Existence Results for a Nonlocal Coupled System of Differential Equations Involving Mixed Right and Left Fractional Derivatives and Integrals

In this paper, we study the existence and uniqueness of solutions for a new kind of nonlocal four-point fractional integro-differential system involving both left Caputo and right Riemann–Liouville fractional derivatives, and Riemann–Liouville type mixed integrals. The Banach and Schaefer fixed point theorems are used to obtain the desired results. An example illustrating the existence and uniqueness result is presented.


Introduction
Fractional-order boundary value problems involving different kinds of fractional derivatives and boundary conditions have been investigated by many researchers in recent years. The literature on the topic is now much enriched and contains a wide variety of results, for instance, see the texts [1][2][3] and articles [4][5][6][7][8][9][10]. On the other hand, there has also been shown a great interest in the fractional differential systems in view of the occurrence of such systems in the mathematical models of physical and engineering problems. In [11], the authors carried out dynamical analysis of time fractional order phytoplankton-toxic phytoplankton-zooplankton system. A delay fractional order model was proposed for the co-infection of malaria and HIV/AIDS in [12]. Chaos synchronization in fractional differential systems was explained in the article [13]. For details on diffusion and reactions in fractals and disordered Systems, we refer the reader to the text [14]. Using the Riemann-Liouville fractional operator, the unsteady axial Couette flow of fractional second grade fluid and fractional Maxwell fluid between two infinitely long concentric circular cylinders was studied in [15]. In a survey [16], the authors collected the results on the fractional analogue of Bhalekar-Gejji system, Lorenz system, Liu system, Chen system and Rössler system as a characteristic representative of fractional order autonomous dynamical system. For more applications of fractional calculus on bioengineering, anomalous diffusion of contamination, earth system dynamics, open channel flow, transient flow, physical models, fluid mechanics, viscoelastic fluids, etc., we refer the reader to the articles [17][18][19][20][21][22][23][24][25][26]. In view of extensive occurrence of couples fractional differential systems in a variety of mathematical models, many authors turned to the theoretical development of such systems, for example, see [27][28][29][30][31].
However, there are fewer works on boundary value problems containing both left and right fractional derivatives. Such problems constitute a special class of Euler-Lagrange equations, and facilitate the study of variational principles [32]. In [33], the authors applied a probabilistic approach to study equations involving both left-sided and right-sided generalized operators of Caputo type, and showed a relationship between these equations and two-sided exit problems for certain Levy processes. In [34], the author related the study of fully mixed and multidimensional extensions of the Caputo and Riemann-Liouville derivatives with Markov processes. The left-sided and right-sided fractional derivatives were used to formulate the fractional diffusion-advection equation to study anomalous superdiffusive transport phenomena in [35]. For further details, we refer the reader to the articles [36][37][38][39]. In a more recent work [40], the authors investigated the existence of solutions for a new kind of integro-differential equation involving right-Caputo and left-Riemann-Liouville fractional derivatives of different orders and right-left Riemann-Liouville fractional integrals equipped with nonlocal boundary conditions.
In this paper, motivated by aforementioned work on mixed fractional differential equations, we introduce and study a new coupled system of nonlinear fractional differential equations, involving left Caputo and right Riemann-Liouville fractional derivatives of different orders and a pair of nonlinearities with one of them in terms of mixed fractional integrals in each equation of the system, equipped with four-point nonlocal coupled boundary conditions given by where C D 0+ denote the right and left Riemann-Liouville fractional integrals of orders p 1 , p 2 , q 1 , q 2 > 0 respectively, f 1 , f 2 , h 1 , h 2 : [0, 1] × R × R → R are given continuous functions and δ, ρ, λ 1 , λ 2 ∈ R.
The rest of the paper is organized as follows. In Section 2 we outline the basic concepts from fractional calculus. In Section 3 we first prove an auxiliary lemma for the linear variant of the problem (1). Then we derive the existence and uniqueness result for the problem (1) by applying Banach's fixed point theorem, while the existence result is established via Schaefer's fixed point theorem. An example illustrating the uniqueness result is also presented.

Preliminaries
In this section we recall some related definitions of fractional calculus needed in our study.

Definition 1 ([41]
). The left and right Riemann-Liouville fractional integrals of order β > 0 for g ∈ L 1 [a, b], existing almost everywhere on [a, b], are respectively defined by Also, according to the classical theorem of Vallee-Poussin and the Young convolution theorem, I β a+ g, Of course, if g ∈ C[a, b] or q 1 + q 2 > 1, then the above relations hold for each x ∈ [a, b].

Definition 2 ([1]
). For g ∈ AC n [a, b], the left Riemann-Liouville and the right Caputo fractional derivatives of order β ∈ (n − 1, n], n ∈ N, existing almost everywhere on [a, b], are respectively defined by

Existence and Uniqueness Results
The following lemma, dealing with a linear variant of the problem (1), plays an important role in the forthcoming analysis.
In view of Lemma 2, we define an operator K : X × X → X × X as where we have used the fact that For computational convenience, we set , , , , , , , where Now we are in a position to prove the existence and uniqueness of solutions to the system (1) by Banach contraction mapping principle. (B 1 ) There exist L 1 , L 2 > 0 such that ∀t ∈ [0, 1] and x i , y i ∈ R, i = 1, 2, (B 2 ) There exist K 1 , K 2 > 0 such that ∀t ∈ [0, 1] and x i , y i ∈ R, i = 1, 2, Then the system (1) has a unique solution on [0, 1] provided that where Ω i , i = 1, 2, . . . , 8, are defined by (13).
Proof. Let us set where f 0 ,f 0 , h 0 ,ĥ 0 are finite numbers defined by |h 2 (t, 0, 0)| =ĥ 0 , and Ψ is defined by (14). Next we consider a closed ball B r = {(x, y) ∈ X × X : (x, y) ≤ r} and show that KB r ⊂ B r . Then, in view of the assumption (B 1 ), we have Similarly, we can find that Then, for (x, y) ∈ B r , we have which implies that Similarly, we can find that Consequently, we get which implies that K(x, y) ∈ B r for any (x, y) ∈ B r . Therefore KB r ⊂ B r . Now, we prove that K is a contraction. Let (x 1 , y 1 ), (x 2 , y 2 ) ∈ X × X for each t ∈ [0, 1]. Then, by the conditions (B 1 ) and (B 2 ), we get In a similar manner, one can find that Using (15) and (16), we obtain K(x 2 , y 2 ) − K(x 1 , y 1 )) ≤ Ψ x 2 − x 1 + y 2 − y 1 .
From the above inequality, it follows by the assumption Ψ < 1 that K is a contraction. Hence we deduce by the Banach fixed point theorem that the operator K has a unique fixed point, which corresponds to a unique solution of system (1). The proof is completed.
Let us now recall Schaefer's fixed point theorem [42], which plays a key role in proving the next existence result.
Proof. Observe that continuity of the functions f 1 , f 2 , h 1 , h 2 implies that the operator K is continuous. Next, we show that the operator K is uniformly bounded. Let Q ⊂ X × X be a bounded set. Then, for all (x, y) ∈ Q, there exist constants Analogously, we can find that From the foregoing inequalities, it follows that Thus the operator K is uniformly bounded. Next, we show that K is equicontinuous. For 0 < t 1 < t 2 < 1, we have Consequently, we get Therefore, the set V is bounded. Hence, by Lemma 3, the operator K has at least one fixed point, which is indeed a solution of the system (1) on [0, 1]. The theorem is proved.

Conclusions
We have presented the criteria for the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations, involving left Caputo and right Riemann-Liouville fractional derivatives of different orders and a pair of nonlinearities, equipped with four-point nonlocal boundary conditions. In order to achieve the desired criteria, we have applied the fixed point theorems due to Banach and Schaefer. Our results are new and enrich the literature on nonlocal boundary value problems of mixed fractional-order coupled integro-differential systems. The work presented in this paper is expected to improve the study carried out in [34,35] as it conveys the idea of introducing a nonlinear forcing term involving the two-sided Riemann-Liouville fractional integrals in addition to the usual nonlinear forcing term.