The pseudospectrum of the convention-diffusion operator with a variable reaction term Текст научной статьи по специальности «Математика»

Том 24, № 128 2019

© Guebbai H., Segni S., Ghiat M., Merchela W., 2019 DOI 10.20310/2686-9667-2019-24-128-354-367 УДК 517.984.5

The pseudospectrum of the convention-diffusion operator with a variable reaction term

Hamza GUEBBAI1, Sami SEGNI1, Mourad GHIAT1, Wassim MERCHELA2

1 Universite 8 Mai 1945 B.P. 401, Guelma, Algerie 2 Derzhavin Tambov State University 33 Internatsionalnaya St., Tambov 392000, Russian Federation

Псевдоспектр оператора конвенции-диффузии с переменным членом реакции

Хамза ГЕББАЙ1, Сами СЕГНИ1, Морад ГИАТ1, Вассим МЕРЧЕЛА2

1 Университет 8 мая 1945, 24000, Алжир, Гельма, В.Р. 401 2 ФГБОУ ВО «Тамбовский государственный университет им. Г.Р. Державина» 392000, Российская Федерация, г. Тамбов, ул. Интернациональная, 33

Abstract. In this paper, we study the spectrum of non-self-adjoint convection-diffusion operator with a variable reaction term defined on an unbounded open set Q of к". Our idea is to build a family of operators that have the same convection-diffusion-reaction formula, but which will be defined on bounded open sets {Qn}nej0 i[ of к". Based on the relationships that link this family to Q, we obtain relations between the spectrum and the pseudospectrum. We use the notion of the pseudospectrum to build relationships between convection-diffusion operator and its restrictions to bounded domains. Using these relationships we are able to find the spectrum of our operator in r+. Also, the techniques developed to obtain the spectrum allow us to study the properties of the spectrum of this operator when we go to the limit as the reaction term tends to zero. Indeed, we show a spectral localization result for the same convection-diffusion-reaction operator when a perturbation is carried on the reaction term and no longer on the definition domain.

Keywords: differential operator; spectrum; pseudospectrum; convention-diffusion operator

For citation: Guebbai H., Segni S., Ghiat M., Merchela W. Pcevdocpektr operatora konvek-tsii-diffuzii s peremenim chlenom reaktsii [The pseudospectrum of the convention-diffusion operator with a variable reaction term]. Vestnik rossiyskikh universitetov. Matematika - Russian Universities Reports. Mathematics, 2019, vol. 24, no. 128, pp. 354-367. DOI 10.20310/ 2686-9667-2019-24-128-354-367.

Аннотация. В статье исследуется спектр несамосопряженного оператора конвекции-диффузии с переменным членом реакции, определенным на неограниченном открытом множестве Q С к". Идея исследования состоит в том, чтобы построить семейство операторов, имеющих такую же формулу конвекции-диффузии-реакции, но определенных

на ограниченных открытых множествах }nej0 i[ С к". Основываясь на соотношениях, которые связывают это семейство с Q, получены соотношения между спектром и псевдоспектром. Для построения соотношений между оператором конвекции-диффузии и его сужениями на ограниченные области используется понятие псевдоспектра. Полученные соотношения используются для определения спектра исходного оператора в к+. Методы, разработанные для нахождения спектра заданного оператора, позволяют также изучить некоторые свойства этого спектра при переходе к пределу, когда член реакции стремится к нулю. В частности, показано, как определить спектр заданного оператора конвекции-диффузии-реакции при возмущении члена реакции, а не области определения.

Ключевые слова: дифференциальный оператор; спектр; псевдоспектр; оператор конвенции-диффузии

Для цитирования: Геббай Х., Сегни С., Гиат М., Мерчела В. Псевдоспектр оператора конвенции-диффузии с переменным членом реакции // Вестник российских университетов. Математика. 2019. Т. 24. № 128. С. 354-367. DOI 10.20310/2686-9667-2019-24-128354-367. (In Engl., Abstr. in Russian)

Introduction

The study of the spectrum of convection-diffusion operator is one of the most complicated problems in functional analysis. In this paper, we study the spectrum of the following operator:

, A i /TT

i= 1

Au = -Au + f -Vh(JJ Xi)\ •Vu + Vu, (0.1)

^ i=i '

where h E C2 (К, К) such that h" is positive and

n n \ 2

V = Vi + V2, Vi(x) = £ (h'(n Xj) П

i=1 4 j=1 j=1 j=i

Non-negative potential V2 is considered as a reaction to the convection-diffusion phenomena

n

represented by A0 = — A + ( — Vh(J^[ xi)) -V + Vi. The operator A in the case h(x) = x

i=1

was studied in [1].

We recall that, for an unbounded operator T defined on D(T) C H to H and for e > 0, the pseudospectrum is given by (see [2])

sp£(T) = {z E re(T) : \\(zl — T)-1\\ > e-1} y sp(T).

The resolvent set is given by

re(T) = {z E C : (zl — T)-1 exists and bounded} ;

sp(T) denotes the spectrum of T and is defined as sp(T) = C \ re(T). An equivalent definition of the pseudospectrum has been given in [3]:

sp£ (T)= y sp (T + D).

D:H^H, linear and ||D||<£

Pseudospectrum is easier to calculate and more efficient than spectrum when dealing with unbounded operators [4,5]. In fact, it has been established that an approximation of the spectrum of differential operators may be unstable when going to the limit, unlike the pseudospectrum which shows to be stable (see [6-8]). For example, if T is a normal operator, its pseudospectrum is equal to the e -neighborhood of its spectrum. The e -neighborhood of S С C is given by

N (S) = {s + z : s e S, |z| < e} .

It is clear that, for all S С C, Q N (S) = S.

£>0

Moreover we take advantage of the fact that the spectrum of an operator is divided into two sets: the pointwise spectrum, spp (T), which consists of all the eigenvalues of T; the essential spectrum, spess (T) which consists of all Л e C such that the operator (XI — T) is injective, but not surjective. In addition, we define the limit of a sequence of sets as follows: for all 9 > 0, So С C, lim<9^0 So = {s e C : 3 {so}o>0 , so e So, limo^0 so = s} .

In this article, we study in detail the pseudospectrum and the spectrum of the operator A to establish that its spectrum is real positive. We conclude this work with a result on the stability of the spectrum obtained by the pseudospectral theory.

1. Convection-diffusion operator

Let Q CRn be an unbounded open set. Let h E C2(R, R) be such that h" is positive. Let A be the convection-diffusion operator (see [9]) defined on L2(Q, C) into itself by (0.1). We define the hermitian form p on L2(Q, C) as

<p(f,g)= Vf ■Vgdx + (-Vh(n Si)) ■V fgdx + V/gdx, Jn Jn y i=1

where the quadratic form associated with p is given by

Q(u)= \\Vu\\2L2(n) + / f-Vh(nxi)\ ■ Vuudx + V|u|2dx.

Jn ^ i=\ ^n

For u E CCf(Q), we set z = fn (-Vh^™=1 x»)) u ■Vudx, so

r. n r. n

z = f-Vh(n x^) |u|2dx + (Ah(H x^) |u|2dx JanK i=i JnK i=1 J

=0

+ I (Vh(n Xi)) •Vuudx = Xi)) |u|2dx — z.

i=i ' Jn v i=i J

m

Using z + z = 2Re(z) = fn (Ah^™=1 Xi,)) |u|2dx yields

1 n n

(2M(П Xi) + ||Vh(n Xi)||2(n) + ВД) |u|2dx > 0,

i=1 i=1

Re(Q(u)) = ||Vu||!2(Q) + /

Jn

z — Re(z )

i

and

r. n

\Im(Q(u))\ = Im^J x¿}) u • Vudx)

1 f n 1 f n < 2 (HVu|L2(n) + ^ ||Vh(nXi)!L2(n)\u\2d^ + -| yn (Ah(nxi)) \u\2dx

Hence, p is a sectorial form defined on the vector space R given by the following expression:

R = H^(Q, C) Q{u G L2(Q, C) : Vu G L2(Q, C)} .

We recall that A is the operator associated with p [10, Theorem 2.1, p. 322], and the domain of A is given by D(A) = H2(Q, C) f| R.

Our goal is to determine the spectrum of A. We notice that D(A) is a Dirichlet integral boundary condition. Consider the eigenvalue problem: find X G C and u G D(A) \ {0} such that

-Au + ( - Vh(^ Xi)) ■ Vu + Vu = Xu on Q, u = 0 on dQ.

i=1

Let {Qn}ne]o,i[ be a sequence of bounded open sets such that Q^ C Qn, for all n < V, and Une]0 1[ Qn = Q. For all n s]0,1[, we identify the hermitian form on L2(Qn, C) by

(f,9)= f Vf-Vgdx +f i-Vh(J] Xi)\ ■ Vfgdx + /" Vfgdx.

J V i_1 J

We recall that ipv is a sectorial form defined on Rv = H1(QV, C). We denote by Av the differential operator associated with [10, Theorem 2.1, p. 322]. The domain of An is given by D(An) = H2(Qn, C) P| H(Qn, C). We notice that An is defined by the same formula as A. Let Bn be the differential operator which is defined by the same formula as A, but is given on D(Bn) = H2(Qn, C). The extension of each function in D(BV/) to Qn by zero belongs to D(BV) [11, Lemma 3.22, p. 57]. Thus D(B^) C D(BV).

To achieve our goal we will define the spectrum of An, after that we will establish a relation between the pseudospectrum and spectrum of An, Bn and A.

2. Spectrum of An and Bn

In this section, we will explain some characteristics of the operator An which allow us to locate the spectrum of A.

2.1. Spectrum of An

Define the following scalar product on L2(Qn) :

„ n

V(u,v) G L2(Qv) x L2(Qv), (u,v)n = exp x^juvdx.

\ w r2r), (u,v)n = J exp ym

i=1

Theorem 2.1. For all n g]0, 1[, An is self-adjoint with respect to (, )n.

1 n

Proof. For all u G D(An), we define u = exp x^)u. Hence, we get

j=1

n 1 n 1 n

AU = (Au + (vh(nXi)) ■ Vu) +1 ||Vh(nXi)HL2(nnex^1 M^QXj) j=1 i=1 i=1

Then, for all (u,v) G D(An) x D(An),

„ n „ n

(An u,v)n = exp x^) An uvdx = ( — Au + ( — Vh(^ Xj)) -Vu +

•/On j=1 ./On i=1

n / 1 n

+ (l|Vh(HXj)lL2(On) + V>(x))u) exp 1 h(nXi) ) Udx

^) + "2(x)y "y exP I 2'

j=1 V j=1

—Auvdx + / (J |Vh(n Xi) 11 L2(On) + V2 (x)) uudx

J°n j=1

f duv f 5 n

VuVudx — da + (4 llVh(^ Xj)||2(Qn ) + V2(x)) uudx.

So, for all n g]0, 1[, (An■,-)n is also a scalar product, which means that An is self-adjoint [10, Theorem 2.7]. □

As a consequence, sp(An) is real for all n g]0, 1[. Since we cannot extend the scalar product (, )n over L2(H), we cannot guarantee that A is self-adjoint. We define

1 n n

K = inf {1Ah(nXj) + ||Vh(nXj)|L2 + V2(x) : (X1,....,Xn) G ^},

j=1 j=1

K1 = inf {|Vh(H Xj)|L2 : (X1,....,xn) G O,},

xi)||2

j=1

m = + K-

^ + K — ,

E = {n E ]0,1] : M < 0} , where CPF is the Poincarre-Friedrichs constant (see [10]). Theorem 2.2. For all n e]0, 1[,

• If n E E, the essential spectrum spess(An) is included in ]5K^C-F + K[, and the point spectrum spp (An) is included in [C-F + K,

• If n E E, An has no essential spectrum, and the point spectrum is included in

[c-2 + k,

P r o o f. For all n e]0, 1[ and all u E D(An),

Re ((An u, u) ) = ^((An u, u) + (u, An u)) = ^((An u,u) + (u, An u)).

On

However

f -Auudx = — f „ju^do + f VuVudx = f |Vu|2dx.

J On J dOn _o On On

n n n n n n

Vh(J^[ Xj) •Vuudx = — Vh(J^[ Xj) •Vuudx — Ah(J^[ Xj) |u|2 dx.

0n j_1 ^0n j_1 ^0n j_1

Then

Re((Avu,u)) = ||Vu||2(Qn) + / ( ^(J]xj) + xj)^^) + V>(x) ) |u|2dx

> lVu|L2(Qn ) + K llullhvn ). By the theorem of Poincarre-Friedrich (see [10]),

Re((Anu,u)) > C-F ||u|L2 (On) + K ||u|L2 (On) > (C-F + K)|u|L2(Qn;

Thus, for all A G R,

||(A — AI)u||L2(0n) > (C-2 + K — A)|u|L2(0n).

Hence (An — AI) is injective for all A < CPF + K, and thus sp(An) is included in

5 ^

H by 4

[C-F + K, +œ[. Let H = H1 (Qv) and A G [—œ, 4K1[. The sesquilinear form is defined on

Px(u,v) = J (VuVv + (4 V1 + V2 — A)uv)dx,

J On

which verifies

(u,v)| < ||Vu|L2(0n )|Vv|L2(0n ) + C ||u||L2 (On ) ||v||L2(0n ),

where

C = sup ^ -V1 (x) + V2 (x) : x G Qnf + |A|,

and

^A(u,v)| > min |1, (5K1 — A)J ||u||H.

Since, for all g G L2(QV), the semilinear form L : H M C, v M f0 gvdx is continuous, it follows from the Lax-Miligram theorem that the equation

px(u,v) = L(v)

has a unique solution u in H for all v G H. Take into consideration the problem

for g G L2(QV), find u G L2(QV) such that (p) { Anu — Au = g on Qv,

u = 0 on d.

We use the same variable change as in the previous theorem: we multiply the equation by exp(h(Hn=iXi)) and set <7 = gexp(Xi)), < = uexp(Xi)). We see that (p) is equivalent to

for < E L2(Qn), find w E L2(Qn) such that

-Aw + (5V1 + V2)w - Aw = < on Qn,

w = 0 on 5Qn

v

Therefore, the sesquilinear form ^A(u,v) = fn VuVv + (5K1 — A)uv) is an inner product in L2(Qn) for A< 5K1. Hence, (p) has a unique solution w, and (p) has a unique solution u defined by u = wexp ( — ^^nn=1 x»)). D

2.2. Relation between An and Bn

It is known that, for every n e]0, 1[, Bn C An, i. e. D(Bn) C D(An) and, for all / E D(Bn),

Bnf = Anf.

Furthermore, for every n e]0, 1[, Bn C A. In fact, for any f E D(Bn) = H0!(Qn), extending f to Q by 0 gives f E D(A). This proves that

U sPp(B

n) C spp (An).

0<n<1

To determine the spectrum of A the study of the difference spp(An)\spp(Bn) is required. In this section, we prove that this difference is empty for n e]0, 1[.

Lemma 2.1. For all £ > 0 and for all n e]0, 1[,

sp£ (Bn) = sp£(An).

Proof. Let A E sp£ (Bn). Then there exists f E D(Bn) such that

\\Bn f — Af \\ L2(n) \\f \\ L2(n^)

But f E D(An), so

f — Af \\L2(nn)

< £.

\\f \\ L2(n^)

and A E spe(An).

Inversely, let A E spe(An). Then there is f E D(An) such that

\\An f — Af \\L2(nn)

\\f \\ L2(n^)

n) < £.

Since the space of infinitely differentiable functions with compact support C°(Qn) is dense in D(A) with respect to the graph norm defined by \\-\\A = ||A-\\L2(n) + \H\L2(n) , for all

f e D(AV), there is a sequence (fn)neN in ) such that \\fn - f \\A = 0. So,

for all 9 > 0, there exists N e N such that, for all n > N, we have

\\An fn - Afn \\l2(Q) \\Anf - xf \\l2 (q)

\\fn\\l2(q) \\f \\l2(q)

< 9.

We set

9 = £_ \\Anf - Af wl2^) > 0

\\f\l2(qv )

Then there exists n0 e N such that

\An fno - Afno \\L2(Q) < £

\\ fno\\l2(q)

However fn0 e D(Bn), then A e spe(Bn). □

Corollary 2.1. For all £ > 0, if 0 < r < j < 1 then

SPe (A^') C Spe(Av). Proof. Let A e spe (Bn'). Then there exists f e D(Bn') such that

\\Bn'f - Af\L2(Hn') \\f \\L2(nv)

Extending f to Qv by 0,

\\Bn f - Af \\ L2(n^)

\\f \\ L2(Q^)

It follows that A belongs to sp£(Bn). Now we can apply Lemma 2.1 to complete the proof. □ We proved that the family {sp£(An)}o<n<i is decreasing, and this makes us to say that the family {sp(An)}0<n<1 is decreasing with respect to inclusion. In fact, for all 0 < n < n' < 1,

sp(An') — sps(An') — sps(An).

But An is self-adjoint, i. e. sp£(An) = N£ (sp(An)) , for all n g]0,1[. Then

sp(Av') c ff N (sp(Av)) = sp(Av). £>0

Theorem 2.3. For all n G E,

sp(Bv) = sp(Av). Proof. Since An is self-adjoint for all n G E, we have

p| SPe(Av) = SP(AV). e>0

Then, by using Lemma 2.1, we find ) C

sp£(Bn) = sp£(An

) ^ sp(Bn) C p| sp£(An) = sp(An). £>0

Reciprocally, let A E sp(An) for some n E E. According to Theorem 2.2, there exists f E D(A),f = 0 such that

An f = Af.

Since C°(^n) is dense in D(An) with respect to the graph norm, there is a sequence (fn)neN in Cc°°(nn) which converges to f in the graph norm. We define the sequence

=_^_, n E N.

For all ne N,

and

r II D л II llAnf - A/) n

lim y^ngn - Agn^L2(Qn) =-¡iTii-- = 0.

Then A E sp(Bn). In fact, if (Bn — А/)-1 exists and is bounded, then

1 = llg™llL2(Qn) < ||(Bn — A1 ) llBn— Agn^L2(Qn) -У 0.

Therefor, the operator (Bn — AI)-1, if it exists, can not be bounded, which means that Bn — AI can not be surjective. □

3. Pseudospectrum and spectrum of A

The pseudospectrum has better stability than the spectrum. Pseudospectrum is easier to be controlled and can be considered as the finest stable for the passage to the limit.

3.1. Pseudospectrum

In this subsection, we establish a relation between the spectrum and the pseudospectrum of A seen as limits of sp (An) and sp (Bn) respectively.

Theorem 3.1. For all £ > 0,

SPe(A) = U sp£ (An) = U sp£(Bn).

n€E n^E

Proof. Let A E U sp£(Bn). Then there exist n1 E E and / E D(Bni ) such that 0<n<i

lBnif - A/|ь2(Пп!)

||f 1к2(Пп )

ni) < e.

Extending f to Q by 0,

\\Af — Af\\l2(n) <£ \\f \ \ L2 (n)

So, it follows that A belongs to spe(A), and

J Spe(Bn) C Spe(A).

nee

Reciprocally, let A E spe(A). Then there is f E D(A) such that

\\Af — Af \ \ l2 (n)

llf Il2(H)

< e.

Since Cc°°(n) is dense in D(A) with respect to the graph norm, for all f E D(A), there is a sequence (fn)neN in such that

lim ||Afn - AfJ^n) = ||Af - Af |l2(Q)

||fjL2(Q) ||f ¡L2(Q)

Like in the proof of Lemma 2.1, we choose n0 such that

||Afno - Afno ¡L2(Q)

l|fno ¡L2(H)

< e.

There is n small enough for which the support of g = fn0 is included in Qn. It follows that A belongs to spe(An). Thus,

spe(A) C (J sp£(An). nee

Now, we use Lemma 2.1 to conclude the proof. □

From the previous theorem, we deduce that

sp£(A) C N£(R+) = {z E C : Rez > 0, |1mz| < £} J {z E C : Rez < 0, |z| < £}.

In fact, for all n E E, An is self-adjoint. Then spe(An) = Ne(sp(An)). But, sp(An) C R+. We obtain

J sp(An) C R+.

nee

3.2. Spectra

In this part, we will set a new relation between the spectrum of A and that of An. First, we begin with a topological result that will allow us to obtain the desired property.

Proposition 3.1. For all £ > 0,

J sp£ (An) = Nj J sp(An)

nee Vnee

Proof. Let A G (J sp£(An). There is n1 G E such that

neE

A G sp£(Am) = N (sp(Am)). So, A = s + z, where s G sp(Ani) and |z| < e. But s G (J sp(An) implies

neE

A G Ne | U sp(An)

\neE

Reciprocally, let A G N£ I (J sp(An) I . Then A = s + z, where s G (J sp(An) and |z| < e.

\n£E J neE

So, there is n1 G E such that

A = s + z G N£(sp(Am)) = sp£(Am).

Thus, A G U sp£(An). □

neE

Theorem 3.2.

sp(A) = U sp(An).

neE

Proof. Let A G (J sp(An). There is n1 G E such that A G sp(Ani). Then there is neE1

f G D(Ani), where Ani f — Af = 0. Since ) is dense in D(Ani) with respect to the

graph norm, there is a sequence (fn)neN in CC°(Qni) which converges to f in the graph norm. We define the sequence

g= i on ^ni, n G N,

n \ 0 on Q/Qni, n G N.

We have, for all n G N,

g,n G D(A), WgnWh^n) = 1,

then

lim WAgn — Agn||L2(o ) = 0. Thus A G sp(A). By Theorem 3.1, we have

sp(A) c sp£(A) = U spMn),

neE

and by Proposition 3.1,

sp(A) C Nj U sp(An)) .

neE

So, we obtain

sp(A) — U sp(An)

neE

as e tends to 0. □

4. Formula perturbation

This section is devoted to the study of pseudospectrum stability when the perturbation is applied directly to the operator's formula. For this purpose, we define for a > 0,

Aa = Ao + aV3,

where V3 is a continuous function over Q such that

K = sup |V3(x)| < = inf |V3(x)| ,

xen xen

which means that, for all a > 0, D(Aa) = D (A0). Our aim is to compare sp (Aa) and sp (A0). For n > 0, Aa,n, A0,n are defined in the same way as An. It is clear, that for

£ > 0,

Va > 0, sp£ (Aa) = y sp£

neEa

where E = {n G ]0,1] : Cp2 + K - ^f - aK3 < 0} , and

Va > 0, sp (Aa) = U sp(Aa,n).

neEa

Theorem 4.1.

lim sp (Aa) C sp (Ao) C lim SPaf2 (Aa) .

Proof. Let £ > 0, n > 0 and A G sp£ (A0,n). Then there exists f G D (A0,n) such that

l|Ac,nf - Af ||

L2(n) < £ ||f ^L2(nn) .

Therefore, for a > 0,

HA«,,/ - A/1| L2(Qn) < + aK2) ||f ¡L2^)

which means that A G sp£+aK2 (Aa,n). However, Ao,n and Aa,n are self-adjoint operators, then

sp (Ao,n) C SPaK2 (A«,n) ^ Sp (Ao) C ^J Sp«K2

neEo

We use the fact that E0 C Ea for all a > 0 to get

sp (Ao) C sp«K2 (A«) ^ sp (Ao) C lim spaX2 (Aa). Inversely, it is clear that, for all a > 0 and all n > 0,

sp (Aa,^) C sp„K2 (Ao,n).

Then

sp (Aa) = I I sp (Aa,v) С I I spaK2 (Aon) ^ lim sp (Aa) С lim I I spaK2 (Aq,v)

V^-Ea V^Ea V^Ea

but

U sp*K2 (A0 ) = NaK2 I l^J sp (A0,n)

a

We use lim Ea = E0 to get

lim sp (Aa) С sp (Ao)

□

Acknowledgements. We are very grateful to the editor and reviewer for their remarks proposed to improve our paper. We thank Mr. Ammar Khellaf for his effort and help.

References

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Information about the authors

Hamza Guebbai, Associate Professor of Mathematics Department. University 8 Mai 1945, Guelma, Algeria. E-mail: [email protected]; [email protected] ORCID: https://orcid.org/0000-0001-8119-2881

Информация об авторах

Геббай Хамза, доцент, кафедра математики. Университет 8 мая 1945, г. Гельма, Алжир. E-mail: [email protected]; [email protected] ORCID: https://orcid.org/0000-0001-8119-2881

Sami Segni, PhD Student of Mathematics. University 8 Mai 1945, Guelma, Algeria. E-mail: [email protected]; [email protected] ORCID: https://orcid.org/0000-0002-5330-1822

Mourad Ghiat, Associate Professor of Mathematics Department. University 8 Mai 1945, Guelma, Algeria.

E-mail: [email protected];

[email protected]

ORCID: https://orcid.org/0000-0002-4484-2504

Wassim Merchela, PhD Student of Mathematics. Derzhavin Tambov State University, Tambov, the Russian Federation. E-mail: [email protected] ORCID: https://orcid.org/0000-0002-3702-0932

There is no conflict of interests.

Corresponding author:

Wassim Merchela

E-mail: [email protected]

Received 22 August 2019 Reviewed 17 October 2019 Accepted for press 29 November 2019

Сегни Сами, аспирант, кафедра математики. Университет 8 мая 1945, г. Гельма, Алжир. E-mail: [email protected]; [email protected] ORCID: https://orcid.org/0000-0002-5330-1822

Гиат Морад, доцент, кафедра математики. Университет 8 мая 1945, г. Гельма, Алжир. E-mail: [email protected]; [email protected] ORCID: https://orcid.org/0000-0002-4484-2504

Мерчела Вассим, аспирант, кафедра функционального анализа. Тамбовский государственный университет им. Г.Р. Державина,г. Тамбов, Российская Федерация.

E-mail: [email protected] ORCID: https://orcid.org/0000-0002-3702-0932

Конфликт интересов отсутствует.

Для контактов:

Мерчела Вассим

E-mail: [email protected]

Поступила в редакцию 22 августа 2019 г. Поступила после рецензирования 17 октября 2019 г. Принята к публикации 29 ноября 2019 г.