SEMI-BLOCH PERIODIC FUNCTIONS, SEMI-ANTI-PERIODIC FUNCTIONS AND APPLICATIONS

Chelyabinsk Physical and Mathematical Journal. 2020. Vol. 5, iss. 2. P. 243-255.

DOI: 10.24411/2500-0101-2020-15211

SEMI-BLOCH PERIODIC FUNCTIONS, SEMI-ANTI-PERIODIC FUNCTIONS AND APPLICATIONS

B. Chaouchi1", M. Kostic2,b, S. Pilipovic2,c, D. Velinov3,d

"chaouchicukm@gmail.com, bmarco.s@verat.net, cpilipovic@dmi.uns.ac.rs,

dvelinovd@gf.ukim.edu.mk

We introduce the notions of semi-Bloch periodic functions and semi-anti-periodic functions. Stepanov semi-Bloch periodic functions and Stepanov semi-anti-periodic functions are considered, as well. We analyze the invariance of introduced classes under the actions of convolution products and briefly explain how one can use the obtained results in the qualitative analysis of solutions of abstract inhomogeneous integro-differential equations.

Let p > 0 and k G R. The study of Bloch (p, k)-periodic functions is an important subject of applied functional analysis. In fact, this kind of functions is often encountered in quantum mechanics and solid state physics. Also, they are very attractive topic in the qualitative theory of differential equations (see the papers [1] by M.F. Hasler, G.M.N’. Guerekata, [2] by M.F. Hasler, [3] by M. Kostic, D. Velinov and references cited therein). As it is well known, the notion of an anti-periodic function is a special case of the notion of a Bloch (p, k)-periodic function. For more details about anti-periodic type functions and their applications, we refer the reader to [4-9] and references cited therein.

The genesis of this paper is motivated by reading the paper [10] by J. Andres and

D. Pennequin, where the authors have introduced and analyzed the class of semi-periodic functions (sequences) and related applications to differential (difference) equations; see also [11]. For the sake of brevity, in this paper we will consider only functions and differential equations.

The main ideas and organization of paper can be briefly described as follows. After collecting necessary definitions and results needed for our further work, in Section 2 we introduce and study the classes of semi-Bloch k-periodic functions and semi-antiperiodic functions (cf. Definition 2 and Definition 3). Any semi-anti-periodic function is semi-periodic and almost anti-periodic [9], while any anti-periodic function is anti-semiperiodic (cf. Proposition 2). In actual fact, the space consisting of all semi-anti-periodic functions is nothing else but the closure of space of all anti-periodic functions in the space

This research is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.

B. Chaouchi, M. Kostic, S. Pilipovic, D. Velinov

of bounded continuous functions (cf. Proposition 3). Similarly, the space consisting of all semi-Bloch k-periodic functions is, in fact, the closure of the space consisting of all Bloch (p, k)-periodic functions, where the parameter p > 0 runs through the positive real axis (cf. Proposition 1). In the same proposition, we show that a function f(■) is semi-Bloch k-periodic iff the function e~lk&f (■) is semi-periodic; with k = 0, this space is therefore reduced to the space of semi-periodic functions. We provide several new observations about Bloch (p, k)-periodic functions; for example, any Bloch (p, k)-periodic function needs to be almost periodic; see also Remark 1. Finally, in Definition 4 we introduce the notions of asymptotical semi-periodicity (asymptotical semi-Bloch k-periodicity, asymptotical semi-anti-periodicity).

In [10], the authors have considered only continuous functions. The main aim of Section 3 is to introduce and analyze the classes of Stepanov semi-Bloch k-periodic functions and Stepanov semi-anti-periodic functions. The invariance of introduced classes under the actions of convolution products are considered in Section 4, where it is shortly explained, without giving full details, how one can use the obtained results in the qualitative analysis of solutions of abstract inhomogeneous integro-differential equations and inclusions. In addition to the above, we present several illustrative examples of Stepanov semi-Bloch k-periodic functions and Stepanov semi-periodic-functions in Section 2 and Section 3.

We use the standard terminology throughout the paper. Unless stated otherwise, we will always assume henceforth that (E, || ■ ||) and (X, || ■ ||X) are two complex Banach spaces. By L(E,X) we denote the space consisting of all linear continuous mappings from E into X; L(E) = L(E,E). Assume that I = R or I = [0, to). By Cb(I : E) we denote the space consisting of all bounded continuous functions from I into E; C0([0, to) : E) denotes the closed subspace of Cb(I : E) consisting of all functions vanishing at infinity. By BUC(I : E) we denote the space consisted of all bounded uniformly continuous functions f : I ^ E. Finally, by Pc(I : E) we denote the space consisting of all continuous periodic functions f : I ^ E. The sup-norm turns these spaces into Banach’s. We will use the symbol * to denote the infinite convolution product; generally, if the functions a : R ^ C and f : R ^ E are measurable, then we denote (a * f )(x) D a(x — y)f (y) dy, x G R, if this integral is well-defined.

The concept of almost periodicity was introduced by Danish mathematician H. Bohr around 1924-1926 and later generalized by many other authors. Suppose that a function f : I ^ E is continuous. Given t > 0, we call т > 0 an e-period for f (■) iff

llf(t + т) — f(t)|l < t t G L

The set constituted of all t-periods for f (■) is denoted by d(f, t). It is said that f (■) is almost periodic iff for each t > 0 the set d(f, t) is relatively dense in [0, to), which means that there exists l > 0 such that any subinterval of [0, to) of length l meets d(f, t). The space consisted of all almost periodic functions from the interval I into E will be denoted by AP(I : E). Equipped with the sup-norm, AP(I : X) becomes a Banach space. Bohr’s transform of any function f G AP (I : X), defined by

lim - e irsf (s) ds,

exists for all r G R. The element Pr(f) is called the Bohr’s coefficient of f (■) and we know that, if Pr(f) = 0 for all r G R, then f (t) = 0 for all t G I as well as that Bohr’s spectrum a(f) := {r G R : Pr(f) = 0} is at most countable. By APa(I : X), where Л is

Semi-Bloch periodic functions, semi-anti-periodic functions and applications

a non-empty subset of I, we denote the vector subspace of AP(I : X) consisting of all functions f £ AP (I : X) for which the inclusion a(f) С Л holds good.

For the sequel, we need some preliminary results from the pioneering paper [12] by H. Bart and S. Goldberg, who introduced the notion of an almost periodic strongly continuous semigroup there. The translation semigroup (W(t))t>0 on AP([0, to) : E), given by [W(t)f](s) := f (t + s), t > 0, s > 0, f £ AP([0, to) : E) is consisted solely of surjective isometries W(t) (t > 0) and can be extended to a C0-group (W(t))teR of isometries on AP([0, to) : E), where W(—t) := W(t)-1 for t > 0. Furthermore, the mapping E : AP([0, to) : E) ^ AP(R : E), defined by

[Ef](t) := [W(t)f](0), t £ R, f £ AP([0, to) : E),

is a linear surjective isometry and Ef(■) is the unique continuous almost periodic extension of function f (■) from AP([0, to) : E) to the whole real line. We have that [E(Bf)] = B(Ef) for all B £ L(E) and f £ AP([0, to) : E).

Let 1 < q < to. A function f £ Lqoc(I : E) is Stepanov q-bounded iff

llf lls* :=stu^^ |f (s)ll? dsl < to.

Equipped with the above norm, the space LqS (I : E) consisted of all Sq-bounded functions is the Banach space. A function f £ LqS (I : E) is said to be Stepanov q-almost periodic, Sq-almost periodic shortly, iff the function f : I ^ Lq([0,1] : E), defined by

Z(t)(s) := f(t + s) t £ ^ s £ [0,1], (1)

is almost periodic. It is said that f £ LqS([0, to) : E) is asymptotically Stepanov q-almost periodic iff the function f : [0, to) ^ Lq([0,1] : E) is asymptotically almost periodic.

Further on, let us recall that f (■) is anti-periodic iff there exists p > 0 such that f (x + p) = —f(ж), x £ I. Any such function needs to be periodic, as it can be easily proved. Given c > 0, we call т > 0 an c-antiperiod for f (■) iff

llf (t + т) + f (t)ll < c t £ I.

By dap(f, c) we denote the set of all c-antiperiods for f (■). The notion of an almost anti-periodic function has been recently introduced in [9, Definition 2.1] as follows:

Definition 1. It is said that f (■) is almost anti-periodic iff for each c > 0 the set dap(f, c) is relatively dense in [0, to).

We know that any almost anti-periodic function needs to be almost periodic. Denote by ANP0(I : E) the linear span of almost anti-periodic functions from I into X. Then [9, Theorem 2.3] implies that ANP0(I : E) is a linear subspace of AP (I : E) as well as that the linear closure of ANP0(I : E) in AP (I : E), denoted by ANP (I : E), satisfies

ANP(I : E) = APr\\{o}(I : E). (2)

Let 1 < q < to. We refer the reader to [9, Definition 3.1] for the notions of an asymptotically almost anti-periodic function f : [0, to) ^ E and an asymptotically Stepanov q-almost anti-periodic function f : [0, to) ^ E.

For more details about almost periodic and almost automorphic type functions, we refer the reader to the monographs [13] by A. S. Besicovitch, [14] by G.M.N’. Guerekata, [15] by T. Diagana and [16] by M. Kostic.

B. Chaouchi, M. Kostic, S. Pilipovic, D. Velinov

We will always assume that I = [0, to) or I = R; set S := N if I = [0, to), and S := Z if I = R.

For the convenience of the reader, we must recall that a bounded continuous function f : I ^ E is said to be Bloch (p, k}-periodic, or Bloch periodic with period p and Bloch wave vector or Floquet exponent k iff f (x + p) = eikpf (x), x G I, with p > 0 and k G R. The space of all functions f : I ^ E that are Bloch (p, k)-periodic will be denoted by Bp,k (I : E}. If f G Bp,k (I : E}, then we have

f (x + mp) = eikmpf (x), x G I, m G S.

Given k G R, we set Bk(I : E) := Up>o Bp,k(I : E). Observing that f G Pc(I : X} satisfies f (x + p) = f (x) for all x G I and some p > 0 iff the function F(x) := eikxf (x), x G I satisfies F(x + p) = eikpF(x), x G I, we may conclude that

Bk (I : E ):= {eikf (■} : f G Pc(I : E}}. (3)

For more details on the Bloch (p, k}-periodic functions, see [1; 3] and references cited therein.

Let us define the notion of a semi-Bloch k-periodic function as follows:

Definition 2. Let k G R. A function f G Cb(I : E} is said to be semi-Bloch k-periodic iff

Ve > 0 3p > 0 Vm G S Vx G I ||f (x + mp) — eikmpf (x}|| < e. (4)

The space of all semi-Bloch k-periodic functions will be denoted by SBk (I : E}.

It is clear that Definition 2 provides a generalization of [10, Definition 2 and Definition 3], given only in the case that I = R. Speaking-matter-of-factly, a function f : R ^ E is semi-periodic in the sense of above-mentioned (equivalent) definitions iff f : R ^ E is semi-Bloch 0-periodic. Further on, it can be easily seen that for each k G R any constant function f = c belongs to the space SBk(I : E}; for this, it is only worth noticing that for each e > 0 and k = 0 we can take p = 2n/k and (4) will be satisfied.

Remark 1. It is not so easy to introdude the concept of almost Bloch k-periodicity, where k G R. In order to explain this in more detail, assume that a function f G Cb(I : E} and a number e > 0 are given. Let us say that a real number p > 0 is an (e, k}-Bloch period for f (■} iff

f (x + p) — eikpf (x)

and f (■} is almost Bloch k-periodic iff for each e > 0 the set constituted of all (e, k}-Bloch periods for f (■} is relatively dense in [0, to). But, then we have f (■} is almost Bloch k-periodic iff f (■} is almost periodic. To see this, it suffices to observe that (5) is equivalent with

e-ik(x+p)f (x + p) — e-ikxf (x)

so that, actually, the function f (■} is almost Bloch k-periodic iff the function e-ik&f (■} is almost periodic, which is equivalent to say that the function f (■} is almost periodic due to [16, Theorem 2.1.1(ii)]. Further on, let f (■} G SBk(I : E}. Then for each number e > 0 we have that the set constituted of all (e,k}-Bloch periods for f (■} is relatively dense in [0, to) since it contains the set {mp : m G N}, where p > 0 is determined by (4). In

Semi-Bloch periodic functions, semi-anti-periodic functions and applications

view of our previous conclusions, we get that f (■) is almost periodic. In particular, any Bloch (p, k)-periodic function needs to be almost periodic, which has not been observed in the researches of Bloch periodic functions carried out so far (see, e.g., [1-3; 7]).

Now we will prove the following fundamental result:

Proposition 1. Let k G R and f G Cb(I : E). Then the following holds:

(i) f (■) is semi-Bloch k-periodic iff e~lk& f (■) is semi-periodic;

(ii) f (■) is semi-Bloch k-periodic iff there exists a sequence (fn) in Pc(I

limn^^ eikxfn(x) = f (x) uniformly in I;

(iii) f (■) is semi-Bloch k-periodic iff there exists a sequence (fn) in Bk(I limn^ fn(x) = f (x) uniformly in I.

Доказательство. The proof of (i) follows similarly as in Remark 1. Since [10, Lemma 1 and Theorem 1] hold for the functions defined on the interval I = [0, to), we have that (i) implies that f (■) is semi-Bloch k-periodic iff there exists a sequence (fn) in Pc(I : E) such that limn^^ eikxfn(x) = f (x) uniformly in I. This proves (ii). For the proof of (iii), it suffices to apply (ii), (3) and the conclusion preceding it. □

Let k G R. Using Proposition 1 and [10, Proposition 2], we may construct a

substantially large class of semi-Bloch k-periodic functions, which do not form a vector

space due to a simple example in the second part of [10, Remark 3]; [10, Lemma 2] can be straightforwardly reformulated for semi-Bloch k-periodic functions, while the function given in [10, Example 1] can be simply used to provide an example of a scalar-valued semi-Bloch k-periodic function which is not contained in the space Bk (I : C). If we define Bloch k-quasi-periodic functions

Bk;q(I : E) := {e-k7(■) : f G QP°(I : E)}, (6)

where QP°(I : E) denotes the space of all quasi-periodic functions from I into E (see [10] and references cited therein for the notion), then [10, Theorem 2] can be also reformulated in our context; this also holds for [10, Example 2, Example 3].

By the foregoing, we have:

Bk(I : E) C SBk(I : E) C AP(I : E) C BUC(I : E), k G R.

Example 1. The function f (x) := cos x, x G R is anti-periodic. Now we will prove that f G SBk(I : E) iff k G Q. For k G Q, this is clear because we can take p in (4) as a certain multiple of 2n. Let us assume now that k G Q. Then it suffices to show that the function e-lk&f(■) is not semi-periodic. Towards see this, let us observe that o(e-lk&f (■)) = {1 — k, —1 — k} so that there does not exist a positive real number в > 0 such that o(e-lk&f (■)) C в ■ Q, which can be simply approved and which contradicts [10, Lemma 2].

Remark 2. Let a G AP (I : C). Based on Proposition 1, we can introduce and analyze the following notion: A function f G Cb(I : E) is said to be semi-a-periodic iff there exists a sequence (fn) in Pc(I : E) such that limn^^ a(x)fn(x) = f (x) uniformly in I.

Any such function needs to be almost periodic due to [16, Theorem 2.1.1(ii)]. We will analyze this notion somewhere else.

Further on, if f : I ^ E is anti-periodic and f(x + p) = — f(x), x G I for some p > 0, then we have f (x + mp) = (—1)mf (x), x G I, m G S. Therefore, it is meaningful to consider the following notion:

: E) such that : E) such that

B. Chaouchi, M. Kostic, S. Pilipovic, D. Velinov

Definition 3. Let f G Cb(I : E). Then we say that f (■) is semi-anti-periodic iff

Ve > 0 3p> 0 Vm G S Vx G I ||f (x + mp) — (—1 )mf (x)| < e.

The space of all semi-anti-periodic functions will be denoted by SANP(I : E).

It immediately follows from the foregoing that any anti-periodic function is semi-antiperiodic. Furthermore, any semi-anti-periodic function f : I ^ E is both semi-periodic and almost anti-periodic. To see this, let e > 0 be given in advance. Then we can find p > 0 such that (4) holds with e replaced by e/2 therein. Then (4) holds with p replaced therein with 2p and k = 0, which follows from the simple estimates

|f (x + 2mp) — f (x)|| < |f (x + 2mp) — (—1)mf (x + mp)| +

+ l(—1)mf (x + mp) — f (x)| < e, x G I, m G S.

Hence, f (■) is semi-periodic. To see that f (■) is almost anti-periodic, notice only that (4) implies

|f (x + (2m + 1)p) + f (x)| < e, x G I, m G S

as well as the set {(2m + 1)p : m G N} is relatively dense in [0, to). Therefore, we have proved the following

Proposition 2. Let f G Cb(I : E).

(i) If f (■) is anti-periodic, then f (■) is semi-anti-periodic.

(ii) If f (■) is semi-anti-periodic, then f (■) is semi-periodic and almost anti-periodic.

Furthermore, applying the same arguments as in the proofs of [10, Lemma 1 and Theorem 1], we may conclude that the following holds:

Proposition 3. Let f G Cb(I : E). Then f (■) is semi-anti-periodic iff there exists a sequence (fn) of anti-periodic functions in Cb(I : E) such that limn^^ fn(x) = f (x) uniformly in I.

Let p > 0 and k G R. If f : R ^ E is periodic (anti-periodic, Bloch (p, k)-periodic) and a G L^R), then the function a * f (■) is likewise periodic (anti-periodic, Bloch (p, k)-periodic). Using the Young inequality, Proposition 1 and Proposition 3, it immediately follows that the space of semi-Bloch k-periodic functions and the space of semi-antiperiodic functions are convolution invariant.

We continue by providing several illustrative examples:

Example 2. Let f = c = 0. Due to (2), f G ANP(R : E) and clearly f (■) is not semi-anti-periodic. On the other hand, f (■) is periodic and therefore semi-periodic.

Example 3. It can be simply verified that the function f(x) := sinx + sinnxv^2, x G R is almost anti-periodic but not semi-periodic (see, e.g., [10, Remark 3] and [9, Example 2.1]).

Example 4. (A slight modification of [10, Example 1]). The function

& gir/(2n+1)

f (x) :=S —32----------, x G R

is semi-anti-periodic because it is a uniform limit of [n■ (2n + 1)!!]-anti-periodic functions

eix/(2n+1)

fN(x):=J] ^2 , x G R (N G N).

Semi-Bloch periodic functions, semi-anti-periodic functions and applications

On the other hand, the function f (■) cannot be periodic.

Example 5. Set Qn := {(2n +l)/(2m + 1) : m, n G Z}. If 9 > 0 and ||a\\(f )|| <

to, then the function

f (t) := ^ aA(f)e%Xt, t G R XeO-Qn

is semi-anti-periodic. This can be inspected as in the proof of [10, Proposition 2] since the function fN(■) used therein is anti-periodic with the anti-period nqiq2 ■ ■ ■ qN/9.

Example 6. Roughly speaking, it is well known that the unique solution of the heat equation ut(x,t) = uxx(x,t), x G R, t > 0, accompanied with the initial condition u(x, 0) = f (x), is given by

(x-s)2 41

f (s) ds,

x G R, t > 0.

By the conclusion from [1, Example 2.1], we know that, if the function f(■) is Bloch (p, k)-periodic, then the solution u(x, ■) is likewise Bloch (p, k)-periodic (p > 0, k G R). Using this fact, the dominated convergence theorem and Proposition 1, we get that, if f (■) is semi-Bloch k-periodic, then the solution u(x, ■) will be likewise semi-Bloch k-periodic.

It is necessary to note that any of the above introduced function spaces becomes a metric space when equipped with the metric

d(f,g) := sup ||f(x) - g(x)||

inherited from Cb(1 : E).

Let us recall that, for every almost anti-periodic function f : [0, to) ^ E, we have that the function Ef : R ^ E is likewise almost anti-periodic; the same assertion holds for the function f : [0, to) ^ E which belongs to the space ANP0([0, to) : E) or ANP([0, to) : E); see [9]. Now we will state and prove the following proposition:

Proposition 4. Let k G R, let p > 0, and let a function f G Cb([0, to) : E) be given. If f (■) is Bloch (p, k)-periodic (semi-Bloch k-periodic, semi-anti-periodic), then the function Ef(■) is likewise Bloch (p,k)-periodic (semi-Bloch k-periodic, semi-anti-periodic).

Доказательство. Suppose first that f (■) is Bloch (p, k)-periodic. Then f (x + p) = eikpf (x), x > 0; we need to show that (Ef)(x + p) = eikp(Ef)(x), x G R, i.e.,

[W(x + p)f](0) = eikp[W(x)f](0), x G R.

Since W(x + p) = W(x)W(p), x G R, it suffices to show that [W(x)f(■ + p)](0) = eikp[W(x)f](0), x G R, i.e., [W(x)eikpf (-)](0) = eikp[W(x)f](0), x G R, which is true.

If f(■) is semi-Bloch k-periodic, then Proposition 1(iii) yields that there exists a sequence (fn) in Bk([0, to) : E) such that limn^^ fn(x) = f (x) uniformly in [0, to). Due to the supremum formula clarified in [16, Theorem 2.1.1(xi)], we have that limn^^(Efn)(x) = (Ef)(x) uniformly in R. By the first part of proof, we know that for each n G N the function (Efn)(-) belongs to the space Bk(R : E). Applying again Proposition 1(iii), we get that Ef (■) is likewise semi-Bloch k-periodic. The proof is quite similar in the case that f (■) is semi-anti-periodic. □

B. Chaouchi, M. Kostic, S. Pilipovic, D. Velinov

The proof of following simple proposition is left to the interested reader:

Proposition 5. Let к G R, let p > 0, and let f : I ^ X. Then we have:

(i) if f (■) is Bloch (p, к)-periodic (semi-Bloch к-periodic, semi-anti-periodic), then cf (■) is Bloch (p, к)-periodic (semi-Bloch к-periodic, semi-anti-periodic) for any c G C;

(ii) if X = C, infxeR |f (x)| = m > 0 and f (■) is Bloch (p, к)-periodic (semi-Bloch к-periodic, semi-anti-periodic), then 1/f (■) is Bloch (p, -к)-periodic (semi-Bloch (-к)-periodic, semi-anti-periodic).

We close this section by introducing the following definition.

Definition 4. Let f G Cb([0, to) : E) and к G R. Then we say that f (■) is asymptotically semi-periodic (semi Bloch к-periodic, semi-anti-periodic) iff there exist a function ф G Co([0, to) : X) and a semi-periodic (semi Bloch к-periodic, semi-anti-periodic) function g : R ^ E such that f (t) = g(t) + ф(к) for all t > 0.

As mentioned in the introduction, the notion of Stepanov semi-periodicity has not been analyzed in [10]. We will use the following definitions:

Definition 5. Let к G R and 1 < q < to. Then we say that a function f G LqS(I : E) is Stepanov q-semi-periodic (Stepanov q-semi-Bloch к-periodic, Stepanov q-semi-anti-periodic), iff the function f : I ^ Lq([0,1] : E), defined by (1), is semi-periodic (semi-Bloch к-periodic, semi-anti-periodic).

Definition 6. Let к G R and 1 < q < to. Then we say that a function f G LS(I : E) is asymptotically Stepanov q-semi-periodic (asymptotically Stepanov q-semi-Bloch к-periodic, asymptotically Stepanov q-semi-anti-periodic), iff the function f : I ^ Lq([0,1] : E), defined by (1), is asymptotically semi-periodic (asymptotically semi-Bloch к-periodic, asymptotically semi-anti-periodic).

If a function f G LqS(I : E) belongs to some of spaces introduced in Definition 5 (Definition 6), then f (■) is Stepanov q-almost periodic (asymptotically Stepanov q-almost periodic). It should be noticed that any of the spaces introduced so far is translation invariant in the sense that if a function f : I ^ E belongs to this space, then for each т G I the function f (■ + т) likewise belongs to this space.

Let p > 0 and к G R. It should be noted that, if f : I ^ E is Bloch (p, к)-periodic (anti-periodic, periodic), then f : I ^ Lq([0,1] : E) is likewise Bloch (p, ^-periodic (anti-periodic, periodic). Further on, it immediately follows from the corresponding definitions that, if f : I ^ E is semi-Bloch к-periodic (semi-antiperiodic, semi-periodic), then f(■) is Stepanov q-semi-Bloch к-periodic (Stepanov q-semi-anti-periodic, Stepanov q-anti-periodic) for every number q G [1, to); a large class of non-continuous periodic or Bloch (p, ^-periodic functions can be used to provide that the converse statement does not hold in general. If 1 < q < q& < to and f : I ^ E is (asymptotically) Stepanov q&-semi-periodic ((asymptotically) Stepanov q&-semi-Bloch к-periodic, (asymptotically) Stepanov q&-semi-anti-periodic), then f(■) is (asymptotically) Stepanov q-semi-periodic ((asymptotically) Stepanov q-semi-Bloch к-periodic, (asymptotically) Stepanov q-semi-anti-periodic). To see that the converse statement does not hold in general, we will provide only one illustrative example:

Semi-Bloch periodic functions, semi-anti-periodic functions and applications

Example 7. Suppose that 1 < q < ж. Let us recall that H. Bohr and E. Fplner have constructed an example of a Stepanov 1-almost periodic function F : R ^ R that is not Stepanov q-almost periodic (see [17, p. 70]). Moreover, for each n G N there exists a bounded periodic function Fn : R ^ R with at most countable points of discontinuity such that

f&t+1

lim sup / |Fn(s) — F(s) ds = 0.

n^ro t£R Jt

Therefore, Fn : R ^ L:([0,1] : E) is a bounded periodic function and, in addition to the above, Fn(-) is continuous (n G N). Due to (7), we have that limn^^ Fn(t) = F(t) uniformly in t G R. This implies that the function F(■) is Stepanov 1-semi-periodic but not Stepanov q-semi-periodic because it is not Stepanov q-almost periodic.

Example 8. Assume that a,3 G R and a[3 1 is a well-defined irrational number. Then the functions

f (t) = sin

are Stepanov q-almost periodic but not almost periodic (1 < q < ж); see, e.g., [16]. We would like to raise the question whether the functions f (■) and g(-) are Stepanov q-semi-periodic for any 1 < q < ж?

Example 9. Define sign(0) := 0. Then, for every almost periodic function f : R ^ R, we have that the function sign(f (■)) is Stepanov q-almost periodic for any finite real number q > 1 (see [16, Example 2.2.2, Example 2.2.3]). Let us consider again the function f (x) := sin x + sinnx\\[2, x G R. Then a simple analysis involving the identity f (x) = 2sinxcosx—, x G R shows that the function sign(f (■)) is identically equal to a function F(■) of the following, much more general form: Let (an)neZ be a strictly increasing sequence of real numbers satisfying limn^+^(an+1 — an) = +ж, limn^+TO an = +ж and limn^_^ an = —ж. Suppose that (bn)neZ is a sequence of nonzero real numbers satisfying that the sets {n G Z : bn < 0} and {n G Z : bn > 0} are

infinite, as well as that there exists a finite positive constant c > 0 such that c < |bn — bi| for any n, l G Z such that bnbl < 0. Define the function F : R ^ R by F(x) := bn if x G [an,an+1), for any n G Z. Then F(■) cannot be Stepanov q-semi-periodic for any finite real number q > 1. Otherwise, for c G (0,cq) we would be able to find a number p > 0 such that for each m G Z and x G R one has:

F (x + mp + s) — F (x + s)

ds < (1/2)q.

Let n G Z be such that [x,x + 1] C [an, an+1) and bn < 0, say. Without loss of generality, we may assume that the set {n G N : bn > 0} is infinite. Then the contradiction is obvious because, for every sufficiently large numbers l G N with bl > 0, we can find m G N such that [x + mp, x + mp +1] C [al, al+1) so that

F (x + mp + s) — F (x + s)

ds > |bn — bl|q > cq.

B. Chaouchi, M. Kostic, S. Pilipovic, D. Velinov

and semi-anti-periodicity under the actions of convolution products

In this section, we will investigate the invariance of semi-Bloch k-periodicity and semi-anti-periodicity under the actions of infinite convolution products and finite convolution products.

For a given function g : R ^ E, which is Stepanov q-semi-Bloch k-periodic or Stepanov q-semi-anti-periodic for some number q G [1, to), we will first investigate the qualitative properties of function G : R ^ X, given by

G(t) := f R(t — s)g(s) ds, t G R, (8)

J — °

where (R(t))t>0 C L(E,X) is a strongly continuous operator family satisfying some growth assumptions. Concerning this question, we have the following result:

Proposition 6. Suppose that k G R, 1 < q < to, 1/q&+1/q = 1 and (R(t))t>0 C L(E,X) is a strongly continuous operator family satisfying that M := ^ °=0 ||R(&)IIl«& [k,k+1] < to. If g : R ^ E is Stepanov q-semi-Bloch k-periodic (Stepanov q-semi-anti-periodic), then the function G(-), given by (8), is well-defined and semi-Bloch k-periodic (semi-antiperiodic).

Доказательство. It can be easily seen that, for every t G R, we have G(t) = /0° R(s)g(t — s) ds. Due to [18, Proposition 2.11], we have that G(-) is well-defined and almost periodic. It remains to be proved that G(-) is semi-Bloch k-periodic (semi-antiperiodic). For the sake of brevity, we will consider only semi-Bloch k-periodicity below because the proof for semi-almost-periodicity is analogous (see also [9, Proposition 3.1]). Let a number e > 0 be given in advance. Then we can find a finite number p > 0 such that, for every m G Z and t G R, we have

Г t+1

||g(s + mp) — etkmpg(s)\\\\q ds < eq, t G R.

Applying Holder inequality and this estimate, we get that

\\\\G(t + mp) — eikmpG(t)\\\\ < ||R(r)|| ■ \\\\g(t + mp — r) — eikmpg(t — r)\\ dr =

° r k+1

= ^Д I l|R(r)| ■ \\\\g(t + mp — r) — eikmpg(t — r)|| dr <

fk+1

k=0J k °

< llR(-) IIl»&[k,k+1]

= llR(-) IIl«&[k,k+1]

< Д IIR(&)|L«/[k,k+1]e = Me, t G R,

g(t + mp — r) — eikmpg(t — r)||q dr

rt—k \\ 1/q

\\\\g(mp + s) — eikmpg(s)\\\\q ds <

lt—k—1 J

which clearly implies the required.

Semi-Bloch periodic functions, semi-anti-periodic functions and applications

The above result can be simply applied in the study of existence and uniqueness of semi-Bloch k-periodic (semi-anti-periodic) solutions of fractional Cauchy inclusion

D]+u(f) e Au(t) + f (t), t e R,

where Dt+ denotes the Riemann — Liouville fractional derivative of order 7 e (0,1], f : R ^ E satisfies certain properties, and A is a closed multivalued linear operator (see [9; 16] for the notion and more details).

In the following proposition, which can be deduced with the help of Proposition 6 and the proof of [18, Propostion 2.13], we analyze the invariance of asymptotical semi-Bloch k-periodicity and asymptotical semi-anti-periodicity under the actions of finite convolution products.

Proposition 7. Suppose that k e R, 1 < q < ж, 1/q&+1 /q = 1 and (R(t))t>0 C L(E,X) is a strongly continuous operator family satisfying that, for every s > 0, we have that

<X>

ms :=J2 llR(-)ll L* [s+k,s+k+1] < Ж k=0

Suppose, further, that the Stepanov q-bounded functions g : R ^ X and q : [0, ж) ^ X satisfy that g(-) is Stepanov q-semi-Bloch k-periodic (Stepanov q-semi-anti-periodic), q e C0([0, ж) : Lq([0,1] : E)) and f (t) = g(t) + q(t) for all t > 0. Let there exist a finite number M > 0 such that the following holds:

(i) limt^+TO llR(r)llllq(s - r)ll dr]q ds = 0;

(ii) limt^+TO ftt+1 m4s ds = 0.

Then the function H(■), given by

H(t) := f R(t — s)f (s) ds, t > 0,

is well-defined, bounded and asymptotically Sq-semi-Bloch k-periodic (asymptotically Sq-semi-anti-periodic).

The above result can be applied in the qualitative analysis of asymptotically (Stepanov q-)semi-Bloch k-periodic solutions and asymptotically (Stepanov q-)semi-anti-periodic solutions of the following abstract Cauchy inclusion

(DFP) : f DYu(t) e Au(t) + f (t), t > 0,

(DFP)f’Y : \\ u(0) = Xo,

where Dj denotes the Caputo fractional derivative of order 7 e (0,1], x0 e E, f : [0, ж) ^ E satisfies certain properties, and A is a closed multivalued linear operator (see [9; 16] for the notion and more details).

Finally, let B be a subset of Rs and f : R x B ^ Rs. Then we say that the function f (■) is uniformly semi-Bloch k-periodic function iff for any compact subset K of B, we have

Ve > 0 3p > 0 Vm e Z Vx e R У a e K ||f (x + mp,a) — eikmp f (x,o)|Rs < e.

We close the paper with the observation that we can simply reformulate [10, Proposition 3] for uniformly semi-Bloch k-periodic functions (uniformly semi-antiperiodic functions) and provide certain applications to matrix differential equations, as it has been done in [10, Theorem 4] for semi-periodic functions.

B. Chaouchi, M. Kostic, S. Pilipovic, D. Velinov

References

Accepted article received 25.02.2020

Corrections received 10.05.2020

Semi-Bloch periodic functions, semi-anti-periodic functions and applications

Челябинский физико-математический журнал. 2020. Т. 5, вып. 2. С. 243-255.

УДК 517.51+517.9 DOI: 10.24411/2500-0101-2020-15211

ПОЛУБЛОХОВЫ ПЕРИОДИЧЕСКИЕ ФУНКЦИИ, ПОЛУ-АНТИПЕРИОДИЧЕСКИЕ ФУНКЦИИ И ПРИЛОЖЕНИЯ

Б. Чаучи1’", M. Костич2,6, С. Пилипович2с, Д. Велинов3^

Введены понятия полублоховых периодических функций и полу-антипериодических функций. Рассматриваются также степановские полублоховы периодические функции и степановские полу-антипериодические функции. Проанализирована инвариантность введённых классов функций относительно действия свёрточных произведений и дано краткое описание возможного использования полученных результатов при проведении качественного анализа решений абстрактных неоднородных интегро-дифференциальных уравнений.

Поступила в редакцию 25.02.2020 После переработки 10.05.2020

Сведения об авторах

Чаучи Белкасем, профессор, лаборатория энергетики и интеллектуальных систем, Университет Хемис Милианы, Хемис Милиана, Алжир; e-mail: chaouchicukm@gmail.com. Костич Марко, профессор, факультет технических наук, Университет Нови-Сада, Нови-Сад, Сербия; e-mail: marco.s@verat.net.

Пилипович Стеван, профессор, отделение математики и информатики, Университет Нови-Сада, Нови-Сад, Сербия; e-mail: pilipovic@dmi.uns.ac.rs.

Велинов Даниэль, доцент, факультет гражданского строительства, Университет Святых Кирилла и Мефодия, Скопье, Северная Македония; e-mail: velinovd@gf.ukim.edu.mk.

Исследование частично поддержано грантом 174024 Министерства науки и технологического развития Республики Сербия.