Existence of positive solutions for nonlinear four-point Caputo fractional differential equation with p-Laplacian

In this article, the existence of positive solutions is considered for nonlinear four-point Caputo fractional differential equation with p-Laplacian operator. We use the monotone iterative technique to acquire the existence of positive solutions for the boundary value problem and get iterative schemes for approximating the solutions. An example is also presented to illustrate the effectiveness of the main results.


Introduction
The differential equation arises in the modeling of different physical and natural phenomena, control system, nonlinear flow laws and many other branches of engineering. Fractional calculus is the extension of integer order calculus to arbitrary order calculus. With the development of fractional calculus, fractional differential equations have wide applications. In these years, there are many papers concerning integer order differential equations with p-Laplacian [-] and fractional differential equations with p-Laplacian [-].
In [], Liu et al. discussed the four-point problem for a class of fractional differential equation with mixed fractional derivatives and with p-Laplacian operator Based on the method of upper and lower solutions, they study the existence of positive solutions of the above boundary problem. Motivated by the aforementioned work, this work discusses the existence of positive solutions for the following fractional differential equation: )] + f (t, u(t)) = , t ∈ (, ), [φ p (D α  + u())] (i) = , i = , , . . . , m -, u() -aD α  + u(ξ ) = , (D α  + u()) (j) = , j = , , . . . , n -,  + are the Caputo fractional derivatives. We use the monotone iterative technique to obtain the existence of positive solutions for the boundary value problem and get iterative schemes for approximating the solutions. A function u(t) is a positive solution of the boundary value problem (.) if and only if u(t) satisfies the boundary value problem (.), and u(t) ≥  for t ∈ [, ]. We will always suppose the following conditions are satisfied:

Preliminaries
To show the main result of this work, we give the following some basic definitions, which can be found in [, ].
Definition . The fractional integral of order α >  of a function y : (, +∞) → R is given by provided that the right side is pointwise defined on (, +∞), where Definition . For a continuous function y : (, +∞) → R, the Caputo derivative of fractional order α >  is defined as where n = [α] + , provided that the right side is pointwise defined on (, +∞).

Main result
Lemma . The boundary value problem (.) is equivalent to the following equation: where σ is the unique solution of the equation In view of [φ p (D α  + u())] (i) = , i = , , . . . , m -, we obtain c  = c  = · · · = c m- = , that is, For t ∈ [, ], integrating from  to t, we get In view of (D α  + u()) (j) = , j = , , . . . , n -, we obtain By (.) and (.), we get obviously, F(t) is continuous and nondecreasing for t ∈ [, ]. From a direct calculation, we get so we see that the equation F(t) =  has a unique solution for t ∈ (, ). Let σ be the unique solution of the equation F(t) = , then by (.) and (.), we obtain Consequently, we get The proof is complete.
Here σ is defined by (.). Obviously, u(t) is a solution of problem (.) if and only if u(t) is a fixed point of T.
The following theorem is the main result in this paper.
Theorem . Assume there exists r >  such that: Then the problem (.) has two positive solutions w * and v * such that Proof We take four steps to prove the theorem.
If  < σ < ξ , then In view of (C), we get a contradiction. So we can obtain σ > ξ . If  > σ > η, then In view of (C), we get a contradiction. So we can obtain σ < η. Therefore we have ξ < σ < η.
Step . We prove that T : P → P is completely continuous. For any u ∈ P, if  ≤ t ≤ σ , If σ ≤ t ≤ , So we get T : P → P. Obviously, T is continuous for the continuity of f (t, u).
Let ⊂ P be bounded, that is, there exists a positive constant l for any u ∈ , and letting M  = max ≤t≤,≤u≤l f (t, u) + , then, for any u ∈ , if  ≤ t ≤ σ , Hence, T( ) is uniformly bounded. Now, we will prove that T( ) is equicontinuous.
therefore, T( ) is equicontinuous. Applying the Arzelá-Ascoli theorem, we conclude that T is a completely continuous operator.
Step . Let P r = {u ∈ P :  ≤ u ≤ r}, then we prove T : P r → P r . For any u ∈ P r , if  ≤ t ≤ σ , Consequently, we get T : P r → P r .
Step . We prove w * and v * are two positive solutions of the problem (.). Since w  (t) = max{ rL  L  + , rL  L  + }, obviously, w  (t) ∈ P r . Since w k+ (t) = Tw k (t), k = , , , . . . and T : P r → P r , we get w k (t) ∈ P r .
If  ≤ t ≤ σ , So we get w  (t) ≤ w  (t), and on the basis of the definition of T, we obtain w  (t) = Tw  (t) ≤ Tw  (t) = w  (t).
Thus, we get w * ∈ P r such that w k → w * . Applying the continuity of T and w k+ (t) = Tw k (t), we get w * (t) = Tw * (t), hence w * (t) is a positive solution of problem (.).
If  ≤ t ≤ σ , So we get v  (t) ≥ v  (t), and on the basis of the definition of T, we obtain v  (t) = Tv  (t) ≥ Tv  (t) = v  (t).
Thus, we get v * ∈ P r such that v k → v * . Applying the continuity of T and v k+ (t) = Tv k (t), we get v * (t) = Tv * (t), hence v * (t) is a positive solution of problem (.).