#### Abstract

In this paper, we are concerned with the existence of the maximum and minimum iterative solutions for a tempered fractional turbulent flow model in a porous medium with nonlocal boundary conditions. By introducing a new growth condition and developing an iterative technique, we establish new results on the existence of the maximum and minimum solutions for the considered equation; at the same time, the iterative sequences for approximating the extremal solutions are performed, and the asymptotic estimates of solutions are also derived.

#### 1. Introduction

Tempered stable laws were introduced to model turbulent velocity fluctuations of physics [1]. Normally, tempered stable laws retain their signature power-law behaviour at infinity and infinite divisibility [2]. By multiplying by an exponential factor for the usual second derivative, one can obtain tempered fractional derivatives and integrals. In [3], an exponential tempering factor was applied to the particle jump density in random walk and stochastic model for turbulence in the inertial range, which is the fractional derivative of Brownian motion exhibiting semi-long range dependence with a power law at moderate time scales.

Tempered stable laws are useful in statistical physics and provide a basic physical model such as turbulent flow for the underlying physical phenomena. Motivated by these physical backgrounds and the sources, in this paper, we focus on the existence of the maximum and minimum iterative solutions for the following tempered fractional turbulent flow equation with nonlocal boundary conditions:where , is a constant, is the -Laplacian operator defined by , is the tempered fractional derivative, denotes a Riemann–Stieltjes integral, is a function of bounded variation, is continuous, and .

Turbulent flow is a fundamental fluid mechanics problem which can be described by a -Laplacian equation with a suitable boundary condition; for details, see [4]. Particularly, if the model is of fractional order, then it can describe turbulent flow in a porous medium [5–10]. On the contrary, fractional-order derivative has nonlocal characteristics; based on this property, the fractional differential equation can also interpret many abnormal phenomena that occur in applied science and engineering, such as viscoelastic dynamical phenomena [11–29], advection-dispersion process in anomalous diffusion [30–34], and bioprocesses with genetic attribute [35, 36]. As a powerful tool of modeling the above phenomena, in recent years, the fractional calculus theory has been perfected gradually by many researchers, and various different types of fractional derivatives were studied, such as Riemann–Liouville derivatives [16, 37–62], Hadamard-type derivatives [63–71], Katugampola–Caputo derivatives [72], conformable derivatives [73–76], Caputo–Fabrizio derivatives [77, 78], Hilfer derivatives [79–82], and tempered fractional derivatives [83]. These works also enlarged and enriched the application of the fractional calculus in impulsive theories [84–89], chaotic system [90–93], and resonance phenomena [94–96]. Among them, by using the fixed point theorem of the mixed monotone operator, Zhang et al. [9] established the result of uniqueness of the positive solution for the Riemann–Liouville-type turbulent flow in a porous medium:where , denote the Riemann–Liouville derivatives, and indicates the Riemann–Stieltjes integral, and is a function of bounded variation; the nonlinear term may be singular at both first variable and second variable. Recently, Zhou et al. [83] investigated a class of tempered fractional differential equations with Riemann–Stieltjes integral boundary conditions; by using the fixed point theorem of the sum-type mixed monotone operator, the existence and uniqueness of positive solutions were established, and iterative sequences for approximating the unique positive solution were also constructed.

However, to the best of our knowledge, there are relatively few results on fractional turbulent flow in a porous medium with nonlocal Riemann–Stieltjes integral boundary conditions, and no work has been reported on the maximal and minimal solutions for the tempered-type fractional turbulent flow equation. Thus, following the previous work, this paper will pay attention to the extremal solutions for the tempered fractional turbulent flow equation in a porous medium with nonlocal Riemann–Stieltjes integral boundary conditions by developing iterative technique, also see [97–100]. Different from [9, 83], in this paper, we will give a new type of growth condition for the nonlinear term to guarantee equation (1) has the extremal solutions. At the same time, the iterative sequences for approximating the extremal solutions are performed, and the asymptotic estimates of solutions are also obtained.

#### 2. Preliminaries and Lemmas

Before starting our work, we firstly recall the definition of the tempered fractional derivative which is an extension of the Riemann–Liouville derivative and integral.

Let ; the -order left tempered fractional derivative is defined bywhere denotes the standard Riemann–Liouville fractional derivative which can be found in [101].

Let

The following results have been proven in [83].

Lemma 1. *Given ; then, the boundary value problem,has the unique solutionwhere is defined by (4) and denotes the Green function as follows:*

In order to obtain the positive extremal solutions of tempered fractional turbulent flow equation (1), it is necessary to preserve nonnegativity of the Green function.

:

Lemma 2. *Assume holds; then, functions and have the following properties:*(1)* and are nonnegative and continuous for .*(2)*For any satisfies** **where**(3)** **where*

Let

In order to obtain the existence of positive extremal solutions of tempered fractional turbulent flow equation (1), we introduce the following new control conditions.

: is continuous and nondecreasing, and there exists a positive constant such that

:

*Remark 1. *Assumption (14) we introduced is a new type of growth condition, which includes a large number of basic functions such as(1), where .(2), where and(3).(4) is continuous and nondecreasing, and there exists a positive constant such that is increasing on , and(5) is continuous and nondecreasing, and there exists a positive constant such that is nonincreasing on .

*Proof. *For cases (1)–(3), takerespectively; obviously,For cases (4) and (5), it is clear; we omit the proof.

Denote as all continuous functions equipped the maximum normDefine a cone ,and an operator in :Then, the fixed point of operator in is the solution of tempered fractional turbulent flow equation (1).

Lemma 3. *Assume that – hold. Then, is a continuous, compact operator.*

*Proof. *It follows from the definition of that, for any , there exists a number such thatSince is increasing with respect to , by (14), (23), and Lemma 2, we havewhereThus, it follows from (24) thatwhich implies that is well defined and uniformly bounded, and .

On the contrary, according to the Arzela–Ascoli theorem and the Lebesgue dominated convergence theorem, it is easy to know that is completely continuous.

#### 3. Main Results

Before we begin to state our main result, we first give the following lemma.

Lemma 4. *Suppose and hold; then, the equationhas unique solution in .*

*Proof. *LetIt follows from and thatOn the contrary, is a continuous function in satisfyingThus, (29)–(31) imply equation (27) has unique zero point in .

Theorem 1. *Suppose – hold. Then, the following is obtained:*(i)*Existence: equation (1) has a positive minimal solution and a positive maximal solution .*(ii)*Asymptotic estimates: there exist positive numbers , such that**(iii)**Iterative sequences: for initial values , construct the iterative sequences*

Then,uniformly, for , where is the unique solution of equation (27) in .

*Proof. *Firstly, let ; we shall show .

For any and for any , we haveConsequently, it follows from and Lemma 4 thatwhich implies that .

Next, take the initial value , and letIt follows from that .

DenoteBy , we have for . It follows from the fact of being a compact operator that is a sequentially compact set.

On the contrary, since and is increasing on , we haveBy induction, one hasConsequently, there exists such that . Noticing that and letting , by the continuity of , we have , which implies that is a nonnegative solution of equation (1), and then is a positive solution of equation (1) since .

Now, we take as the initial value and letIt follows from that . Thus, construct the iterative sequenceWe havesince . It follows from Lemma 3 that is a sequentially compact set.

Now, since and is increasing, one hasand thenIt follows from induction thatwhich implies that there exists such that . Letting , from the continuity of and , we have , which implies that is another positive solution of equation (1).

Next, we prove that and are the maximum and minimum positive solutions of equation (1). In fact, suppose is any positive solution of equation (1); then, we haveThus, it follows from induction thatTaking the limit, we havewhich implies that and are the maximal and minimal positive solutions of equation (1), respectively.

In the end, since , there exist constants such that

#### 4. Example

Since the fractional-order derivative possesses long-memory characteristics, in fluid mechanics, equation (1) can describe a turbulent flow in a porous medium. Here, we give a specific example to illustrate the main results.

Example: consider the following nonlocal tempered fractional turbulent flow equation:where

Then, equation (51) has the positive minimal and maximal solutions and , and there exist constants such that

Let

Firstly, we have

Thus, holds.

Obviously, is continuous and nondecreasing. Take ; then, we havewhich implies that is satisfied.

Now, we compute :

Thus, we have

Consequently,

So, condition holds.

Thus, by Theorem 1, equation (51) has a positive minimal solution and a positive maximal solution , and there exist constants such that

#### 5. Conclusion

In this work, we establish a new result on the existence of the maximum and minimum solutions for a class of tempered fractional-order differential equations with nonlocal boundary conditions. This type of equation can describe a turbulent flow of a porous medium in fluid mechanics and diffusive interaction. In order to obtain the extremal solutions of the equation, a new type of growth condition is introduced, and the iterative sequences with explicit initial values are constructed which converge uniformly to the maximum and minimum solutions; in addition, the estimations of the upper bounds of the maximum and minimum solutions are also derived.

#### Data Availability

No data were used to support the findings of this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Authors’ Contributions

The study was carried out in collaboration among all authors. All authors read and approved the final manuscript.

#### Acknowledgments

The authors were supported financially by the National Natural Science Foundation of China (11871302 and 11571296).