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Existence and Attractivity Results for Coupled Systems of Nonlinear Volterra-Stieltjes Multidelay Fractional Partial Integral Equations.

1. Introduction

Fractional integral and fractional differential equations are among the most fast growing field in mathematics. They are used to describe many phenomena, especially the ones with long memory. Examples include but are not limited to viscoelasticity, viscoplasticity, biochemistry, control theory, mathematical psychology, mechanics, modeling in complex media (porous, etc.), and electromagnetism [1-4]. In recent years, there has been a significant development in ordinary and partial fractional integral equations; see, for instance, the monographs of Abbas et al. [5-7], Agarwal et al. [8], Kilbas et al. [9], Miller and Ross [10], Podlubny [11], Samko et al. [12], and the papers [13-18] and the references therein.

In this paper we study the existence and attractivity of solutions to the following coupled system of nonlinear fractional Riemann-Liouville-Volterra-Stieltjes quadratic multidelay partial integral equations:

[mathematical expression not reproducible], (1)

[mathematical expression not reproducible], (2)

where [mathematical expression not reproducible], are continuous and bounded functions with [mathematical expression not reproducible] is the (Euler's) Gamma function defined by

[mathematical expression not reproducible]. (3)

2. Preliminaries

In this section, we recall some notations, definitions, and preliminary facts which will be used in this paper. [L.sup.1]([0, p] x [0, q]), p, q > 0, will denote the space of all Lebesgue-integrable functions u : [0, p] x [0, [right arrow] R equipped with the norm

[mathematical expression not reproducible]. (4)

BC := BC([-T,[infinity]) x [-[xi],b]) will denote the usual Banach space of all bounded and continuous functions from [-T, [infinity]) x [-[xi], b] into R equipped with the standard norm

[mathematical expression not reproducible]. (5)

It is clear that the product space BC := SC x BC turns out to be a Banach space if equipped with the norm

[mathematical expression not reproducible]. (6)

Definition 1 (see [19]). Let r = ([r.sub.1], [r.sub.2]) [member of] (0, [infinity]) x (0, [infinity]), [theta] = (0,0) and m [member of] [L.sup.1]([0,p] x [0,q]). The left-sided mixed Riemann-Liouville integral of order r of u is defined by

[mathematical expression not reproducible], (7)

provided the integral exists.

Example 2. Let [lambda], [omega] [member of] (0, [infinity]) and r = ([r.sub.1], [r.sub.2]) [member of] (0, [infinity]) x (0, [infinity]), then

[mathematical expression not reproducible]. (8)

If m is a real-valued function defined on the interval [a, b], then we will use the symbol [[disjunction].sup.b.sub.a] u to denote the variation of u on [a, b]. We say that u is of bounded variation on the interval [a, b] whenever [[disjunction].sup.b.sub.a] u is finite. If w : [a, b] x [c, d] [right arrow] R, then the symbol [[disjunction].sup.q.sub.t=p] w(t,s) indicates the variation of the function t [right arrow] w(i, s) on the interval [p, q] [subset] [a, b], where s is arbitrarily fixed in the interval [c, d]. Analogously we define [[disjunction].sup.q.sub.s=p] w (t, s). For more details on the properties of functions of bounded variation we refer the reader to [20].

If u and [phi] are two real-valued functions defined on the interval [a, b], then under some appropriate conditions (see [20]) we can define the Stieltjes integral (in the Riemann-Stieltjes sense)

[[integral].sup.b.sub.a] u(t) d[phi](t) (9)

of the function u with respect to [phi]. In this case we say that m is Stieltjes integrable on [a, b] with respect to [phi]. Several conditions are known to ensure Stieltjes integrability [20]. One of the most frequently used requires that u is continuous and [phi] is of bounded variation on [a, b].

Now we recall a few properties of the Stieltjes integral included in the lemmas below.

Lemma 3 (see [20, 21]). If u is Stieltjes integrable on the interval [a, b] with respect to a function [phi] of bounded variation, then

[mathematical expression not reproducible]. (10)

Lemma 4 (see [20, 21]). Let u and v be Stieltjes integrable functions on the interval [a, b] with respect to a nondecreasing function [phi] such that w(i) [less than or equal to] v(i) for t [member of] [a, b]. Then

[[integral].sup.b.sub.a] u(t) d [phi](i) [less than or equal to] [[integral].sup.b.sub.a] v (t) d[phi](t). (11)

From now on, we will also consider Stieltjes integrals of the form

[[integral].sup.b.sub.a] u(t) [d.sub.s]g(t,s) (12)

and Riemann-Liouville-Stieltjes integrals of fractional order of the form

[mathematical expression not reproducible], (13)

where g : [R.sub.+] x [R.sub.+] [right arrow] R, r [member of] (0, [infinity]) and the symbol [d.sub.s] indicates the integration with respect to s.

Let 0 [not equal to] [OMEGA] [subset] BC, and let G : [OMEGA] [right arrow] [OMEGA], and consider the solutions of equation

(Gu) (t, x) = u (t, x). (14)

In light of the definition of the attractivity of solutions of integral equations (for instance, [15]), we will introduce the following concept of attractivity of solutions for (14).

Definition 5. A solutions of (14) is said to be locally attractive if there exists a ball B([u.sub.0], [eta]) in the space SC such that, for arbitrary solutions v = v(t, x) and w = w(t, x) of (14) belonging to B([u.sub.0], [eta]) [intersection] [OMEGA], we have that, for each x [member of] [0, b],

[mathematical expression not reproducible]. (15)

When the limit (15) is uniform with respect to B([u.sub.0], [eta]) [intersection] [OMEGA], solutions of (14) are said to be uniformly locally attractive (or equivalently that solutions of (14) are locally asymptotically stable).

Definition 6 (see [15]). The solution v = v(t, x) of (14) is said to be globally attractive if (15) holds for each solution w = w(t, x) of (14). If condition (15) is satisfied uniformly with respect to the set [OMEGA], solutions of (14) are said to be globally asymptotically stable (or uniformly globally attractive).

Lemma 7 (see [22], p. 62). Let D [subset] BC. Then D is relatively compact in BC if the following conditions hold:

(a) D is uniformly bounded in BC.

(b) The functions belonging to D are almost equicontinuous on [-T,[infinity]) x [-[xi],b], i.e., equicontinuous on every compact subset of [-T, [infinity]) x [-[xi],b].

(c) The functions from D are equiconvergent; that is, given [mathematical expression not reproducible].

3. Existence and Attractivity Results

Definition 8. By a solution to problem (1)-(2), we mean every coupled functions (u, v) [member of] BC such that (u, v) satisfies (1) on J and (2) on [??].

We will use the following assumptions in the sequel:

([H.sub.1]) There exist positive functions [p.sub.j] [member of] BC; j = 1,2 such that

(1 + [alpha](t))[absolute value of [[mu].sub.j] ([alpha](t),x)] [less than or equal to] [p.sub.j] (t,x); (t,x) [member of] J. (16)

([H.sub.2]) For all [t.sub.1],[t.sub.2] [member of] [R.sub.+] such that [t.sub.1] < [t.sub.2], the function s [??] g([t.sub.2], s) - g([t.sub.1],s) is nondecreasing on [R.sub.+].

([H.sub.3]) The function s [??] g(0, s) is nondecreasing on [R.sub.+].

([H.sub.4]) The functions s [??] g(t,s) and t [??] g(t,s) are continuous on [R.sub.+] for each fixed t [member of] [R.sub.+] or s [member of] [R.sub.+], respectively.

([H.sub.5]) There exist continuous functions [q.sub.ji] : J' [right arrow] [R.sub.+]; i = 1,...,m, j = 1,2 such that

[mathematical expression not reproducible]. (17)

Moreover, assume that

[mathematical expression not reproducible]. (18)

Remark 9. Set [mathematical expression not reproducible],

[mathematical expression not reproducible]; (19)

for i = 1, ..., m and j = 1,2. From the above assumptions, we infer that [[PHI].sup.*.sub.j], [p.sup.*.sub.j], [q.sup.*.sub.ji] are finite.

Theorem 10. Assume that hypotheses ([H.sub.1]) - ([H.sub.5]) hold. Then problem (1)-(2) has at least one solution in the space BC. Moreover, solutions to problem (1)-(2) are uniformly globally attractive.

Proof. Define the operators [N.sub.j] : BC [right arrow] BC; j = 1,2 by

[mathematical expression not reproducible], (20)

and consider the operator N : BC [right arrow] BC such that, for any ([u.sub.1],[u.sub.2]) [member of] BC,

(N ([u.sub.1],[u.sub.2])) (t, x) = (([N.sub.1][u.sub.1]) (t, x), ([N.sub.2][u.sub.2]) (t, x)). (21)

From the hypotheses above, we deduce that N(u) is continuous on [-T, [infinity]) x [-[xi], b]. Now let us prove that N([u.sub.1], [u.sub.2]) [member of] BC for any [u.sub.j] [member of] BC; j = 1,2. For arbitrarily fixed (t,x) [member of] J, we have

[mathematical expression not reproducible], (22)

and for all (t, x) [member of] [??] and each [u.sub.j] [member of] BC, j = 1,2, we have

[mathematical expression not reproducible]. (23)

Thus,

[mathematical expression not reproducible]. (24)

Hence

[mathematical expression not reproducible]. (25)

Therefore N(u) [member of] BC. The problem of finding the solutions of the coupled system (1)-(2) is reduced to finding the solutions of the operator equation N([u.sub.1],[u.sub.2]) = ([u.sub.1],[u.sub.2]). From (25), we infer that N transforms the ball [mathematical expression not reproducible] into itself. Now we will show that N : [B.sub.[eta]] [right arrow] [B.sub.[eta]] satisfies the Schauder's fixed point theorem [23]. The proof will be presented in several steps and cases.

Step 1 (N is continuous). Let [{([u.sub.n], [v.sub.n])].sub.n[member of]N] be a sequence such that [u.sub.n] [right arrow] u and [v.sub.n] [right arrow] v in [B.sub.[eta]]. Then, for each (t, x) [member of] [-T,[infinity]) x [-[xi],b], we have

[mathematical expression not reproducible]. (26)

Case 1. Assume that (t,x) [member of] [??] [union] ([0,a] x [0,b]); a > 0, then, since ([u.sub.n], [v.sub.n]) [right arrow] (u,v) as n [right arrow] m and [f.sub.1],g,[gamma] are continuous, (26) implies

[parallel]N ([u.sub.n]) - N (u)[parallel][sub.BC] [right arrow] 0 as n [right arrow] [infinity]. (27)

Case 2. Let (t,x) [member of] (a,[infinity]) x [0,b]; a > 0, then from ([H.sub.5]) and (26) we obtain

[mathematical expression not reproducible]. (28)

Since t [right arrow] m, then (28) gives

[parallel][N.sub.1] ([u.sub.n]) - [N.sub.1] (u)[parallel][sub.BC] [right arrow] 0 as n [right arrow] [infinity]. (29)

Let us show that [N.sub.2] is continuous in the same way as continuity of [N.sub.1].

Step 2 (N([B.sub.[eta]]) is uniformly bounded). This fact is obvious because N([B.sub.[eta]]) [subset] [B.sub.[eta]] and [B.sub.[eta]] is a bounded set.

Step 3 (N([B.sub.[eta]]) is equicontinuous on every compact subset [0,a] x [0,b] of J, a > 0). Let ([t.sub.1],[x.sub.1]),([t.sub.2],[x.sub.2]) [member of] [0,a] x [0,b], [t.sub.1] < [t.sub.2], [x.sub.1] < [x.sub.2], and let (u, v) [member of] [B.sub.[eta]]. Without loss of generality, let us assume that [beta]([t.sub.1]) [less than or equal to] [beta]([t.sub.2]). Then we obtain

[mathematical expression not reproducible]. (30)

Thus

[mathematical expression not reproducible]. (31)

Using continuity of the functions [mathematical expression not reproducible], the right-hand side of the above inequality tends to zero. The equicontinuity of [N.sub.1] for the cases [mathematical expression not reproducible] is immediate.

We can also prove that

[mathematical expression not reproducible]. (32)

Hence

[mathematical expression not reproducible]. (33)

Step 4 (N([B.sub.[eta]]) is equiconvergent). Let (i, x) [member of] / and m [member of] [[beta].sub.[eta]], then we get

[mathematical expression not reproducible]. (34)

Now, since [alpha](t) [right arrow] [infinity] as t [right arrow] [infinity], we conclude that, for each x [member of] [0, b], we obtain

[absolute value of (N(u, v))(t,x)] [right arrow] 0, as t [right arrow] +[infinity]. (35)

Also, for each x [member of] [-[xi], 0], we get

[mathematical expression not reproducible]. (36)

Then, for each x [member of] [-[xi], b], we get

[absolute value of (N(u, v)) (t, x)] [right arrow] 0, as t [right arrow] +[infinity]. (37)

Hence,

[mathematical expression not reproducible]. (38)

In view of Steps 1 to 4, along with the Lemma 7, we deduce that N : [B.sub.[eta]] [right arrow] [B.sub.[eta]] is continuous and compact. From an application of Schauder's theorem [23], we conclude that N has a fixed point (u, v) which is a solution of the coupled system (1)-(2).

Step 5 (the uniform global attractivity of solutions). Now let us study the stability of solutions of the coupled system (1)-(2). Let ([u.sub.1],[u.sub.2]) and ([v.sub.1], [v.sub.2]) be two solutions of (1)-(2). Then, for each (t, x) [member of] [-T, [infinity]) x [-[xi], b], we obtain

[mathematical expression not reproducible]. (39)

Thus

[mathematical expression not reproducible]. (40)

Hence

[mathematical expression not reproducible]. (41)

By using (41) and ([H.sub.5]), we obtain

[mathematical expression not reproducible]. (42)

Therefore, all solutions of the coupled system (1)-(2) are uniformly globally attractive.

Let [bar.BC] := B[C.sup.n] (product space) be the Banach space equipped with the following norm:

[mathematical expression not reproducible]. (43)

From the above theorem, we deduce the following consequence.

Corollary 11. Consider the system of nonlinear fractional Riemann-Liouville-Volterra-Stieltjes quadratic multidelay partial integral equations of the form

[mathematical expression not reproducible]. (44)

[mathematical expression not reproducible]. (45)

where [mathematical expression not reproducible].

Suppose that ([H.sub.2])-([H.sub.4]) and the following assumptions are verified:

([H'.sub.1]) There exist positive functions [p.sub.j] [member of] BC, j = 1,..., n, such that

[mathematical expression not reproducible]. (46)

([H'.sub.2]) There exist continuous functions [q.sub.ji] : J' [right arrow] [R.sub.+], i = 1,..., m, j = 1, ..., n, such that

[mathematical expression not reproducible]; (47)

Moreover, assume that

[mathematical expression not reproducible]. (48)

Then problem (44)-(45) has at least one solution in the space [bar.BC]. In addition, the solutions are uniformly globally attractive.

4. An Example

To illustrate our results, we consider the following coupled system of nonlinear fractional order Riemann-Liouville-Volterra-Stieltjes quadratic multidelay partial integral equation

[mathematical expression not reproducible], (49)

[mathematical expression not reproducible], (50)

where [mathematical expression not reproducible],

[mathematical expression not reproducible], (51)

[mathematical expression not reproducible], (52)

First, we can see that [lim.sub.t[right arrow][infinity]] [alpha](t) = [infinity] and [lim.sub.t[right arrow][infinity]] [[PHI].sub.j](i, x) = 0; j = 1,2. Next, the assumption ([H.sub.1]) is satisfied with [p.sub.j](t, x) = [chi square][e.sup.-t] and consequently [p.sup.*.sub.j] = 1. Also, it is clear that the function g satisfies assumptions ([H.sub.2]) - ([H.sub.4]).

Finally, the functions [f.sub.j], j = 1,2, satisfy the assumption ([H.sub.5]). Indeed, [f.sub.j] are continuous and satisfy the inequality

[mathematical expression not reproducible]. (53)

(t, x, s, y) [member of] J', [u.sub.1], [u.sub.2], [v.sub.1], [v.sub.2] [member of] R. Also, we have

[mathematical expression not reproducible]. (54)

and

[mathematical expression not reproducible]. (55)

For i = 1,2, we have also

[mathematical expression not reproducible], (56)

and

[mathematical expression not reproducible]. (57)

Consequently, Theorem 10 implies that the coupled system (49)-(50) has a solution defined on [-1/2, [infinity]) x [-2,1]; moreover solutions of this system are uniformly globally attractive.

https://doi.org/10.1155/2018/8735614

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Said Abbas (iD), (1) Mouffak Benchohra, (2) Naima Hamidi, (2) and Gaston N'Guerekata (iD) (3)

(1) Laboratory of Mathematics, Geometry, Analysis, Control and Applications, Tahar Moulay University of Salda, P.O. Box 138, EN-Nasr, 20000 Saida, Algeria

(2) Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes, P.O. Box 89, Sidi Bel-Abbes 22000, Algeria

(3) Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21252, USA

Correspondence should be addressed to Gaston N'Guerekata; nguerekata@aol.com

Received 7 March 2018; Accepted 20 September 2018; Published 4 October 2018

Academic Editor: Jozef Banas
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Author:Abbas, Said; Benchohra, Mouffak; Hamidi, Naima; N'Guerekata, Gaston
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Date:Jan 1, 2018
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