Solvability of Some Fractional Boundary Value Problems with a Convection Term

Abstract

This paper is devoted to the research of some Caputo’s fractional derivative boundary value problems with a convection term. By the use of some fixed-point theorems and the properties of Green function, the existence results of at least one or triple positive solutions are presented. Finally, two examples are given to illustrate the main results.

1. Introduction

Fractional differential equations (FDEs) present new models for many applications in physics, biomathematics, environmental issues, control theory, image processing, chemistry, mechanics, and so on [1–17]. Recently, researchers focus on studying various aspects of fractional differential equations, such as stability analysis, existence, multiplicity, and uniqueness of solutions [1–40]. Among all these topics, the existence and multiplicity results of positive solutions represent a topic of high interest in fractional calculus.

Some authors studied the existence and uniqueness of solutions for fractional differential equations with Caputo or Riemann-Liouville derivatives based on the Banach contraction principle and investigate the stability results for various fractional problems [4, 5, 16, 17]. Others studied the existence and multiplicity results of positive solutions or the iterative scheme. By the use of the Krasnoesel-skii’s fixed-point theorem, Zhang [34] obtained some existence results of positive solutions of following problem without the convection term: Wang and Liu [29] deduced the Green function and some interesting properties for the Dirichlet BVPs where is the Riemann-Liouville (R-L) fractional derivative, , and . And they established an iterative scheme to approximate the unique positive solution under the singular conditions.

Meng and Stynes [23] considered the following linear two-point fractional differential equation BVPs with general Robin type boundary condition: where denotes the Caputo derivative, , are constant, and . Meng used two parameter Mittag-Leffler functions to establish explicitly Green’s function for the problems. They obtained the nonnegativity of Green’s function.

This paper is devoted to the research of the solvability of the following nonlinear fractional BVPs:where and are constants. is the Caputo’s fractional derivative. Compared to the existing literature, the interesting point here is that the convection term is involved in the study of the solvability of fractional differential boundary value problems. By applying some fixed-point theorems, some existence and multiplicity results of positive solutions are given. Some examples are presented in last section to illustrate the main theorems.

In the sequel, the following conditions will be used:

(H₁) ;

(H₂) the constants . The function is defined in Section 2.

2. Background Material and Definitions

In order to solve the BVPs (4), (5), the following definitions and lemmas are needed.

Definition 1. Define the two-parameter Mittag-Leffler function by

Definition 2. Assume and . The Caputo fractional derivative of order is defined asIn particularly, for

From the above equation, for there is and

Remark 3. For simplicity, letthe particular,

Lemma 4 (see [23]). Suppose and the conditions and hold. For , the problemhas a unique solutionwhere is Green’s function and is defined by Remark 3.

By the use of some interesting properties of the Mittag-Leffler function, Meng and Stynes proved the following result.

Lemma 5 (see [23]). Suppose . Then Green’s function for if and only if

We pointed out here that this lemma comes from Theorem 5.1 and Remark 5.2 of Ref. [23] with some equivalent changes.

The following theorems are fundamental for proving the main results.

Lemma 6 (see [41]). Let be a Banach space, a cone, and two bounded sets of with . Suppose that is completely continuous such that either and , or
and holds.
Then the operator has a fixed point in .

Lemma 7 (see [42]). Let be a cone in a real Banach space , , is a nonnegative continuous concave functional on such that , for , and . Suppose is completely continuous and there exist four constants satisfying
and for ;
for ;
for with .
Then the operator has three fixed-points such that

Remark 8. Specially, if , then condition of Lemma 7 implies condition .

3. Existence and Multiplicity

In this section, by applying Lemma 6 and Lemma 7, some solvability results for BVPs (4), (5) will be obtained.

Let be a Banach space; . The cone is defined as Let the concave functional be defined by Define the operator by It is clear that is a positive solution of BVPs (4), (5) equivalent to that is a fixed-point of .

Lemma 9. The operator is completely continuous.

Proof. Taking into account that the functions and are all continuous and nonnegative, the operator is continuous and nonnegative, i.e., . Suppose is a bounded set, and for all there holds for some . Let ; then, for , there is Thus, the set is bounded.
By the fact that the Green function is continuous on , one has the fact that it is uniformly continuous. Therefor, for given , there exists such that for each , and ; there holds Thus Hence, the set is equicontinuous. Thus, using the Arzela-Ascoli theorem, we claim that is a completely continuous operator.

Let

Theorem 10. Assume conditions and hold. If there exist two different positive constants such that
(A1) , for ;
(A2) , for ,
then the BVPs (4), (5) have one positive solution such that .

Proof. Define two open sets For , there is for . The condition yields that SoFor , there is for . According to , for , So And the proof also holds when . Therefore, by Lemma 6, the proof is complete.

Theorem 11. Assume conditions and hold. If there exist constants such that
(B1) , for ;
(B2) , for ;
(B3) , for ,
then the BVPs (4), (5) have three positive solutions with

Proof. We just need to prove that all the conditions of Lemma 7 hold.
For , there is . Assumption shows for . Taking into account the definition of , there is Hence . Similarly, if , then assumption shows that for . Therefore, of Lemma 7 holds.
For , choose It is clear that , . Thus the set is nonempty. If , then for . And assumption shows for . So That is to say that for . I.e., the condition of Lemma 7 holds. By Remark 8, one has the fact that the condition holds, too.
Then by Lemma 7, the BVPs (4), (5) have three positive solutions such that The theorem is proven.

4. Two Examples

Now, we present two examples to check the main results. Readers can easily find that the following problem cannot be solved with existing literature. Choose , and

Example 1. Consider the following problem:Clearly, conditions and are satisfied. A simple computation shows . Let , and there areBy Theorem 10, the BVPs (35), (36) have one positive solution and .

Example 2. Consider the following fractional problem with the convection term:where Clearly, conditions and hold. A simple computation shows that . Choose , and there hold By Theorem 11, the BVPs (38), (39) have three positive solutions such that

Data Availability

No data were used to support this study.

Additional Points

Results and Discussion. Under the nonlinear term satisfying some growth condition, we studied some fractional differential equation boundary value problems in Caputo sense with the convection term. By applying fixed-point theory, some solvability criteria of positive solution are obtained. Two examples are given to illustrate the existence and multiplicity results.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper.

Acknowledgments

This work is supported by NSFC (11571207), SDNSF (ZR2018MA011), and the Taishan Scholar project.