In this paper, we are going to investigate the existence and uniqueness of solutions of a coupled system of nonlinear fractional hybrid equations with nonseparated type integral boundary hybrid conditions. We are going to use Banach’s and Leray-Schauder alternative fixed point theorems to obtain the main results. Lastly, we are giving two examples to show the effectiveness of the main results.

1. Introduction

Fractional differential equations appear naturally in a number of fields bymany fields of scince such as physics, engineering, biophysics, blood flow phenomena, aerodynamics, electron-analytical chemistry, biology, and economy; for more details, we refer the readers to [1–4] and many other references therein which give an excellent account on the study of fractional differential equations.

Hybrid fractional differential equations have also been studied by several researchers. This class of equations involves the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. Some recent results on hybrid differential equations can be found in a series of papers (see [5, 6]).

On the other hand, coupled systems of fractional differential equations are very important to study byattract the attention of many researches, because they appear naturally in many problems (see [3, 7–10]).

In [11], Sitho et al. discussed the following existence bysome results for the following hybrid fractional integrodifferential equations:

(1)

where D^α denotes the Riemann-Liouville fractional derivative of order α, 0 < α ≤ 1, I^ϕ is the Riemann-Liouville fractional integral of order ϕ > 0, ϕ ∈ {β₁, β₂, ⋯, β_m}, f ∈ C(J × ℝ, ℝ\{0}), g ∈ C(J × ℝ, ℝ), with h_i ∈ C(J × ℝ, ℝ) with h_i(0, 0) = 0, i = 1, 2, ⋯, m.

In [12], Hilal and Kajouni considered the following boundary value problems for hybrid differential equations with fractional order (BVPHDEF for short) involving Caputo differential operators of order 0 < α < 1:

(2)

where f ∈ C(J × ℝ, ℝ\{0}),

and a, b, c are real constants with a + b ≠ 0.

Dhage and Lakshmikantham [13] discussed the following first-order hybrid differential equation:

(3)

where f ∈ C(J × ℝ, ℝ\{0}) and

. They established the existence, uniqueness results, and some fundamental differential inequalities for hybrid differential equations initiating the study of theory of such systems and proving the utilization of the theory of inequalities, the existence of extremal solutions, and comparison results.

Zhao et al. [5] discussed the following fractional hybrid differential equations involving Riemann-Liouville differential operators:

(4)

where f ∈ C(J × ℝ, ℝ\{0}) and

. They established the existence byof solutions and some fundamental differential inequalities are also established and the existence of extremal solutions.

Benchohra et al. [14] studied the following boundary value problems for differential equations with fractional order:

(5)

where ^cD^α is the Caputo fractional derivative, f : [0, T] × ℝ⟶ℝ is a continuous function, and a, b, c are real constants with a + b ≠ 0.

Hannabou and Hilal [15] considered the boundary value problem of a class of impulsive hybrid fractional coupled differential equations:

(6)

where D^α, D^β stand for Caputo fractional derivatives of order α, β, respectively, f_i ∈ C([0, 1] × ℝ × ℝ, ℝ\{0}), g_i ∈ C([0, 1] × ℝ × ℝ, ℝ), (i = 1, 2) and ϕ, ψ : C([0, 1], ℝ)⟶ℝ are continuous functions defined by

where ξ_i, η_j ∈ (0, 1) for i = 1, 2, ⋯, n, j = 1, 2, ⋯, m, and I_k : ℝ⟶ℝ ,

and

represent the right, left limits of u(t) at t = t_k, (k = i, j).

The present paper is a continuation of the work in [16] in order to study the existence and uniqueness of solutions for a coupled system of fractional hybrid equation of the following forme:

(7)

subject to the fractional nonseparated integral boundary hybrid conditions

(8)

where 0 < α_i < 1, 1 < β_i ≤ 2, λ_i, μ_i, σ_1i, σ_2i, σ_3i ∈ ℝ^∗ with μ_i ≠ −1 for i = 1, 2,

are the Caputo fractional derivatives, f₁, f₂ : [0; 1] × ℝ²⟶ℝ\{0}, ω₁, ω₂ : [0; 1] × ℝ²⟶ℝ and g₁, h₁, k₁, g₂, h₂, k₂ : [0; 1] × ℝ⟶ℝ are given continuous functions.

By a solution of the problems (7)–(8), we mean a function x ∈ C(J, ℝ) such that

(i)
the function t ↦ x/f_i(t, x₁, x₂)(i = 1, 2) is continuous for each (x₁, x₂) ∈ ℝ²,
(ii)
(x₁, x₂) satisfies the equations in (7)–(8).

This paper is organized as follows: in the second section, we recall some notations and several known results. In the third section, we show the existence and uniqueness of solutions of problem (7)–(8), these results can be viewed as extension of the result given in [12]. In the fourth section, we give some examples to demonstrate the application of our main results.

2. Preliminaries and Notations

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let X be a Banach space of all continuous functions defined frome endowed with norm. Then, the product space is also a Banach space equipped with the norm.

We denote byL¹(J, ℝ) the space of Lebesgue integrable real-valued functions on J equipped with the norm

defined by

(9)

Definition 1 (see [17].)The fractional integral of the function h ∈ L¹([a, b], ℝ⁺) of order α ∈ ℝ⁺ is defined by

(10)

where Γ is the gamma function.

Definition 2 (see [17].)Let h be a fonction difened on [a, b], the Riemann-Liouville fractional derivative of order α is defined by

(11)

where n = [α] + 1 and [α] denotes the integer part of α.

Definition 3 (see [17].)Let h be a function defined on [a, b], the Caputo fractional derivative of order α is defined by

(12)

where n = [α] + 1 and [α] denotes the integer part of α.

Lemma 4 (see [2].)Let α, β ≥ 0, then the following relations hold:

(13)

Lemma 5 (see [2].)Let n ∈ ℕ and n − 1 < α < n. If f is a continuous function, then we have

(14)

Lemma 6 (Leray-Schauder alternative, see [18]). Let be a completely continuous operator (i.e., a map that is restricted to any bounded set in G is compact). Let . Then, either the set is unbounded or has at least one fxed point.

We make the following assumption:

(H₀) The function x ↦ x/f_i(t, x₁, x₂)(i = 1, 2) is increasing in ℝ almost everywhere for t ∈ J.

Lemma 7. Assume that hypothesis (H₀) holds. Then, for any y₁, y₂ ∈ C(J, ℝ²). The function x ∈ C(J, ℝ) is a solution of the coupled system,

(15)

subject to the boundary condition (8), has a solution given by

(16)

where

(17)

Proof. Using Lemma 5, we obtain

(18)

where a₀, a₁, a₂ ∈ ℝ.

According to the condition

we find that

(19)

Using the facts that

and

we have

(20)

Substituting the values of a₀, a₁, and a₂, we obtain

(21)

Analogously, we can deduce that

(22)

By a direct computation, the converse of the lemma can be easily verified.

3. Main Results

In view of Lemma 7, we define the operator U : X × X⟶X × X by U(x₁, x₂) = (U₁(x₁, x₂), U₂(x₁, x₂)).

Where,

(23)

(24)

For computational convenience, we set

(25)

Before giving the main results, we impose the following assumptions:

(H₁) The functions f_i are continuous and bounded; that is, there exist positive numbers L_i > 0 such that |f_i(t, u, v)| ≤ L_i for all (t, u, v) ∈ [0, 1] × ℝ × ℝ(i = 1, 2).

(H₂)w₁, w₂ : [0, 1] × ℝ²⟶ℝ and h₁, g₁, k₁, h₂, g₂, k₂ : [0, 1] × ℝ⟶ℝ are continuous functions.

(H₃) There exist positive constants q₁₁, q₂₁, q₃₁, q₁₂, q₂₂, q₃₂ such that for all t ∈ [0, 1] and x₁, x₂, y₁, y₂ ∈ ℝ, we have

(26)

(H₄) There exist positive constants q₄₁, q₅₁, q₆₁, q₄₂, q₅₂, q₆₂ such that

(27)

(H₅)|f_i(t, x, y)| ≤ m_i(t); |h_i(t, x)| ≤ ρ_i(t); |k_i(t, x)| ≤ ψ_i(t); |g_i(t, x)| ≤ ϕ_i(t), ∀(t, x, y) ∈ [0, 1] × ℝ × ℝ with m_i, ρ_i, ϕ_i, ψ_i ∈ C([0, 1]; ℝ⁺), for i = 1, 2.

3.1. First Result

In our first result, we discuss the existence of solutions for system (7)–(8) by means of the Banach fixed point theorem.

Theorem 8. Suppose that (H₁)–(H₃) are satisfied.

Then, there exist a unique solution for systems (7)–(8) provided that r₁₁ + r₁₂ < 1.

Proof. We put , , , , for i = 1, 2.

Let B_r = {(x₁, x₂) ∈ X × X : ‖(x₁, x₂)‖ ≤ r} with r ≥ (r₂₁ + r₂₂)/(1 − (r₁₁ + r₁₂)), where

(28)

We prove that U(B_r)⊆B_r.

For (x₁, x₂) ∈ B_r, t ∈ [0, 1], we have

(29)

Consequently,

(30)

In the same way, we obtain that

(31)

Therefore, we have

(32)

Now, for

and t ∈ [0; 1], we get

(33)

Analogously, one has

(34)

and thus

(35)

Since r₁₁ + r₁₂ < 1, by thene the operator U is a contraction mapping. Hence, we deduce that systems (7)–(8) have a unique solution.

3.2. Second Result

In our second result, we discuss the existence of solutions for system (7)–(8) by means of the so-called Leray-Schauder alternative.

Theorem 9. Assume that conditions (H₁)–(H₃) and (H₅) hold. Furthermore, it is assumed that

(36)

Then, system (7)–(8) have at least one solution.

Proof. We will show that the operator Y : X × Y⟶X × Y satisfes all the assumptions of Lemma 6.

In the first step, we prove that the operator Y is completely continuous.

Clearly, it follows by the continuity of functions f₁, f₂, g₁, and g₂ that the operator Y is continuous.

Let S ⊂ X × Y be bounded. Then, we can find positive constants H₁ and H₂ such that

(37)

Thus, for any x₁, x₂ ∈ S, we get

(38)

In a similar manner, we have

(39)

From the inequalities above, we deduce that the operator Y is uniformly bounded.

Next, we prove that Y is equicontinuous.

The continuity of f₁, f₂, h₁, h₂, g₁, g₂, k₁, k₂ implies that the operator Y₁ is continuous. Moreover, Y₁ is uniformly bounded on B_r′.

Suppose that 0 ≤ t₁ < t₂ ≤ 1. Then we have

(40)

Similarly, one has

(41)

which tend to 0 independently of (x₁, x₂). This implies that the operator Y(x₁, x₂) is equicontinuous. Thus, by the above fndings, the operator Y(x₁, x₂) is completely continuous.

In the next step, we will prove that the set P = {(x₁, x₂) ∈ X × Y/x₁, x₂ = λY(x₁, x₂), 0 ≤ λ ≤ 1} is bounded.

Let (x₁, x₂) ∈ P. Then, we have (x₁, x₂) = λY(x₁, x₂). Thus, for any t ∈ [0, 1], we can write

(42)

Then,

(43)

In consequence, we have

(44)

then ‖(x₂, x₁)‖ ≤ π₁ + π₂(η₀ + η₁‖x₁‖ + η₂‖x₂‖) + π₃‖x₁‖ + π₄‖x₂‖ with

(45)

This shows that the set P is bounded. Hence, all the conditions of Lemma 6 are satisfied, and consequently, the operator Y has at least one fxed point, which corresponds to a solution of system (7)–(8). This completes the proof.

4. Examples

4.1. Example 1

Consider the following system of fractional hybrid differentiel equation:

(46)

where t ∈ [0, 1], β₁ = β₂ = 4/3, α₁ = α₂ = 1/3, λ₁ = 1/700, λ₂ = 1/300, μ₁ = μ₂ = 1, σ_i1 = 1/300, σ_i2 = 1/200, i = 1, 2, 3 and

(47)

Clearly, q₁₁ = q₂₁ = 1/400, q₁₂ = q₂₂ = 1/500, q₄₁ = q₅₁ = q₆₁ = 1/300, q₄₂ = q₅₂ = q₆₂ = 1/200, and q₆₁ = q₆₂ = 1/400; furthermore, we have

(48)

Thus, by Theorem 8, system (46) has an unique solution.

4.2. Example 2

Consider the following system:

(49)

where t ∈ [0, 1], β₁ = 3/2, α₁ = 1/2, β₂ = 4/3, α₂ = 1/3, λ₁ = 1/600, λ₂ = 1/700, μ₁ = 1, μ₂ = 1, σ₁₁ = σ₂₁ = σ₃₁ = σ₁₂ = σ₂₂ = σ₃₂ = 1/300, and

(50)

In this conrete application, we have

(51)

The riview of Theorem 9, problem (49) has a least one solution.

5. Conclusion

It is known that most natural phenomena are modeled by different types of fractional differential equations. This diversity in investigating complicated fractional differential equations increases our ability for exact modeling of different phenomena. This is useful in making modern software which helps us to allow for more cost-free testing and less material consumption. For our contribution in this present work, we investigate a fractional hybrid differential system with mixed integral hybrid and boundary hybrid conditions. We investigatedtwo numerical examples to illustrate our main results

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Open Research

Data Availability

No data were used to support this study.

References

1 Miller K. S. and Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, 1993, John Wiley & Sons, New York, NY, USA.
Google Scholar
2 Podlubny I., Fractional Differential Equations, 1999, Academic Press, San Diego.
Web of Science® Google Scholar
3 Ntouyas S. K. and Obaid M., A coupled system of fractional differential equations with non-local integral boundary conditions, Advances in Difference Equations. (2012) 2012, no. 1, https://doi.org/10.1186/1687-1847-2012-130, 2-s2.0-84873332796.
10.1186/1687-1847-2012-130
PubMed Google Scholar
4 Zhou Y., Basic Theory of Fractional Differential Equations, 2014, Xiangtan University, China, https://doi.org/10.1142/9069.
10.1142/9069
Google Scholar
5 Zhao Y., Sun S., Han Z., and Li Q., Theory of fractional hybrid differential equations, Computers & Mathematcs with Applications. (2011) 62, no. 3, 1312–1324, https://doi.org/10.1016/j.camwa.2011.03.041, 2-s2.0-79961001020.
10.1016/j.camwa.2011.03.041
Web of Science® Google Scholar
6 Dhage B. C., Basic results in the theory of hybrid differential equations with mixed perturbations of second type, Functional Differential Equations. (2012) 19, 1–20.
Google Scholar
7 Wang J. and Zhang Y., Analysis of fractional order differential coupled systems, Mathematical Methods in the Applied Sciences. (2015) 38, no. 15, 3322–3338, https://doi.org/10.1002/mma.3298, 2-s2.0-84941662195.
10.1002/mma.3298
Web of Science® Google Scholar
8 Shah K., Ali A., and Khan R. A., Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems, Boundary Value Problems. (2016) 2016, no. 1, https://doi.org/10.1186/s13661-016-0553-3, 2-s2.0-84958048148.
10.1186/s13661-016-0553-3
Google Scholar
9 Houas M., Existence results for fractional differential equations with integral and multi-point boundary conditions, Mediterrenean Journal of Modeling and Simulation. (2018) 2018, no. 1, 45–59, https://doi.org/10.1186/s13661-017-0924-4, 2-s2.0-85040255613.
10.1186/s13661-017-0924-4
Google Scholar
10 Agarwal P. and Singh R., Modelling of transmission dynamics of Nipah virus (Niv): a fractional order approach, Physica A: Statistical Mechanics and its Applications. (2020) 547, article 124243, 10.1016/j.physa.2020.124243.
Web of Science® Google Scholar
11 Sitho S., Ntouyas S. K., and Tariboon J., Existence results for hybrid fractional integro-differential equations, Boundary Value Problems. (2015) 2015, no. 1, https://doi.org/10.1186/s13661-015-0376-7, 2-s2.0-84948445364.
10.1186/s13661-015-0376-7
Google Scholar
12 Hilal K. and Kajouni A., Boundary value problems for hybrid differential equations with fractional order, Advances in Difference Equations. (2015) 2015, no. 1, https://doi.org/10.1186/s13662-015-0530-7, 2-s2.0-84935866638.
10.1186/s13662-015-0530-7
PubMed Google Scholar
13 Dhage B. C. and Lakshmikantham V., Basic Results on Hybrid Differential Equations, Nonlinear Analysis: Hybrid Systems. (2010) 4, no. 3, 414–424, https://doi.org/10.1016/j.nahs.2009.10.005, 2-s2.0-77955583469.
10.1016/j.nahs.2009.10.005
Web of Science® Google Scholar
14 Benchohra M., Hamani S., and Ntouyas S. K., Boundary value problems for differential equations with fractional order, Surveys in Mathematics and its Applications. (2008) 3, 1–12, https://doi.org/10.1016/j.na.2009.01.073, 2-s2.0-67349155890.
10.1016/j.na.2009.01.073
Google Scholar
15 Hannabou M. and Hilal K., Investigation of mild solution to a coupled systems of impulsive hybrid fractional differential equations, International Journal of Differential Equations. (2019) 2019, 9, 2618982, https://doi.org/10.1155/2019/2618982.
10.1155/2019/2618982
Web of Science® Google Scholar
16 Ibnelazyz L., Guida K., Hilal K., and Melliani S., Existence of solution for a fractional Langevin system with nonseparated integral boundary conditions, Journal of Mathematics. (2021) 2021, 16, 3482153, https://doi.org/10.1155/2021/3482153.
10.1155/2021/3482153
Web of Science® Google Scholar
17 Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, 2006, Elsevier, Amesterdam.
10.1016/S0304-0208(06)80001-0
Google Scholar
18 Granas A. and Dugundji J., Fixed Point Theory, 2003, Springer, New York, NY, USA, https://doi.org/10.1007/978-0-387-21593-8.
10.1007/978-0-387-21593-8
Web of Science® Google Scholar

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Existence and Uniqueness of Solutions for a Fractional Hybrid System with Nonseparated Integral Boundary Hybrid Conditions

Abstract

1. Introduction

2. Preliminaries and Notations

3. Main Results

3.1. First Result

3.2. Second Result

4. Examples

4.1. Example 1

4.2. Example 2

5. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

References

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Existence and Uniqueness of Solutions for a Fractional Hybrid System with Nonseparated Integral Boundary Hybrid Conditions

Abstract

1. Introduction

2. Preliminaries and Notations

3. Main Results

3.1. First Result

3.2. Second Result

4. Examples

4.1. Example 1

4.2. Example 2

5. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

References

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