Volume 2018, Issue 1 8735614

Research Article

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

Saïd Abbas

orcid.org/0000-0002-2518-8658

Laboratory of Mathematics, Geometry, Analysis, Control and Applications, Tahar Moulay University of Saïda, P.O. Box 138, EN-Nasr, 20000 Saïda, Algeria univ-saida.dz

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Mouffak Benchohra,

Mouffak Benchohra

Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria univ-sba.dz

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Naima Hamidi,

Naima Hamidi

Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria univ-sba.dz

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Gaston N’Guérékata,

Corresponding Author

Gaston N’Guérékata

[email protected]

orcid.org/0000-0001-5765-7175

Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21252, USA morgan.edu

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Saïd Abbas,

Saïd Abbas

orcid.org/0000-0002-2518-8658

Laboratory of Mathematics, Geometry, Analysis, Control and Applications, Tahar Moulay University of Saïda, P.O. Box 138, EN-Nasr, 20000 Saïda, Algeria univ-saida.dz

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Mouffak Benchohra,

Mouffak Benchohra

Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria univ-sba.dz

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Naima Hamidi,

Naima Hamidi

Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, Sidi Bel-Abbès 22000, Algeria univ-sba.dz

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Gaston N’Guérékata,

Corresponding Author

Gaston N’Guérékata

[email protected]

orcid.org/0000-0001-5765-7175

Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21252, USA morgan.edu

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First published: 04 October 2018

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

Citations: 3

Academic Editor: Jozef Banas

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Abstract

We are concerned with some existence and attractivity results of a coupled fractional Riemann–Liouville–Volterra–Stieltjes multidelay partial integral system. We prove the existence of solutions using Schauder’s fixed point theorem; then we show that the solutions are uniformly globally attractive.

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:

(1)

(2)

where

, i = 1, …, m, T = max_i=1,…,m{τ_i},

, are given continuous functions, lim_t→∞α(t) = ∞, μ_j, j = 1,2, are bounded, J^′ = {(t, x, s, y) ∈ J² : s ≤ t, y ≤ x},

, are continuous and bounded functions with lim_t→∞Φ_j(t, x) = 0, x ∈ [−ξ, b], μ_j(α(t), 0) = Φ_j(t, 0) for each

and μ_j(α(0), x) = Φ_j(0, x), for each x ∈ [0, b], and Γ(·) is the (Euler’s) Gamma function defined by

(3)

2. Preliminaries

In this section, we recall some notations, definitions, and preliminary facts which will be used in this paper. L¹([0, p]×[0, q]), p, q > 0, will denote the space of all Lebesgue-integrable functions

equipped with the norm

(4)

BC≔BC([−T, ∞)×[−ξ, b]) will denote the usual Banach space of all bounded and continuous functions from [−T, ∞)×[−ξ, b] into

equipped with the standard norm

(5)

It is clear that the product space

turns out to be a Banach space if equipped with the norm

(6)

Definition 1 (see [19].)Let r = (r₁, r₂)∈(0, ∞)×(0, ∞), θ = (0,0) and u ∈ L¹([0, p]×[0, q]). The left-sided mixed Riemann–Liouville integral of order r of u is defined by

(7)

provided the integral exists.

Example 2. Let λ, ω ∈ (0, ∞) and r = (r₁, r₂)∈(0, ∞)×(0, ∞), then

(8)

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

If u and φ 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)

(9)

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

(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 φ such that u(t) ≤ v(t) for t ∈ [a, b]. Then

(11)

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

(12)

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

(13)

where

and the symbol d_s indicates the integration with respect to s.

Let ∅ ≠ Ω ⊂ BC, and let G : Ω → Ω, and consider the solutions of equation

(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(u₀, η) in the space BC such that, for arbitrary solutions v = v(t, x) and w = w(t, x) of (14) belonging to B(u₀, η)∩Ω, we have that, for each x ∈ [0, b],

(15)

When the limit (15) is uniform with respect to B(u₀, η)∩Ω, 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 Ω, solutions of (14) are said to be globally asymptotically stable (or uniformly globally attractive).

Lemma 7 (see [22], p. 62.)Let D ⊂ 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, ∞)×[−ξ, b], i.e., equicontinuous on every compact subset of [−T, ∞)×[−ξ, b].
(c)
The functions from D are equiconvergent; that is, given ϵ > 0, x ∈ [−ξ, b], there corresponds λ(ϵ, x) > 0 such that |u(t, x) − lim_t→∞u(t, x)| < ϵ for any t ≥ λ(ϵ, x) and u ∈ D.

3. Existence and Attractivity Results

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

We will use the following assumptions in the sequel:

(H₁) There exist positive functions p_j ∈ BC; j = 1,2 such that
(16)
(H₂) For all such that t₁ < t₂, the function s ↦ g(t₂, s) − g(t₁, s) is nondecreasing on .
(H₃) The function s ↦ g(0, s) is nondecreasing on .
(H₄) The functions s ↦ g(t, s) and t ↦ g(t, s) are continuous on for each fixed or , respectively.
(H₅) There exist continuous functions ; i = 1, …, m, j = 1,2 such that
(17)
for (t, x, s, y) ∈ J^′, ; i = 1, …, m. Moreover, assume that
(18)

Remark 9. Set ,

(19)

for i = 1, …, m and j = 1,2. From the above assumptions, we infer that

are finite.

Theorem 10. Assume that hypotheses (H₁)−(H₅) hold. Then problem (1)-(2) has at least one solution in the space Moreover, solutions to problem (1)-(2) are uniformly globally attractive.

Proof. Define the operators N_j : BC → BC; j = 1,2 by

(20)

and consider the operator

such that, for any

(21)

From the hypotheses above, we deduce that N(u) is continuous on [−T, ∞)×[−ξ, b]. Now let us prove that

for any u_j ∈ BC; j = 1,2. For arbitrarily fixed (t, x) ∈ J, we have

(22)

and for all

and each u_j ∈ BC, j = 1,2, we have

(23)

Thus,

(24)

Hence

(25)

Therefore N(u) ∈ BC. The problem of finding the solutions of the coupled system (1)-(2) is reduced to finding the solutions of the operator equation N(u₁, u₂) = (u₁, u₂). From (25), we infer that N transforms the ball into itself. Now we will show that N : B_η → B_η satisfies the Schauder’s fixed point theorem [23]. The proof will be presented in several steps and cases.

Step 1 (N is continuous). Let be a sequence such that u_n → u and v_n → v in B_η. Then, for each (t, x)∈[−T, ∞)×[−ξ, b], we have

(26)

Case 1. Assume that , then, since (u_n, v_n)→(u, v) as n → ∞ and f₁, g, γ are continuous, (26) implies

(27)

Case 2. Let (t, x)∈(a, ∞)×[0, b]; a > 0, then from (H₅) and (26) we obtain

(28)

Since t → ∞, then (28) gives

(29)

Let us show that N₂ is continuous in the same way as continuity of N₁.

Step 2 (N(B_η) is uniformly bounded). This fact is obvious because N(B_η) ⊂ B_η and B_η is a bounded set.

Step 3 (N(B_η) is equicontinuous on every compact subset [0, a]×[0, b] of J, a > 0). Let (t₁, x₁), (t₂, x₂)∈[0, a]×[0, b], t₁ < t₂, x₁ < x₂, and let (u, v) ∈ B_η. Without loss of generality, let us assume that β(t₁) ≤ β(t₂). Then we obtain

(30)

Thus

(31)

Using continuity of the functions μ₁, α, β, γ, f₁, g, h, q_i, i = 1, …m, and since t₁ → t₂ and x₁ → x₂, the right-hand side of the above inequality tends to zero. The equicontinuity of N₁ for the cases t₁ < t₂ < 0, x₁ < x₂ < 0 and t₁ ≤ 0 ≤ t₂, x₁ ≤ 0 ≤ x₂ is immediate.

We can also prove that

(32)

Hence

(33)

Step 4 (N(B_η) is equiconvergent). Let (t, x) ∈ J and u ∈ B_η, then we get

(34)

Now, since α(t) → ∞ as t → ∞, we conclude that, for each x ∈ [0, b], we obtain

(35)

Also, for each x ∈ [−ξ, 0], we get

(36)

Then, for each x ∈ [−ξ, b], we get

(37)

Hence,

(38)

In view of Steps 1 to 4, along with the Lemma 7, we deduce that N : B_η → B_η 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 (u₁, u₂) and (v₁, v₂) be two solutions of (1)-(2). Then, for each (t, x)∈[−T, ∞)×[−ξ, b], we obtain

(39)

Thus

(40)

Hence

(41)

By using (41) and (H₅), we obtain

(42)

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

Let (product space) be the Banach space equipped with the following norm:

(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

(44)

(45)

where r₁, r₂ ∈ (0, ∞), τ_i, ξ_i ≥ 0; i = 1, …, m, T = max_i=1,…,m{τ_i},

, j = 1, …, n are given continuous functions, lim_t→∞α(t) = ∞, μ_j; j = 1, …, n are bounded, J^′ = {(t, x, s, y) ∈ J² : s ≤ t, y ≤ x},

; j = 1,2 are continuous and bounded functions with lim_t→∞Φ_j(t, x) = 0; x ∈ [−ξ, b], μ_j(α(t), 0) = Φ_j(t, 0) for each

and μ_j(α(0), x) = Φ_j(0, x); for each x ∈ [0, b].

Suppose that (H₂)−(H₄) and the following assumptions are verified:

There exist positive functions p_j ∈ BC, j = 1, …, n, such that
(46)
There exist continuous functions , i = 1, …, m, j = 1, …, n, such that
(47)
for (t, x, s, y) ∈ J^′, , i = 1, …, m, j = 1, …, n. Moreover, assume that
(48)

Then problem (44)-(45) has at least one solution in the space

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

(49)

(50)

where

, r₁ = 1/4, r₂ = 1/2, α(t) = β(t) = γ(t) = t;

(51)

if (t, x, s, y) ∈ J^′,

(52)

First, we can see that lim_t→∞α(t) = ∞ and lim_t→∞Φ_j(t, x) = 0; j = 1,2. Next, the assumption (H₁) is satisfied with p_j(t, x) = x²e^−t and consequently

Also, it is clear that the function g satisfies assumptions (H₂)−(H₄).

Finally, the functions f_j, j = 1,2, satisfy the assumption (H₅). Indeed, f_j are continuous and satisfy the inequality

(53)

(t, x, s, y) ∈ J^′,

Also, we have

(54)

and

(55)

For i = 1,2, we have also

(56)

and

(57)

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Open Research

Data Availability

No data were used to support this study.

References

1 Baleanu D., Diethelm K., Scalas E., and Trujillo J. J., Fractional Calculus Models and Numerical Methods, 2012, 3, World Scientific Publishing, New York, USA, Series on Complexity, Nonlinearity and Chaos, https://doi.org/10.1142/9789814355216, MR2894576.
10.1142/8180
Google Scholar
2 Diethelm K., The Analysis of Fractional Differential Equations, 2010, 2004, Springer, Berlin, Germany, Lecture Notes in Mathematics, https://doi.org/10.1007/978-3-642-14574-2, MR2680847.
10.1007/978-3-642-14574-2
Web of Science® Google Scholar
3 Diethelm K. and Ford N. J., Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications. (2002) 265, no. 2, 229–248, 2-s2.0-0037081673, https://doi.org/10.1006/jmaa.2000.7194, ZBL1014.34003, Zbl1014.34003.
10.1006/jmaa.2000.7194
Web of Science® Google Scholar
4 Tarasov V. E., Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, 2010, Springer, Heidelberg; Higher Education Press, Beijing, https://doi.org/10.1007/978-3-642-14003-7, MR2796453.
10.1007/978-3-642-14003-7
Google Scholar
5 Abbas S., Benchohra M., Graef J. R., and Henderson J., Implicit Fractional Differential and Integral Equations; Existence and Stability, 2018, 26, De Gruyter, Berlin, Germany, De Gruyter Series in Nonlinear Analysis and Applications, MR3791511.
10.1515/9783110553819
Google Scholar
6 Abbas S., Benchohra M., and N′Guérékata G. M., Topics in Fractional Differential Equations, 2012, Springer, New York, NY, USA, https://doi.org/10.1007/978-1-4614-4036-9, MR2962045, Zbl1273.35001.
10.1007/978-1-4614-4036-9
Google Scholar
7 Abbas S., Benchohra M., and N′Guérékata G. M., Advanced Fractional Differential and Integral Equations, 2015, Nova Science Publishers, New York, NY, USA, MR3309582, Zbl1314.34002.
Google Scholar
8 Agarwal R. P., Meehan M., and O′Regan D., Fixed Point Theory and Applications, 2001, 141, Cambridge University Press, Cambridge, UK, https://doi.org/10.1017/cbo9780511543005, MR1825411.
10.1017/CBO9780511543005
Google Scholar
9 Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, 2006, Elsevier Science Publisher B.V., ‎Amsterdam, Netherland, MR2218073.
10.1016/S0304-0208(06)80001-0
Google Scholar
10 Miller K. S. and Ross B., An Introduction to the Fractional Calculus and Differential Equations, 1993, John Wiley, New York, USA.
Google Scholar
11 Podlubny I., Fractional Differential Equations, 1999, 198, Academic Press, San Diego, Calif, USA, Mathematics in Science and Engineering, MR1658022, Zbl0924.34008.
Web of Science® Google Scholar
12 Samko S. G., Kilbas A. A., and Marichev O. I., Fractional Integrals and Derivatives, Theory and Applications, 1993, Gordon and Breach, Yverdon, Switzerland, MR1347689.
Google Scholar
13 Banaś J., Existence results for Volterra-Stieltjes quadratic integral equations on an unbounded interval, Mathematica Scandinavica. (2006) 98, no. 1, 143–160, https://doi.org/10.7146/math.scand.a-14988, MR2221549.
10.7146/math.scand.a-14988
Web of Science® Google Scholar
14 Banaś J. and Cabrera I. J., On existence and asymptotic behaviour of solutions of a functional integral equation, Nonlinear Analysis: Theory, Methods and Applications. (2007) 66, 2246–2254.
10.1016/j.na.2006.03.015
Web of Science® Google Scholar
15 Banaś J. and Dhage B. C., Global asymptotic stability of solutions of a functional integral equation, Nonlinear Analysis: Theory, Methods & Applications. (2008) 69, no. 7, 1945–1952, https://doi.org/10.1016/j.na.2007.07.038, MR2437598, Zbl1154.45005, 2-s2.0-45349090141.
10.1016/j.na.2007.07.038
Web of Science® Google Scholar
16 Banaś J. and Rzepka B., On existence and asymptotic stability of solutions of a nonlinear integral equation, Journal of Mathematical Analysis and Applications. (2003) 284, no. 1, 165–173, https://doi.org/10.1016/S0022-247X(03)00300-7, MR1996125.
10.1016/S0022-247X(03)00300-7
Web of Science® Google Scholar
17 Darwish M. A., Henderson J., and O′Regan D., Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument, Bulletin of the Korean Mathematical Society. (2011) 48, no. 3, 539–553, https://doi.org/10.4134/BKMS.2011.48.3.539, MR2827764, Zbl1220.45011.
10.4134/BKMS.2011.48.3.539
Web of Science® Google Scholar
18 Dhage B. C. and Bellale S. S., Local asymptotic stability for nonlinear quadratic functional integral equations, Electronic Journal of Qualitative Theory of Differential Equations. (2008) no. 10, 1–13, https://doi.org/10.14232/ejqtde.2008.1.10, MR2385414.
10.14232/ejqtde.2008.1.10
Web of Science® Google Scholar
19 Vityuk A. N. and Golushkov A. V., Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscillations. (2004) 7, no. 3, 318–325, https://doi.org/10.1007/s11072-005-0015-9, MR2151816.
10.1007/s11072-005-0015-9
Google Scholar
20 Natanson I. P., Theory of Functions of a Real Variable, 1960, Frederick Ungar Publishing Company, New York, NY, USA, MR0067952.
Google Scholar
21 Appell J., Banaś J., and Merentes N., Bounded Variation and Around, 2014, 17, Walter de Gruyter, Berlin , Germany, Series in Nonlinear Analysis and Applications, MR3156940, Zbl1282.26001.
Google Scholar
22 Corduneanu C., Integral Equations and Stability of Feedback Systems, 1973, Academic Press, New York, NY, USA, MR0358245, Zbl0273.45001.
Web of Science® Google Scholar
23 Granas A. and Dugundji J., Fixed Point Theory, 2003, Springer-Verlag, New York, NY, USA, https://doi.org/10.1007/978-0-387-21593-8, MR1987179.
10.1007/978-0-387-21593-8
Web of Science® Google Scholar

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

Abstract

1. Introduction

2. Preliminaries

3. Existence and Attractivity Results

4. An Example

Conflicts of Interest

Open Research

Data Availability

References

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

Abstract

1. Introduction

2. Preliminaries

3. Existence and Attractivity Results

4. An Example

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

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