Positive Solutions for Three-Point Boundary Value Problem of Fractional Differential Equation with p-Laplacian Operator

We investigate the existence ofmultiple positive solutions for three-point boundary value problemof fractional differential equation with p-Laplacian operator −Dt β (φp(Dt α x))(t) = h(t)f(t, x(t)), t ∈ (0, 1), x(0) = 0,Dt γ x(1) = aDt γ x(ξ),Dt α x(0) = 0, where Dt β ,Dt α ,Dt γ are the standard Riemann-Liouville derivatives with 1 < α ≤ 2, 0 < β ≤ 1, 0 < γ ≤ 1, 0 ≤ α − γ − 1, ξ ∈ (0, 1) and the constant a is a positive number satisfying aξ ≤ 1 − γ; p-Laplacian operator is defined as φp(s) = |s| p−2 s, p > 1. By applying monotone iterative technique, some sufficient conditions for the existence of multiple positive solutions are established; moreover iterative schemes for approximating these solutions are also obtained, which start off a known simple linear function. In the end, an example is worked out to illustrate our main results.

In [15], Li et al. were concerned with the nonlinear differential equation of fractional order D t   () +  (,  ()) = 0,  ∈ (0, 1) , 1 <  ≤ 2 subject to the boundary conditions By using some fixed point theorems, the existence and multiplicity results of positive solutions were established.On the other hand, the differential equations with -Laplacian have also been widely studied owing to the fact that -Laplacian boundary value problems have important application in theory and application of mathematics and physics.For example, in [16], by using the fixed point index, Yang and Yan investigated the existence of positive solution for the third-order Sturm-Liouville boundary value problems with -Laplacian operator: (  (  ())) +  (,  ()) = 0,  ∈ (0, 1) , However, there are few articles dealing with the existence of solutions to boundary value problems for fractional differential equation with -Laplacian operator.In [17], the authors investigated the nonlinear nonlocal problem =  () , where 0 <  ≤ 2,0 <  ≤ 1,0 ≤  ≤ 1,0 <  < 1.By using Krasnoselskii's fixed point theorem and Leggett-Williams theorem, some sufficient conditions for the existence of positive solutions to the above BVP are obtained.
In [18], by using upper and lower solutions method, under suitable monotone conditions, Wang et al. investigated the existence of positive solutions to the following nonlocal problem: =  () , where 1 < ,  ≤ 2, 0 ≤ ,  ≤ 1, 0 < ,  < 1.Recently, Chai [19] investigated the two-point boundary value problem of fractional differential equation with -Laplacian operator: By means of the fixed point theorem on cones, some existence and multiplicity results of positive solutions are obtained.Motivated by the above mentioned works, in this paper, we consider the multiplicity results of positive solutions for the three point boundary value problem of fractional differential equation with -Laplacian operator.Difference to [15][16][17][18][19], by using monotone iterative technique, we not only establish the existence of multiple positive solutions but also obtain the iterative sequences of these positive solutions.

Preliminaries and Lemmas
In this section, we introduce some preliminary facts which are used throughout this paper.
And if  −−2 ≤ 1 − , the Green function (, ) also satisfies where Let N be the set of positive integers, let R be the set of real numbers, and let R + be the set of nonnegative real numbers.Let  = [0, 1].Denote by (, R) the Banach space of all continuous functions from  into R with the norm Define the cone  in (, R + ) as Let  > 1 satisfy the relation (1/) + (1/) = 1, where  is given by (1).
Now, for any  ∈ , define one operator  as follows: Then by ( 20) and ( 23), the BVP ( 1) is equivalent to the fixed point problems of the operators .
Next, supposing  ⊂  is a bounded set, then for any  ∈ , there exists a constant  > 0 such that |||| ≤ .Thus for any  ∈ , we have (, )) which implies () is bounded.On the other hand, according to the Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we easily see  :  →  is completely continuous.In the end, noticing the monotonicity of  on  and the definition of , we also have that the operator  is nondecreasing.
This means that  * is also a positive solution of boundary value problem (1).
In the end, let  * be any fixed point of  in [0, ], then and then By induction, we have Taking the limit, we have This implies that  * and  * are maximal and minimal solutions of the BVP (1).Let  1 = || * ||,  2 = || * ||, then we have The proof is completed.
Remark 11.If ℎ() ≡ 1, then (H2) holds naturally, and in this case we take Thus we have the following Corollary 12.
Remark 14.In Corollary 13, we obtain that the BVP (1) has the maximal and minimal solutions  * and  * only by comparing −1 to .But note that  and  are irrelative, so (62) is easy to be satisfied; this implies that Corollary 13 is very interesting.(71)