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6
Chelyabinsk Physical and Mathematical Journal. 2020. Vol. 6, iss. 2. P. 190-207.
DOI: 10.24411/2500-0101-2021-16205
c-ALMOST PERIODIC TYPE DISTRIBUTIONS
M. Kostic1", S. Pilipovic1b, D. Velinov2c, V.E. Fedorov34d
1Novi Sad University, Novi Sad, Serbia
2Ss. Cyril and Methodius University, Skopje, North Macedonia 3 Chelyabinsk State University, Chelyabinsk, Russia
4South Ural State University (National Research University), Chelyabinsk, Russia "marco.s@verat.net, bpilipovic@dmi.uns.ac.rs, cvelinovd@gf.ukim.edu.mk, dkar@csu.ru
We introduce and systematically analyze various classes of c-almost periodic type distributions and asymptotically c-almost periodic type distributions with values in complex Banach spaces. We provide an interesting application in the study of existence of asymptotically c-almost periodic type solutions for a class of ordinary differential equations in the distributional spaces.
Keywords: c-almost periodic type distribution, asymptotically c-almost periodic type distribution, generalized function, ordinary differential equation, Banach space.
1. Introduction and preliminaries
The concept of almost periodicity was introduced by H. Bohr [1] around 1924-1926 and later generalized by many other mathematicians. Let I = R or I = [0, to), let (X, || ■ ||) denote a complex Banach space, and let f : I ^ X be a continuous function. Given e > 0, we call t > 0 an e-period for f (■) if and only if ||f (t + t) — f (t)|| < e, t E I. The set of all e-periods for f (■) is denoted by $(f, e). It is said that f (■) is almost periodic if and only if for each e > 0 the set $(f, e) is relatively dense in [0, to), which means that there exists l > 0 such that any subinterval of [0, to) of length l meets $(f, e). The vector space consisting of all almost periodic functions is denoted by AP(I : X); see [2-11] and references cited therein for more details on the subject.
Let w > 0 and c E C\{0}. The class of (w, c)-periodic functions has been introduced and investigated by E. Alvarez, A. Gomez and M. Pinto in [12] (see also the research article [13] by E. Alvarez, S. Castillo and M. Pinto; in a series of our recent research studies, M.T. Khalladi, M. Kostic, A. Rahmani and D. Velinov have introduced and investigated generalized (w,c)-almost periodic type functions). Let us recall that a continuous function f : I ^ X is said to be c-periodic if and only if there exists w > 0 such that f (■) is (w, c)-periodic, i. e., f (x+w) = cf (x) for all x E I (here, c E C\{0}). The space consisting of all (w, c)-periodic functions and the space consisting of all c-periodic functions are denoted by Pw,c(/ : X) and Pc(1 : X), respectively; if c = —1, then we also say that a (—1)-periodic function is anti-periodic. The space Pc(1 : X) generalizes the space of Bloch (p, k)-periodic functions; let us recall that a bounded continuous function f : I ^ X is said to be Bloch (p, k)-periodic, or Bloch periodic with period p and Bloch wave vector or Floquet exponent k (here, p > 0 and k E R) if and only if
The research is partially supported by Ministry of Science and Technological Development, Republic of Serbia (Grant no. 451-03-68/2020/14/200156), Bilateral project between MANU and SANU, by the Russian Foundation for Basic Research (Grant no. 21-51-54003) and by Act 211 of Government of the Russian Federation, contract 02.A03.21.0011.
/ (x + p) = eikpf(x), x G I. On the other hand, any anti-periodic function is Bloch (p, k)-periodic with certain real parameters p > 0 and k G R. It should be recalled that any Bloch (p, k)-periodic function is almost periodic [14] but not necessarily periodic since the function /(x) := eix + , x g R is Bloch (p, k)-periodic with p = 2n + v^n
and k = -\/2 — 1 but not periodic. For more details about (generalized) Bloch (p, k)-periodic functions, see the research articles [15] by M. Hasler and [16] by M.F. Hasler, G.M. N'Guerekata.
In [17], M.T. Khalladi, M. Kostic, M. Pinto, A. Rahmani and D. Velinov have analyzed the notion which depends on only one parameter, c G C\{0}. More precisely, in this paper, the authors have introduced and analyzed various notions of c-almost periodicity for a continuous function / : I ^ X:
(i) we call a number t > 0 a (e,c)-period for /(■) if and only if ||f (t + t) — cf (t)|| < e for all t G I (e > 0); by 0c(f, e) we denote the set of all (e, c)-periods for /(■). It is said that /(■) is c-almost periodic if and only if for each e > 0 the set 0c(f, e) is relatively dense in [0, ro). The space of all c-almost periodic functions from I into X will be denoted by APc(I : X);
(ii) /(■) is called c-uniformly recurrent if and only if there exists a strictly increasing sequence (an) of positive real numbers such that lim an = +ro and
n—+<
lim ||f(■ + an) — cf(.)L = 0.
n—^- + <^0 11 <
If c = —1, then /(■) is said to be uniformly anti-recurrent. The space of c-uniformly recurrent functions from I into X will be denoted by URc(I : X);
(iii) set S := N if I = [0, ro) and S := Z if I = R; then it is said that:
(a) /(■) is semi-c-periodic (of type 1) if and only if
Ve > 0 3w> 0 Vm G S Vx G I ||f (x + mw) — cmf (x)|| < e;
the space of all semi-c-periodic functions will be denoted by SAPc(I : X);
(b) /(■) is semi-c-periodic of type 2 if and only if
Ve > 0 3w> 0 Vm G S Vx G I ||c-mf (x + mw) — /(x)|| < e;
(c) /(■) is semi-c-periodic of type 1+ if and only if
Ve > 0 3w > 0 Vm G N Vx G I ||f (x + mw) — cmf (x)|| < e;
(d) /(■) is semi-c-periodic of type 2+ if and only if
Ve > 0 3w > 0 Vm G N Vx G I ||c-mf (x + mw) — /(x)|| < e.
The space Pc (I : X) is contained in the space SAPc(I : X), which is further contained in the space APc(I : X). In [17], we have proved the following: Let |c| = 1, i G {1, 2} and / : I ^ X be a continuous function. Then /(■) is semi-c-periodic of type i (i+) if and only if /(■) is semi-c-periodic of type 1. We will also employ the following result whose proof will appear somewhere else.
Lemma 1. Let |c| = 1, i G {1, 2} and / : I ^ X. Then /(■) is semi-c-periodic of type i (i+) if and only if /(■) is c-periodic.
A continuous function / : I ^ X is called asymptotically c-almost periodic (asymptotically c-uniformly recurrent, asymptotically semi-c-periodic) if and only if there are a c-almost periodic function (c-uniformly recurrent function, semi-c-periodic function) g : R ^ X and a function h G C0(I : X) such that /(t) = g(t) + h(t),
t E I; here, Co(I : X) denotes the vector space consisting of all continuous functions f : I ^ X such that ||f (t)|| = 0. Suppose that I = R. In this case, it is worth
noting that the notion of asymptotical almost periodicity introduced above (c = 1) is different from the corresponding notion used in [18] (see also [10, Section 2.7]), where we have assumed that the decomposition f (t) = g(t) + h(t) holds only for t > 0 as well as that h E C0([0, to) : X) (cf. also [19, Remark 2.5]). In order to be more consistent henceforth, we will say that f : R ^ X is half-asymptotically c-almost periodic (half-asymptotically c-uniformly recurrent, half-asymptotically semi-c-periodic) if and only if there are a c-almost periodic function (c-uniformly recurrent function, semi-c-periodic function) g : R ^ X and a function h E C0([0, to) : X) such that f (t) = g(t) + h(t), t > 0.
If c = 1, then we also say that f (■) is ((half-)asymptotically) uniformly recurrent (((half-)asymptotically) semi-periodic, ((half-)asymptotically) almost periodic); if c = — 1, then we also say that f(■) is ((half-)asymptotically) almost anti-periodic (((half-)asymptotically) uniformly anti-recurrent, ((half-)asymptotically) semi-anti-periodic). Note, if f (■) is c-almost periodic, then f (■) is almost periodic and therefore bounded (see [17]).
We will use the following lemma, which can be deduced with the help of [19, Theorem 2.6] and [9, Theorem 3.36, Theorem 3.47; pp. 97-98].
Lemma 2. Suppose that the sequence (fn : R ^ X) of asymptotically almost periodic functions (half-asymptotically almost periodic functions) converges uniformly to a function f : R ^ X. Then f (■) is asymptotically almost periodic (half-asymptotically almost periodic).
The classes of scalar-valued bounded and almost periodic distributions have been introduced by L. Schwartz [20] and later extended to the vector-valued distributions by I. Cioranescu in [21]. On the other hand, the class of scalar-valued asymptotically almost periodic distributions has been introduced by I. Cioranescu in [22], while the notion of a vector-valued asymptotically almost periodic distribution has been analyzed by D.N. Cheban [23] following a different approach (cf. also I.K. Dontvi [5] and A. Halanay, D. Wexler [24]). For more details about the subject, we refer the reader to [25-32] as well as the recent research studies [33] by C. Bouzar, F. Z. Tchouar, [18] by M. Kostic and [34] by M. Kostioc, S. Pilipovioc, D. Velinov.
As mentioned in the abstract, in this paper we introduce and investigate various classes of vector-valued c-almost periodic type distributions and vector-valued asymptotically c-almost periodic type distributions. The organization and main ideas of this paper, which is created as a certain continuation of studies [17] and [18], can be briefly described as follows. In Subsection 1.1, we remind ourselves of the basic definitions and results about vector-valued (asymptotically) almost periodic distributions (concerning original contributions of ours, we would like to say that, in this subsection, we introduce a new distributional space B+ (X) and provide a simple structural characterization of B+ (X) in Proposition 1). Section 2 introduces and thoroughly analyzes the above-mentioned classes of c-almost periodic type distributions; the main results of paper are Theorem 9 and Theorem 11. In the last section, we analyze the asymptotically c-almost periodic type solutions for the systems of ordinary differential equations in the distributional spaces.
1.1. Vector-valued (asymptotically) almost periodic distributions
Denote by D(X) = D(R : X) the Schwartz space of all infinitely differentiable functions / : R ^ C with compact support in X. By S (X) = S (R : X) we denote the Schwartz space of all rapidly decreasing functions with values in X, and by E (X) = E (R : X) we denote the space of all infinitely differentiable functions with values in X; D = D(C), S = S (C) and E = E (C). The spaces of all linear continuous mappings from D, S and E into X are denoted by D'(X), S'(X) and E'(X), respectively [20]; D0 stands for the subspace of D consisting of all functions with the support contained in [0, ro). If T G D'(X) and p G D, then we define T * p G E(X) by (T * p)(x) := (T, p(x — ■)>. If / : R ^ X, then we define / : R ^ X by /(t) := /(—t), t G R; for any T G D'(X), we define T G D'(X) by (T, p> := (T, p>, p G D.
Let 1 < p < ro. By DLp (R : X) we denote the vector space consisting of all infinitely differentiable functions / : R ^ X such that /(j) G Lp (R : X) for all j G N0. The Frechet topology on DLp(R : X) is induced by the following system of norms ||f ||k := ^j=0||/(j)||LP(R), k G N. If X = C, then the above space is simply denoted by DLp. The space of all linear continuous mappings / : DL1 ^ X is denoted by D'L1 (X). Endowed with the strong topology, D'L 1 (X) becomes a complete locally convex space; D'L 1 (X) is a well known space of bounded X-valued distributions. In the sequel, we will use the fact that a vector-valued distribution T G D'(X) is bounded if and only if the function T * p is bounded for all p G D; see e. g., [21, Theorem 1.1].
Let T G DL 1 (X). Then the following assertions are equivalent [21]:
(i) T * p G AP(R : X), p G D;
(ii) there exist an integer k G N and almost periodic functions fj(■) : R ^ X (0 < j < k) such that T = £j=o fj(j)
in the distributional sense. We say that a bounded distribution T G D'L 1 (X) is almost periodic if and only if T satisfies any of the above two equivalent conditions; if this is the case, then we know that the restriction of T to the space S is an X-valued tempered distribution [10]. By B'AP(X) we denote the space consisting of all almost periodic distributions.
Define the space of bounded distributions tending to zero at plus infinity as follows:
B+,o(X) := {T G DL1 (X) ; lim <Th, p> = 0, p G d),
' I h—+< J
where (Th, p> := (T, p(- — h)>, T G D'(X), h > 0. A bounded distribution T G D'L 1 (X) is said to be asymptotically almost periodic if and only if there exist an almost periodic distribution Tap G BAP (X) and a bounded distribution tending to zero at plus infinity Q G B+0(X) such that (T, p> = (Tap,p> + (Q,p>, p G D0. By B'AAP(X) we denote the vector space consisting of all asymptotically almost periodic distributions (see e.g., [18, Definition 1]).
Let T G DL 1 (X). Then we know that the following assertions are equivalent (see e.g., [18, Theorem 1]):
(i) T G BA AP (X);
(ii) the function T * p is half-asymptotically almost periodic for all p G D0;
(iii) the function T * p is half-asymptotically almost periodic for all p G D;
(iv) there exist an integer k G N and half-asymptotically almost periodic functions fj(■) : R ^ X (0 < j < k) such that T = £j=0 fj(j) on [0, ro), i. e.,
k r <
(T,p> = £(—1)j / p(j)(t)/j(t) dt, p GD0; (1)
j=0 ^
(v) there exists a sequence (Tn) of half-asymptotically almost periodic functions from E(X) such that limn^^ Tn = T in V'Li(X).
For the first time in the existing literature, we consider here the space B+ (X):={T GDLi(X); lim <Th,p> = 0, p G d\,
which is slightly different from the space B+0(X) used above. For example, the regular distribution determined by the locally integrable function f : R ^ R, given by f (t) := 1 for t < 0 and f (t) := 0 for t > 0, belongs to the space B+0(X) but not to the space
B+ (X). Since for every fixed test function p gD and for every real number h G R we have < > < > < >
<T, p(- - h)> = <t, p(- ■ -h)> = <t, p(- - h)>,
it follows that T G B+ (X) if and only if T G B+,0(X) and T G B+,0(X). Therefore, [22, Proposition 1] immediately implies the following result (see also [33, Proposition 10]):
Proposition 1. Suppose that T G VLi (X). Then the following statements are equivalent:
(i) T G B+ (X);
(ii) the restrictions of functions T * p and T * p to the non-negative real axis belong to the space C0([0, to) : X) for all p G D;
(iii) there exist an integer k G N and functions fj G C0(R : X) (0 < j < k) such that
T = v^k f (j);
T Z^ j=0 f j ;
(iv) there exists a sequence (Tn) in E'(X) which converges to T for topology of D'Li (X).
2. c-Almost periodic type distributions
and asymptotically c-almost periodic type distributions
We start this section by introducing the following notion:
Definition 1. Let T G D'(X) and c G C \ {0}.
(i) T is said to be a c-almost periodic (c-uniformly recurrent, semi-c-periodic) distribution, (APc) ((URc), (SAPc)) distribution in short, if and only if T* p G APc(R : X) (T * p G URC(R : X), T * p G SAPC(R : X)) for all p G D. By B'APc (X) (BURc(X), BSAPc(X)) we denote the space of all c-almost periodic (c-uniformly recurrent, semi-c-periodic) distributions;
(ii) T is said to be a (half-)asymptotically c-almost periodic ((half-)asymptotically c-uniformly recurrent, (half-)asymptotically semi-c-periodic) distribution if and only if the function T * p is (half-)asymptotically c-almost periodic ((half-)asymptotically c-uniformly recurrent, (half-)asymptotically semi-c-periodic) for all p G D;
(iii) T is said to be a (half-)asymptotically (D0, c)-almost periodic ((half-)asymptotically (D0, c)-uniformly recurrent, (half-)asymptotically semi-(D0,c)-periodic) distribution if and only if the function T * p is (half-)asymptotically c-almost periodic ((half-)asymptotically c-uniformly recurrent, (half-)asymptotically semi-c-periodic) for all p G D0.
Remark 1. In [14, Definition 2], we have introduced the notion of a semi-Bloch k-periodic function (k G R). The class of semi-Bloch k-periodic distributions can be also introduced but we will skip all related details concerning this notion for simplicity.
All distribution spaces introduced in Definition 1 are closed under differentiation. It is also clear that, if T G D'(X) belongs to any of the spaces introduced above, then the
distribution aT belongs to the same space, where a G C and (aT, p) := (T, ap), p G D; but, if c = 1, then the spaces of c-almost periodic functions (c-uniformly recurrent functions, semi-c-periodic functions) are not closed under pointwise addition, which continues to hold for corresponding distribution spaces. Further on, since every c-almost periodic (semi-c-periodic) function is almost periodic and therefore bounded continuous, an application of [21, Theorem 1.1] shows that any c-almost periodic (semi-c-periodic) distribution is a bounded distribution. This is no longer true for the c-uniformly recurrent distributions because there exists an unbounded uniformly recurrent function [35] and therefore the regular distribution determined by this function is a uniformly recurrent distribution which is not a bounded distribution (c = 1). We continue by stating the following:
Proposition 2. Suppose that T is a c-uniformly recurrent distribution and c G C\{0} satisfies |c| = 1. Then T = 0.
Proof. By definition, we have T * p G URc(R : X) for all p G D. Since |c| = 1, Proposition 2.6 from [17] yields that T * p = 0 for all p G D. This immediately implies T = 0. □
A distribution T G D'(X) is called c-periodic if and only if T * p G Pc(R : X) for all p G D. Keeping in mind Lemma 1, we can similarly prove the following:
Proposition 3. Suppose that T is a semi-c-periodic distribution and c G C\{0} satisfies |c| = 1. Then T is c-periodic.
Keeping in mind Proposition 2 and Proposition 3, it seems reasonable to impose the following condition:
Blank Hypothesis. Unless stated otherwise, we will always assume henceforth that c G C and | c| = 1.
Now we will state and prove the following proposition:
Proposition 4. The following statements are equivalent:
(i) T G B'APc(X) (T G B'URc(X), T G B'SAPc(X));
(ii) T G DApVc(X) (T G D'URi/c(X), T G D'SAP/C((X)).
Proof. Clearly, it suffices to show that (i) implies (ii). We will do that only for c-almost periodicity. Let p G D be fixed; we need to show that T * p G APi/c(R : X). Keeping in mind [17, Proposition 2.7], it suffices to show that
T * p = T * p, p G D. (2)
To prove this equality, fix a real number t G R. Then (2) follows from the next simple computations:
(T * p)(t) = <T, p(t - •)> = <t, p(t - •)) = (T, p(t + ■)),
T * p(t) = (T * p) (-t) = <T, p(-t - ■)> = <T, p(t - ■)> = (T, p(t + ■)).
□
We continue by observing that [17, Proposition 2.8, Proposition 2.9] directly imply the following: if T G B'Apc(X) (T G B^(X), T G B^(X)), then ||T * p|| : I ^ [0, to) is almost periodic (uniformly recurrent, semi-periodic) for all p G D as well as T G BAP ,(X) (T G BUR (X), T G B'SAP (X)) for any positive integer l G N.
Furthermore, [17, Corollary 2.10, Proposition 2.11, Proposition 2.12] directly imply the following:
(i) suppose that p G Z\{0}, q G N, (p, q) = 1 and arg(c) = pn/q and T G B'AP (X) (T G B^RC(X), T G BSAPc(X)); c
(a) if p is even, then T G B'AP(X) (T G B^R(X), T G BSAP(X));
(b) if p is odd, then T is almost anti-periodic (uniformly anti-recurrent, semi-anti-periodic) distribution;
(ii) suppose that arg(c)/n G Q and T G B'APr(X), then T G B'AP ,(X) for all d G Si = {z G C : |z| = 1}; ' '
(iii) suppose that arg(c)/n G Q and T G B'SAPc(X), then T G B'AP ;(X) for all d G {d : l G N}. C
(iv) suppose that arg(c)/n G Q and T G B^APc(X), then T G B'AP ;(X) for all d G S1. The following statements known for functions can be simply deduced for
distributions, as well (see [17, Example 2.22] for more details):
(i) suppose that c =1. Then the set consisting of all c-almost periodic distributions is a vector space together with the usual operations, while the set consisting of d-uniformly recurrent distributions and the set consisting of semi-d-periodic distributions are not vector spaces together with the usual operations;
(ii) suppose that c =1. Then the set consisting of all c-almost periodic (c-uniformly recurrent, semi-d-periodic) distributions is not a vector space together with the usual operations.
In [17, Example 2.23], we have considered the pointwise products of c-almost periodic type functions with the scalar-valued functions. It is worthwhile to mention that all established statements can be reformulated for the pointwise products of c-almost periodic type distributions with the scalar-valued infinitely differentiable functions; concerning Stepanov classes of c-almost periodic type functions, it should be noticed that [18, Proposition 1] continues to hold in our new framework. Details can be left to the interested readers.
We proceed by stating the following simple result:
Proposition 5. Let h G R, b G R \ {0} and T G B'AP (X) (T G B^R (X), T G B'sapc(X)). Then: ° °
(i) any translation Th of T G B'APc(X) (T G B^Rc(X), T G B'SAPc(X)) belongs to
BAPc(X) (BURc(X)> bSAPc(X));
(ii) define the distribution Tb by (Tb, p) := (T, p(b^)), p G D. Then Tb G B'AP (X) (Tb G B^Rc(X), Tb G BSAPc(X)). c
Proof. We will prove the proposition only for c-almost periodicity. To show (i), suppose that T G BAPc (X) and p G D. Then we know that T * p G APc(I : X). Due to the first part of [17, Theorem 2.13(iv)], the above implies that the function x M- (T, p(x + h — •)), x G R is c-almost periodic (c-uniformly recurrent, semi-c-periodic). Now the conclusion follows from the calculation
(Th * p) (x) = (Th, p(x — •)) = (Th, p(x — •)) = (T, p(x + h — •)>, x G R.
To show (ii), define the test function pb(-) by pb(t) := p(bt), t G R. Then Gb := T* pb G APc(R : X) and the required conclusion follows from the second part of [17, Theorem 2.13(iv)] and the calculation
(Tb * p)(t) = (Tb,p(t — •)) = (T,p(t — b-)> =
= (T,p(b((t/b) — •))) = (T, pb((t/b) — •)) = Gb(t/b), t G R.
□
The following result is a distributional analogue of [17, Proposition 2.18]:
Proposition 6. Let T G B'URc(X) (T G B'SAPc(X)) and T = 0. Then T G B+0(X).
Proof. Since T = 0, there exists p G D such that T * p = 0. Clearly, T * p is a c-uniformly recurrent function (semi-c-periodic function), so that [17, Proposition 2.18] implies T * p G Co(R : X). Assume to the contrary that T G B+,0(X). Then we have
(T * p) (t) = (T, p(- - t)) ^ 0 as t ^ +to,
which is a contradiction. □
The following result will be important in our further analyses:
Theorem 5. Suppose that there exist an integer k G N and c-almost periodic (c-uniformly recurrent, semi-c-periodic) functions fj : R ^ X (0 < j < k) such that the function
t ^ f (t) = (fo(t), •••,ffc(t)), t G R (3)
is c-almost periodic (c-uniformly recurrent, semi-c-periodic). Define T := j=0 fjj). Then T G BApc(X) (T G B^Rc(X), T G B^(X)).
Proof. We will prove the theorem only for c-almost periodicity because the proofs for c-uniform recurrence and semi-c-periodicity are quite analogous. It is clear that T G D'(X) as well as that (1) yields that for each p G D and t G R we have:
k
(T * p)(t) = <T,p(t -•)> = £ / p(j)(t - v)f(v) dv
j=o ^
k r
= £ / p(j) (v)f(t - v) dv.
j=0
Let e > 0 be given. Then the set 0c(/, e) is relatively dense in [0, to); let t G 0c(/, e) be arbitrary. Then the above computation shows that
II (T * p)(t + t) - c(T * p)(t)|| < ^ / p(j)(v) ■ ||/j(t + t - v) - cf (t - v)|| dv <
k î>
< P(j)(v)
dv, p G D, t G R,
which simply implies the required statement. □
The following counterexample demonstrates the fact that Theorem 5 does not generally hold if the function /(•), defined by (3), is not c-almost periodic (c-uniformly recurrent, semi-c-periodic); we will provide a direct non-trivial calculation showing this:
Example 1. Suppose that c = -1, k =1, /0(t) = cos t and /i(t) = cos(2t) for all t G R. Then the function /(•), defined by (3), is not almost anti-periodic (see [36, Example 2.2(ii)]) and we have
p(v)cos vdv — / p'(v) cos(2v) dv, p gD.
-OO J — OD
Due to (4), we have:
(T * p)(t + t) + (T * p)(t)
p(v)[cos(t + t — v) + cos(t — v)] dv+
+ p'(v)[cos(2(t + t — v)) + cos(2(t — v))] dv, p G D, t G R.
Applying the partial integration, the above implies
(T * p)(t + t) + (T * p)(t) = p(v) cos(t + t — v) + cos(t — v) — 2 sin(2(t + t — v)) — 2 sin(2(t — v))
dv, (5)
for any p G D and t G R. Suppose that p G D is non-negative and its support belongs to the interval [1/6,1/4] C [0,1/4] as well as that
r 1/4
0 <e< / p(v)min((sin v)/2, cos v, 2sin(2v)) dv (6)
and
2 f1/4 (n 1 \ 11 f1/4 0 <e<- p(v) dv • sin( — — - ), 0 <e< 2 sin — • cos - p(v) dv. (7) 3 In V4 8/ 12 4 ,/n
We will prove that the set 0-1(T * p,e) is empty in the following, a rather technical, way. Suppose to the contrary that t G 0-1(T * p,e). Then (5) implies
1/4
p(v) cos(t + t — v) + cos(t — v) — 2 sin(2(t + t — v)) — 2 sin(2(t — v))
dv
<e (8)
for all t G R. Plugging t = —t, t = n — t and t = (n/2) — t in (8), we get: /•1/4 ,
p(v) cos(v) + cos(t + v) + 2 sin(2v) + 2 sin(2(T + v)) dv < e,
1/4
p(v) — cos(v) — cos(t + v) + 2 sin(2v) + 2 sin(2(T + v))
dv
< e
and
1/4
p(v) sin(v) + sin(T + v) — 2 sin(2v) — 2 sin(2(T + v))
dv
respectively. Adding and subtracting of (9) and (10), we get:
/•1/4 p
' p(v) cos(v) + cos(t + v) dv
and
1/4
p(v) sin(2v) + sin(2(T + v))
dv
<e
< e/2,
respectively. Inserting (13) in (11), we get
/•1/4
p(v) sin(v) + sin(T + v)
dv
< 2e.
< e,
10)
11)
12)
13)
14)
0
0
0
0
0
0
0
Further on, there exist k G N U { —1, 0} and a G [0, 2n) such that t = (2k + 1)n + a. Then (12)-(14) gives
•1/4
p(v) cos(v) — cos(v + a)
1/4
p(v) sin(2v) + sin(2a + 2v)
dv
dv
< e,
< e/2
:i5)
:i6)
and
1/4
p(v) sin(v) — sin(v + a)
dv
< 2e.
:i7)
If a G [0, (n/2) - (1/4)], then 2a + 2v G [0,n/2] for all v G [0,1/4] and the contradiction is obvious due to our choice of number e in (6) and the estimate (16); if a G [n, 2n — (1/4)], then a+v G [n, 2n] for all v G [0,1/4] and the contradiction is obvious due to our choice of number e in (6) and the estimate (17). Further on, if a G [n/2, n], then a+v G [n/2, 3n/2] for all v G [0,1/4] and the contradiction is obvious due to our choice of number e in (6) and the estimate (15). If a G [(n/2) — (1/4),n/2], then the estimates (15) and (17) imply
1/4
p(v) sin(v + (a/2)) ■ sin(a/2) dv
and
1/4
p(v) cos(v + (a/2)) ■ sin(a/2) dv
< e/2
< e,
respectively. By adding, we get
1/4
p(v) [sin(v + (a/2)) + cos(v + (a/2))] dv
< 3e ■ [sin(a/2)]-1 <
2sin( п — 8
which is a contradiction due to our choice of e in (7) and the fact that a + v < (n + 1)/4 for all v G [0,1/4] and therefore sin(v + (a/2)) + cos(v + (a/2)) > 1 for all v G [0,1/4]. Finally, if a G [2n — (1/4), 2n), then
f1/4 r i f1/4
/ p(v) sin(2v) + sin(2a + 2v) dv = 2 / p(v) sin(2v + a) ■ cos(a) dv 00
r1/4 1 1 r1/4
= 2 p(v) sin(2v + a) ■ cos(a) dv > 2si^—- ■ cos - p(v) dv,
'1/6
12
4
'0
which contradicts the second inequality in (7).
We continue by stating the following structural characterization of space B'AP (X)
(B^Rc(X), B'SAPC(X)): c
Theorem 7. Let T e B'APc (X) (T G B^Rc (X), T G B^APc (X)) and let T be a bounded distribution. Then there exist an integer p G N and a c-almost periodic (bounded c-uniformly recurrent, semi-c-periodic) function F : R ^ X such that
T = (—j Л F
:is)
0
0
0
0
0
3
0
in the distributional sense.
Proof. The proof essentially follows from the argumentation contained in the proof of [31, Theorem 1]; we will only outline the main details for c-almost periodicity because the proofs for c-uniform recurrence and semi-c-periodicity are quite analogous. Let us consider a fundamental solution G of the differential operator (1 — d2/dx2)p for a certain sufficiently large natural number p G N depending on T. By the proof of the above-mentioned theorem, we have that the convolution F := T * G exists as a continuous function and (18) holds in the distributional sense; furthermore, there exists a sequence (pk) in D such that limk^+ro(T * pk)(t) = F(t), uniformly in t G R. Since for each integer k G N the function (T * pk)(■) is c-almost periodic (apply also [21, Theorem 1.1] for c-uniform recurrence), an application of [17, Theorem 2.13(iii)] shows that F(■) is c-almost periodic, as well. This completes the proof. □
Now we are able to formulate and prove the following result:
Theorem 9. Suppose that T G D^(X). Then the following statements are equivalent:
(i) we have T G B'APc(X) (T G B'URc(X), T G B'SAPc(X));
(ii) there exist an integer p G N and a c-almost periodic (bounded c-uniformly recurrent, semi-c-periodic) function F : R ^ X such that (18) holds in the distributional sense;
(iii) there exist an integer k G N and c-almost periodic (bounded c-uniformly recurrent, semi-c-periodic) functions fj : R ^ X (0 < j < k) such that the function f (■), defined through (3), is c-almost periodic (c-uniformly recurrent, semi-c-periodic) and T = j j
(iv) there exists a sequence (Tn) of c-almost periodic functions (bounded c-uniformly recurrent functions, semi-c-periodic functions) from E(X) such that limn^ro Tn = T in D'li (X).
Proof. The implication (i) ^ (ii) is proved in Theorem 7, while the implication (ii) ^ (iii) is trivial. The implication (iii) ^ (i) follows from Theorem 5; therefore, we have proved the equivalence of statements (i), (ii) and (iii). Their equivalence with (iv) essentially follows from the argumentation contained in the proof of [31, Proposition 7]; see also the proof of Theorem 11 below. □
As a direct consequence of Theorem 9 (see also [18, Remark 1(ii)]), we have the following:
Corollary 1. Let (Tn) be a sequence in B'APc(X) (B^Rc(X) n DLi(X), B'SAPc(X)), and let limn^ro Tn = T in D'Li (X). Then T G B'AP (Xc (T G B^R (X) n d£i (X), T G B'SAPc(X)). c c
For the sequel, we need the following definition:
Definition 2. Suppose that T G D'(X).
(i) We say that T is an asymptotically c-almost periodic distribution of type 1 (asymptotically c-uniformly recurrent distribution of type 1, asymptotically semi-c-periodic distribution of type 1) if and only if there exist a c-almost periodic (c-uniformly recurrent, semi-c-periodic) distribution Tapc G B'AP (X), (Turc G B^Rc(X), Tsapc G BSAPc(X)) and a distribution Q G B+ (X) such that (T, p) = {Tapc, p) + (Q,p), p G D, ((T, p) = (TUrc,p) + (Q,p), p G D,
(T p) = (Tsapc, p) + (Q p^ p G D).
(ii) We say that T is an asymptotically (D0, c)-almost periodic distribution of type 1 (asymptotically (D0, c)-uniformly recurrent distribution of type 1, asymptotically semi-(D0, c)-periodic distribution of type 1) if and only if there exist a c-almost periodic (c-uniformly recurrent, semi-c-periodic) distribution Tapc e B'AP (X), (Turc e BURc(X), Tsapc e B'SAPc(X)) and a distribution Q e B+ (X) such that (T, p) = (Tapc,^) + (Q,p), p e Do, ((T, p) = (TOTc,p) + (Q,p), p e Do,
(T, p) = (Tsapc, p) + (Q, p), p e Do).
Remark 2. Concerning Definition 2 (ii), it should be noted that it is completely irrelevant whether we will write Q e B+ (X) or Q e B+,0(X) here because any element Q e B+0(X) can be extended to an element Q e B+ (X) by the formula Q := F • Q, where F e C^(R) is any fixed function satisfying F(t) = 1 for all t > 0 and F(t) = 0 for all t < -1.
Remark 3. We note that the decompositions in Definition 2 are unique in the case of consideration of c-almost periodicity (semi-c-periodicity) because they are unique for almost periodicity [18].
Now we will prove the following asymptotical analogue of Theorem 9, which gives some new insights at the assertion of [18, Theorem 1] and [33, Theorem 2] (in the last mentioned theorem, C. Bouzar and F. Z. Tchouar have recently established a structural characterization for the space of asymptotically almost automorphic distributions following the approach of I. Cioranescu from [22] (see also [18, Theorem 2]); our novelty here is the use of approach obeyed in the proof of [31, Proposition 7], with a direct proof of implication (i) ^ (ii) and a new characterization (iii) for the class of vector-valued asymptotically almost automorphic distributions):
Theorem 11. Suppose that T e D'Li (X). Then the following statements are equivalent:
(i) T is (half-)asymptotically (D0, c)-almost periodic ((half-)asymptotically semi-(D0, c)-periodic);
(ii) T is (half-) asymptotically c-almost periodic ((half-) asymptotically semi-c-periodic);
(iii) there exist an integer p e N and a bounded (half-)asymptotically c-almost periodic (bounded (half-)asymptotically semi-c-periodic) function F : R ^ X such that (18) holds in the distributional sense;
(iii)' there exist an integer p e N and a bounded (half-)asymptotically c-almost periodic (bounded (half-)asymptotically semi-c-periodic) function F : R ^ X such that (18) holds in the distributional sense ((18) holds in the distributional sense on [0, to));
(iv) there exist an integer k e N and bounded (half-) asymptotically c-almost periodic (bounded (half-)asymptotically semi-c-periodic) functions fj : R ^ X (0 < j < k) such that the function f (•), defined through (3), is (half-)asymptotically c-almost periodic ((half-) asymptotically semi-c-periodic) and T = fjj);
(iv)' there exist an integer k e N and bounded (half-)asymptotically c-almost periodic (bounded (half-)asymptotically semi-c-periodic) functions fj : R ^ X (0 < j < k) such that the function f (•), defined through (3), is (half-)asymptotically c-almost periodic ((half-) asymptotically semi-c-periodic) and T = ^jj=0 fjj) (T = jj=0 fjj) on [0, to));
(v) T is an asymptotically c-almost periodic distribution of type 1 (asymptotically semi-c-periodic distribution of type 1), in the case of consideration of asymptotical c-almost periodicity (asymptotical semi-c-periodicity), resp. T is an asymptotically
(D0, c)-almost periodic distribution of type 1 (asymptotically semi-(D0,c)-periodic distribution of type 1), in the case of consideration of half-asymptotical c-almost periodicity (half-asymptotical semi-c-periodicity);
(vi) there exists a sequence (Tn) of bounded (half-) asymptotically c-almost periodic functions (bounded (half-)asymptotically semi-c-periodic functions) from E(X) such that limn^ro Tn = T in D'L1 (X). Proof. We will prove the implication (i) ^ (ii) only for half-asymptotical (D0, c)-almost periodicity. Let p G D be given and let supp(p) C [a,b]. If a > 0, then p G D0 and therefore the function T * p is half-asymptotically c-almost periodic, as required. If a < 0, then we consider the function po(-) := p(- + a) G D0. Since the convolution mapping is translation invariant, we have that the function (T * p)o(-) = (T * po)(-) is half-asymptotically c-almost periodic, so that there exist a c-almost periodic function g : R M X and a function h G C0([0, to) : X) such that (T * p)0(t) = (T * p0)(t) = g(t) + h(t) for all t > 0. This implies (T * p)(t) = g(t - a) + h(t - a) := go(t) + ho(t), t > a. It is clear that the restriction of function h0(-) to the non-negative real axis belongs to the space C0([0, to) : X), so that the statement (ii) follows by applying [17, Theorem 2.13(iv)] with I = [0, to) and the number a replaced therein with the number —a > 0. The implication (ii) ^ (iii) can be proved following the lines of proof of Theorem 9; we will use the same notation. As in the proof of the above-mentioned result, we have that limk^+ro(T * pk)(t) = F(t), uniformly in t G R; due to [21, Theorem 1.1], the function F(■) is bounded. In the newly arisen situation, the function (T * pk)(■) is (half-)asymptotically c-almost periodic ((half-)asymptotically semi-c-periodic) for all integers k G N. Therefore, there exist a c-almost periodic function (semi-c-periodic function) gk : R M X and a function hk G C0(R : X), resp. hk G C0([0, to) : X), such that (T * pk)(t) = gk(t) + hk(t), t G R, resp. (T * pk)(t) = gk(t) + hk(t), t > 0 (k G N). Since for each integer k G N the function gk(■) is almost periodic, the use of Lemma 2 yields that there exists an almost periodic function g : R M X and a function 0 G C0(R : X), resp. 0 G C0([0, to) : X), such that F(t) = g(t) + 0(t) for all t G R, resp. F(t) = g(t) + 0(t) for all t > 0. But, the argumentation contained in the proofs of [9, Theorem 3.36, Theorem 3.47; pp. 97-98] also shows that the sequence of functions (gk) converges to the function g(-), uniformly on R. Since for each integer k G N the function gk(■) is c-almost periodic (semi-c-periodic), an application of [17, Theorem 2.13(iii)] shows that the function g(-) is also c-almost periodic (semi-c-periodic). This implies (iii). The implications (iii) ^ (iv) ^ (iv)' are trivial. We will prove that (iv)' implies (v) only for half-asymptotical c-almost periodicity. It simply follows that there exist c-almost periodic functions gj : R M X and functions hj G C0([0, to) : X) (0 < j < k) such that the function t M- (g0(t), ■ ■ -,gk(t)), t G R is c-almost periodic as well as that fj(t) = gj(t) + hj(t) for all t > 0. Define Topc G B'APc(X) by
k
Topc(p) 1)M p(j)(v)gj(v) dv, p G D
- opc\
j=0
(see Theorem 5) and
k
Q(p) := 1)j / p(j)(v)he(v) dv, p GD, j=0
where he(-) denotes the even extension of the function hj(■) to the whole real axis. It is clear that we have (T, p) = (Tapc, p) + (Q, p), p G D0. In order to see that Q G B+ (X), it suffices to observe that, for every test function p G D with supp(p) C [a,b], we have that
_ pb
(Q,p(—h)) :=^/ p(j)(v)he(v + h) dv, p gD, h G R
j=oJ a
and therefore p( — h)) = 0, p G D. In order to see that (v) implies
(i), it suffices to repeat verbatim the argumentation given in [18, Remark 2]. We will prove that (vi) implies (i) only for half-asymptotical c-almost periodicity. Using the argumentation contained in the proof of [31, Proposition 7], it suffices to show that, for every fixed function p G D with supp(p) C [0, b] and for every fixed bounded half-asymptotically c-almost periodic function f : R ^ X, the function p * f is bounded and half-asymptotically c-almost periodic. This is clear for boundedness; in order to see that the function p * f is half-asymptotically c-almost periodic, we can argue as follows. Let g : R ^ X be a c-almost periodic function and let h G C0([0, to) : X) such that f (t) = g(t) + h(t) for all t > 0. Then we have
t-b
p(s)g(t — s) ds +/ p(s)h(t — s) ds, t > b, Jo
so that the final conclusion follows from the fact that the space consisting of all c-almost periodic functions is convolution invariant as well as that the function
p(s)g(t — s) ds, t > 0
belongs to the class C0([0, to) : X), which a simple consequence of the last equality. The implication (i) ^ (vi) follows directly from the corresponding part of the proof of [31, Proposition 7]. Therefore, we have proved the equivalence of all statements (i)-(vi). Since (iii)' implies (iv)' and (iv)' implies (v), we have that (iii)' or (iv)' implies all other statements (i)-(vi). On the other hand, it is clear that (iii) implies (iii)', finishing the proof. □
Corollary 2. Let (Tn) be a sequence of bounded (half-)asymptotically c-almost periodic ((half-) asymptotically semi-c-periodic) distributions, and let limn^^ Tn = T in D'li(X). Then T is (half-)asymptotically c-almost periodic ((half-)asymptotically semi-c-periodic).
Remark 4.
(i) It is worth noting that the implications (i) ^ (ii) and (iii) ^ (iv) ^ (v) ^ (i) and the equivalence (vi) ^ (i) can be formulated for bounded (half-)asymptotically c-uniformly recurrent functions, but it is not clear how one can prove that (ii) implies (iii) in this framework.
(ii) Using the idea from the proof of implication (i) ^ (ii) of Theorem 11, we may conclude that a distribution T G D'(X) is c-periodic (c-almost periodic, c-uniformly recurrent, semi-c-periodic) if and only if the function T * p is c-periodic (c-almost periodic, c-uniformly recurrent, semi-c-periodic) for all p G D0.
(iii) If c = 1, then it is not clear how we can introduce and analyze the classes of c-almost automorphic functions and c-almost automorphic distributions.
3. An application
Let n G N, and let A = [a^-]i<ij<n be a given complex matrix such that a(A) C {z G C : Re z < 0}. Following the analysis of C. Bouzar and M.T. Khalladi [28], we will
provide here a small application in the analysis of the existence of half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic) solutions of equation
T' = AT + G, T gD'(Xn) on [0, to), (19)
where G is a half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic) X"-valued distribution. By a solution of (19), we mean any element T G D'(Xn) such that (19) holds in the distributional sense on [0, to). Since the spaces of half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic) distributions are not closed under the pointwise addition of functions, some obvious unpleasant difficulties occur in the case that c = 1. In the one-dimensional case, these difficulties can be overcomed, fortunately:
Theorem 13. Suppose that F is a half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic) distribution, n =1 and an = A < 0. Then there exists a half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic) distributional solution of (19). Furthermore, any distributional solution T of (19) is half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic).
Proof. By Theorem 11, we know that there exist an integer p G N and a bounded half-asymptotically c-almost periodic (bounded half-asymptotically semi-c-periodic) function F : R M X such that (18) holds in the distributional sense, with T replaced with G therein. By the proof of [18, Theorem 4], given in the ultradistributional case, we get the existence of a positive integer m G N, continuous functions Fj : [0, to) m X (0 < j < m) and a function Q G C0([0, to) : X) such that any function Fj(■) has the form
Fj(t) = ci,j(A)F(t) + c2,j(A) [ ex(t-s)F(s) ds, t > 0,
0
for certain complex numbers c1;j- (A) and c2j (A) (0 < j < m) and T = Q + j=0 F(j on [0, to). By the proofs of [10, Proposition 2.6.11] and [8, Lemma 4.1] (see also [17, Proposition 2.32]), we have that the function t M (F0(t), ■ ■ ■, Fm(t)), t > 0 is half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic) so that it can be uniquely extended to a half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic) function t M (F0(t), ■ ■ ■,Fm(t)), t G R due to [17, Proposition 2.25]. Define
T0 := Yl™0 Fj(j) and T1 := Qe, where Qe denotes the even extension of function Q(-) to the whole real axis. Then T = T0 + T1 on [0, to), T is c-almost periodic (semi-c-periodic) and T1 G B+ (X), so that T is half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic). The existence of solutions is proved in a similar fashion. □
Unfortunately, the use of [28, Lemma 1] and the arguments contained in the proof of [28, Theorem 3, pp. 117-118] do not enable us to extend Theorem 13 to the multidimensional case. Keeping in mind the proofs of [18, Theorem 4] and Theorem 13, we can only prove the following:
Theorem 15. Let there exist an integer m G N and half-asymptotically c-almost periodic (half-asymptotically semi-c-periodic) Xn-valued functions Gj(■) (0 < j < m) such that G = Sj=0 Gj on [0, to). Then there exists a solution T of (19) which has the same form as G; furthermore, any distributional solution T of (19) has the same form as G (with the meaning clear).
We close the paper with the observation that the various classes of c-periodic type (ultra-)distributions and various classes of c-almost periodic (ultra-)distributions will be considered somewhere else.
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Article received 14.02.2021. Corrections received 16.04.2021.
Челябинский физико-математический журнал. 2021. Т. 6, вып. 2. С. 190-207.
УДК 517.5+517.9 DOI: 10.47475/2500-0101-2021-16205
РАСПРЕДЕЛЕНИЯ c-ПОЧТИ ПЕРИОДИЧЕСКОГО ТИПА
М. Костич1'", C. Пилипович1,6, Д. Велинов2,с, В. Е. Федоров3,4^
1 Университет Нови-Сада, Нови-Сад, Сербия
2 Университет св. Кирилла и Мефодия, Скопье, Северная Македония
3 Челябинский государственный университет, Челябинск, Россия 4Южно-Уральский государственный университет (национальный исследовательский университет), Челябинск, Россия
"marco.s@verat.net, bpilipovic@dmi.uns.ac.rs, cvelinovd@gf.ukim.edu.mk, dkar@csu.ru
Вводятся и систематически анализируются различные классы распределений c-почти периодического типа и распределений асимптотически c-почти периодического типа со значениями в комплексных банаховых пространствах. Предложено интересное приложение для изучения существования решений асимптотически c-почти периодического типа для одного класса обыкновенных дифференциальных уравнений в пространствах распределений.
Keywords: 'распределение c-почти периодического типа, 'распределение асимптотически c-почти периодического типа, обобщённая функция, обыкновенное дифференциальное уравнение, банахово пространство.
Поступила в редакцию 14.02.2021. После переработки 16.04.2021.
Сведения об авторах
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. Федоров Владимир Евгеньевич, доктор физико-математических наук, профессор, профессор кафедры математического анализа, Челябинский государственный университет; научный сотрудник лаборатории функциональных материалов, Южно-Уральский государственный университет (национальный исследовательский университет), Челябинск, Россия; e-mail: kar@csu.ru.
Работа частично поддержана Министерством науки и технологического развития Республики Сербия (грант № 451-03-68/2020/14/200156), двусторонним проектом между МАНУ и САНУ, поддержана Российским фондом фундаментальных исследований (грант № 21-51-54003) и Постановлением № 211 Правительства Российской Федерации, договор 02.A03.21.0011.
