Some Properties of Solutions to Multiterm Fractional Boundary Value Problems with -Laplacian Operator

Abstract

In this paper, we study some properties of positive solutions to a class of multipoint boundary value problems for nonlinear multiterm fractional differential equations with -Laplacian operator. Using the Banach contraction mapping principle, the existence, the uniqueness, the positivity, and the continuous dependency on -point boundary conditions of the solutions to the given problem are investigated. Also, two examples are presented to demonstrate our main results.

1. Introduction

This paper deals with a class of multipoint boundary value problems for nonlinear multiterm fractional differential equations with -Laplacian operator: where and are the standard Riemann-Liouville derivatives. And it is assumed that the nonlinear source term satisfies , and is defined as .

Fractional differential equations have developed into a powerful tool to mathematically model and solve many real-world problems that arises in various fields such as physics, chemistry, biology, and mechanics (see [1, 2] and references therein). Especially, since the turbulent flow was one of the fundamental problems in the field of fluid dynamics, Leibenson [3] and Esteban and Vazquez [4] proposed a mathematical model for one-dimensional, polytropic, and turbulent flow of a gas in a porous medium as where was a scaled density at every point and time . According to them, if , the equation refers to the laminar filtration, in which case, it is often known as the fast diffusion equation if and the linear heat equation if ; otherwise, the porous media equation. On the other hand, the case has been widely studied in non-Newtonian fluid dynamics (see [5] and the references therein). By some substitutions and tricks, this nonlinear problem was reduced into the following -Laplacian equation to investigate the existence and properties of solutions to it [6, 7]:

Due to the significance of equation (3), many works have been studied to establish a lot of valuable existence and multiplicity results for various classes of higher-order and generalized -Laplacian differential equations with integer-order derivatives [8–10]. For example, Guo et al. [8] dealt with the existence of at least three positive solutions to the quasilinear second-order differential equation subject to one of the following -point boundary conditions: or by using the five functional fixed point theorem. Generalizing the results of the above studies for integer-order differential equations, many researchers have obtained great outcomes of the existence of solutions to single-term fractional differential equations with -Laplacian operator (see [11–15]). From the representative results, Lv [11] established the existence and multiplicity of positive solutions to -point boundary value problems of -Laplacian fractional differential equations with a parameter: by using the theory of the fixed-point index in a cone and the monotone iterative technique. In addition, Li and Qi [12] improved the works of Guo et al. [8] to show the existence of positive solutions for the following multipoint boundary value problems of nonlinear fractional differential equations with -Laplacian operator: where are integers and are the Caputo fractional derivatives. It is noted that the notations and appearing throughout the paper refer to the Riemann-Liouville and Caputo fractional derivatives, respectively, unless otherwise stated.

However, since the differential equation studied by Leibenson [3] is a multiterm one, it is important to extend the research results for single-term fractional differential equations to the case of multiterm. To do this, the existence of solutions to nonlinear multiterm fractional differential equations with a -Laplacian operator have been studied by employing useful techniques in a nonlinear functional analysis such as the Banach contraction mapping principle, the Schauder fixed-point theorem, and the Krasnoselskii fixed-point theorem (see [16–20]). In particular, it is of great significance to study the case where are in or , when considering the following fractional differential equation: generalized equation (3) because of the inclusion of first-order derivatives outside and inside of in it. Here, and refer to fractional derivatives in several senses such as the Riemann-Liouville and the Caputo. Therefore, by using the coincidence degree theory, Chen et al. [16] studied two-point boundary value problems of fractional differential equations with -Laplacian operator: to obtain new results on the existence of solutions to them. Besides, according to Liu et al. [17], the following four-point boundary value problems of -Laplacian fractional differential equations with mixed fractional derivatives: were considered to establish a lower and upper solution method which was used to prove the existence of positive solutions to them. After analyzing the results for nonlinear multiterm fractional differential equations with -Laplacian operator, we have found that most of these previous works only dealt with some kinds of -Laplacian equations that had certain rules for the fractional derivatives in nonlinear source terms and were limited to the cases of two-point, three-point, and four-point boundary conditions.

On the other hand, Dishlieva [21] and Li et al. [22] investigated the continuous dependency of solutions to their differential equations on initial conditions, barrier curves, and source terms in their studies but not on boundary conditions. This also implies that as far as we know, no works concerned with this issue for fractional differential equations with -Laplacian operator can be done.

Motivated by the analysis mentioned above, this paper is aimed at studying some properties of positive solutions to -point boundary value problems of nonlinear multiterm fractional differential equations with -Laplacian operator including the existence, the uniqueness, and the continuous dependency on boundary conditions.

This study is organized as follows. In Section 2, we give some necessary definitions and preliminary results which will be used to prove our main results. In Section 3, we prove the existence and uniqueness of positive solutions to -Laplacian fractional boundary value problem (1) and continuous dependence of them on the perturbations with respect to the coefficients in -point boundary conditions. Finally, in Section 4, we illustrate our results by giving two examples.

Throughout the whole paper, it is also supposed that

2. Preliminaries

For the sake of convenience of the readers, some necessary definitions and lemmas will be presented here.

The Riemann-Liouville fractional integral and the Riemann-Liouville fractional derivative of order of a function are defined as where , provided that the right-hand sides are pointwise defined on (see [23]).

Lemma 1 (see [24]). Assume that with a fractional derivative of order that belongs to . Then, for some , where is the smallest integer greater than or equal to .
Since , it is obvious that . And putting and , the following lemmas hold.

Lemma 2 (see [11]). Let . Then, the fractional differential equation has a unique solution which is given by where ,
in which

Lemma 3 (see [11]). If , then the function in Lemma 2 satisfies the following conditions: (i), for any (ii), for any where

Lemma 4 (see [25]). Let . Then, the fractional differential equation has a unique solution which is given by where , in which

Lemma 5 (see [25]). If , then the function in Lemma 4 satisfies the following conditions: (i), for any (ii), for any where The following properties of which will be used later can be found in [26]: (i)If , and , then (ii)If , then

3. Main Results

3.1. Existence and Uniqueness of Positive Solutions to Problem (1)

Definition 6. A function is called a solution of problem (1) if it satisfies the fractional differential equation and the boundary conditions of (1).

Lemma 7. If the function is a solution of problem (1), then is a solution of the integral equation in and conversely, if is a solution of the integral equation (25), then is a solution of problem (1), where is a number such that .

Proof. Let be a solution of the problem (1). Then, since , by using Lemma 1, it can be easily seen that The upper continuities of the functions at provide us that . Applying on both sides, we have Combining with , we can see that . Since , Lemma 1 yields that Therefore, it follows from the upper continuities of the functions at that . This implies that by putting , it holds that . Thus, we can know that is a solution of the following problem: Put . Lemma 2 shows that Also, by applying Lemma 4, the notation gives us that Since , by equations (30) and (31), we obtain Conversely, let the function be a solution of integral equation (25). From this, by putting , it can be also obtained that . Denote as follows: From the continuities of the functions , and , it is obvious that . The definition of the function implies that so, we have This means that . Also, combining the definition of the function with the relation we can get This yields that and therefore, it holds that . Since equations (15) and (19) have unique solutions by Lemmas 2 and 4, , the solution of integral equation (25), satisfies the fractional differential equation and the boundary conditions of the problem (29). To sum up, it can be proved that is a solution of problem (1).

Let be the Banach space equipped with the norm and put . Since , it holds that . Therefore, define an operator as then, it satisfies . This implies that integral equation (25) has a solution in if and only if operator has a fixed point. It is also obvious that the fixed point of operator is a solution of integral equation (25) in .

Put . The following hypotheses will be used throughout the paper.

(H1) There exist such that where and

(H2) There exist such that for any and any ,

(H3) There exist such that for any ,

Lemma 8. Assume that the hypothesis (H1) is satisfied. Then, it holds that where .

Proof. Since , it is sufficient to prove that . It can be easily evaluated that for any , By using Lemma 5 and hypothesis (H1), we have Applying the properties of the function indicated in Lemma 3, the following holds: The relation shows that is an inverse power of . So, it can be rewritten as Since it is satisfied for that we can get This concludes the lemma.

Lemma 9. If the hypothesis (H3) holds, then for any and any , where .

Proof. It follows from Lemma 4 and hypothesis (H3) that By simple calculation, the integration terms in the right side of inequality (51) can be estimated as These estimations provide us the conclusion (50).

Denote . Then, it can be found in [25] that for any ,

Theorem 10. Suppose that and hypotheses (H1)-(H3) are satisfied. If then, integral equation (25) has a unique solution in .

Proof. To prove this theorem, operators are defined as Obviously, we can see that for any , , and . Because , it is satisfied that . So, by employing Lemma 8, Lemma 9, and property (23), we can obtain that for any and any , Substituting inequality (53) into (56), we have Thus, it holds that Since , by Lemma 3, we can get This yields that Combining inequalities (54) and (60) with Lemma 8, the operator is a contraction mapping and the Banach contraction mapping principle provides us that it has a unique fixed point in . So, integral equation (25) has a unique solution in .

Lemma 11. Assume that the hypothesis (H1) holds. Then, for any and any , it is satisfied that where

Proof. In a similar way to the proof of Lemma 8, it can be easily proved that

Theorem 12. Suppose that and hypotheses (H1) and (H2) hold. If then, integral equation (25) has a unique solution in .

Proof. In the case , due to , it holds that . Similar to Theorem 10, defining operators with Lemma 8, Lemma 11, and property (24) give us that for any and any , So, we have Obviously, for , the following holds: Therefore, by employing inequality (64), equality (67), and Lemma 8, the operator is a contraction mapping. It follows from the Banach contraction mapping principle that the operator has a unique fixed point in . This completes the proof.

Remark 13. Theorems 10 and 12 show the existence and uniqueness of solutions to integral equation (25) in the cases and , respectively. The Banach contraction mapping principle, employed in the proofs of these theorems, guarantees that the iterative sequences can be constructed to converge to the exact solution of integral equation (25). In other words, the iterative sequence initialized at any point and defined by converges to the exact solution of integral equation (25), , in . This provides the possibility to obtain the approximate solution of integral equation (25) by using Theorems 10 and 12.

Theorem 14. Problem (1) has a unique solution in if and only if the solution to integral equation (25) uniquely exists in , where . In particular, if hypothesis (H3) is satisfied, then the solution to problem (1) is positive.

Proof. By Lemma 7, for , a solution to problem (1), and satisfying integral equation (25), the following relation holds: This proves the first part of the theorem.
For the second part, it is necessary that if hypothesis (H3) holds, then To prove this, we must see if the following holds: Since , using Lemma 9, it can be easily obtained that Considering that and , we can get From the definition of function , we have