View MathML</a> which is equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M202">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M217">View MathML</a> and which is linear on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M218">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M212">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M220">View MathML</a>

and subsequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M221">View MathML</a>

 □

Remark 6 Assume that f is L-Lipschitzian w.r.t. its second variable. It follows from the proof that so is φ, from which we can deduce the same for ψ. So, a u.s.p. function Lipschitzian w.r.t. its second variable can be approximated uniformly on each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M200">View MathML</a> (K compact) by a sequence of functions which are periodic w.r.t. their first variables and Lipschitzian (with the same constant L) w.r.t. their second variables.

Remark 7 It is possible to define the same for the discrete case and to obtain analogous results. This will be omitted here, because the proofs are quite similar.

Concerning the Nemytskii operator, in the continuous case, it is not true that if f is u.s.p. and ϕ is s.p., then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M223">View MathML</a> is s.p. As an example, take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M224">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M225">View MathML</a>. On the other hand, it is true in the discrete case.

Proposition 4Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M226">View MathML</a>is s.p. and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M227">View MathML</a>is s.p. with the range in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M228">View MathML</a>. Then the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M229">View MathML</a>is s.p.

Proof Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M230">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M55">View MathML</a> is a.p., K is a compact subset of M. So, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M23">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M233">View MathML</a> s.t.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M234">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M235">View MathML</a>. We know that we can find two integers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M237">View MathML</a> s.t.

Let T be a common multiplier of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M237">View MathML</a> (for instance, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M241">View MathML</a>). The last inequalities remain true, when replacing every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M242">View MathML</a> by T. Thus, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M243">View MathML</a>,

 □

For an alternative proof, one can employ the approximation by periodic sequences.

Semi-periodic solutions of difference equations

In this section, we are interested in semi-periodic solutions of the difference equation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M245">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M246">View MathML</a>

(1)

where A is a real square <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M247">View MathML</a> matrix.

Theorem 3Assuming thatAhas no eigenvalues with modulus one and thatfis u.s.p. and Lipschitzian w.r.t. the second variable with a sufficiently small constant, there exists a unique semi-periodic solution for the difference equation (1).

Proof We know (see, e.g., Proposition 2.2 in [23]) that, for each a.p. sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M248">View MathML</a> with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M245">View MathML</a>, there exists a unique a.p. solution to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M250">View MathML</a>

(2)

Denoting by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M251">View MathML</a> the Banach space of a.p. sequences (cf.[23]), the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M252">View MathML</a>, determined by the left-hand side of (2), is obviously invertible. Since T is continuous satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M253">View MathML</a>, we know from the well-known Banach theorem that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M254">View MathML</a> must be continuous as well.

Now, consider a s.p. sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M255">View MathML</a> with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M245">View MathML</a>. We are firstly interested in the a.p. solution to the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M257">View MathML</a>

(3)

By the hypothesis imposed on f and in view of Proposition 4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M258">View MathML</a> is s.p. Therefore, there exists a unique a.p. solution of (3) (see again Proposition 2.2 in [23]). We can now consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M259">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M254">View MathML</a> maps the space of periodic sequences into itself, by the unique solvability of (3) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M261">View MathML</a> and by the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M254">View MathML</a>, the mapping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M263">View MathML</a>

maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M264">View MathML</a> into itself. Denote by L the Lipschitz constant to all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M265">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M266">View MathML</a> is a Lipschitz constant for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M267">View MathML</a>.

Assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M268">View MathML</a>, the mapping T is a contraction in the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M264">View MathML</a>. So it has a unique fixed point representing the desired s.p. solution of (1). □

Remark 8 Using a triangular form of −A (like Jordan’s one) (see, e.g., [[4], Proposition 6.14 and Remark 6.26]), it is possible to compute explicitly a constant c s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M270">View MathML</a>. For such a constant, it is sufficient to assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M271">View MathML</a> in order to justify Theorem 3.

Semi-periodic solutions of differential equations

Let us consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M272">View MathML</a>

(4)

We assume that a real square <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M273">View MathML</a> matrix A has an exponential dichotomy property, i.e., that there exist a projection matrix P (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M274">View MathML</a>) and constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M275">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M276">View MathML</a>, such that

where X is the fundamental matrix of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M278">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M279">View MathML</a>, i.e., the unit matrix (see, e.g., [[8], Chapter III.5]). Furthermore, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M280">View MathML</a> be u.s.p. with respect to the variable x.

Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M281">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M282">View MathML</a>

is the Green function associated to A, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M284">View MathML</a> stand for the corresponding spectral projections on the invariant subspaces of A, we can formulate the following theorem.

Theorem 4Assume still thatfisL-Lipschitzian w.r.t. the second variable with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M285">View MathML</a>. Then there exists a unique semi-periodic solution of the equation (4).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M100">View MathML</a> be a sequence of periodic functions w.r.t. their first variables s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M287">View MathML</a>, uniformly. We can assume without any loss of generality (see Remark 3) that each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M107">View MathML</a> is L-Lipschitzian w.r.t. its second variable. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M289">View MathML</a> be the unique bounded (in fact, periodic) solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M290">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M291">View MathML</a> be the unique bounded solution of (4). Such solutions exist; for more details, see, e.g., [[8], Chapter III.5].

It will be sufficient to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M292">View MathML</a>, uniformly.

We have the integral representations (see again, e.g., [[8], Chapter III.5])

It can be easily checked that, in view of uniqueness of bounded solutions, the periods <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M106">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M107">View MathML</a> are also periods of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M289">View MathML</a>. It holds

Now, let us prove that there exists a uniform estimate to all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M289">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M299">View MathML</a>

and

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M301">View MathML</a>

and, according to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M302">View MathML</a>, still

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M303">View MathML</a>

where R is the desired bound. Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M304">View MathML</a>, we arrive at

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M306">View MathML</a>. Thus, we finally get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M307">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M289">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M106">View MathML</a>-periodic, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M291">View MathML</a> is semi-periodic. □

Concluding remarks

Remark 9 Because of the right-hand side <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M311">View MathML</a> in (4), even in the scalar case, Theorem 4 cannot be deduced from the results in [14], where the scalar equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/141/mathml/M312">View MathML</a> was considered.

Remark 10 Since Theorem 3 and Theorem 4 represent only illustrative examples, the obtained existence and uniqueness criteria were tendentiously very simple. More sophisticated situations will be considered by ourselves elsewhere.

Remark 11 Analogously as in [24,25], where almost-periodic solutions were under consideration, it would be interesting to obtain similar results concerning semi-periodic solutions of monotone systems or those treated by means of variational methods.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in this article. They read and approved the final manuscript.

Acknowledgements

The first author was supported by the project A-Math-Net Applied Mathematics Knowledge Transfer Network No CZ.1.07/2.4.00/17.0100.

References

  1. Berg, JD, Wilansky, A: Periodic, almost-periodic, and semiperiodic sequences. Mich. Math. J.. 9, 363–368 (1962)

  2. Besicovitch, AS: Almost Periodic Functions, Dover, New York (1954)

  3. Levitan, BM: Almost-Periodic Functions, GITTL, Moscow (1953) in Russian

  4. Corduneanu, C: Almost Periodic Oscillations and Waves, Springer, Berlin (2009)

  5. Bell, H, Meyer, KR: Limit periodic functions, adding machines, and solenoids. J. Dyn. Differ. Equ.. 7, 409–422 (1995). Publisher Full Text OpenURL

  6. Schwarz, W, Spilker, J: Arithmetical Functions, Cambridge University Press, Cambridge (1994)

  7. Andres, J: Periodic-type solutions of differential inclusions. In: Baswell AR (ed.) Advances in Mathematical Research, pp. 295–353. Nova Sciences Publishers, New York (2009)

  8. Andres, J, Górniewicz, L: Topological Fixed Point Principles for Boundary Value Problems, Kluwer Academic, Dordrecht (2003)

  9. Ichihara, N, Ishii, H: Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians. Commun. Partial Differ. Equ.. 33, 784–807 (2008). Publisher Full Text OpenURL

  10. Lorenz, EN: Noisy periodicity and reverse bifurcation. Nonlinear Dynamics. 282–291 (1980)

  11. Andres, J, Bersani, AM, Grande, RF: Hierarchy of almost-periodic function spaces. Rend. Mat. Appl.. 26, 121–188 (2006)

  12. Corduneanu, C: A scale of almost periodic function spaces. Differ. Integral Equ.. 24, 1–28 (2011)

  13. Andres, J, Pennequin, D: On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations. Proc. Am. Math. Soc.. 140, 2825–2834 (2012). Publisher Full Text OpenURL

  14. Alonso, AI, Obaya, R, Ortega, R: Differential equations with limit-periodic forcings. Proc. Am. Math. Soc.. 131, 851–857 (2002)

  15. Johnson, RA: On almost-periodic linear differential systems of Milionshchikov and Vinograd. J. Math. Anal. Appl.. 85, 452–460 (1982). Publisher Full Text OpenURL

  16. Millionshchikov, VM: Proof of the existence of irregular systems of linear differential equations with almost periodic coefficients. Differ. Equ.. 4, 203–205 (1968)

  17. Blot, J, Pennequin, D: Spaces of quasi-periodic functions and oscillations in differential equations. Acta Appl. Math.. 65, 83–113 (2001). Publisher Full Text OpenURL

  18. Goes, G: Fourier-Stieltjes transforms of discrete measures; periodic and semiperiodic functions. Math. Ann.. 174, 148–156 (1967). Publisher Full Text OpenURL

  19. Jiménez, MN: Multipliers on the space of semiperiodic sequences. Trans. Am. Math. Soc.. 291, 801–811 (1985)

  20. Andres, J, Pennequin, D: On Stepanov almost-periodic oscillations and their discretizations. J. Differ. Equ. Appl.. 18, 1665–1682 (2012). Publisher Full Text OpenURL

  21. Mortola, S, Peirone, R: The sum of periodic functions. Boll. Unione Mat. Ital.. 8, 393–396 (1999)

  22. Andres, J, Bednařík, D, Pastor, K: On the notion of derivo-periodicity. J. Math. Anal. Appl.. 303, 405–417 (2005). Publisher Full Text OpenURL

  23. Pennequin, D: Existence of almost periodic solutions of discrete time equations. Discrete Contin. Dyn. Syst.. 7, 51–60 (2001)

  24. Blot, J, Cieutat, P, Mawhin, J: Almost periodic oscillations of monotone second-order systems. Adv. Differ. Equ.. 2, 693–714 (1997)

  25. Mawhin, J: Bounded and almost periodic solutions of nonlinear differential equations: variational vs nonvariational approach. In: Ioffe A, Reich S, Shafrir I (eds.) Calculus of Variations and Differential Equations, pp. 167–184. Chapman & Hall/CRC, Boca Raton (1999)