Existence and Uniqueness of Solutions for a Singular System of Higher-order Nonlinear Fractional Differential Equations with Integral Boundary Conditions *

In this paper, we study the existence and uniqueness of solutions for a singular system of nonlinear fractional differential equations with integral boundary conditions. We obtain existence and uniqueness results of solutions by using the properties of the Green's function, a nonlinear alternative of Leray–Schauder-type, Guo–Krasnoselskii's fixed point theorem in a cone and the Banach fixed point theorem. Some examples are included to show the applicability of our results.


Introduction
Fractional differential equations have been of great interest recently.It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various fields of sciences and engineering such as control, porous media, electromagnetic, and other fields.There are many papers deal with the existence and multiplicity of solution of nonlinear fractional differential equations (see [1][2][3][4][5][6][7][8] and the references therein).
Inspired by the work of the above papers and many known results, in this paper, we study the existence of positive solutions of BVP (1).The existence and uniqueness results of solutions are obtained by a nonlinear alternative of Leray-Schauder-type, Guo-Krasnoselskii's fixed point theorem in a cone and the Banach fixed point theorem.We consider the singular system of nonlinear fractional differential equations with integral boundary conditions where n − 1 < α, β n, n 3, 0 < η, c 1, i, j ∈ N , 0 i, j n − 2 and i, j are fixed constants, f, g : (0, 1] × [0, +∞) → [0, +∞) are two given continuous functions and singular at t = 0 (that is, lim t→0+ f (t, •) = +∞, lim t→0+ g(t, •) = +∞), and D α 0+ , D β 0+ are the standard fractional Riemann-Liouville's derivatives.
The paper is organized as follows.Firstly, we present some necessary definition and preliminaries, and derive the corresponding Green's function known as fractional Green's function and argue its properties.Secondly, the existence results of positive solutions are obtained by a nonlinear alternative of Leray-Schauder-type, Guo-Krasnoselskii's fixed point theorem in a cone and the Banach fixed point theorem.Finally, we construct some examples to demonstrate the application of our main results.

Background materials and Green's function
For the convenience of the reader, we present here the necessary definitions, lemmas and theorems from fractional calculus theory to facilitate analysis of BVP (1).These definitions, lemmas and theorems can be found in the recent literature, see [1][2][3][4][5][6][7][8].
Definition 1.The Riemann-Liouville fractional integral of order α > 0 of a function y : (0, ∞) → R is given by provided the right-hand side is pointwise defined on (0, ∞).
Remark 1. (See [6].)The following properties are useful for our discussion: In the following, we present Green's function of the fractional differential equation boundary value problem.
Proof.We may apply Lemma 2 to reduce (2) to an equivalent integral equation for some C 1 , C 2 , . . ., C n ∈ R. Consequently, the general solution of (2) is On the other hand, u (i) (1) = λ η 0 u(s) ds combining with Nonlinear Anal.Model.Control, 2013, Vol. 18, No. 4, 493-518 Therefore, the unique solution of the problem (2) is For t η, one has www.mii.lt/NA For t η, one has The proof is complete.
Similarly to the proof of Case 2, so we omit it.The proof is complete.
From Lemma 3 we can write the system of BVPs ( 1) as an equivalent system of integral equations which can be proved in the same way as Lemma 3.3 in [6].For convenience, the proof is omitted.
Proof.For any (u, v) ∈ P , we have that we get that A 1 : P 1 → P 1 by Lemma 7 and the nonnegativity of f .Set v 0 ∈ P 1 and v 0 = c 0 .If v ∈ P 1 and v − v 0 < 1, then v < 1 + c 0 := c.By the continuity of t σ1 f (t, y), we get that t σ1 f (t, y) is uniformly continuous on [0, 1] × [0, c], namely, for all ε > 0, exists δ > 0 (δ < 1), when . Hence, we have It follows from ( 4), we can get Owing to the arbitrariness of v 0 , we know that A 1 : P 1 → P 1 is continuous.Similarly, we can get that A 2 : P 2 → P 2 is continuous.So, we proved A : P → P is continuous.Let M ⊂ P be bounded.That is to say there exists a constant l > 0 such that (u, v) l for all (u, v) ∈ M .Since t σ1 f (t, y), t σ2 g(t, y) are continuous on .
Hence, we have .
Similarly, we have Thus, Therefore, A(M ) is bounded.Next, we prove that A is equicontinuous.Let, for all ε > 0, .
Then BVP (1) has at least one positive solution.
Then the BVP (1) has a positive solution.
we have that 0 u(t), v(t) ρ.By condition (ii) and Lemma 4, we get Then we obtain Similarly, Therefore, A(u, v) ρ = (u, v) .Besides, by Lemma 8, operator A : P → P is completely continuous.Then with Lemma 5, our proof is complete.
Again by (iv) we know (u, v) = r which contradicts that (u, v) ∈ ∂U .Then based on Lemma 6, there is a fixed point (u, v) ∈ U .Therefore the BVP (1) has a positive solution.