A study of fractional-order coupled systems with a new concept of coupled non-separated boundary conditions

In this paper, we introduce a new concept of coupled non-separated boundary conditions and solve a coupled system of fractional differential equations supplemented with these conditions. The existence results obtained in the given configuration are not only new but also yield some new special results corresponding to particular values of the parameters involved in the problem. For the illustration of the existence and uniqueness result, an example is constructed. MSC: 34A08; 34B15


Introduction
Fractional differential equations have gained considerable importance due to their varied applications in many problems of physics, chemistry, biology, applied sciences and engineering. The tools of fractional calculus are found to be of great support in developing a more realistic mathematical modeling of the applied problems in terms of fractional differential equations. Fractional-order models are regarded as better than the classical ones (based on differential equations) as fractional derivatives can take care of the hereditary properties of materials and processes involved in the problem at hand. For details and explanations, we refer the reader to the texts [-]. In particular, a great interest has been shown by many authors in the subject of fractional-order boundary value problems (BVPs), and a variety of results for BVPs equipped with different kinds of boundary conditions have been obtained, for instance, see [-] and the references cited therein.
Coupled systems of fractional-order differential equations constitute an interesting and important field of research in view of their applications in many real world problems such as anomalous diffusion [], disease models [-], ecological models [], synchronization of chaotic systems [-], etc. For some theoretical works on coupled systems of fractional-order differential equations, we refer the reader to a series of papers [-].
In this paper, we consider a new boundary value problem of coupled Caputo type fractional differential equations: subject to the following non-separated coupled boundary conditions: where c D α , c D β denote the Caputo fractional derivatives of order α and β, respectively, f , g : [, T] × R × R → R are appropriately chosen functions, and λ i , μ i , i = , , are real constants with λ i μ i = , i = , .
Here we emphasize that our problem is new in the sense of non-separated coupled boundary conditions introduced here. To the best of our knowledge, fractional-order coupled system (.) has yet to be studied with the boundary conditions (.). In consequence, our findings of the present work will be a useful contribution to the existing literature on the topic. The existence and uniqueness results for the given problem are new, though they are proved by applying the well-known method based on Banach's contraction principle and Leray-Schauder's alternative.
The rest of the contents of the paper is organized as follows. In Section , we recall some basic definitions of fractional calculus and present an auxiliary lemma, which plays a pivotal role in obtaining the main results presented in Section . We also discuss an example for illustration of the existence-uniqueness result. The paper concludes with some interesting observations.

Preliminaries
First of all, we recall some basic definitions of fractional calculus.
Definition . The fractional integral of order r with the lower limit zero for a function f is defined as provided the right-hand side is point-wise defined on [, ∞), where (·) is the gamma function, which is defined by (r) = ∞  t r- e -t dt.
Definition . The Riemann-Liouville fractional derivative of order r > , n - < r < n, n ∈ N, is defined as where the function f (t) has an absolutely continuous derivative up to order (n -).
Definition . The Caputo derivative of order r for a function f : [, ∞) → R can be written as Now we present an auxiliary lemma which 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 Proof We know that the general solution of fractional differential equations in (.) can be written as where a i , b  , i = , , are arbitrary real constants.
Using the boundary conditions in (.) and (.), we have From the last two relations we find Substituting a  and b  in the first two relations, we find Inserting the values of a i , b i , i = , , in (.) and (.), we get solutions (.) and (.). The converse follows by direct computation. This completes the proof.

Main results
Let us introduce the space In view of Lemma ., we define the operator T : where For convenience, we put In the first result, we prove the existence and uniqueness of solutions of boundary value problem (.)-(.) via Banach's contraction principle.  We show that

Theorem . Assume that:
Hence In the same way, we can obtain that Consequently, Now, for (u  , v  ), (u  , v  ) ∈ X × X and for any t ∈ [, T], we get and consequently we obtain Similarly, It follows from (.) and (.) that Since (M  + M  )  + (M  + M  )  < , therefore, T is a contraction operator. So, by Banach's fixed point theorem, the operator T has a unique fixed point, which is the unique solution of problem (.)-(.). This completes the proof.
The second result is based on the Leray-Schauder alternative.
Lemma . (Leray-Schauder alternative [], p.) Let F : E → E be a completely continuous operator (i.e., a map restricted to any bounded set in E is compact). Let Then either the set E(F) is unbounded or F has at least one fixed point.

Theorem . Assume that:
(H  ) f , g : [, T]×R×R → R are continuous functions and there exist real constants k i , γ i ≥  (i = , , ) and k  > , γ  >  such that ∀x i ∈ R (i = , ), we have Proof First we show that the operator T : X × X → X × X is completely continuous. By the continuity of functions f and g, the operator T is continuous.
Let ⊂ X × X be bounded. Then there exist positive constants L  and L  such that Then, for any (u, v) ∈ , we have which implies that Similarly, we get Thus, it follows from the above inequalities that the operator T is uniformly bounded, since T(u, v) ≤ (M  + M  )L  + (M  + M  )L  . Next, we show that T is equicontinuous. Let t  , t  ∈ [, T] with t  < t  . Then we have Analogously, we can obtain Therefore, the operator T(u, v) is equicontinuous, and thus the operator T(u, v) is completely continuous.