Positive Solutions to a Fractional-Order Two-Point Boundary Value Problem with -Laplacian Operator
This paper systematically investigates positive solutions to a kind of two-point boundary value problem (BVP) for nonlinear fractional differential equations with -Laplacian operator and presents a number of new results. First, the considered BVP is converted to an operator equation by using the property of the Caputo derivative. Second, based on the operator equation and some fixed point theorems, several sufficient conditions are presented for the nonexistence, the uniqueness, and the multiplicity of positive solutions. Finally, several illustrative examples are given to support the obtained new results. The study of illustrative examples shows that the obtained results are effective.
Fractional differential equation has recently attracted many scholars’ interest due to its wide applications [1–3] in engineering, technology, biology, chemical process, and so on. The first issue for the theory of fractional differential equations is the existence of solutions to kinds of boundary value problems (BVPs), which has been studied recently by many scholars, and lots of excellent results have been obtained [4–17] by means of fixed point theorems, upper and lower solutions technique, and so forth.
As an important branch of BVPs, -Laplacian equation was firstly introduced in  to model the following turbulent flow in a porous medium: where , , , and . Then, it was investigated in both integer-order BVPs [19, 20] and fractional BVPs [21–25]. In , Chen and Liu considered the antiperiodic boundary value problem of fractional differential equation with -Laplacian operator and obtained the existence of one solution by using Schaefer’s fixed point theorem under certain nonlinear growth conditions. Han et al.  investigated a class of fractional boundary value problem with -Laplacian operator and boundary parameter and presented several existence results for a positive solution in terms of the boundary parameter. It is noted that although there exist several results on the existence of one solution to fractional -Laplacian BVPs, there are, to our best knowledge, relatively few results on the nonexistence, the uniqueness, and the multiplicity of positive solutions to fractional -Laplacian BVPs.
In this paper, we study the following two-point boundary value problem of nonlinear fractional differential equations with -Laplacian operator: where , , , , , , , , , , and is the Caputo derivative. We first convert BVP (2) into an equivalent operator equation by using the property of the Caputo derivative and then present some sufficient conditions for the nonexistence, the uniqueness, and the multiplicity of positive solutions to this problem. Finally, we give several illustrative examples to support our new results.
The main contributions of this paper are as follows. (i) We systematically study the nonexistence, the uniqueness, and the multiplicity of positive solutions to BVP (2) and propose some techniques to deal with fractional -Laplacian BVPs, which enriches this academic area. (ii) We present a sufficient condition for the existence of three positive solutions to BVP (2) by introducing a free constant to construct a proper concave functional. The main feature of this free constant is that one can weaken the conditions by regulating the free constant (see Remark 12).
Throughout this paper, we consider BVP (2) in the real Banach space with the norm . Let , for all . Then, is a normal solid cone of with , for all . A solution is said to be a positive solution to BVP (2), if and . We will study the existence of positive solutions to BVP (2) in .
The rest of this paper is structured as follows. Section 2 contains some preliminaries on the Caputo derivative. Section 3 investigates the nonexistence, the uniqueness, and the multiplicity of positive solutions to BVP (2) and presents the main results of this paper. In Section 4, three illustrative examples are worked out to support our obtained results.
Definition 1 (see ). The Riemann-Liouville fractional integral of order of a function is given by provided that the right side is pointwise defined on .
Definition 2 (see ). The Caputo fractional derivative of order of a continuous function is given by where is the smallest integer greater than or equal to , provided that the right side is pointwise defined on .
One can easily obtain the following property from the definition of the Caputo derivative.
Proposition 3 (see ). Let . Assume that , . Then, the following equality holds: for some , , where is the smallest integer greater than or equal to .
3. Main Results
In this section, we first convert BVP (2) into an equivalent operator equation and then present some new results on the nonexistence, the uniqueness, and the multiplicity of positive solutions to BVP (2).
Firstly, we convert BVP (2) into an equivalent operator equation.
Theorem 4. Given that , the unique solution of is where
Proof. Assume that satisfies (6). Then, from Proposition 3, we have
From the boundary value condition , one can see that Thus, we have which together with the boundary value condition yields that
The proof is completed.
Lemma 5. Consider that is a solution to BVP (2), if and only if , where and .
Proof. The necessity is obvious. Next, we prove the sufficiency.
In fact, if , a straightforward calculation shows that .
Since we have , and
Therefore, is a solution to BVP (2).
In the following, we consider the nonexistence of positive solutions to BVP (2) and present the following result.
Definition 7. Let be a solid cone in a real Banach space , an operator, and . Then, is called a -concave operator if
Lemma 8. Assume that is a normal solid cone in a real Banach space , , and is a -concave increasing operator. Then has only one fixed point in .
Based on Lemma 8, we have the following result.
Proof. Since , one can see that .
For any with , from the monotonicity of and , we have Thus, is increasing in .
Next, we prove that is a -concave operator.
In fact, from (19), for any , , it is easy to see that which implies that is a -concave operator.
By Lemma 8, BVP (2) has a unique positive solution.
Finally, we investigate the multiplicity of positive solutions to BVP (2).
We first recall the famous Leggett-Williams fixed point theorem .
Let be a Banach space and be a cone on . A continuous mapping is said to be a concave nonnegative continuous functional on , if satisfies for all and .
Let , , be constants. Define , , and .
Lemma 10. Let be a Banach space, a cone of , and a constant. Suppose that there exists a concave nonnegative continuous functional on with for all . Let be a completely continuous operator. Assume that there are numbers , , and with , such that(i) and for all ;(ii) for all ;(iii) for all with .
Then, has at least three fixed points , , and in . Furthermore, ; ; .
Let be a constant. Define , for all . Then, one can easily see that is a concave non-negative continuous functional. We have the following result.
Theorem 11. Consider BVP (2). Suppose that . Assume that there exist positive constants and with , such that (H1) there exists a constant , such that (H2) there exists a constant , such that (H3) there exists a constant , such that Then, BVP (2) has at least three positive solutions.
Proof. Let us divide the proof into 4 steps.
Step 1. By (H1) and Lemma 5, for any , we have
which implies that . Thus, .
Next, let us show that is completely continuous.
By the continuity of and , we can get that is continuous and is bounded. Moreover, there exists a constant such that , for all , .
In addition, for any and , we have
Thus, the Arzela-Ascoli theorem guarantees that is compact. Therefore, is completely continuous.
Step 2. Choose a constant . Let , for all . Then, and . Thus, .
Now, let us prove that holds for all . In fact, implies that , for all . For any , one can obtain from (H2) that Hence, condition (i) of Lemma 10 holds.
Step 3. It is easy to see from (H3) that for all , for all , there exists , such that Let . Now, we prove that for all . As a matter of fact, for all , one can see that which implies that . Thus, , for all .
Step 4. Let us prove that holds for all with .
For with , we have , for all . From (H2), one can see that Therefore, condition (iii) of Lemma 10 is satisfied.
To sum up, all conditions of Lemma 10 hold. By Lemma 10, BVP (2) has at least three positive solutions.
Remark 12. It is noted that (H2) is dependent of the selection on . Specifically, to obtain a weaker condition, one can choose which maximizes
4. Illustrative Examples
In this section, we give three illustrative examples to support our new results.
Example 2. Consider the following BVP: where , , , and , are arbitrary.
Example 3. Consider the following BVP: where
The authors would like to thank the anonymous reviewers for their constructive comments and suggestions which improved the quality of the paper. This work is supported by the National Natural Science Foundation of China (G61174036), the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education, and the Graduate Independent Innovation Foundation of Shandong University (yzc10064).
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