Existence–uniqueness and monotone iteration of positive solutions to nonlinear tempered fractional differential equation with p-Laplacian operator

In this paper, without requiring the complete continuity of integral operators and the existence of upper–lower solutions, by means of the sum-type mixed monotone operator fixed point theorem based on the cone Ph\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P_{h}$\end{document}, we investigate a kind of p-Laplacian differential equation Riemann–Stieltjes integral boundary value problem involving a tempered fractional derivative. Not only the existence and uniqueness of positive solutions are obtained, but also we can construct successively sequences for approximating the unique positive solution. As an application of our fundamental aims, we offer a realistic example to illustrate the effectiveness and practicability of the main results.


1.4)
A is a function of a bounded variation, Riemann-Stieltjes integral with respect to A. By using the sum-type mixed monotone fixed theorem based on the cone P h , we show the existence and uniqueness of positive solutions for the p-Laplacian differential system (1.1).
In recent years, many theories and experiments have shown that a large number of abnormal phenomena that occurs in the applied science and engineering can be well described by fractional calculus. Especially, fractional differential equations have been proved to be powerful tools in the modeling of various phenomena in various fields of science and engineering, for example fluid mechanics, physics and heat conduction; see for instance [1][2][3][4][5][6]. Meanwhile, it is well known that the p-Laplacian operator is also used in analyzing biology, physics, mechanics and the related fields of mathematical modeling; see [7][8][9][10][11][12][13][14]. In [7], for studying the turbulent flow in porous media, Leibenson introduced the p-Laplacian differential equation as follows: where ϕ p (s) = |s| p-2 s, p > 1. Motivated by Leibenson's work, Guo et al. [8] studied the existence of a solution for an ordinary differential equation m-point boundary value problem with p-Laplacian operator. Lu et al. [9] investigated a fractional differential equation for a two points boundary value problem involving the p-Laplacian operator as follows: where 2 < α ≤ 3, 1 < β ≤ 2 and ϕ p (s) = |s| p-2 s. D α 0+ , D β 0+ are standard Riemann-Liouville fractional derivatives. By employing the Guo-Krasnosel'skii fixed-point theorem and upper-lower solutions method, the existence of positive solutions was obtained.
In [10], Ren, Li and Zhang studied the existence of maximum and minimum solutions for the following nonlocal p-Laplacian fractional differential system: denotes a Riemann-Stieltjes integral and A i is a function of bounded variation, ϕ p i is a p-Laplacian operator. By using the monotone iterative technique, some new results as regards the existence of maximal and minimal solutions were established, and the estimation of the lower and upper bounds of the maximum and minimum solutions was also derived.
Recently, in [15], we investigated the conformable differential equation with p-Laplacian operator as follows: where n -1 ≤ α < n and T 0 + α is a new fractional derivative called "the conformable fractional derivative". By using the Guo-Krasnosel'skii fixed point theorem, some new existence conclusions of positive solutions were obtained to the boundary value problem (1.8).
In [16], we continued to investigate the existence of multiple positive solutions for high order Riemann-Liouville fractional differential equation involving the p-Laplacian operator as follows: where n -1 < α ≤ n, R 0 D α t is the standard Riemann-Liouville fractional derivative, ϕ p is the p-Laplacian operator. By means of the Leggett-Williams fixed point theorem and a functional-type cone expansion-compression fixed point theorem, not only the existence of two positive solutions was obtained, but also some sufficient conditions for the existence of at least three positive solutions was established.
In addition, Zhang et al. [17] investigated the eigenvalue problem for a kind of singular fractional differential equation Riemann-Stieltjes integral boundary value problem involving the p-Laplacian operator as follows: (1.10) where D β t and D α t are standard Riemann-Liouville fractional derivatives with 0 < β ≤ 1, 1 < α ≤ 2, 1 0 x(s) dA(s) is the standard Riemann-Stieltjes integral and A is a function of the bounded variation. By using the Schauder fixed point theorem and upper and lower solution methods, some new theorems on existence were obtained. Inspired by the above work, in this paper, we investigate the existence and uniqueness of positive solutions for a p-Laplacian differential equation Riemann-Stieltjes integral boundary value problem involving a tempered fractional derivative (1.1). To the best of our knowledge, this kind of integral boundary value problem involving a tempered fractional derivative has seldom been researched up to now. Compared with other references, the present article has the following characteristics. Firstly, the tempered fractional deriva- Secondly, the Riemann-Stieltjes integral boundary conditions involving a tempered fractional derivative are more general cases, which cover the common integral boundary conditions as special cases. Thirdly, compared with the p-Laplacian differential system (1.8) and (1.9), in this paper, the integral operator need not be completely continuous or compact. Fourthly, in this paper, by employing the sum-type mixed monotone operators fixed points theorem, our conclusions cannot only guarantee the existence of a unique positive solution, but also construct successively sequences for approximating the unique positive solution. Finally, it is worth mentioning that some important properties of two different kernel functions rely on the parameter λ.
The rest of this paper is organized as follows. In Sect. 2, we briefly introduce some necessary basic definitions and preliminary results which will be used to prove our main results. In Sect. 3, we study the existence and uniqueness and monotone iteration of a positive solution to the p-Laplacian differential system (1.1) by means of sum-type mixed monotone fixed points theorems based on the cone P h . At last, in Sect. 4, we demonstrate the effectiveness and feasibility of the main results by an example.

Preliminaries
In the section, we first list some basic notations, concepts in ordered Banach spaces. For convenience, we refer the reader to [18,19] for details.
Suppose that (E, · ) is a real Banach space which is partially ordered by a cone P ⊂ E, that is, x ≤ y if and only if yx ∈ P. If x ≤ y and x = y, then we denote x < y or y > x. By θ we denote the zero element of E. A nonempty closed convex set P ⊂ E is a cone if it satisfies: In addition, for a given h > θ , we denote by P h the set P h = {x ∈ E | x ∼ h}, in which ∼ is an equivalence relation, i.e., x ∼ y means that there exist λ > 0 and μ > 0 such that λx ≥ y ≥ μx for all x, y ∈ E.

Lemma 2.4
If g ∈ C[0, 1] is given, then the p-Laplacian tempered fractional differential equation integral boundary value problem

8)
has a unique integral formal solution

9)
where H(t, s) is given as (2.3), G(t, s) is a Green function and

10)
in which Proof From Lemma 2.1, integrating both sides of the first equation of (2.8), we obtain Furthermore, applying the tempered fractional derivative operators R 0 D γ i ,λ t (i = 1, 2) on both sides of Eq. (2.11), we have (2.13)

G(t, s) g(s) ds.
By employing the p-Laplacian operator ϕ q on both sides of the above equation, we have Setting g(t) : ϕ q ( 1 0 G(t, s) g(s) ds), thus, the p-Laplacian tempered fractional differential system (2.8) is equivalent to the integral boundary value problems as follows:   H(t, s) If 0 ≤ s ≤ t ≤ 1, clearly, we can see that Hence, the proof is complete.

Main results
In this section, we will work in the Banach space C[0, 1], the space of all continuous functions on [0, 1]. It is obvious that this space can be equipped with a partial order 1]} and h(t) = e -λt t α 1 -1 , then we see that P is a normal cone in C[0, 1]. From Lemma 2.4, we can recognize that the p-Laplacian differential equation integral boundary value problem (1.1) is equivalent to the integral formulation given by For convenience, we define an operator T by It is evident that u * is a solution of p-Laplacian differential equation integral boundary value problem (1.1) if and only if T(u * , u * ) = u * .
Proof Firstly, we define two operators A : P × P → E and B : P → E by and u is a solution of the p-Laplacian differential system (1.1) if and only if T(u, u) = u. We show that the operator A satisfies the condition (2.17) in Lemma 2.7 and the operator B is a sub-homogeneous operator. From (H 1 ), Lemma 2.5 and Lemma 2.6, we know that A : P × P → P and B : P → P. In addition, it follows from (H 1 ) and (H 2 ) that A is a mixed monotone operator and B is an increasing operator. For ∀γ ∈ (0, 1) and u, v ∈ P, from (3.2), we obtain That is, A(γ u, γ -1 v) ≥ γ ξ A(u, v) for ∀γ ∈ (0, 1), u, v ∈ P. Furthermore, for ∀γ ∈ (0, 1) and u ∈ P, from (3.     From L 2 > l 2 > 0 and l 2 h ≤ B(h) ≤ L 2 h, we get Bh ∈ P h . Since h ∈ P h , letting h 0 = h, we see that the condition (I 1 ) of Lemma 2.7 is satisfied.