Local existence–uniqueness and monotone iterative approximation of positive solutions for p-Laplacian differential equations involving tempered fractional derivatives

In this paper, we are concerned with a kind of tempered fractional differential equation Riemann–Stieltjes integral boundary value problems with p-Laplacian operators. By means of the sum-type mixed monotone operators 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 obtain not only the local existence with a unique positive solution, but also construct two successively monotone iterative sequences for approximating the unique positive solution. Finally, we present an example to illustrate our main results.

In the past decades, fractional calculus and all kinds of fractional differential equations have been proved to be powerful tools in the modeling of various phenomena in a great deal of fields of science and engineering, such as chemical physics, fluid mechanics, heat conduction, control theory, economics, and so on; see, for example, [1][2][3][4]. Abdullah and Zeynep [5] investigated the generalized fractional integral inequalities for continuous random variables and obtained new generalized integral inequalities for the generalized dispersion and the generalized fractional variance functions of a continuous random variable having the probability density function. Muhammad et al. [6] considered one of the important classes of Caputo fractional-order evolution equations by using fixed point theorems of Banach and Krasnoselskii type, obtained the existence and uniqueness of the solution, and studied Ulam-Hyer-type stability of the numerical solution.
In fact, a standard Riemann-Liouville (or Caputo) fractional derivative is a convolution with power law, so does fractional integration, and the difference between the two fractional derivatives only lies in the order of derivation and integration. Based on the definition of classical fractional derivative, the tempered fractional derivative multiplies the power law kernel by exponential factor, and various differential equation models based on tempered fractional derivative open up a new possibility for robust mathematical modeling of anomalous phenomena and complex multiscale problems; we refer the readers to [7][8][9][10]. In [11], we studied two kinds of tempered fractional differential systems involving the following Riemann-Stieltjes integral boundary value conditions: and t u (i = 1, 2, 3) are the tempered fractional derivatives. By using a class of sum-type mixed monotone operators fixed point theorems and increasing ϕ-(h, σ )-concave operators fixed point theorems, respectively, we constructed sufficient conditions to guarantee the existence-uniqueness of positive solutions for Riemann-Stieltjes integral boundary value problems (1.2) and (1.3), respectively.
It is well known that the p-Laplacian operator is used in analyzing various complex problems in physics, mechanics, and the related fields of mathematical modeling; see [12][13][14]. In [12], for studying the turbulent flow in a kind of porous media, Leibenson introduced the p-Laplacian differential equation where ϕ p (s) = |s| p-2 s, p > 1. Motivated by Leibenson's work, Ren, Li, and Zhang [15] studied the existence of maximum and minimum solutions for the nonlocal p-Laplacian fractional differential system (1.5) where ϕ p i denotes the p-Laplacian operator, D t are the standard Riemann-Liouville derivatives with 1 < α i , β i < 2, 1 0 x i (t) dA i (t) denotes the Riemann-Stieltjes integral, and A i is a function of bounded variation. By employing the cone theory and monotone iterative technique, some new existence results on maximal and minimal solutions were established. Furthermore, the estimation of the bounds of maximum and minimum solutions was derived.
In [16], we investigated the existence of multiple positive solutions for the following p-Laplacian fractional differential equations with two-point boundary values: (1.6) where n -1 < α ≤ n, R 0 D α t is the standard Riemann-Liouville fractional derivative, and ϕ p is the p-Laplacian operator. By employing the functional-type cone expansion-compression fixed point theorem and Leggett-Williams fixed point theorem, we obtained the existence of multiple positive solutions for p-Laplacian differential systems (1.6).
To study more boundary value problems for complex fractional differential equations, we combine the Riemann-Stieltjes integral boundary value conditions with p-Laplacian operators, where the nonlinear terms are sum-type nonlinear terms in (1.1). Comparing with the previous references, this paper has the following characteristics. Firstly, the tempered fractional derivative R 0 D α,λ t is more general than the standard Riemann-Liouville Secondly, Riemann-Stieltjes integral boundary conditions are more general and cover the common integral boundary conditions as particular cases. Finally, comparing with p-Laplacian differential systems (1.6), the integral operator in this paper need not be completely continuous or compact. Furthermore, we not only obtain the local existence with a unique positive solution, but also construct a Cauchy sequence to approximate the unique positive solution.
The organization of the paper is as follows. In Sect. 2, we list some concepts, symbols, definitions, and lemmas in abstract Banach spaces, which need to be used in the subsequent proof process. In Sect. 3, by employing the sum-type mixed monotone operators fixed point theorem based on cone P h we show that the existence-uniqueness and monotone iteration of positive solutions of the two-point boundary value problems for the p-Laplacian differential equation (1.1). In Sect. 4, we present an example to demonstrate our main results.

Preliminaries
A nonempty closed convex set P ⊂ E is called a cone if it satisfies the following conditions: In addition, let (E, · ) be a real Banach space that is partially ordered by a cone P ⊂ E, that is, yx ∈ P implies that x ≤ y. If x ≤ y and x = y, then we write x < y or y > x. We denote the zero element of E by θ . If for all x, y ∈ E, there exists M > 0 such that θ ≤ x ≤ y implies x ≤ y , then the cone P is called normal; in this case, M is the infimum of such constants and is called the normality constant of P.
where ∼ is an equivalence relation, that is, for all x, y ∈ E, x ∼ y means that there exist λ > 0 and μ > 0 such that λx ≥ y ≥ μx.
Proof Firstly, applying the fractional integral operator 0 I α t on both sides of the first equation of integral boundary value problems (3.1), we have Furthermore, applying the tempered fractional derivative operator R 0 D γ ,λ t on both sides of (3.3), we have t u))(0) = 0 and 1 < αγ < 2 we deduce that d 2 = 0, that is, Secondly, applying the tempered fractional derivative operator R 0 D β i ,λ t (i = 1, 2) on both sides of (3.4), we get (3.6)

Combining (3.6) with the Riemann-Stieltjes integral boundary value condition
t u(s))] dA(s), we obtain (3.7) Substituting (3.7) into (3.4), we obtain Furthermore, applying the p-Laplacian operator ϕ q on both sides of (3.8), we get Finally, setting g(t) : ϕ q ( 1 0 G(t, s) g(s) ds), we easily see that p-Laplacian fractional differential system (3.1) is equivalent to the following fraction differential equation integral boundary value problem: (3.10) By means of Lemma 2.3 we get that the tempered fractional differential system with p-Laplacian operator (3.10) has a unique integral solution This completes the proof.
Proof To begin with, we define two operators A : P × P → E and B : P → E by From p = 2 and 1 p + 1 q = 1 we easily see that q = 2. Evidently, we have T(u, v) = A(u, v) + B(u). In addition, u * is a solution of the Riemann-Stieltjes integral boundary value problem (1.1) if and only if T(u * , u * ) = u * . From Lemma 2.4 we get A : P × P → P and B : P → P. Furthermore, it follows from (H 1 ) and (H 2 ) that A is a mixed monotone operator and B is an increasing operator. For all γ ∈ (0, 1) and u, v ∈ P, from (3.13) we obtain (3.18) Hence the mixed monotone operator A satisfies condition (2.10) in Lemma 2.5. In addition, for all γ ∈ (0, 1) and u ∈ P, from (3.14) we have  It is clear that L 1 > l 1 > 0. Hence  From L 2 > l 2 > 0 and l 2 h(t) ≤ B(h) ≤ L 2 h(t) we get Bh ∈ P h . Since h ∈ P h , letting h 0 = h, we get that condition (I 1 ) in Lemma 2.5 holds.
Finally, for all u, v ∈ P, from (3.15) we have Then we have: (I) the p-Laplacian differential equation Riemann-Stieltjes integral boundary value problem we obtain x n → u * and y n → u * as n → ∞.
Proof Setting g(t, u(t)) ≡ 0, by means of Theorem 3.1 we get the conclusions.