DENSITY PROBLEM SOME OF THE FUNCTIONAL SPACES FOR STUDYING DYNAMIC EQUATIONS ON TIME SCALES Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2022-15-1-46-55 УДК 517.4

Density Problem Some of the Functional Spaces for Studying Dynamic Equations on Time Scales

Fatima Zohra Ladrani*

Higher Training Teacher's School of Oran (ENSO)

Oran, Algeria

Amin Benaissa Cherif"

University of Science and Technology of Oran "Mohamed-Boudiaf" (USTOMB)

Oran, Algeria

Abderrahmane Beniani*

Laboratory of Analysis and Control of Partial Differential Equations

University Center of Ain Temouchent Ain Temouchent, Algerian

Khaled Zennir§

College of Sciences and Arts, Al-Ras, Qassim University

Kingdom of Saudi Arabia

Svetlin Georgiev^

Faculty of Mathematics and Informatics University of Sofia Sofia, Bulgaria

Received 10.04.2021, received in revised form 10.06.2021, accepted 20.08.2021 Abstract. In this paper we study some topological density properties of certain functional spaces on the time scales and its relationships to Lebesgue spaces in the sense of V-integrals on time scales. Our results are provided with applications.

Keywords: time scale, density, measure.

Citation: F.Z. Ladrani, A.B. Cherif, A.Beniani, K. Zennir, S. Georgiev, Density Problem some of the Functional Spaces for Studying Dynamic Equations on Time Scales, J. Sib. Fed. Univ. Math. Phys., 2022, 15(1), 46-55. DOI: 10.17516/1997-1397-2022-15-1-46-55.

1. Introduction and some preliminaries

The theory of time scales was introduced by Stefan Hilger in his PhD thesis [9] in 1988, in order to unify and generalize continuous and discrete analysis, see [2,6]. Bohner and Guseinov have introduced the Lebesgue A-integral [6, Chapter 5]. Recent results, by A.Cabada, D.Vivero in [8], are devoted on fundamental relations between Riemann and Lebesgue A-integrals. In

* f.z.ladrani@gmail.com

tamin.benaissacherif@univ-usto.dz

ia.beniani@yahoo.fr

§ khaledzennir4@gmail.com ^svetlingeorgiev1@gmail.com © Siberian Federal University. All rights reserved

2006, R.Agarwal et al. in [1], define and study Sobolev spaces in the sense of A-integrals on time scales. After such pioneer work, the study of density properties in the Lebesgue spaces and Sobolev spaces in the sense of the A-derivative on time scales was continued by A. Benaissa et al. in [3,4]. B. Bendouma et al. in [5] presented a relationship between Riemann and Lebesgue V-integrals.

The main purpose of this paper is to be investigated some important functional spaces. We deduct some topological density properties of the considered functional spaces.

The paper is organized as follows. In the next section, we give some auxiliary results needed to be proved our main results. Our results are represented in Section 3. We study functional subspaces on the time scales by report Lebesgue spaces in the sense of V-integrals on time scales. For example, the space of continuous functions C (T, R), the space of ld-continuous functions Cld (T, R), and so on. In the last section, we present the use of density properties and we present diagrams that summarizes our main results.

2. Auxiliary results

A time scale is an arbitrary nonempty closed subset of the real numbers. We will denote it by T. We define the forward and backward jump operators a, g : T ^ T as follows

a(t) = inf{s G T : s > t} and g(t) = sup{s G T : s <t},

respectively, where supT = inf 0, inf T = sup0. The point t G T is said to be left-dense, if g(t) = t and t > inf T, left-scattered if p(t) < t, right-dense if a(t) = t and t < sup T, and right-scattered if a(t) > t. The graininess function n : T ^ [0, x>) is defined by n (t) = a (t) — t and the backward graininess function v : T ^ [0, x>) is defined by v (t) = t — g (t). If T has a right-scattered infimum m, define Tk = T — {m}, otherwise, set Tk = T. If T has a left-scattered supremum M, define Tk = T — {M}, otherwise, set Tk = T.

Definition 2.1 (Nabla derivative [2,6]). Assume f : T ^ R and let t G Tk. We define

f ^ (t)= lim f 6 (t) — f (S), s^t,s£T g (t) — s

provided the limit exists. We call fv (t) the nabla derivative of f at t. Moreover, we say that f is nabla differentiable on Tk provided fv (t) exists for all t G Tk. The function fv : Tk ^ R is then called the (nabla) derivative of f on Tk.

Definition 2.2 (Delta derivative [2,6]). Assume f : T ^ R and let t G Tk. We define

f ^ (t)= lim f ^ (t) — f (s), s^t,seT a (t) — s

provided the limit exists. We call fA (t) the delta derivative of f at t. Moreover, we say that f is nabla differentiable on Tk provided fA (t) exists for all t G Tk. The function fA : Tk ^ R is then called the (delta) derivative of f on Tk.

Definition 2.3 ([2,6]). The function f : T ^ R is called

1) ld-continuous provided it is continuous at left-dense points in T and right-sided limits exist at right-dense points in T. The space of all ld-continuous functions on T will be denoted by Cid (T, R) or Cid(T).

2) rd-continuous provided it is continuous at each right-dense points and has a left-sided limit at each point. The space of all rd-continuous functions on T will be denoted by Crd(T, R) or Crd(T).

Lemma 2.1 ([5,8]). The set of all right-scattered points and the set of all left-scattered points of T are at most countable.

We recall some notions and results related to the theory of V-measure (respectively A-measure) and Lebesgue V-integration (respectively A-integration) for an arbitrary bounded time scale T, where —to < a = inf T < supT = b < to. For more details we refer the reader to [3,4,8].

Lemma 2.2 ([5,8]). Let A C T. Then the following properties are equivalent

1) A is a V-measurable,

2) A is a A-measurable,

3) A is Lebesgue measurable. Notation 2.1. For simplification, we note

R = {te T,a(t) >t} and L = {t e T, g(t) <t}.

Proposition 2.1 ([5,8]). Let A C T be a Lebesgue measurable set. Then the following properties hold.

1) If ae A, then (A) = pl (A) + £ v(s),

seLnA

2) If be a, then MA (A) = PL (A) + £ p (s),

seRnA

3) (A) = pL (A) if and only if a e A and A has no left-scattered points,

4) ma (A) = pL (A) if and only if b e A and A has no right-scattered points.

Theorem 2.1 ([8]). Let A C T be a V-measurable set such that a e A. Let also, f : T ^ R be a V-measurable function. Then

if (t) Vt = f f (t) dt + £ v (s) f (s). (1)

Ja Ja seLnA

Theorem 2.2 ([5]). Let A C T be a A-measurable set such that b e A. Let also, f : T ^ R be a A-measurable function. Then

f f (t) At = i f (t) dt + £ p (s) f (s). (2)

Ja Ja seRnA

We state some of their properties.

Definition 2.4 ([3]). Let p e [1, +to). Then, the set LA (T, R) is a Banach spaces together with the norm defined for every f e LA (T, R) as follows

f \f(s)|pAs< to.

J T

Theorem 2.3 ([4]). Let pG [1, to). Then C (T, R) is dense in LA (T,

A it

3. Main results

In this section, assume that T is a bounded time scale with a = sup T < to and b = inf T > —to. For simplification, we note Ta = Tn (a, b] and Tb = Tn [a, b).

Definition 3.1. Let p € [1, +to) and f : T ^ R be a V-measurable function. We say that f € Lpv (T, R) provided

L = / \f (s)\p Vs < +TO. (3)

V J T„

Remark 3.1. Let p € [1, +to). Then the spaces Lp (T,R) and LpA (T,R) are not even spaces. Really, let T = {ai, a2, a3, a4} be such that ai < ... < a4 and let fi, f2 : T ^ R be two functions such that fi (t) = (t — ai)-1, f2 (t) = (t — a4)-i. Then fi € Lpv (T, R), fi € LPA (T,R), f2 € Lvv (T, R) and f2 € LP (T, R).

Lemma 3.1. Let p € [1, +to), and f : T ^ R be a V-measurable function. Then f € Lp (T, R) if and only if fa € LP (T, R).

Proof. Let s is left-scattered. Then p (s) is right-scattered and a(p(s)) = s. Hence,

v(s) = s — p(s) = a(p(s)) — p(s) = p(p(s))

and p(L) C R. Take f € Lp (T, R) arbitrarily. By Definition 3.1, Theorem 2.1 and Theorem 2.2, we have

L =

i \f(s)\p dt + £ v (s) \f(s)\p =

sec

= f \r(s)\p dt + £ p (g (s)) \r (g (s))\p .

sec

If s € R, then a (s) € L and g(a(s)) = s. Therefore R C g(L) and g(L) = R. Consequently

WfWL^ = / f (t)\p dt + £ p (t) f (T)\p = wr \\pLl.

JTb t eR

This completes the proof. □

Corollary 3.1. Let p € [1, to). Then Lp (T, R) is a Banach spaces equipped with the norm (3).

Proof. It is clear that Lp (T, R) is a normed space. Let (f )£ be a Cauchy sequence in Lp (T, R). By Lemma 3.1, we deduce that f )£ is a Cauchy sequence in LP (T,R), which is a Banach space. Then there exists ga € LP (T, R) such that limff = ga in LP (T, R). Hence, limfe = g

in Lp (T, R). This completes the proof. □

Lemma 3.2. Let p € [1, +to) and f € Cld(T). For £ > 0, there is a continuous function f£ : T ^ R such that \\f — f£WLp ^ 0, as e ^ 0.

Proof. Let s be right-dense. Then there exist an e > 0 and an element sE € T such that 0 < sE — s < £. Consider the function f : T ^ R defined by

fe (t)= ^ f (s)+ f (s) — fs) (t — s), t € [s,se]T , s € T\R, f (t) , otherwise .

<

Note that fe £ CW(T). Let t £ [s, s£]T and s £ T\R. Then

t — s

f (t) - f (t)| < |f(s)| + |f(t)| + |f(s) - f (se)| -

s — se

< f (s)| + f (t)| + f (s) — f(se)| <

< f (s)| + f (t)| + f (s)| + f (se)| <

< 4Hf\U

Set Be = U sec [s se]T- By ProPosition 2.1, we get

W (Be) = £ W ([s, se]T)^£ A ([s,s£]) + £ £ V (t) <

seT\R sec sec te£n[s,se]

< E e + E e2 =

seT\R seT\R

= E (e+e2) <

seT\R

< (e + e2)(b — a).

As a result, we find

\\f — fe\\K < 4 \\f [MA (Be)]p < 4Hf \\ ^o (e + e2)p (b — a)p ^ 0, as e ^ 0. This completes the proof. □

For to £ T, define the generalized polynomials as follows

ho (t,to) = 1 and hn (t,to) = i hn-1 (r,to) Vt for n£ N, t £ T. (4)

J to

Lemma 3.3. Let to,t £ T be such that t ^ to. Then, for all n £ N, there are positive constants an and /3n such that

\U

ßn (t - to)n < hn (t,to) < an (t — to)n for all t > to. (5)

Proof. For n = 1, we have hi(t,to) = t —10 and for ai = ßi = 1, we get (5). Now, assume that (5) holds for n — 1 for some n £ N, n ^ 2. We will prove (5) for n. From Theorem 2.1, we have

et

hn (t,to) < an-i ! (T — to)n 1 Vr = (6)

J to

= an-i (r — to)n-i dr + an-i ^^ v (r) (r — to)n-1 <

■'to re[to,t]nC

n +1

< an_i-(t — to)n for n£ N, t > to. (7)

n

Moreover, we have

hn (t,to) > ßn-i f (r — to)n-iVr to t

= ßn-i f (r — to)n-i dr + ßn-i Y. v (r) (r — to)n-i >

■'to re[to,t]nC

^ ßn-l (t — to)n for n£ N, t > to.

By the last inequality and by (7), we get that (5) holds for n. Hence, we conclude that (5) holds for any n G N. This completes the proof. □

Lemma 3.4. Let p G [1, and f : T ^ R be V-differentiate such that fv G Cld(T). For any £ > 0, there is a function f E : T ^ R that is V-differentiable and fv G C(T), \\f — f e\\Lp ^ 0 and llfv — fvLp ^ 0, as £ ^ 0.

ii nLv

Proof. Let s be right-dense. Then for any e > 0 there exists s E G T such that 0 < sE — s < £. Consider the function Qe (s,.) : [s, se] ^ R defined by

Qe (s, t) = f (s) + fv (s) (t — s) + Xeh2 (t, s) + Yshs (t, s)

and the function f E : T ^ R given by

( Qe (s,t), t G [s,se]T , s G T\R,

f (t) = \ , , N (8) [ f (t), otherwise.

It is clear that the function f E is continuous at all points t G R\{sE : s G T\R}.We choose the constants Ae,ye such that lim f (t) = lim Qe (s,t). We find

t^s-

KM (sE, s) + Yeh3 (sE, s) = f (s e) — f (s) — fv (s) hi (se, s) . (9)

Moreover, f E is V-differentiable at all points t G R\ {sE : s G T\R} and fj is given by

f v (t) \ fV (s) + Kh (t, s) + leh2 (t, s), t G [s, se]T , s G T\R, fv (t) , otherwise.

Note that fv is continuous at all points t G R\ {sE : s G T\R}. If lim fv (t) = lim Qv (s,t),

t^s+ t^s-

we find

Kh (sE, s) + Yeh2 (sE, s) = fv (sE) — fv (s). (10)

By (9) and (10), we have

" Ae ' h2 (se, s) h3 (se, s) -l " f (se )

. ye _ _ hi (se, s) h2 (se, s)

— f (s) — fv (s) hi fv (se ) — fv (s)

By Lemma 3.3, we obtain

I < r7 [\\f \\ ^o + ||f V||J and |7e | < ^ [\\f \J + ||f V||J (11)

for some positive constant r. Thus,

Ifs (t) -f(t)i < e = c [\\f\j + ||fV|J] and \fV (t) - fV (t)i^ e

for some positive constant C. Then

\\f - fs\\lv < e [ma (Bs)]p ^ 0, as e ^ 0,

and

||fV - fVHL* < e [ma (Bs)]p ^ 0, as e ^ 0,

ii nLv

where Bs is defined as in the proof of Lemma 3.2. This completes the proof. □

Lemma 3.5. Let f : T ^ R be continuous. For any £ > 0, there is a function f£ : T ^ R that is V-differentiable and f^Cld(T), \\f — f£\\x ^ 0, as £ ^ 0.

Proof. Define the function g : T ^ R as follows

= ( f (s)+ fV (s)(t — s) , t£ (g (s) , s] , s £ L, g (t) = f (t) , otherwise .

Then g £ C(T). Since C1 (T) is dense in C (T), for £ > 0, we have BE (g) n C1 (T) = 0, where Be (g) = {h£C (T) : \\g — h\\^ < £}. Then there is ge £ C1 (T) such that \\ge — g\\^ < £. Now, consider the function fe : T ^ R, defined by fe = ge. Then fe is V-differentiable on Tk and fV is given by

( g'e (t), t£ T\L,

fV (t)= { ge (g (s)) — ge (s)

v (s)

yi

t.

We have that fe £ C[d (T, R) and \\fe — f \\^^ < \\ge — g\\^ < £. This completes the proof. □

Lemma 3.6. Let f : T ^ R be continuous. For any £ > 0, there is a function f e : T ^ R that is V-differentiable and fV £ Cld(T) and \\f — fe\\L^ ^ 0, as £ ^ 0.

Proof. Let f : T ^ R be continuous and £ > 0. From Lemma 3.5, it follows that there is f e £ Cid (T,R) such that \\f — fe\\^ 0, as £ ^ 0. Hence, \\f — fe\\l$ < (b — a) \\f — fe\\^^ ^ 0, as £ ^ 0. This completes the proof. □

Remark 3.2. Note that C (T, R), Crd (T, R) and Cld (T, R) are Banach spaces together with the norm defined by

\\f \ \ ^o = sup {f (t)| : t£ T} .

Lemma 3.7. Let p £ [1, +to), f : T ^ R is V-differentiable and fV £ Cld(T). For any £ > 0 there is a function f e : T ^ R that is V-differentiable and fV is continuous such that \\f — fe\\^ 0, as £ ^ 0.

Proof. Consider the function f e : T ^ R given by (8). By the proof of Lemma 7, we find

\\fe — f \ \ ^o < A£ [\\f \ \ ^q + |f V||J for some positive constant A. This completes the proof. □

The next result is a generalization of the theoretical density(see [7, Theorem 4.3]). Theorem 3.1. Let p£ [1, to). Then Cld (T, R) is dense in LpV (T, R).

Proof. Let f £ Lp (T, R). By Lemma 3.1, we have fa £ LPA (T,R). Then there exists a sequence (gn)n £ C (T, R) that converges to fin LA (T, R). Set ( fX = (gn)n ■ We have (U)n Q Cld (T, R). Therefore

\\f — fn\Lv = \\fa — gn L.

Hence, (fn)n converges to f in Lp (T, R). This completes the proof. □

Cv (T, R) = {f : T ^ R : such as fv G C (Tk, R)} ,

We denote by

fV

Cld (T, R) = {f : T ^ R : such as fV e Crd (Tk

The following result is a consequence of Theorem 3.1.

Theorem 3.2. Let p e [1, to). Then CV (T, R) is dense in LpV (T, R).

Proof. Let f e LpV (T, R) and e > 0. From Theorem 3.1, we have B4 (f) n Cld (T, R) = 0,

where BE (f) := {g e LpV (T, R) : \\g - f \\Lv < e}. Then there is fs,i e Cld (T, R) such that

e

\\fs ,i - f \\lp < 4. By Lemma 3.2, we conclude that there is fs,2 e C (T, R) such that

e

\\fs,2 - fs,i\\Lv < 4. By Lemma 3.6, there exists is fs,3 e Cfd (T,R) such that \\fs,2 - fs,3\\Lv <

ee 4. From Lemma 3.4, there exists is fs,4 e C1 (T,R) so that \\fs,4 - fs,3\\lp < 4. Therefore, we

find \\fs,4 - f \\lp < e, which implies that BE (f) n CV (T, R) = 0. Then CV (T, R) is dense in Lp (T, R). This completes the proof. □

Remark 3.3. Let E,F,G be three spaces such that E C F C G and (G, T) be a topological space. If E is dense in G, then F is dense in G.

By the previous result, we deduce the following corollary.

Corollary 3.2. Let p e [1, to). Then the spaces Cd (T, R) ,C (T, R) and Crd (T,R) are dense in LpV (T, R).

Proof. Let p e [1, to). We have CV (T, R) C Cfd (T, R) C C (T, R) C Cu (T, R). From Theorem 3.1 and Remark 3.3, we conclude Cfd (T, R), C (T, R) and Crd (T, R) are dense in LpV (T, R). This completes the proof. □

The next result shows that the spaces Cjd (T, R) and Cp (T, R) are dense in C (T, R). Theorem 3.3. The spaces Cd (T, R) and CV (T, R) are dense C (T, R). Proof. Let f e C (T, R) and e > 0. By Lemma 3.5, we have B2 (f) n Cfd (T, R) = 0, where

e

Bs (f) = {g e C (T, R) : \\f - g\\j < e}. Then there is gE e Cd (T, R) such that \\gE - f \ J < -. By Lemma 3.7, we conclude that there is hs e C1 (T,R) such that \\hs - gs\\Lv < -. Therefore

\\hs - f \\lp < e and hence, we conclude that Bs (f) n CV (T,R) = 0. This completes the proof. □

4. Conclusion and application

Use of density properties : To show some results concerning a given function f, it is sometimes useful to look at the problem with hindsight by placing yourself in a suitable functional spaces and using density properties of certain function subclasses. Thus, we are led to demonstrate the desired property for simpler functions. We give an application that can be attacked in the following way.

Lemma 4.1. If f e L1V (T, R) is such that

i f (t) y (t) Vt = 0, for y e Cid (T, R), (12)

J T

then f (t) = 0; V-a.e in T.

Proof. Let f e Cid (T, R) be such that J f (t) y (t) Vt = 0. For y e Cid (T, R), take y = f. We

T

obtain \\f \\L2 (TR) = / \f (t)| Vt = 0, which implies f (t) = 0, V-a.e in T. So, the property (12)

v T

is verified for f e Cld (T, R). As Cld (T, R) is dense in L^ (T, R), if f e L^ (T, R) and e > 0, then there is f£ e Cid (T, R) such that limf = f in L^ (T, R) and / f£ (t) y (t) Vt ^ 0, as e ^ 0.

This completes the proof. □

We give the results found in the paper in the following diagram density between some of the functional spaces on time scales.

Cld (T, R) ^ Cid (T, R)

I \ I

Cid (T, R) ^ Lpv (T, R) ^ L1V (T, R) (13)

t S t

Cy (T, R) Crd (T, R).

References

[1] R.P.Agarwal, V.Otero-Espinar, K.Perera, D.R.Vivero, Basic properties of Sobolev's spaces on time scales, Adv. Differ. Equ., 2006, Article ID, 38121.

[2] M.Bohner, A.C.Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkauser Boston, Inc., Boston, MA, 2001.

[3] A.Benaissa cherif, A.Hammoudi, F.Z.Ladrani, Density problems in L£ (T,R) space, Elec. J. Math. Anal. Appl., 1(2013), no. 2, 178-187.

[4] A.Benaissa cherif, F.Z.Ladrani, Density problems in Sobolev's spaces on time scales, Kragu-jevac J. Math., 45(2021), 215-223.

[5] B.Bendouma, A.Benaissa Cherif, A.Hammoudi, Systems of first-order nabla dynamic equations on time scales, Malaya J. Mat., 6(2018), 619-624.

[6] M.Bohner, A.C.Peterson, Advances in Dynamic Equations on Time Scales, Birkauser Boston, Inc., Boston, MA, 2003.

[7] H.Brezis, Analyse Fonctionnelle: Theorie et Applications, Masson, Paris, 1996.

[8] A.Cabada, D.Vivero, Expression of the Lebesgue A-integral on time scales as a usual Lebesgue integral, application to the calculus of A-antiderivatives, Math. Comp. Modelling, 43(2006), 194-207. DOI: 10.1016/j.mcm.2005.09.028.

[9] S.Hilger, Ein Mafikettenakalk-iil mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universitat Wiirzburg, 1988.

[10] F.Z.Ladrani, A.Benaissa Cherif, Hardy-Sobolev-Mazya inequality on time scale and application to the boundary value problems, Elec. J. Math. Anal. Appl., 6(2018), 137-143.

[11] V.Lakshmikantham, S.Sivasundaram, B.Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Boston, 1996.

Проблема плотности некоторых функциональных пространств для изучения динамических уравнений на временных масштабах

Фатима Зохра Ладрани

Кафедра математики Высшей педагогической школы г. Оран

(ЭНСО) Оран, Алжир

Амин Бенаисса Шериф

Оранский университет науки и технологий "Мохамед-Будиаф"

(УСТАМБ) Оран, Алжир Абдеррахман Бениани

Лаборатория анализа и управления уравнениями с частными производными

Университетский центр Айн Темушент, Айн Темушент, Алжир

Халед Зеннир

Колледж наук и искусств, Аль-Рас, Университет Касима Королевство Саудовская Аравия

Светлин Георгиев

Факультет математики и информатики Софийский университет София, Болгария

Аннотация. В этой статье мы изучаем некоторые свойства топологической плотности некоторых функциональных пространств на временных масштабах и их отношения с пространствами Лебега в смысле У-интегралов на временных масштабах. Наши результаты снабжены приложениями.

Ключевые слова: шкала времени, плотность, мера.