A study of coupled systems of mixed order fractional differential equations and inclusions with coupled integral fractional boundary conditions

In this work we investigate existence and uniqueness of solutions for new coupled systems of mixed order fractional differential equations and inclusions supplemented with coupled nonlocal fractional boundary conditions. We apply the Leray–Schauder alternative and the Banach contraction mapping principle to obtain the existence and uniqueness results, while in the multi-valued case we use the nonlinear alternative for Kakutani maps and Covitz and Nadler’s fixed point theorem.

Differential inclusions are found to be of great utility in studying dynamical systems and stochastic processes. For some recent results on boundary value problems for fractional differential inclusions, see [30][31][32][33].
Recently, in [34], the authors studied a boundary value problem of coupled Caputo type fractional differential inclusions of the form: (t, u(t), v(t)), t ∈ [0, T], 1 < α ≤ 2, (t, u(t), v(t)), t ∈ [0, T], 1 < β ≤ 2, where c D α , c D β denote the Caputo fractional derivatives of orders α and β respectively, F, G : [0, T] × R × R → P(R) are given multi-valued maps, P(R) is the family of all nonempty subsets of R, and ν i , μ i , i = 1, 2, are real constants with ν i μ i = 1, i = 1, 2. By applying standard fixed point theorems for multi-valued maps, some new existence results for the given problem are derived when the multi-valued maps involved in the given problem have convex as well as non-convex values.
In this work, motivated by [34], we consider the following systems of Caputo and Riemann-Liouville type mixed order coupled fractional differential equations and inclusions: where c D α , c D p are the Caputo fractional derivatives of order α and p respectively, RL D β is the Riemann-Liouville fractional derivative of order β, I q is the Riemann-Liouville fractional integral of order q, f , g :  .3), we rely on the Leray-Schauder alternative and the Banach contraction mapping principle to obtain the exis-tence and uniqueness results, which are presented in Sect. 3. Section 4 contains the existence results for convex and non-convex valued multi-valued maps F and G involved in multi-valued system (1.2)-(1.3), which are respectively derived with the aid of the nonlinear alternative for Kakutani maps and Covitz and Nadler's fixed point theorem. The background material related to our work is outlined in Sect. 2. Here we remark that the tools of the fixed point theory employed in our analysis are standard, however their exposition to the problems at hand is new.

Preliminaries
Let us begin this section with some basic definitions of multi-valued maps [35,36].
Let (X , · ) be a normed space and that P cl (X ) = {Y ∈ P(X ) : Y is closed}, P cp,c (X ) = {Y ∈ P(X ) : Y is compact and convex}.
Next, we outline some preliminary concepts of fractional calculus.

Definition 2.1
The fractional integral of order σ with the lower limit zero for a function ζ is defined as

Definition 2.2
The Riemann-Liouville fractional derivative of order σ > 0, n -1 < σ < n, n ∈ N, is defined as follows: where the function ζ has absolutely continuous derivative up to order (n -1).

Definition 2.3
The Caputo derivative of order σ for a function ζ : [0, ∞) → R can be written as In the rest of the paper, we use c D σ instead of c D σ 0+ for the sake of convenience.
The following auxiliary lemma, which concerns the linear variant of system (1.1), plays a key role in the sequel.
. Then the solution of the linear fractional differential system is equivalent to the system of integral equations and where it is assumed that Proof Applying the Riemann-Liouville operators I α and I β to the Caputo and Riemann-Liouville fractional differential equations respectively in (2.1) and using the composition laws of fractional order integral and differential operators [2], we obtain where c 0 , c 1 , c 2 are arbitrary constants. By the boundary conditions of (2.1) in (2.5), we get c 2 = 0 and a system of algebraic equations in the unknown constants c 0 and c 1 : Solving the above system, we get and Substituting the values of c 0 , c 1 , c 2 in (2.5), we get solutions (2.2) and (2.3). We can prove the converse of the lemma by direct computation. The proof is completed.

Single-valued system (1.1)-(1.3)
where and . For convenience, we set the notations: Our first existence result is based on the Leray-Schauder alternative [37, p. 4].
Theorem 3.1 Assume that:

continuous functions and that there exist real constants
Proof Firstly we show that the operator H : X × X → X × X defined by (3.1) is completely continuous. Notice that continuity of the operator H follows from that of the functions f and g.
Let Ω ⊂ X × X be bounded. Then there exist positive constants L 1 and L 2 such that which implies that In a similar way, we can find that From the above inequalities we conclude that the operator H is uniformly bounded, since Analogously, we can obtain Thus the operator H(x, y) is equicontinuous. In view of the foregoing arguments, we deduce that the operator H(x, y) is completely continuous.
Finally, it will be verified that the set Then and In consequence, we have Thus we have   Then we show that HB r ⊂ B r , where B r = {(x, y) ∈ X × X : (x, y) ≤ r} and H is defined by (3.1).
By assumption (A 2 ), for (u, v) ∈ B r , t ∈ [0, T], we have In consequence, we obtain which implies that In the same way, we can find that From the above inequalities, it follows that Next, for (x 2 , y 2 ), (x 1 , y 1 ) ∈ X × X and for any t ∈ [0, T], we get which leads to Similarly, one can obtain From (3.7) and (3.8), we deduce that Since (M 1 + M 3 ) 1 + (M 2 + M 4 ) 2 < 1, therefore, H is a contraction. So, by Banach's contraction mapping principle, the operator H has a unique fixed point, which corresponds to a unique solution of problem (1.1)-(1.3). This completes the proof.

Multi-valued system (1.2)-(1.3)
and is called a solution of coupled system (1. In view of Lemma 2.5, we define the operators K 1 , K 2 : X × X → P(X × X) as follows: and where Q 1 (x, y)(t) and Then we define an operator K : X × X → P(X × X) by where K 1 and K 2 are defined by (3.12) and (3.13).

The Carathéodory case
Our first result dealing with convex values F and G is proved via the Leray-Schauder nonlinear alternative for multi-valued maps [37]. Proof Consider the operators K 1 , K 2 : X × X → P(X × X) defined by (3.12) and (3.13). From (B 1 ), it follows that the sets S F,(x,y) and S G,(x,y) are nonempty for each (x, y) ∈ X × X.
Then there exist f ∈ S F,(x,y) , g ∈ S G,(x,y) such that and Then we have Analogously, we can obtain Therefore, the operator K(x, y) is equicontinuous, and thus, by the Ascoli-Arzelá theorem, the operator K(x, y) is completely continuous. We know from [35, Proposition 1.2] that a completely continuous operator is upper semicontinuous if it has a closed graph. Thus we need to prove that K has a closed graph. Let (x n , y n ) → (x * , y * ), (h n ,h n ) ∈ K(x n , y n ) and (h n ,h n ) → (h * ,h * ), then we need to show (h * ,h * ) ∈ K(x * , y * ). Observe that (h n ,h n ) ∈ K(x n , y n ) implies that there exist f n ∈ S F,(x n ,y n ) and g n ∈ S G,(x n ,y n ) such that andh n (x n , y n )(t) = I β g n (t) + t β-1 Λ I β g n (T)γ I q+α f n (ξ )λγ ξ q Γ (1 + q) I β-p g n (η) .
Finally, we establish the a priori bounds on solutions. Let (x, y) ∈ νK(x, y). Then there exist f ∈ S F,(x,y) and g ∈ S G, (x,y) such that and For each t ∈ [0, T], we obtain following the same arguments as in the second step. Thus In view of (B 3 ), there exists N such that (x, y) = N . Let us set Note that the operator K : U → P cp,cv (X) × P cp,cv (X) is completely continuous and upper semicontinuous. There is no (x, y) ∈ ∂U such that (x, y) ∈ νK(x, y) for some ν ∈ (0, 1) by the choice of U. Hence, by the nonlinear alternative of Leray-Schauder type [37], we deduce that K has a fixed point (x, y) ∈ U, which is a solution of coupled system (1.2)-(1.3). This completes the proof.

The Lipschitz case
This subsection is concerned with the case when the multi-valued maps in system (1.2) have non-convex values. Let (X, d) be a metric space induced from the normed space (X; · ), and let H d : (u, v) and d(u, V ) = inf v∈V d (u, v). Then (P b,cl (X), H d ) is a metric space and (P cl (X), H d ) is a generalized metric space (see [39]). Definition 3.6 A multi-valued operator G : X → P cl (X) is called (i) γ -Lipschitz if and only if there exists γ > 0 such that H d (G(a), G(b)) ≤ γ d(a, b) for each a, b ∈ X; and (ii) a contraction if and only if it is γ -Lipschitz with γ < 1.
In the forthcoming result, we make use of the fixed point theorem for multi-valued maps due to Covitz and Nadler [40].  (3.14) Proof The sets S F,(x,y) and S G,(x,y) are nonempty for each (x, y) ∈ X × Y by assumption (B 3 ), so F and G have measurable selections (see Theorem III.6 in [41]). Now we show that the operator K satisfies the assumptions of Covitz and Nadler's fixed point theorem [40]. First we show that K(x, y) ∈ P cl (X)×P cl (X) for each (x, y) ∈ X ×X. Let (h n ,h n ) ∈ K(x n , y n ) such that (h n ,h n ) → (h,h) in X × X. Then (h,h) ∈ X × X and there exist f n ∈ S F,(x n ,y n ) and g n ∈ S G,(x n ,y n ) such that h n (x n , y n )(t) andh n (x n , y n )(t) = I β g n (t) + t β-1 Λ I β g n (T)γ I q+α f n (ξ )λγ ξ q Γ (1 + q) I β-p g n (η) .
Therefore K is a contraction in view of assumption (3.14). Hence it follows by Covitz and Nadler's fixed point theorem [40] that K has a fixed point (x, y), which is a solution of problem (1.2)-(1.3). This completes the proof.

Conclusion
In the present research we studied the existence of solutions for coupled fractional differential equations and inclusions involving fractional derivatives of different orders and supplemented with nonlocal boundary conditions containing fractional derivative and integral. In the single-valued case we establish existence and uniqueness of solutions by applying the Leray-Schauder alternative and the Banach contraction mapping principle respectively. In the multi-valued case we proved existence results for both convex and non-convex multi-valued maps via the nonlinear alternative for Kakutani maps and Covitz and Nadler's fixed point theorem. Examples illustrating the obtained results are also constructed.