Positive Solutions of a Singular Nonlocal Fractional Order Differential System via Schauder’s Fixed Point Theorem

and Applied Analysis 3 It follows from u(t) = Iα1x(t) and V(t) = Iβ1y(t) that x(t) = D α 1 t u(t) and y(t) = Dβ t V(t). So substituting the above formulas into (3), we obtain (7). On the other hand, if (x, y) is a solution of (7), (8) yields −D α t u (t) = − d n dt I n−α I α 1x (t) = − d n

However, the research on the systems of fractional differential equations has not received much attention.So motivated by the results mentioned above, in this paper, we study the existence of positive solutions for the following singular where are the standard Riemann-Liouville derivatives, and  and  are positive parameters. : (0, 1) × (0, +∞) → [0, +∞) and  : [0, 1] × [0, +∞) → (0, +∞) are continuous and (, ) may be singular at  = 0, 1 and  = 0.The system ( 3) is an abstract model arising from biological dynamic system, which was introduced by Perelson [5] to describe the primary infection with HIV in integerorder version, and was extended to a fractional order version of HIV-1 infection of CD4 + T-cells by Arafa et al. [11].
The present paper has several interesting features.Firstly, the system depends on two parameters and the nonlinear terms  and  are allowed to have different nonlinear character; that is,  is decreasing on  and  is increasing on ; secondly,  may be singular at  = 0 and  = 0, 1; so far fewer work was done when  can be singular at  = 0; thirdly, the boundary conditions of the system are nonlocal and involve fractional derivatives of the unknown functions.

Preliminaries and Lemmas
In this section, we firstly define an appropriate invariant set and then make a change of variables for the system (3) so that Schauder's fixed point theorem can be applied.Our work is based on fractional framework; for further background knowledge of fractional calculus, we refer readers to the monographs [1][2][3][4] or the papers [6,8,17,18] and the references therein.
Throughout this paper, we mean by [0, 1] the Banach space of all continuous functions on [0, 1] with the usual norm then  is a normal cone in the Banach space .Thus the space [0, 1] can be equipped with a partial order given by Now define a subcone of  as follows: Obviously,  is nonempty since  − 1 −1 ∈ .
Proof.By using semigroup property of the fractional integration operator (see [1] page 73, Lemma 2.3 or [4] Sections 2.3 and 2.5), one has And then, it follows from (8) that In the same way, we also have V().So substituting the above formulas into (3), we obtain (7).
On the other hand, if (, ) is a solution of ( 7), (8) yields So from ( 11), we have Moreover Consequently, (  1 (),   1 ()) is a positive solution of (3).Now we recall some useful lemmas by [18], which are important to the proof of our main results.
It is well known that (, ) ∈ (0, 1) × [0, 1] is a solution of the system (7) if and only if (, ) is a solution of the nonlinear integral system of equations  () =  ∫ Then the existence of solutions to the system of ( 7) is equivalent to the existence of fixed point of the nonlinear operator ; that is, if  * () is a fixed point of  in [0, 1], then system (7) has at least one solution ( * (),  * ()) which can be written by and then system (3) has at least one solution: In order to find the fixed point of , we need the definitions of the upper solution and lower solution for the following integrodifferential equation:   (, )  (,  ()) ) ,  (0) ≤ 0, (  ) . (30)

Main Result
For the convenience in presentation, we now present some assumptions to be used in the rest of the paper.
(A2) For any real numbers ,  > 0, ∫ Proof.We start by showing that (3) has at least one positive solution ( * , V * ).For this purpose, we firstly prove that the operator  is well defined and () ⊂ .
Next we define an operator  in [0, 1] by and consider the following boundary value problem: It is easy to see that the fixed point of  is the solution of (46).
is uniformly bounded.From the uniform continuity of   (, ) and Lebesgue dominated convergence theorem, we get  is equicontinuous.So  is completely continuous.It follows from Schauder's fixed point theorem that  has at least one fixed point  * () such that  * () = ( * )().