SOLVABILITY FOR NONLINEAR SINGULAR FRACTIONAL DIFFERENTIAL SYSTEMS WITH MULTI-ORDERS (cid:3)

In this paper, we consider the existence of positive solutions for a class of nonlinear singular fractional diﬀerential systems with multi-orders. Our analysis relies on (cid:12)xed point theorems on cones. Some suﬃcient conditions for the existence of at least one or two positive solutions for boundary value problem of nonlinear singular fractional diﬀerential systems with multi-orders are established. As an application, an example is presented to illustrate the main results.


Introduction
Fractional calculus has been of great interest recently.It is caused both by the intensive development of the theory of fractional calculus itself and by the applications; see [11].Recently, there have appeared a large number of papers dealing with the existence of solutions of nonlinear fractional differential equations by the use of techniques of nonlinear analysis; see [2,3,13,15,17,18,[20][21][22].
Yu et al. [17] examined the existence of positive solutions for the following problem D α 0 + u(t) + f (t, u(t)) = 0, 0 < t < 1, where 2 < α ≤ 3 is a real number, f ∈ C([0, 1] × [0, +∞), (0, +∞)) and D α 0 + is the Riemann-Liouville fractional differentiation.By using the properties of the Green function, some existence criteria for one or two positive solutions for singular and nonsingular boundary value problems were obtained by means of the Krasnosel'skii fixed point theorem and a mixed monotone method.
Xu et al. [15] considered the existence of positive solutions for the following problem where 3 < α ≤ 4 is a real number, f ∈ C([0, 1] × [0, +∞), (0, +∞)) and D α 0 + is the Riemann-Liouville fractional differentiation.By using the properties of the Green function, some multiple positive solutions for singular and nonsingular boundary value problems were given by means of Leray-Schauder nonlinear alternative, a fixed point theorem on cones and a mixed monotone method.
On the other hand, the study of singular and nonsingular systems involving fractional differential equations is also important as such systems occur in various problems; see [1,4,5,9,10,12,14,16,19].
Bai et al. [1] considered the existence of positive solutions of singular coupled system where 0 < s, p < 1, and f, g : [0, 1) × [0, +∞) → [0, +∞) are two given continuous functions, lim t→0 + f (t, •) = +∞, lim t→0 + g(t, •) = +∞ and D s , D p are two standard Riemann-Liouville fractional derivatives.The existence results of positive solutions were established by a nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed point theorem on a cone.Su [12] discussed a boundary value problem for a coupled differential system of fractional order are given functions and D is the standard Riemann-Liouville fractional derivative.By means of Schauder fixed point theorem, an existence result for the solution was obtained.Zhao et al. [19] examined the existence of positive solutions for a coupled system of nonlinear differential equations of mixed fractional orders where 2 < α ≤ 3, 3 < β ≤ 4, D α 0 + , D β 0 + are the standard Riemann-Liouville fractional derivative, and f, g : [0, 1] × [0, +∞) → [0, +∞) are given continuous functions, f (t, 0) ≡ 0, g(t, 0) ≡ 0. Their analysis relied on fixed point theorems on cones.Some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem were established.
From the above works, we can see a fact, although the coupled systems of fractional boundary value problems have been investigated by some authors, the singular coupled systems due to multi-order fractional orders are seldom considered.On the one hand, the orders α and β of the nonlinear singular fractional differential systems witch are considered in the existing papers belong to the same interval (n, n+1] (n ∈ N + ).On the other hand, in Remark 3.2 ( [19]), conditions f (t, 0) ≡ 0 and g(t, 0) ≡ 0 are too strong for the boundary value problem (1.1).Therefore, we will give some new existence criteria for the boundary value problem (1.1) without conditions f (t, 0) ≡ 0 and g(t, 0) ≡ 0 in this paper.Our results in this paper improve some known results in [19].
Motivated by all the works above, in this paper we investigate the existence of positive solutions for the boundary value problem (1.1) of the nonlinear singular fractional systems with multi-orders, where f, g : (0, 1] × [0, +∞) → [0, +∞) are continuous, and lim t→0 + f (t, •) = +∞, lim t→0 + g(t, •) = +∞ (that is, f and g are singular at t = 0).Our analysis relies on fixed point theorems on cones.Some sufficient conditions for the existence of at least one or two positive solutions for boundary value problem of the nonlinear singular fractional systems with multiorders are established.Finally, we present an example to demonstrate our results.
The plan of the paper is as follows.In Section 2, we shall give some definitions and lemmas to prove our main results.In Section 3, using Leray-Schauder nonlinear alternative theorem and Guo-Krasnosel'skii fixed point theorem, we obtain some new existence criteria for boundary value problem (1.1) of nonlinear singular fractional systems with multi-orders.Section 4 gives an illustrative example to support our new results, which is followed by a brief conclusion in Section 5.

Preliminaries
For the convenience of readers, we give some background materials from fractional calculus theory to facilitate analysis of the problem (1.1).These materials can be found in the recent literatures; see [6-8, 15, 17].
From the definition of the Riemann-Liouville derivative, we can obtain the following statements.
In the following, we present the Green's function for the boundary value problem of fractional differential equations. ) where here G 1 (t, s) is called the Green's function of the boundary value problem (2.1) and (2.2).

Lemma 2.6 ( [15]
).The function G 2 (t, s) defined by (2.6) satisfies the following conditions: The following two lemmas are fundamental in the proofs of our main results.

Lemma 2.7 ( [7]
).Let E be a Banach space, and let P ⊂ E be a cone in E. Assume and let S : P → P be a completely continuous operator such that, either Then S has a fixed point in P ∩ (Ω 2 \Ω 1 ).

Main Results
In this section, we establish some new existence results for the boundary value problem (1.1) of the nonlinear singular fractional systems with multi-orders.Consider the following coupled system of integral equations: Proof.By the continuity of t σ1 F 1 (t) and It is easily to check that H 1 (0) = 0.The proof is divided into three cases: Case 1: where B(•, •) denotes the beta function.
Case 2: The proof is similar to that of Case 2, so we omit it.
The proof is completed.
Proof.This proof is similar to that of Lemma 3.1, so is omitted.
Proof.For each (u, v) ∈ P , we have that u, v ∈ P 1 = {y ∈ X : )ds.By Lemma 3.1 and the fact that f, G 1 (t, s) are nonnegative, we have T 1 : P 1 → P 1 .
By the arbitrariness of v 0 , T 1 : P 1 → P 1 is continuous.Similarly, by Lemma 3.2, we have Then, T 2 : P 2 → P 2 is continuous.That is, we get the the operator T : P → P is continuous.
Let M ⊂ P be bounded, i.e., there exists a positive constant b such that For each (u, v) ∈ M , then we have ) . Thus, .
Similarly, we have So, Hence, T (M ) is bounded.
In the following, we proof that T is equicontinuous.In fact, ∀ ϵ > 0, let where ).
In order to estimate t α−σ1 , we can apply a method used in [2].In the following, we divide the proof into three cases.
Case 1: Thus, we obtain Similarly, we get We can prove that if t 1 , t 2 ∈ [0, 1] are such that 0 < t 2 − t 1 < δ, then we have Hence, for the Euclidean distance d on R 2 , we have that if Therefore, T (M ) is equicontinuous.By means of the Arzela-Ascoli theorem, T (M ) is compact.Thus, the operator T : P → P is completely continuous.This completes the proof. where q 1 (t), q 2 (t), such that Then the boundary value problem (1.1) has at least one positive solution.
Proof.From Lemma 3.4, we have T : P → P is completely continuous.By assumptions of the theorem, we have We divide the proof into the following two steps. Step1: q 1 (t) and q 2 (t) Step 2: and Therefore, by Lemma 2.7 and 3.3, we complete the proof.
Then the boundary value problem (1.1) has one positive solution.

Example
In this section, as an application, an example is given to illustrate the main results.

Conclusion
In this paper, we have considered existence of positive solutions for a class of the boundary value problem of the nonlinear singular fractional differential systems with multi-orders.Some sufficient conditions for the existence of positive solutions for the boundary value problem of the nonlinear singular differential systems with multiorders have been established by Leray-Schauder nonlinear alternative theorem and Guo-Krasnosel'skii fixed point theorem.The main results have been well illustrated with the help of examples.

Lemma 2 . 8 (
[6]).Let E be a Banach space with C ⊂ E close and convex.Assume U is a relatively open subset of C with 0 ∈ U and A : U → C is a continuous compact map.Then either (E1) A has a fixed point in U ; or (E2) there exists a u ∈ ∂U , and a λ ∈ (0, 1) with u = λAu.