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LINEAR FRACTIONAL SYSTEM OF INCOMMENSURATE TYPE WITH DISTRIBUTED DELAY AND BOUNDED LEBESGUE MEASURABLE INITIAL CONDITIONS.

1. INTRODUCTION

To obtain a deep understanding about the fractional calculus and respectively the fractional differential equations in details see the monographs of Kilbas et al. [12], Kiryakova [13], Podlubny [26], Feckan et al. [9] and Abbas et al. [1]. For distributed order fractional differential equations see [11], for an application-oriented exposition Diethelm [7] and for fractional evolution equations in Banach spaces Bajlecova [4]. The impulsive differential and functional differential equations with fractional derivatives and some applications are considered in the monograph of Stamova and Stamov [28]. Also it is worth noting some new interesting results for fractional differential equations and systems obtained in [2], [20], [29], [35], [38] and [39].

The first detailed study of linear differential equations and system with distributed delay (fundamental theory, variation of constants formula, stability, etc.) was done by A.D. Myshkis in his fundamental monograph [22]. It may be noted that fractional systems of retarded and neutral type with distributed delays are studied (basically existence and stability) in [14], [21], [31]-[34] for single order fractional derivatives and in [5] for Caputo-type distributed order fractional derivatives. Note that a lot of results are obtained from many authors, using the definition of Caputo type derivative applicable only in the particular case when the functions are absolutely continuous. In this work, we use the definition of Caputo-type derivative without the assumption that the functions are absolutely continuous.

It is well known that the problem of establishing a formula for the general solution for linear fractional differential equations and/or systems with delay, as well as its integral representation (variation of constants formula) need theorem for existing of fundamental matrix, i.e. theorem for existence and uniqueness of the solutions of initial problem (IP) in the case of discontinuous initial functions (see for example [6], [10], [37]). It must be noted that this problem is more complicated in compare with the integer order differential equations with delay. We point out that this is conditioned that a distinguishing feature of the fractional differential equations with delay is that the evolution of the processes described by such equations depends on the past history inspired from two sources, first of them is the impact conditioned of the delays and the other one the impact conditioned from the availability of Volterra type integral in the definitions of the fractional derivatives, i.e. the memory of the fractional derivative. It must be noted that the first of them (conditioned by the delays) is independent from the derivative type (integer or fractional).

In the present work, we consider a nonautonomous linear fractional system with distributed delay and derivatives in Caputo sense of incommensurate type. For this system, we study the important problem for existence and uniqueness of the solutions of initial problem (IP) in the case of bounded Lebesgue measurable initial conditions.

As far as we know, there are only a few results concerning IP for fractional differential equations with delay and discontinuous initial function. In [15] is studied an IP with bounded Lebesgue measurable initial conditions for autonomous linear system with distributed delay in the case when all differentiations orders are equal. In [6] are obtained results for existence and uniqueness for nonautonomous fractional system with distributed delay and piecewise continuous initial functions with finite many jumps. The results in [6] are generalized in [37] for neutral systems. It must be noted, that the technique of the proofs in the present paper (inspired from [15]) is different in compare with the technique used in [6]. Since in our obtained results the fractional differentiation orders are of incommensurate type then our result extends the corresponding one in [15] even in the autonomous case too. The proposed conditions coincide with the conditions which guaranty the same result in the case of integer order linear differential equations with distributed delay.

The results obtained in this article would be a good basis for building models of different processes from the real world. Good examples of new studies with application in modeling are [3], [16]-[19], [23]-[25], [27], [30].

The paper is organized as follows. In Section 2, we recall some needed definitions of Riemann-Liouville and Caputo fractional derivatives, as well as the needed part of their properties. In this section also we present a slightly modified version of the Weissinger generalization of the Banach's fixed point theorem ([36], Fixpunktsatz, p. 195) which will be used by the proof of the main results and is presented the problem statement too. In Section 3 as main result are obtained sufficient conditions for the existence and the uniqueness of the solutions of the Cauchy problem for linear incommensurate fractional differential system with distributed delays in the cases of Caputo derivatives and with Lebesgue measurable, bounded initial function.

2. PRELIMINARIES AND PROBLEM STATEMENT

For convenience and to avoid possible misunderstandings, below we recall only the definitions of Riemann-Liouville and Caputo fractional derivatives and some needed their properties. For details and other properties we refer to [12, 13, 26].

Let [alpha] [member of] (0, 1) be an arbitrary number and denote by [L.sup.loc.sub.1] (R, R) the linear space of all locally Lebesgue integrable functions f : R [right arrow] R. Then for each t > a, a [member of] R and f [member of] [L.sup.loc.sub.1] (R, R) the left-sided fractional integral operator and the corresponding left side Riemann-Liouville and Caputo fractional derivatives of order a are defined by

[mathematical expression not reproducible]

respectively. We will use also the following relations (see [12]):

(i) ([D.sup.0.sub.[alpha]+]f)(t) = f(t);

(ii) c[D.sup.[alpha].sub.a+] [D.sup.[-alpha].sub.a] f(t) = f(t);

(iii) [D.sup.[-alpha].sub.a+] c[D.sup.[alpha].sub.a+] f(t) = f(t)-f(a).

We will need a slightly modified version of the Weissinger generalization of the Banach's fixed point theorem (see [36], Fixpunktsatz, p. 195).

Theorem 1. Let [OMEGA] be a complete metric space with metric [d.sub.[OMEGA]] and let the following conditions hold:

1. There exists a sequence [[gamma].sub.q] [greater than or equal to] 0, q [member of] N with [[infinity].summation over q = 1] [[gamma].sub.q] < [infinity].

2. The operator T : [OMEGA] [right arrow] [OMEGA] satisfies for each q [member of] N and for arbitrary x, y [member of] [OMEGA] the inequality

[[d.sub.[OMEGA]] ([T.sup.q] x, [T.sup.q] y) [less than or equal to] [[gamma].sub.q] [d.sub.[OMEGA]] (x, y)

Then T has a unique fixed point [x.sup.*] [member of] [OMEGA] and for every x [member of] [member of] we have that

[mathematical expression not reproducible].

Remark 1. This modification of the Weissinger generalization of the Banach's fixed point theorem is not new, it is used in [8] in the case when [OMEGA] is a Banach space. But it is simply to be seen that the original Weissinger proof is correct in the presented in Theorem 1 case too, with some elementary modifications.

Consider the nonautonomous linear nonhomogeneous fractional system of incommensurate type with distributed delay in the following form

[mathematical expression not reproducible], (1)

and the corresponding homogeneous one

[mathematical expression not reproducible], (2)

where [mathematical expression not reproducible] denotes the left side Caputo fractional derivative [mathematical expression not reproducible].

We will use also the following notations: [mathematical expression not reproducible]. With BV [-[sigma], 0] we will denote the linear space of matrix valued functions Y(t, [theta]) with bounded variation in [theta] on [[-sigma], 0] for every t [member of] [J.sub.a] and [Var.sub.[- sigma],0]] Y (t, *) = [n.[summation over k,j=1] [Var.sub.[[-sigma],0] [yk.sub.j] (t, *). For W (t) = ([w.sub.1](t), ..., [[w.sub.n](t)).sup.T] : [J.sub.a] [right arrow] [R.sup.n], [beta] = ([[beta].sub.1], ..., [beta].sub.n]), [[beta].sub.k] [member of] [-1, 1], k [member of] <n> we will use the notation [mathematical expression not reproducible].

With [C.sup.*.sub.a] we denote the Banach space of initial vector functions [PHI] = [([[phi].sub.1], ..., [[phi].sub.n]).sup.T]: [a-[sigma], a] [right arrow] [R.sup.n], a [member of] R, which are bounded and Lebesgue measurable on the interval [a-[sigma], a] with norm

[mathematical expression not reproducible].

Consider the following initial conditions for the system (1) ((2)):

X(t) = [PHI](t) ([x.sub.k] (t) = [[phi].sub.k] (t), k [member of] <n>), t [member of] [a-[sigma], a], [PHI] [member of] [C.sup.*.sub.a] (3)

We say that for the kernels [U.sup.i] : [J.sub.a] * R [right arrow] [R.sup.n*n] the conditions (S) are fulfilled if for each i [member of] [<m>.sub.0] the following conditions hold:

(S1) The function (t, [theta]) [right arrow] [U.sup.i](t, [theta]) is measurable in (t, [theta]) [member of] [J.sub.a] * R and normalized so that [U.sup.i](t, [theta]) = 0 for [theta] [greater than or equal to] 0 and [U.sup.i](t, [theta]) = [U.sup.i](t, [[-sigma].sub.i]) for [theta] [less than or equal to] [[-sigma].sub.i], t [member of] [J.sub.a].

(S2) For each t [member of] [J.sub.a] the kernel [U.sup.i](t, [theta]) is continuous from the left in [theta] on ([[-sigma].sub.i], 0) and [U.sup.i](t, *) [member of] BV[[[-sigma].sub.i]; 0].

(S3) The Lebesgue decomposition of the kernel [U.sup.i](t, [theta]) for t [member of] [J.sub.a] and [theta] [member of] [[[-sigma].sub.i], 0] has the form:

[mathematical expression not reproducible],

where [N.sup.i](t, [theta]) = [{[a.sup.i.sub.kj](t)H([theta] + [[sigma].sub.i]t))}.sup.n.sub.k,j=1], the functions [A.sup.i](t) = {[a.sup.i.sub.kj](t)}.sup.n.sub.k,j=1] [member of] [L.sup.loc.sub.1]([J.sub.a], [R.sub.n]) are locally bounded on [J.sub.a], Y(t, [theta]) = [{[g.sub.kj] (t, [theta])}.sup.n.sub.k,j=1] [member of] C ([J.sub.a] * R, [R.sup.n*n]), [[sigma].sub.i] (t) [member of] C([J.sub.a] [[R.sup.-.sub.+]) for i [member of] <m>, [[sigma].sub.0] (t) = 0 for every t [member of] [J.sub.a], H (t) is the Heaviside function and the function B(t, [theta]) = [{[b.sub.kj](t, [theta])}.sup.n.sub.k,j=1] [member of] [L.sup.loc.sub.1]([J.sub.a] * R, [R.sup.n*n]) is locally bounded on [J.sub.a] * R.

(S4) There exists a locally bounded function [z.sub.u] [member of] [L.sup.loc.sub.1]([J.sub.a], [R.sub.+]) such that [mathematical expression not reproducible] for each t [member of] [J.sub.a].

(S5) For each t [member of] [J.sub.a] the following relation holds: [mathematical expression not reproducible].

(S6) The sets [S.sup.i.sub.[PHI]] = {t [member of] [J.sub.a]|t- [[sigma].sub.i](t) [member of] [S.sub.[PHI]]} for every i [member of] <m> do not have limit points, where with [S.sub.[PHI]] is denoted the set of all jump points of the initial function [PHI].

Definition 1. The vector function X(t) = [([x.sub.1](t), ..., [x.sub.n](t)).sup.T] is a solution of the IP (1), (3) in [a, a+b], b > 0([J.sub.a]) if X [member of] C([a, a+b], [R.sup.n])(X [member of] C([J.sub.a], [R.sub.n])) satisfies the system (1) for all t [member of] (a, a+b](t [member of] (a, [infinity])) and the initial condition (3) for t [member of] [a-[sigma], a].

Consider the following auxiliary system

[mathematical expression not reproducible] (4)

or in more detailed form for k [member of] <n>

[mathematical expression not reproducible] (5)

with the initial condition (3), where [I.sub.-1]([GAMMA]([alpha])) = diag([[GAMMA].sup.-1]([[alpha].sub.i]), ..., [[GAMMA].sup.-1] ([[alpha].sub.n])).

Definition 2. The vector function X(t) = [([x.sub.1](t), ..., [x.sub.n](t)).sup.T] is a solution of the IP (4), (3) in [a, a + b], b > 0([J.sub.a]) if X [member of] C([a, a+b], [R.sup.n])(X [member of] C([J.sub.a], [R.sub.n])) satisfies the system (4) for all t [member of] (a, a + b](t [member of] (a, [infinity])) and the initial condition (3) for t [member of] [a-[sigma], a].

Lemma 1. Let the following conditions hold:

1. Conditions (S) hold.

2. The function F [member of] [L.sup.loc.sub.1]([J.sub.a], [R.sup.n]) is locally bounded.

Then for each initial function [PHI] [member of] [C.sup.*.sub.a] every solution X(t) of IP (1), (3) is a solution of the IP (4), (3) and vice versa.

The proof is analogical of the proof of Lemma 3.3 in [6] and will be omitted.

3. MAIN RESULTS

In this section we will obtain sufficient conditions for existence and uniqueness of the solutions of IP (1), (3). In virtue of Lemma 1 it is enough to study the IP (4), (3).

Let for every initial function [PHI](t) = ([[phi].sub.1](t), ..., [[phi].sub.n](t)) [member of] [C.sup.*.sub.a] define the sets

[mathematical expression not reproducible]

and introduce in them a metric function [d.sup.[PHI].sub.1] : [[OMEGA].sup.[PHI].sub.1] * [[OMEGA].sup.[PHI].sub.1] [right arrow] [R.sup.- .sub.+] for each G, [bar.G] [member of] [[OMEGA].sup.[PHI].sub.1] as follows:

[mathematical expression not reproducible]

Obviously the set [[OMEGA].sup.[PHI].sub.1] equipped with [d.sup.[PHI].sub.1] is a complete metric space in respect to the introduced metric function.

Introduce for each G = [([g.sub.1], ..., [g.sub.n]).sup.T] [member of] [[OMEGA].sup.[PHI].sub.1] the operator (RG)(t) = [([R.sub.1][g.sub.1](t), ..., [R.sub.n][g.sub.n](t)).sup.T] for k [member of] <n> with

[mathematical expression not reproducible] (6)

for t [member of] (a, a + 1] and with [R.sub.k][g.sub.k](t) = [[phi].sub.k](t) for t [member of] [a - [sigma]], a].

Theorem 2. Let the following conditions be fulfilled:

1. Conditions (S) hold.

2. The function F [member of] [L.sup.loc.sub.1] ([J.sub.a], [R.sub.n]) is locally bounded.

Then for every initial function [PHI] [member of] [C.sup.*.sub.a] the IP (4), (3) has a unique solution in [a, a + 1].

Proof. Let [PHI] [member of] [C.sup.*.sub.a] be an arbitrary initial function. Then since [PHI] is bounded and Lebesgue measurable, then from conditions (S) it follows that for every t [member of] [a, a + 1] the functions [mathematical expression not reproducible] are bounded and at least Lebesgue integrable for each k, j [member of] <n>, i [member of] [<m>.sub.0]. Then (6) implies that the functions [R.sub.k][g.sub.k](t) are continuous for each [mathematical expression not reproducible]. Thus [R.sub.k][g.sub.k](t) [member of] C([a, a+1], [R.sup.n]) for k [member of] <n> and hence R[[OMEGA].sup.[PHI].sub.1] [subset][[OMEGA].sup.[PHI].sub.1], i.e. the operator R maps [[OMEGA}.sup.[PHI].sub.1] into [[OMEGA].sup.[PHI].sub.1].

We remind that [GAMMA](z), z [member of] [R.sub.+] has a local minimum at [z.sub.min] [approximately equal to] 1.46163, where it attains the value [GAMMA]([z.sub.min]) [approximately equal to] 0.885603. Introduce the notations [mathematical expression not reproducible], and for every q [member of] <[q.sub.0]> denote with [[alpha].sub.q] that number among the numbers [[alpha].sub.1], ..., [[alpha].sub.n] for which [mathematical expression not reproducible].

Let G, [bar.G] [member of] [[OMEGA].sup.[PHI].sub.1] be arbitrary. Then from (6) for every t [member of] [a, a + 1] and k [member of] <n> we have the estimation

[mathematical expression not reproducible] (7)

where [mathematical expression not reproducible].

Let assume that for k [member of] <n> and every t [member of] [a, a+1] the estimate

[mathematical expression not reproducible] (8)

holds for some q [member of] N. Note that inequality (7) implies that (8) holds for q = 1 and every t [member of] [a, a+1], k [member of] <n>. Using the notations [R.sub.q] G(t) = Y(t) = [([y.sub.1](t), ..., [y.sub.n](t)).sup.T] and [R.sub.q][bar.G](t) = [bar.Y](t) = [([bar.[y.sub.1]](t), ..., [bar.[y.sub.n]](t)).sup.T] we have that

[mathematical expression not reproducible]. (9)

According the induction hypothesis (8) we have

[mathematical expression not reproducible] (10)

Substitute s-a = z(t-a) in the right side of (10) and using the relation between the gamma and beta functions we obtain

[mathematical expression not reproducible] (11)

and hence by mathematical induction we have proved that (8) holds for each q [member of] N, k [member of] <n> and every t [member of] [a, a + 1].

For q [member of] <[q.sub.0]> from (8) it follows that

[d.sup.[PHI].sub.1] (R.sup.q] G, [R.sup.q] [bar.G]) [less than or equal to]n [U.sup.q.sub.1]/[GAMMA] (1 + q[[alpha].sub.q]) [d.sup.PHI.sub.1] (G, [bar.G].

For all q > [q.sub.0] from (8) it follows that

[d.sup.[PHI].sub.1] (R.sup.q] G, [R.sup.q] [bar.G]) [less than or equal to]n [U.sup.q.sub.1]/[GAMMA] (1 + q[[alpha].sub.m]) [d.sup.PHI.sub.1] (G, [bar.G].

Let for q [member of] <[q.sub.0]> denote [[gamma].sub.q] = n[U.sup.q.sub.1]/[GAMMA](1 + q[[alpha].sub.q] and for every q > [q.sub.0] denote [[gamma].sub.q] = [[gamma].sub.q] = n[U.sup.q.sub.1]/[GAMMA](1 + q[[alpha].sub.m)].

Consider the one parameter Mittag-Leffler function

[mathematical expression not reproducible].

It is simply to be seen that the series [[infinity].summation over q=1] [U.sup.q.sub.1]/[GAMMA] (1 + [[alpha].sub.m]q) is convergent because it is the considered Mittag-Leffler function evaluated at z = [U.sub.1]. Then we have

[mathematical expression not reproducible].

and from Theorem 1 it follows that the IP (4), (3) has a unique solution in [a, a + 1].

Theorem 3. Let the following conditions be fulfilled:

1. Conditions (S) hold.

2. The function F [member of] [L.sup.loc.sub.1]([J.sub.a], [R.sup.n]) is locally bounded.

Then for every initial function [PHI] [member of] [C.sub.*.sub.a] the IP (4), (3) has a unique solution in [J.sub.a].

Proof. Let [[PHI].sup.0] [member of] [C.sup.*.sub.a] be an arbitrary initial function, l = 1 and denote by [X.sup.1](t, [[PHI].sup.0]) the unique solution of the IP (4), (3) in the interval [a, a+1], existing according Theorem

2. Then we can define the function [[PHI].sup.1](t) = [([[phi].sup.1.sub.1](t), ..., [[phi].sup.1.sub.n]).sup.T] : [a-[sigma], a+1] [right arrow] [R.sup.n] as follows: [[PHI].sup.1|.sub.[a-[sigma],a]] = [[PHI].sup.0] [C.sup.*.sub.[alpha]], [[PHI].sup.1|.sub.[a,a+1]] = [X.sup.1](t, [[PHI].sup.0]) where [X.sup.1](t, [[PHI].sup.0]) is the unique solution of the IP (4), (3) in the interval [a, a+1] with the initial condition [X.sup.1](t, [[PHI].sup.0]) = [[PHI].sup.0](t) for t [member of] [a-[sigma], a]. By induction if the solution [X.sup.l](t, [[PHI].sup.l-1]) exists, we can define for this 1 [member of] N the function [[PHI].sup.l](t) = [([[phi].sup.l.sub.1](t), ..., [[phi].sup.l.sub.n]).sup.T]: [a-[sigma], a+l] [right arrow] [R.sup.n] (which will be used as initial function in the next step) with [[PHI].sup.l|.sub. [a-[sigma],a+(l-1)]] = [[PHI].sup.l-1], [[PHI].sup.l.|sub.[a+(l-1),a+l]] = [X.sup.l](t, [[PHI].sup.l-1]), where [X.sup.l](t, [[PHI].sup.l-1]) is the unique solution of the IP (4), (3) in the interval [a+(1-1), a+l] with the initial condition [X.sub.l](t, [[PHI].sup.l-1]) = [[PHI].sup.l-1](t) for t [member of] [a-[sigma], a+(1-1)].

For the proof of the statement we will use the mathematical induction. Let [[PHI].sup.0] [member of] [C.sup.*.sub.a] be an arbitrary initial function.

Assume that for some 1 [member of] N the statement holds, i.e. there exists [X.sup.l](t, [[PHI].sup.l-1]) as the unique solution of the IP (4), (3) in the interval [a + (1-1), a+1] under the initial condition [X.sup.l](t, [[PHI].sup.l-1]) = [[PHI].sup.l-1](t) for t [member of] [a-[sigma], a+(1-1)]. Note that according Theorem 2 our assumption is true for 1=1. Moreover our assumption allows us to define the next initial function [[PHI].sup.l](t) with [[PHI].sup.l.|sub.[a-[sigma],a+(l- 1)]] = [[PHI].sup.l-1], [[PHI].sup.l]|.sub.[a+(l-1), a+l]] = [X.sup.l](t, [[PHI].sup.l- 1]).

Define the sets

[mathematical expression not reproducible]

with a metric function [mathematical expression not reproducible] as follows:

[mathematical expression not reproducible].

Obviously the set [mathematical expression not reproducible] equipped with [mathematical expression not reproducible] is a complete metric space in respect to the introduced metric function.

Define the operator [mathematical expression not reproducible] with

[mathematical expression not reproducible] (12)

for t [member of] (a + 1, a + (1 + 1)] and with [[??].sub.k][g.sub.k](t) = [[phi].sup.l.sub.k](t) for t [member of] [a-[sigma], a + l].

Taking into account that [mathematical expression not reproducible] as in the proof of Theorem 2 we can conclude that [[??].sub.k][g.sub.k](t) [member of] C([a, a+(1+1)], [R.sup.n]) for k [member of] <n> and hence [mathematical expression not reproducible] i.e. the operator [mathematical expression not reproducible].

For arbitrary [mathematical expression not reproducible] from (12) for every t [member of] [a + 1, a + (1 + 1)] and k [member of] <n> the same way as in Theorem 2 we have the estimation

[mathematical expression not reproducible] (13)

where [mathematical expression not reproducible].

As in Theorem 2 we assume that for k [member of] <n> and every t [member of] [a + 1, a + (1 + 1)] the inequality

[mathematical expression not reproducible] (14)

holds for some q [member of] N. Note that the inequality (13) implies that (14) holds for every t [member of] [a + 1, a + (1 + 1)], k [member of] <n> and q=1. Then using the same notations as in Theorem 2 we have that

[mathematical expression not reproducible]. (15)

In the next calculations, using the induction hypothesis (14) we obtain

[mathematical expression not reproducible] (16)

Substitute s-(a + 1) = z(t-(a + 1)) in the right side of (16) and using the relation between the gamma and beta functions we obtain

[mathematical expression not reproducible] (17)

and hence by mathematical induction we have proved that (14) holds for each q [member of] N, k [member of] <n> and every t [member of] [a + l, a + (l + 1)].

For q [member of] <[q.sub.0]> from (14) it follows that

[mathematical expression not reproducible].

For all q > [q.sub.0] from (14) it follows that

[mathematical expression not reproducible].

Let for q [member of] <[q.sub.0]> denote [[gamma].sub.q] = n[U.sup.q.sub.l+1]/[GAMMA] (1 + q [[alpha].sub.q] and for every q > [q.sub.0] denote [[gamma].sub.q] = n[U.sup.q.sub.l+1]/[GAMMA] (1 + q [[alpha].sub.m].

Consider the one parameter Mittag-Leffler function

[mathematical expression not reproducible]

It is simply to be seen that the series [[infinity].summation over q=1] [U.sup.q.sub.l+1]/[GAMMA](1+[[alpha].m]q)] is convergent because it is the considered Mittag-Leffler function evaluated at z = [U.sub.l+1]. Then we have

[mathematical expression not reproducible]

and from Theorem 1 it follows that the IP (4), (3) has a unique solution in [a + 1, a + l + 1].

Thus by mathematical induction (in respect to l) we have proved that the IP (4), (3) has a unique solution in [J.sub.a].

Corollary 1. Let the following conditions be fulfilled:

1. Conditions (S) hold.

2. The function F [member of] [L.sup.loc.sub.1]([J.sub.a], [R.sub.n]) is locally bounded.

Then for every initial function [PHI] [member of] [C.sup.*.sub.a] the IP (1), (3) has a unique solution in [J.sub.a].

Proof. The statement follows immediately from Theorem 3 and Lemma 1.

Received: December 1, 2018; Revised: March 27, 2019; Published (online): April 4, 2019

doi: 10.12732/dsa.v28i2.14

4. ACKNOWLEDGMENT

This paper has been supported by Grant FP19-FMI-002 and by the National Scientific Program "Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICTinSES)", financed by the Ministry of Education and Science.

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ANDREY ZAHARIEV (1), HRISTO KISKINOV (2), AND EVGENIYA ANGELOVA (3)

(1,2,3) Faculty of Mathematics and Informatics

University of Plovdiv

(4) Tzar Asen, 4000, Plovdiv, BULGARIA
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Author:Zahariev, Andrey; Kiskinov, Hristo; Angelova, Evgeniya
Publication:Dynamic Systems and Applications
Date:Jan 1, 2019
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