Study on Implicit-Type Fractional Coupled System with Integral Boundary Conditions

This paper is concerned with a class of implicit-type coupled system with integral boundary conditions involving Caputo fractional derivatives. First, the existence result of solutions for the considered system is obtained by means of topological degree theory. Next, Ulam–Hyers stability and generalized Ulam–Hyers stability are studied under some suitable assumptions. Finally, one example is worked out to illustrate the main results.

Another topic which has increasingly received interest in recent years is stability theory, which plays an important role in optimization and numerical point of view. In order to deal with the difficulties that arise from exponential Mittag-Leffler and Lyapunovy stability theory, Ulam [19] introduced another kind of stability, that is Ulam-Hyers stability, which was further studied by Hyers [20] in the subsequent years. Since then, many achievements in various kinds of differential equations with integer order in this field have been investigated ( [21,22], etc.). Nowadays, researchers are focusing on the study of Ulam stability for FDEs--for more details about Ulam stability with FDEs, please see [23][24][25][26] and the references therein.
Integral boundary value conditions (IBCs, for short) of various types of differential equations have also been regarded as an important research field in recent years. IBCs play an important role in the study of computational fluid dynamics studies related to blood flow problems. In order to solve the unreasonable problem that the cross-section of blood vessels is always assumed to be circular, IBCs provide an effective and applicable approach. In addition, IBCs also have an important applications in the dynamic behavior of bioprocesses, such as HIV infection. More details can be found in [27,28]. Additionally, IBCs have useful applications in regularizing ill-posed parabolic backward problems in time partial differential equations-see, for example, mathematical models for bacterial self-regularization [29]. Some recent investigations on coupled systems of FDEs with IBCs can be found in [23,27,[30][31][32][33] and the references cited therein.
As a branch of the research on FDEs, the IFDEs have been given widespread attention, due to the fact that many of the problems around managerial and economics are modeled by IFDEs. For more details on IFDEs, one can see [34][35][36], and references therein. Recently, IFDEs have been properly investigated by many researchers, such as Z. Ali et al. [25], M. Benchohra et al. [24], J. J. Nieto et al. [37], and Samina et al. [38]. They focus on the existence and stability theory. For example, J. J. Nieto et al.,in [37], investigated the following implicit fractional differential equations via the Liouville-Caputo derivative: where η ∈ (0, +∞), J = [0, η], and ϑ, θ : J × R × R → R are nonlinear continuous functions. By applying Banach and Schauder's fixed point theorem, the existence and uniqueness results for the considered problem were obtained under some suitable assumptions.
From all of the above works, we find the fact that their nonlinear terms need to satisfy the Lipschitz condition for two variables. The implicit of the corresponding problem can be overcome by this condition, which plays a key role in obtaining the existence results of solutions. Although this condition is convenient, it is very restrictive and excludes a lot of nonlinear IFDEs. Therefore, we want to investigate whether it works when one "certain variable" satisfies the Lipschitz condition in nonlinear terms.
As far as we know, there are few paper studied Ulam-Hyers stability for IFDEs and there is no paper considering such a problem for (1). Thus, the purpose of the present paper is to fill this gap. Moreover, our nonlinear terms only need to satisfy the Lipschitz condition for one "certain variable", which relax the restriction of the nonlinear terms in the literatures [24,38].
We subdivide this paper into the following Sections. Section 2 contains some preliminaries. In Section 3, some transformations are introduced to convert the problem (1) into the corresponding explicit form and integral form. In Section 4, the existence results of solutions will be given and proved by topological degree theory together with the properties of Green function. In Section 5, we establish Ulam-Hyers stability and generalized Ulam-Hyers stability under some suitable assumptions. Finally, in Section 6, one example is worked out to support our main results.
Next, we list some basic definitions and fundamental results about topological degree theory. For this sake, let E be a Banach space with norm · , D ⊂ E, and the operator A : D → E. χ(·) represents the Kuratowski measure of noncompactness.

Definition 3 ([39]). Assume that A is continuous and bounded.
(1) If there is a constant σ ≥ 0, such that χ(A(S)) ≤ σχ(S) for any bounded S ⊂ E, then A is called a σ-set contraction; a σ-set contraction with σ < 1 is called a strict set contraction. (2) A is said to be condensing if χ(A(S)) ≤ χ(S) for any bounded S ⊂ E, which is not relatively compact.

Proposition 1 ([39]). If A is χ-Lipschitz having constants h, namely
Then A is h-set contraction in D.
Theorem 1 ([39]). Let T : E → E be condensing and If Ω is a bounded set in E, then T has at least one fixed point.

Conversion of the Considered System
In this section, we convert problem (1) into explicit form to overcome the difficulties that arise from implicity. X = C[0, 1] represents a Banach space with norm The product X × X is a Banach space with norm (x, y) = x + y . For simplicity and convenience, set Now let us list the following assumptions satisfied throughout the paper.
For convenience, set In order to convert system (1) into explicit form, we also need to list the following two lemmas.
Proof of Lemma 2. For each y ∈ X , define a mapping F : X → X by Then, we claim that the mapping F is contractive. In fact, by Hypothesis 3, for each ϕ, ϕ 0 ∈ X , by (3), (4), and Hypothesis 3, one can get that where := 3σ 1 · κ 1 ∈ (0, 1). Namely, F is a contractive mapping in X . Applying the contraction mapping principle, one can get that there exists continuous function From a process similar to the above, there exists the continuous function To sum up, the proof is complete.

Lemma 3.
Define the mapping G : X × X → X × X by G = (ϕ 1 , ϕ 2 ). Assume that Hypothesis 1 and Hypothesis 3 hold. Then, the mapping G is continuous.

Proof of Lemma 3.
Let (x n , y n ) → (x, y) ∈ X × X as n → ∞. For convenience, set u n := ϕ 1 (·, y n ), v n := ϕ 2 (·, x n ). Then, we need only to claim that To this end, by Hypothesis 1, Hypothesis 3, and Lemma 2, one can get that, for each t ∈ [0, 1], Taking the maximum for both sides of the above inequality in t ∈ [0, 1], we immediately get that As a result, the conclusion of this lemma follows.
Based on the Lemmas 2 and 3, we know that the existence of nontrivial solutions for problem (1) is equivalent to the existence of the following explicit fractional equations, as Therefore, we need only to investigate the existence of nontrivial solutions for (6).

Lemma 4.
Let h ∈ X be a given function. Then, the unique solution of the following boundary value problem    D α x(t) = h(t), 0 < t < 1; is given by Proof of Lemma 4. By using Lemma 1, we can reduce (6) to the following equivalent equation for some c i ∈ R, i = 0, 1.
Applying boundary conditions to (9), one can obtain that Putting (10) into (9), we have Moreover, by (11) one can derive which indicates that Substituting (13) into (11), we obtain This completes the proof.
From Lemmas 2-4, and Hypothesis 1, we need to only solve the following system of integral equations: Now, define operators A = (A 1 , A 2 ), B = (B 1 , B 2 ), and T = A + B, where A 1 , A 2 : X → X and B 1 , B 2 : X × X → X × X are defined by T (x, y) = A(x, y) + B(x, y).

Then (15) in operator form becomes
Obviously, (16) guarantees that the existence of solutions for (15) is equivalent to the existence of the fixed point of T in X × X . Therefore, we just need to find the fixed point of T in the following work.

Existence Results
In this section, we will present the existence result of solutions for problem (1). To do this, first we give some properties of mapping T .

Consequently, it is a k-set contraction.
Proof of Lemma 5. For (x, y), (x 0 , y 0 ) ∈ X × X , by virtue of Hypothesis 2, we can obtain Therefore, mapping A satisfies the Liptschitz condition. By Proposition 1, we immediately get that mapping A is a k-set contraction. Lemma 6. Assume that Hypothesis 1 and Hypothesis 3 hold. Then, mapping B : X × X → X × X is completely continuous.
To sum up, the conclusion of this lemma follows.
Now we are in a position to prove our existence result of solutions in the following.

Theorem 2.
Assume that Hypotheses 1-3 hold. Then problem (1) has at least one solution in X × X .
Proof of Theorem 2. From Lemmas 5 and 6, we know that A is a k-set contraction and B is a 0-set contraction, respectively. Thus, Definition 3 and Proposition 3 indicate that T is condensing. Define Now, we claim that Λ is bounded in X × X . To this end, by Hypothesis 2, one can get that for (x, y) ∈ Λ, where From (19) and (20), it follows that Thus, Then, Λ is bounded in X × X . Hence, by Theorem 1, T has at least one fixed point. The conclusion of this theorem follows.

Stable Results
In this section, we investigate the Ulam Stability for problem (1). First, let us list the following assumption.

Definition 5. ([40]) Problem
Moreover, we also need the following lemma in stability analysis.

Lemma 7.
Let (x, y) ∈ X × X be a solution of inequalities (21). Then, the following inequalities hold:  Proof of Lemma 7. Obviously, there exist Φ 1 , Φ 2 ∈ X , which depend on x, y, respectively, such that |Φ 1 (t)| ≤ ε 1 , |Φ 2 (t)| ≤ ε 2 ; and By using Lemma 4, one can see that Similarly, The proof is complete. Now, we are in a position to prove the main stable theorem of this section.
Furthermore, the solution of problem (1) is generalized Ulam-Hyers stable.
Proof of Lemma 3. Let (x, y) ∈ X × X be a solution of (21) and (x, y) ∈ X × X be the unique solution to (1). Then, in view of Lemma 4, we have that Now, consider By (H3 ), we have Thus, Similarly, we also have From (27) and (28), it follows that We write (29) in matrix form as Then, we get where Further simplification gives From above, we have Taking ε = max{ε 1 , ε 2 }, one can get that Therefore, the solution of system (1) is Ulam-Hyers stable.
By careful calculation and (2), one can obtain that Obviously, we can see that Hypothesis 1 holds. Now assumptions Hypothesis 2-(H3 ) are satisfied if we set Consequently, system (33) has at least one solution by Theorem 2 and the solution of system (33) is Ulam-Hyers stable and generalized Ulam-Hyers stable by Theorem 3.

Discussion
In this paper,firstly by means of topological degree theory, some existence results are obtained for a class of implicit-type coupled systems with integral boundary conditions, involving Caputo fractional derivatives. As far as we know, there are no existence results available for considered systems (1). Furthermore, the nonlinear terms here need only to satisfy the Lipschitz condition for one variable, which relax the restriction of the nonlinear terms in the literatures. The main reason is that the contraction mapping principle is used to obtain the existence of implicit function (see Lemmas 2 and 3). Secondly, Ulam-Hyers stability and generalized Ulam-Hyers stability are investigated under some suitable assumptions. The method used in present paper can be applied in broadest context, such as many implicit-type systems with various boundary conditions involving integer or fractional derivatives.