Existence results for a class of nonlinear singular p -Laplacian Hadamard fractional di ff erential equations

: Based on properties of Green’s function and the some conditions of f ( t , u ), we found a minimal and a maximal positive solution by the method of sequence approximation. Moreover, based on the properties of Green’s function and fixed point index theorem, the existence of multiple positive solutions for a singular p -Laplacian fractional di ff erential equation with infinite-point boundary conditions was obtained and, at last, an example was given to demonstrate the validity of our main results.


Introduction
Fractional-order differential operator has become one of the most important tools for mathematical modeling of complex mechanics and physical processes because it can describe mechanical and physical processes with historical memory and spatial global correlation succinctly and accurately.Additionally, and the fractional-order derivative modeling is simple, the physical meaning of parameters is clear and the description is accurate.In recent years, fractional derivative has become an important tool to describe all kinds of complex mechanical and physical behaviors, so the study of a positive solutions of fractional differential equations has attracted much attention.For an extensive collection of such literature, readers can refer to [1,2,[2][3][4][5][6][7][8][9][10][11][12].Alsaedi et al. [13] investigated the following equation where φ p (t) = |t| p−2 • t, 1 p + 1 q = 1, p, q > 1, 1 < α, β ≤ 2, ρ > 0, ζ > 0, 0 < µ, η < 1, 0 ≤ λ 1 < 1 µ ρ(β−1) , 0 ≤ λ 2 < 1 (1−η ρ ) α−1 , and ρ D α 0 + u and ρ D β 0 + u denote the right and left fractional derivatives of orders α and β with respect to a power function, respectively.The authors proved the uniqueness of positive solutions for the given problem for the cases 1 < p ≤ 2 and p > 2 by applying an efficient novel approach together with the Banach contraction mapping principle.Li and Liu [8] considered the fractional differential equation C 0 D α g u(t) + f (t, u) = 0, 0 < t < 1 with boundary value condition ), and the existence of multiple solutions for the following system by the fixed point theorem are on cone.
In recent years, more and more scientists have devoted themselves to the study of Hadamard fractional differential systems.For the part of outstanding results of Hadamard's research on fractional differential systems, please refer to [14][15][16].Ardjouni [14] studied the following Hadamard fractional differential equations are given continuous functions, ϕ is not required for any monotone assumption, and φ is nondecreasing on x.The authors get the existence and uniqueness of the positive solution by the method of upper and lower solutions and Schauder and Banach fixed point theorems.In [15], Berhail and Tabouche studied the following fractional differential equation
where 3 < α ≤ 4, g 1 , g 2 ∈ C([1, T ], [0, +∞)), a, b > 0, and H D α 1 + denotes the Hadamard factional order of α.Based on the properties of Green's function, the authors get the existence of a positive solution for the equation in [15] by the Avery-Peterson fixed point theorem.In [16], the authors studied the existence of a positive solution and stability analysis of the following equation with integral boundary conditions where α, β, and µ are three positive real numbers with α ∈ (2, 3], β ∈ (1, 2], and is the Hadamard fractional differential equation.The authors get the positive solutions by using the fixed point methods.
Motivated by the excellent results above, in this paper, we will devote to considering the following infinite-point singular p-Laplacian Hadamard fractional differential equation: with infinite-point boundary condition ) are the standard Hadamard fractional-order derivatives.
In this paper, we investigate the existence of positive solutions for a singular infinite-point p-Laplacian boundary value problem.Compared with [15], the equation in this paper is a p-Laplacian fractional differential equation and the method in which we used is a fixed point index and sequence approximation.Compared with [16,17], value at infinite points are involved in the boundary conditions of the boundary value problem (1.1,1.2) and the minimal positive solution and maximal positive solution are obtained in this paper.

Preliminaries and lemmas
For some basic definitions and lemmas about the theory of Hadamard fractional calculus, the reader can refer to the recent literature such as [6,9,18].Definition 2.1( [18,19]).The Hadamard fractional integral of α(α > 0) order of a function ℏ : (0, ∞) → R 1 + is given by 18,19]).The Hadamard fractional derivative of α(α > 0) order of a continuous function ℏ : (0, ∞) → R 1 + is given by where n = [α] + 1 and [α] denotes the integer part of the number α, provided that the righthand side is pointwise defined on (0, ∞).Lemma 2.1( [18,19]).If α, β > 0, then , then the equation of the BVPs with boundary condition u where Proof.By means of the Lemma 2.3, we reduce (2.1) to an equivalent integral equation From u(0) = 0, we have C n = 0, then taking the first derivative, we have By u ′ (1) = 0, we have C n−1 = 0, and taking the derivative step by step and combining . Consequently, we get By some properties of the fractional integrals and fractional derivatives, we have On the other hand, ), and combining with (2.5), we get where Ξ(s), ∆ are as (2.3), then, Therefore, Ψ(t, s) is as (2.3).□ Lemma 2.5.The Green functions (2.3) have the following properties: Proof.(i) By simple calculation, we have Ξ ′ (s) > 0. For s ∈ [1, e], we have Ξ(s) ≥ Ξ(1), then by the expression of Ξ(s) and ∆, we have For 1 < s ≤ t < e by the preceding formula, we have For 1 < t ≤ s < e, obviously, Ψ(t, s) > 0. Furthermore, by direct calculation, we get . By the similar method, for 1 < s ≤ t < e, we get For 1 ≤ t ≤ s ≤ e, we have where in which then the Eqs (1.1), (1.2) can be changed into the following equation then by the similar method with Lemma 2. Proof.The proof is similar to Lemma 2.5 of this paper and we omit it here.

Main results
Now, we define the Banach space E = C([1, e], R), which is assigned a maximum norm, that is, ∥u∥ = sup 1≤t≤e |u(t)|.Let P = {u ∈ E|u(t) ≥ 0}, then P is a cone in E. Define an operator T : P → P by then equation (1.1, 1.2) has a solution if, and only if, the operator T has a fixed point.Proof.From the continuity and nonnegativeness of Ψ(t, s) and f (t, u(t)), we know that T : P → P is continuous.Let Ω ⊂ P be bounded, then for all t ∈ [1, e] and u ∈ Ω, there exists a positive constant M such that | f (t, u(t))| ≤ M. Thus, where which implies that T (Ω) is equicontinuous.By the Arzela-Ascoli theorem, we have that T : P → P is completely continuous.The proof is complete.
) is nondecreasing in u, and λ ∈ (0, +∞), then BVP(1.1,1.2) has a minimal positive solution v in B r and a maximal positive solution ϱ in B r .Moreover, v m (t) → v(t),ϱ m (t) → ϱ(t) as m → ∞ uniformly on [1, e], where and Ψ(e, s) ds s For u ∈ B r , there exists a positive constant This, together with f (t, u) being nondecreasing in u, yields that Since T is compact, we have that {v m } is a sequentially compact set.Hence, there exists Taking limits as m → ∞ in (3.5), we have v(t) ≤ u(t) for t ∈ [1, e].Thus, v is a minimal positive solution, then, we show BVP(1.1,1.2)has a positive solution in B r , which is a maximal positive solution.
, and the following conditions hold: 2)has at least two positive solutions u 1 and u 2 such that where l 6 B 1 > l 3 L and l 6 B 1 > l 1 L
r → B r By Lemma 3.1, we get that T : B r → B r is completely continuous.Therefore, by the Schauder fixed point theorem, the operator T has at least one fixed point, and so BVP(1.1,1.2) has at least one solution in B r .Next, we show that BVP(1.1,1.2) has a positive solution in B r , which is a minimal positive solution.From (3.1) and (3.2), we have that v m (t) = (T v m−1 )(t), t ∈ [1, e], m = 1, 2, 3, . . .(3.4)