Positive Solutions for a Class of p -Laplacian Hadamard Fractional-Order Three-Point Boundary Value Problems

: In this paper, using the Avery–Henderson ﬁxed point theorem and the monotone iterative technique, we investigate the existence of positive solutions for a class of p -Laplacian Hadamard fractional-order three-point boundary value problems. J.X. D.O..; methodology, J.J., J.X. D.O.; software, J.J.; validation, J.J., J.X. and D.O.; formal analysis, J.J., J.X. and D.O.; investigation, J.J., J.X. and D.O.; resources, J.X.; data curation, J.X.; writing draft preparation, J.J., J.X. and D.O.; writing and editing, J.J., J.X. and

On the other hand, p-Laplacian equations are extensively used in physics, mechanics, dynamical systems, etc (see [15][16][17][18][20][21][22][23]31]). For example, Leibenson [31] introduced p-Laplacian differential equations to study a mechanics problem involving turbulent flow in a porous medium. Recently, G. Wang et al. used the tools of Hadamard type fractional-order equations to study turbulent flow models, see [20,21]. In [20], they studied the uniqueness, the existence and nonexistence of solutions for the following Hadamard type fractional differential equation with the p-Laplacian operator where f ∈ C([1, e] × R 2 , R) and θ ∈ C [1, e]. In [21], they also studied the unique positive solution for a Caputo-Hadamard-type fractional turbulent flow model where CH D is Caputo-Hadamard fractional derivative, I γ,δ η is the generalized Erdelyi-Kober fractional integral operator: In this paper, we study positive solutions for the p-Laplacian Hadamard fractional-order differential Equation (1). Using the monotone iterative technique we show that (1) has two positive solutions, and we establish iterative formulas for the two solutions. In addition from the Avery-Henderson fixed point theorem, we also obtain that (1) has two positive solutions under some appropriate conditions on the nonlinearity f . It is interesting to note that the methods used in this paper can be applied to very general integral equations (and therefore very general differential equations) if the kernel has a suitable behavior as described in Section 2. The behavior of the Greens' function of a problem will indicate whether the theory presented in this paper can be used efficiently.

Preliminaries
In this section, we provide the definition of the Hadamard fractional derivative; for other related detail materials see the book [5]. Definition 1. The Hadamard derivative of fractional order q for a function g : [1, ∞) → R is defined as where n = [q] + 1, [q] denotes the integer part of the real number q and log(·) = log e (·).

Proof. From the first equation in
Substituting this c 1 , we obtain This completes the proof.
Note that Then we have and together with the boundary conditions y(1) = y(e) = δy(1) = δy(e) = 0, we have the following result. (3) is equivalent to the following Hammerstein-type integral equation

Lemma 2. The problem
Proof. For convenience, we put Then by a similar argument as in Lemma 1, we have As a result, we have (1 − log s) β−2 φ(s) ds s = 0.
Solving this system, we obtain Hence, we have Consequently, we find This completes the proof.

Lemma 3.
The functions H 0 and G have the following properties: Lemma 3 (ii) and (iii) are direct results from Lemma 3 in [14]. Moreover, by Lemma 3 (i) we have Let E := C[1, e], y := max t∈[1,e] |y(t)| and P := {y ∈ E : y(t) ≥ 0, ∀t ∈ [1, e]}. Then (E, · ) is a real Banach space and P a cone on E. From Lemma 2, we define an operator A : E → E as follows: Note that the continuity of the functions G, H, f , guarantees that the operator A is a completely continuous operator. Moreover if there is a y ∈ E such that Ay = y, then from Lemma 2 we have that y is a solution for (1). Therefore, in what follows we study the existence of fixed points of the operator A.

Lemma 4.
(see [32]). Let E be a partially ordered Banach space, and x 0 , y 0 ∈ E with x 0 ≤ y 0 , D = [x 0 , y 0 ]. Suppose that A : D → E satisfies the following conditions: (i) A is an increasing operator; (ii) x 0 ≤ Ax 0 , y 0 ≥ Ay 0 , i.e., x 0 and y 0 is a subsolution and a supersolution of A; (iii) A is a completely continuous operator.
Then A has the smallest fixed point x * and the largest fixed point y * in [x 0 , y 0 ], respectively. Moreover, x * = lim n→∞ A n x 0 and y * = lim n→∞ A n y 0 .

Main Results
In this section we let and Now, we state our main results and give their proofs.

Theorem 1.
Suppose that (H1)-(H2) and the following conditions hold: (H5) f (t, y) is an increasing function in the second variable y, i.e., f (t, Then (1) has two positive solutions. Moreover, there exist two iterative sequences uniformly converging to the two solutions.
Proof. For y ∈ P, from Lemm 3 (ii) we have In particular, we have This completes the proof.

Conclusions
In this paper, we first used the monotone iterative technique to show that (1) has two positive solutions, and we established iterative formulas for the two solutions when the nonlinearity f grows (p − 1)-sublinearly. Next, using the Avery-Henderson fixed point theorem, we showed that (1) has two positive solutions under some appropriate conditions on the nonlinearity f .