Iterative Analysis of the Unique Positive Solution for a Class of Singular Nonlinear Boundary Value Problems Involving Two Types of Fractional Derivatives with <i>p</i>-Laplacian Operator

Wang, Fang; Liu, Lishan; Wu, Yonghong; Zou, Yumei

doi:https://doi.org/10.1155/2019/2319062

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 2319062 | https://doi.org/10.1155/2019/2319062

Iterative Analysis of the Unique Positive Solution for a Class of Singular Nonlinear Boundary Value Problems Involving Two Types of Fractional Derivatives with p-Laplacian Operator

Fang Wang,¹Lishan Liu,^1,2Yonghong Wu,²and Yumei Zou³

Academic Editor: Sergey Dashkovskiy

Received04 Apr 2019

Revised24 Aug 2019

Accepted30 Aug 2019

Published27 Oct 2019

Abstract

This article is concerned with a class of singular nonlinear fractional boundary value problems with p-Laplacian operator, which contains Riemann–Liouville fractional derivative and Caputo fractional derivative. The boundary conditions are made up of two kinds of Riemann–Stieltjes integral boundary conditions and nonlocal infinite-point boundary conditions, and different fractional orders are involved in the boundary conditions and the nonlinear term, respectively. Based on the method of reducing the order of fractional derivative, some properties of the corresponding Green’s function, and the fixed point theorem of mixed monotone operator, an interesting result on the iterative sequence of the uniqueness of positive solutions is obtained under the assumption that the nonlinear term may be singular in regard to both the time variable and space variables. And we obtain the dependence of solution upon parameter. Furthermore, two numerical examples are presented to illustrate the application of our main results.

1. Introduction

In the past decades, fractional differential equations arise in many mathematical disciplines as the analogue modeling of systems and processes in many scientific fields, such as control theory and engineering. In fact, fractional-order models have proved to be valuable tools in modeling many physical phenomena (for details, see [1–3] and the references therein). Accordingly, there has been a substantial development in the research for fractional differential equations, the properties of solutions, especially. We refer the readers to the papers [4–54]. For instance, in [15], Xu and Wei investigated the positive solutions of the following fractional differential equations:where is the Riemann–Liouville’s fractional derivative, , and . The existence of positive solutions is obtained by the Krasnoselskii–Zabreiko fixed point theorem. And by using the method of lower and upper solutions, the authors discussed the uniqueness of positive solution of this problem.

On the other hand, the fractional differential equations with the p-Laplacian operator can describe various phenomena, such as the flow of some specific fluid. Hence, the study of fractional differential equations with the p-Laplacian operator is gaining much significance and attention (see [16, 18–21, 37–39, 48–50]). For example, in [48], in order to research the mechanics phenomenon of turbulent flow in a porous medium, the author introduced the following type equation:where is the p-Laplacian operator, i.e., . In the past thirty years, many papers discussed equation (2) with different boundary value conditions and then drew related conclusion in practice. We refer the reader to [49, 50] and the references cited therein. Recently, Liu et al. in [16] investigated the existence of positive solution of the following fractional differential equations with p-Laplacian operator:where is the Riemann–Liouville’s fractional derivative and is the Caputo fractional derivative, , , , is the p-Laplacian operator, and . The existence of lower and upper solutions is obtained by using the monotone iterative technique.

Based on the above illustration, it is of significance to study the fractional differential equations with p-Laplacian operator. In this article, we investigate the following fractional differential equation, which is a generalized form of the problem (2):with the boundary conditionswhere denotes the Caputo fractional derivative; denotes the Riemann–Liouville’s fractional derivative; is the p-Laplacian operator, i.e., , , , , , , , , , , , ; are continuous with ; , denote the Riemann–Stieltjes integral, in which are functions of bounded variation; and is continuous. Furthermore, we also consider the following equation with a parameter:with the boundary conditions (5), where is a parameter. In fact, we regard the problem (6)-(5) as a new problem for modeling the problem (4)-(5) involving a parameter. In this paper, a nonnegative function is called a positive solution of boundary value problem (4)-(5) (resp. (6)-(5)) if it satisfies the problem (4)-(5) (resp. (6)-(5)) for and for .

In this paper, we make the following hypothesis: is continuous and satisfies for all , where are continuous. For any , , , and are nondecreasing in , and for any , , , and are nonincreasing in . For any , there exists such that for any , and : There exists such that for any , and :

Our main results are as follows.

Theorem 1. Suppose that hold, then the problem (4)-(5) has a unique solution , and there exists such thatAnd for any initial values , by structuring the following sequences:we have

Theorem 2. Suppose that hold, then for any , the problem (6)-(5) have a unique solution , and there exists such thatAnd moreover,(1)If there exists such that then is continuous with respect to . That is, for any ,(2)If then implies .(3)If there exists such thatthen

The key argument of the problem (4)-(5) and (6)-(5) is the iterative positive solution by applying the method of reducing the order of fractional derivative and the fixed point theory of mixed monotone operator. The method of reducing the order of fractional derivative is based on certain semigroup properties of the Riemann–Liouville’s fractional integral and derivative. We refer the reader to [3, 24, 25, 34, 36, 46]. In this paper, different orders of Riemann–Liouville’s fractional derivative are involved in the nonlinearity f, which is solved in a more complex space, in most cases. By using the method of reduction, we transform the problem (4)-(5) (resp. (6)-(5)) into an equivalent and low-order problem, in which the nonlinearity f contains no fractional derivative. Therefore, the work could proceed in the space , which is more interesting and meaningful. On the other hand, by using the properties of relevant Green’s function and cone, the theory of mixed monotone operator could be applied on the research of fractional boundary value problems. We suggest that one refers to [13, 17, 24, 31, 32, 34, 47]. In this paper, by structuring a suitable mixed monotone operator, the problem (4)-(5) (resp. (6)-(5)), which contains the p-Laplacian operator, is solved. In [15, 16], the positive solutions are obtained under the assumption that the nonlinear term f is continuous. But, we obtain the uniqueness of positive solution with the assumption that the nonlinear term may be singular in regard to both the time variable and space variables. Compared with [15], our equation contains p-Laplacian operator, which is more general. Compared with [16], our boundary conditions are of general significance, which would be used to describe more phenomena in practice.

The paper is organized as follows. In Section 2, we present some relevant definitions and lemmas. And we transform the problem (4)-(5) (resp. (6)-(5)) into an equivalent and low-order problem (28)-(29) (resp. (44)-(29)) and obtain some properties of the corresponding Green’s function. In Section 3, we give the proof of our main results, as one researches the uniqueness of iterative positive solutions for the problem (4)-(5) and the other one analyses the dependence of solution upon parameter for the problem (6)-(5). In Section 4, we give two examples to illustrate our theoretical results. In Section 5, we give a conclusion of our main results.

2. Preliminaries and Lemmas

Definition 1. Let . The Riemann–Liouville fractional integral of order α of a function is given byprovided that the right-hand side is pointwise defined on .

Definition 2. Let . The Riemann–Liouville fractional derivative of order α of a continuous function is given bywhere , denotes the integer part of the number α, provided that the right-hand side is pointwise defined on . In particular, if , then .

Definition 3. Let . The Caputo fractional derivative of order α of a continuous function is given bywhere , denotes the integer part of the number α, provided that the right-hand side is pointwise defined on . In particular, if , then .

Lemma 1 (see [7]). Let . If we assume , then the equationhasas the unique solution, where n is the smallest integer greater than or equal to α.

Lemma 2 (see [7]). Let and . Then,where n is the smallest integer greater than or equal to α.

Lemma 3 (see [4]). Let . Then,where for and for .

Lemma 4 (see [4]). (1)If , , then(2)Let . If , then .

Let , then the problem (4)-(5) could be transformed to the following fractional differential equation:with the boundary conditionswhere .

Lemma 5. If the problem (4)-(5) have a solution , then is a solution of the problem (28)-(29). And in turn, if the problem (28)-(29) have a solution , then is a solution of the problem (4)-(5).

Proof. Letwhere is a solution of the problem (4)-(5). By using Lemma 2, we havewhere . By the boundary condition , we get . So, we haveFurthermore, we obtainwhich combines with the boundary condition meaning . In a similar manner, we obtain based on the boundary condition . Thus, we haveApplying the operator to both sides of this equality, we haveSimilarly, we knowIn addition, by means of Lemma 4, we can obtainAccording to (30), (34), (36), and (38), we haveIt follows from (35)–(37) thatOn the basis of (39)–(43), we know is a solution of the problem (28)-(29).
On the other hand, if the problem (28)-(29) has a solution , then is a solution of the problem (4)-(5). Since the proof is similar to Lemma 3 in [44], we omit it here.

Remark 5. According to Lemma 5, we conclude that the work on considering the solution of the problem (4)-(5) is equivalent to the search for the solution of (28)-(29). Similarly, we can prove the work on the problem (6)-(5) is equivalent to the following equation:with the boundary conditions (29).

Lemma 6. Let . Then, the fractional differential equationwith the boundary conditionshas a unique solutionwherein whichObviously, is continuous.

Proof. We may apply Lemma 2 to reduce equation (45) to an equivalent integral equation:where . Since , we have . SoApplying the operator to both sides of (51), we haveHence, according to (52) and the boundary conditionwe can obtainwhereThenThe proof is complete.

Lemma 7. Let be a given function. Then, the following equationwith the boundary conditionshas a unique solutionwhere , is defined in Lemma 6, and

Proof. Let . Then, we can strip down the problem (57)‐(58) to the following two problems:Obviously, the problem (61) can be written asThe boundary conditions imply that . Thus,On the other hand, by Lemma 6, we know the problem (62) has a unique solution:Therefore, the problem (57)-(58) has a unique solution:

Lemma 8. Let (defined in Lemma 6), , , , and . Then, the following properties hold:(1), where ,(2), where(3), where ,

Proof. (1)For , For ,(2)Similarly as (1), we can obtain that(3)In view of the conclusions of (1) and (2), we haveThe proof is complete.

Let be a Banach space, P be a cone in E, and θ be the zero element of E. P is said to be normal if there exists a constant such that

The smallest constant, which satisfies the above inequality, is called the normality constant of P. Then, E is a partially ordered Banach space by P; that is to say,

For , the notation shows that there exist and such that . Clearly, is an equivalence relation in E. For any , let . Then, is a component of P. For more details, we suggest readers to refer [10, 11, 13].

Definition 4 (see [11]). Let E be a Banach space and . The operator is called a mixed monotone operator if is increasing in and decreasing in , ,

Lemma 9. (see [13, 17]). Let P be a normal cone in the Banach space E and be mixed monotone operators which satisfy the following:(1)For any , there exists such that(2)or any , ,(3)There exists a constant such that for any , .Then, the equation has a unique fixed point . And for any initial values , by structuring the following sequences:we have and in E, as .

Lemma 10. (see [13, 17]). Assume T and S satisfy all the conditions of Lemma 9. Then, for any , the equation has a unique solution , which satisfies the following:(1)If there exists such that then is continuous with respect to . That is, for any ,(2)If then implies .(3)If there exists such thatthenIn this paper, we denote with the norm . Then is a Banach space. Let be a cone in E. It is easy to check that P is normal in E with the normality constant .

3. Proof of Main Results

In this section, let , where (defined in Lemma 8). Then, is a component of P. Let us define three operators and as follows:

It is easy to check that is a positive solution of the problem (28)-(29) if it is a fixed point of T in .

Proof of Theorem 1. Obviously, by , . And it is easy to check that is a solution of the problem (28)-(29) if it satisfies .
The first work is to prove are well defined. For any , there exists a constant such thatOn the other hand, it is easy to check thatLet Then, . According to – and (87)–(89), we haveSimilarly, we haveOn the basis of (90), (92), and , we know that for any ,That is, are well defined. On the other hand, by means of , for any , satisfying and , we haveHence, are mixed monotone operators.
In the following, we prove that . It follows from (91) and (93) that for any u, and ,where M is a constant, which satisfiesAccording to (94) and (95), we haveOn the basis of (98), (99), (101), and (102) we infer that .
Moreover, it follows from , for any and u, and , we haveFrom , for any u, and , we can obtainSo, based on (103)–(105) and Lemma 9, we know the equation has a unique fixed point , which means is the unique positive solution of the problem (28)-(29), and there exists such thatAnd for any initial values , we can construct the following sequences:such thatFinally, on the basis of Lemma 5, we know is the unique positive solution of problem (4)-(5). By (106), we haveLet and ; by using the monotonicity and continuity of fractional integral, we have

Proof of Theorem 2. For any , let us define three operators by , , and , respectively. It is easy to check that is a positive solution of the problem (44)–(29) if it is a fixed point of in , i.e., . Considering the results of Theorem 1 and Lemma 10 together, we know the following equationhas a unique solution , which implies is the unique positive solution of the problem (44)–(29), and there exists such thatFurthermore, satisfies the following conclusion:(1)If there exists such thatthen is continuous with respect to . That is, for any ,(2)If then implies .(3)If there exists such thatthenFinally, by Lemma 5, we deduce is a unique positive solution of the problem (6)-(5), which satisfiesAnd by using the monotonicity and continuity of fractional integral, we have the following:(1)If there exists such that then is continuous with respect to . That is, for any ,(2)If then implies .(3)If there exists such thatthen

4. Numerical Examples

In this section, we give two simple theoretical numerical examples which justify Theorems 1 and 2.

Example 1. We consider the following equation:with the boundary conditionswhere and

Proof. Let, , , , , , , , , , , , , , , , , and . Then, the problem (124)-(125) can be expressed as the problem (4)-(5). Notice thatWe infer that the properties of Green’s function in Lemma 8 are achieved. LetIt is easy to check the following conditions: For fixed , and , , are increasing in ; for fixed and , and are decreasing in . Let . Then, for , , and , Let . Then, for all , The functions F and G satisfyThus, the assumptions of Theorem 1 are satisfied. By calculation, we obtain the approximate solution of the problem (124)-(125) is . On the other hand, let and . We take and . By iterating the sequences, the numerical results of the iterative process are shown as follows (Figure 1 and Tables 1 and 2).

(a)

(b)

Example 2. We consider the equationwith the boundary conditionswhere and

Proof. Let, , , , , , , , , , , , , , , , , and . Then, the problems (133)-(134) can be expressed as the problems (6)-(5). Notice thatWe infer that the properties of Green’s function in Lemma 8 are achieved. LetIt is easy to check the conditions – are all satisfied with and . On the other hand, let . Then,(1)(2)So, the assumptions of Theorem 2 are satisfied. In the following, we give the graphical simulations and table of the solution with respect to , 1, 1.5, 2, 3, 10, respectively (Figure 2 and Table 3).

Based on the graphical simulations and table, we obtain the following conclusions:(1) is continuous with respect to . That is, for any ,(2) implies .(3)

5. Conclusion

In this paper, we introduce the fixed point theorem of mixed monotone operator for finding the uniqueness of positive solution of a class of fractional boundary value problems, which is a generalized form of turbulent flow problem in a porous medium. Two theoretical numerical examples are given to illustrate Theorems 1 and 2; the results then bring us a step closer to research the characters of solutions. Furthermore, as the application of mixed monotone operator operator, further research and discussion are required in practice.

Data Availability

The datasets used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are supported financially by the National Natural Science Foundation of China (11871302) and the Natural Science Foundation of Shandong Province of China (ZR2017MA036). The support from the Australian Research Council for the research is also acknowledged.

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