Multiple positive solutions for fractional differential systems

In this paper, we study the existence of positive solution to boundary value problem for fractional differential system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{ll}D_{0^+}^\alpha u (t) + a_1 (t) f_1 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1),\\D_{0^+}^\alpha v (t) + a_2 (t) f_2 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1), \;\; 2 < \alpha < 3,\\u (0)= u' (0) = 0, \;\;\;\; u' (1) - \mu_1 u' (\eta_1) = 0,\\v (0)= v' (0) = 0, \;\;\;\; v' (1) - \mu_2 v' (\eta_2) = 0,\end{array}\right.$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D_{0^+}^\alpha}$$\end{document} is the Riemann-Liouville fractional derivative of order α. By using the Leggett-Williams fixed point theorem in a cone, the existence of three positive solutions for nonlinear singular boundary value problems is obtained.

Fractional differential equations have been of great interest recently. This is because of both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various scientific fields such as physics, mechanics, chemistry, engineering, etc. For details, see [1][2][3] and the references therein.
The existence of solutions of initial value problems for fractional order differential equations have been studied in the literature [4][5][6][7][8] and the references therein. Saadi and Benbachir [9] considered the following boundary value problem where η ∈ (0, 1), μ ∈ 0, 1 η α−2 are two arbitrary constants. They applied the Guo-Krasnosel'skii fixed point theorem and Schauder's fixed point theorem to establish some results on the existence, nonexistence and uniqueness of positive solutions (2).
Motivated by the work mentioned above, our purpose in this paper is to show the existence and multiplicity of positive solutions to the problem (1) by using the Leggett-Williams fixed point theorem.
The rest of the article is organized as follows: in Sect. 2, we present some preliminaries that will be used in Sect. 3. The main result and proof will be given in Sect. 3. Finally, in Sect. 4, an example is given to demonstrate the application of our main result.

Preliminaries
In this section, we present some notations and preliminary lemmas that will be used in the proofs of the main results. Definition 2.1 Let X be a real Banach space. A non-empty closed set P ⊂ X is called a cone of X if it satisfies the following conditions: (1) x ∈ P, μ ≥ 0 implies μx ∈ P, (2) x ∈ P, −x ∈ P implies x = 0.

Definition 2.2
The Riemann-Liouville fractional integral operator of order α > 0, of function f ∈ L 1 (R + ) is defined as where (·) is the Euler gamma function.

Definition 2.3
The Riemann-Liouville fractional derivative of order α > 0, n − 1 < α < n, n ∈ N is defined as where the function f (t) have absolutely continuous derivatives up to order (n − 1).
has a unique solution where Proof The proof is similar to that of Lemma 5 in [9], so we omit it here.
Now, we consider the system (1). Obviously, is a solution of the following nonlinear integral system: To establish the existence three positive solutions of system (1), we will employ the following Leggett-Williams fixed point theorem.
For the convenience of the reader, we present here the Leggett-Williams fixed point theorem [11]. Given a cone K in a real Banach space E, a map α is said to be a nonnegative continuous concave (resp. convex) functional on K provided that α : K → [0. + ∞) is continuous and for all x, y ∈ K and t ∈ [0, 1]. Let 0 < a < b be given and let α be a nonnegative continuous concave functional on K . Define the convex sets P r and P(α, a, b) by and P (α, a, b) Theorem 1 (Leggett-Williams fixed point theorem). Let A : P c → P c be a completely continuous operator and let α be a nonnegative continuous concave functional Then A has at least three fixed points x 1 , x 2 , and x 3 and such that x 1 < a, b < α(x 2 ) and x 3 > a, with α(x 3 ) < b.

Main result
For convenience, we introduce the following notations. Let Then, choose a cone K ⊂ E, by It is obvious that K is a cone. Define an operator T by where

Lemma 6 The operator defined in (9) is completely continuous and T : K → K .
Proof For any (u, v) ∈ K , then from properties of G(t, s), G 11 (t, s) and G 12 (t, s), 1], and it follows from (10) that s)a 1 (s) f 1 (s, u(s), v(s))ds Thus, for any (u, v) ∈ K , it follows from Lemma 5 and (11) that min τ ≤t≤1 In the same way, for any (u, v) ∈ K , we have From the above, we conclude that This completes the proof.
It is clear that the existence of a positive solution for the system (1) is equivalent to the existence of a nontrivial fixed point of T in K . Finally, we define the nonnegative continuous concave functional on K by .
Throughout this section, we assume that p i , i = 1, 2, are two positive numbers satisfying 1 p 1 + 1 p 2 ≤ 1. To state our main result, we will assume that the following conditions are satisfied: (H1) a i (t) do not vanish identically on any subinterval of (0, 1), and there exists t 0 ∈ (0, 1) such that a i (t 0 ) > 0 and 0 Now, we can state our main result.

Theorem 2
Assume that (H1) holds. In addition, assume there exist nonnegative numbers a, b, c such that 0 < a < b ≤ min{τ, m 1 p 1 M 1 , m 2 p 2 M 2 }c, and f i (t, u, v) satisfy the following conditions:

Then, the system (1) has at least three positive solutions
Proof First, we show that T : P c → P c is a completely continuous operator. If (u, v) ∈ P c , then by condition (H2), we have Therefore, T (u, y) ≤ c, that is, T : P c → P c . The operator T is completely continuous by an application of the Ascoli-Arzela theorem.