Abstract
The spaces of semiperiodic sequences and functions are examined in the relationship to the closely related notions of almostperiodicity, quasiperiodicity 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 semiperiodic solutions of quasilinear difference and differential equations.
MSC: 34C15, 34C27, 34K14, 39A10, 42A16, 42A75.
Keywords:
semiperiodic sequences; semiperiodic functions; semiperiodic solutions; difference equations; differential equationsIntroduction
In [1], it is observed that although the set of periodic sequences forms a linear space, its uniform closure is not the space of almostperiodic sequences but of semiperiodic sequences. In fact, the space of semiperiodic 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 semiperiodic 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 almostperiodic functions with oneterm ℚbase and, in case of integral oneterm 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 semiperiodic 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., semiperiodic), 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 semiperiodic functions, which is analogous to [1], we prove that the uniform closure of the set of periodic functions is again the one of semiperiodic functions. Unlike in the discrete case, the space of semiperiodic functions is, however, not linear and so not Banach. In order to clarify transparently the position of semiperiodic 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 almostperiodic, semiperiodic, quasiperiodic and periodic functions and sequences and some of their sums (in the continuous case) are compared in this way. For this, the semiperiodicity is considered by means of the FourierBohr coefficients.
There are even more general interesting classes of almostperiodic 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 almostperiodicity 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 semiperiodicity and prove that the associated Nemystkii operators map the set of semiperiodic 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 semiperiodic solutions in the form of theorems, both in the discrete and in the continuous cases. Although many various sorts of periodictype solutions were investigated (for their panorama, see [7]), as far as we know, semiperiodic 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 semiperiodic functions in the next section, it will be convenient to mention some facts about semiperiodic sequences.
Hence, denoting as usually by
Definition 1 A sequence
One can readily check that Definition 1 can be regarded as a discrete version of Definition 2
below for semiperiodic functions. Similarly, the definition of quasiperiodic (q.p.)
sequences can be regarded as a discretized (i.e., restricted to ℤ) version of the one for quasiperiodic functions recalled below.
A q.p. extending function has the FourierBohr expansion with
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 13 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].
Figure 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 almostperiodic (a.p.) sequences were shown in [20] to coincide with Bohr a.p. sequences.
Figure 2. Venn’s diagram: continuous case.
Continuous semiperiodic functions
Let
be the set of periodic functions and
Definition 2 A continuous function
Such a T will be called an εsemiperiod of f.
Let
It is easy to see from the definition that every continuous periodic function is semiperiodic.
Moreover, if f is semiperiodic, then f is uniformly (Bohr) almostperiodic (i.e.,
Definition 3 A (bounded) continuous function
We have
From this, we can consider
As we will see later,
Lemma 1Let
Proof Consider a
Since ψ is not necessarily continuous, consider still
and subsequently
□
Remark 1 For
We are ready to give the first theorem.
Theorem 1
Proof Assume firstly that f is s.p. Taking in Lemma 1
Reversely, assume that f is in the closure of the set of continuous Tperiodic functions. Then, for any
□
Remark 2 In view of Theorem 1, one can now also define a semiperiodic 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
where
Proposition 1Let
1.
2. there exists a s.p. function with a semiperiod in ℕ whose restriction to ℤ is
3.
Proof For (1) ⇒ (2), take
For (3) ⇒ (1), given T as an εsemiperiod of
□
Let us now consider the Fourier expansion of a semiperiodic function. Recall that every a.p. function has the FourierBohr expansion,
where
and
is the mean operator (see, e.g., [3,4,11]). It follows from the above formula that
Set
Proposition 2 (for
Set
for a fixed
Thenfis s.p.
Proof Consider
Clearly, if
which already proves that f is s.p. □
The following result is also, at least for
Lemma 2If
Proof Let us consider λ and μ s.t.
Remark 3
1. This proof also demonstrates that, for a sufficiently large n, the period
2. It indicates that
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:
Moreover, one can readily check that the function f can be obtained as a uniform limit of the sequence
Theorem 2Every s.p. function which is also q.p. is in fact periodic:
Proof Let
Set
Let us show that
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 almostperiodic (a.p.) but neither quasiperiodic nor a sum of semiperiodic functions, consider
where the
We will prove that we cannot find a finite set of numbers
Firstly, assume this has already been proved. Then if f is a sum of semiperiodic functions
which is not true. If f were quasiperiodic, we could find
which is again wrong. Now, we can make the first part of the proof. So, let us assume
We have
Let us now consider the square matrix
If it is invertible, we can express
Assuming that the matrix is singular, its rows are linearly dependent. So, we can
find
Example 3 As an example of a function which is quasiperiodic (q.p.) but not a sum of periodic functions, consider
Here
where
An easy calculation yields
by which
Since ℕ is infinite, we can find two different integers m, n with the same
This implies that
which is not possible.
Remark 5 We know (see, e.g., [4,11]) that every almostperiodic (a.p.) f is a uniform limit of a sequence of a finite sum of periodic functions
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 almostperiodic, semiperiodic and quasiperiodic functions are in circles, while sums of semiperiodic functions are in the ellipse. Sums of periodic functions are in the intersection of the classes of quasiperiodic functions and sums of semiperiodic 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 semiperiodic functions which is quasiperiodic. Periodic functions are, according to Theorem 2, at the same time semiperiodic and quasiperiodic. Purely semiperiodic functions are in the grey strip.
Now, consider the primitives of s.p. functions.
Lemma 3Assume thatfis a.p. and consider
Then
Indeed, φ is necessarily differentiable, and integrating the equality
because φ is bounded. This already proves Lemma 3. It is well known that
Example 4 Let us consider the s.p. function
We have a normal convergence, so the series exists and defines a s.p. function for
which
We have a uniform convergence on each compact set, because
Uniformly semiperiodic functions with respect to a parameter
Definition 4 Let
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
Proposition 3Any u.s.p. function is a uniform limit, on each
Proof Let T be given by the definition and consider a Tperiodic function
uniformly w.r.t.
and subsequently
□
Remark 6 Assume that f is LLipschitzian 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
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
Proposition 4Assume that
Proof Set
Set
Let T be a common multiplier of
□
For an alternative proof, one can employ the approximation by periodic sequences.
Semiperiodic solutions of difference equations
In this section, we are interested in semiperiodic solutions of the difference equation
in
where A is a real square
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 semiperiodic solution for the difference equation (1).
Proof We know (see, e.g., Proposition 2.2 in [23]) that, for each a.p. sequence
Denoting by
Now, consider a s.p. sequence
By the hypothesis imposed on f and in view of Proposition 4,
maps
Assuming that
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.
Semiperiodic solutions of differential equations
Let us consider the equation
We assume that a real square
where X is the fundamental matrix of
Setting
where
is the Green function associated to A, and
Theorem 4Assume still thatfisLLipschitzian w.r.t. the second variable with
Proof Let
and
It will be sufficient to show that
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
Now, let us prove that there exists a uniform estimate to all
and
Thus,
and, according to
where R is the desired bound. Putting
where
Since
Concluding remarks
Remark 9 Because of the righthand side
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 almostperiodic solutions were under consideration, it would be interesting to obtain similar results concerning semiperiodic 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 AMathNet Applied Mathematics Knowledge Transfer Network No CZ.1.07/2.4.00/17.0100.
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