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Positive Solution for the Nonlinear Hadamard Type Fractional Differential Equation with -Laplacian
We study the following nonlinear fractional differential equation involving the -Laplacian operator , , , , where the continuous function , , . denotes the standard Hadamard fractional derivative of the order , the constant , and the -Laplacian operator . We show some results about the existence and the uniqueness of the positive solution by using fixed point theorems and the properties of Green's function and the -Laplacian operator.
Fractional differential equations have attracted more and more attention for their useful applications in various fields, such as economics, science, and engineering; see [1–4]. In the last few decades, much attention has been focused on the study of the existence and uniqueness of solutions for boundary value problems of Riemann-Liouville type or Caputo type fractional differential equations; see [5–19]. There are few papers devoted to the research of the -Laplacian fractional differential equations; see [20–25].
By the use of the fixed point theorem on cones, Chai in  obtained the existence and multiplicity of positive solutions for a class of boundary value problem of fractional differential equation with -Laplacian operator: where , , , , is a positive constant number, and are the standard Riemann-Liouville derivatives. , , , .
Han et al. in  studied the following boundary value problem of nonlinear fractional differential equation with -Laplacian operator: where , , , , is a parameter, and are the standard Caputo fractional derivatives. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results and the uniqueness of positive solutions are acquired.
Liu et al. in  investigated the solvability of a fractional differential equation model involving the -Laplacian operator with boundary value conditions as follows: where , , , and is the standard Caputo derivative. By the means of the Banach contraction mapping principle, they obtained the existence and uniqueness of a solution for the model.
Lu et al. in  considered the following fractional boundary value problem with -Laplacian operator: where , , and are the standard Riemann-Liouville fractional derivatives. By the properties of Green’s function, the Guo-Krasnosel’skii fixed point theorem, the Leggett-Williams fixed point theorem, and the upper and lower solutions method, some new results on the existence of positive solutions are gained.
Motivated by the mentioned papers, we will consider the Hadamard fractional boundary value with -Laplacian operator as below: where , , , and is a positive continuous function. Evidently, for any , , here . Here is the standard Hadamard fractional derivative of order which is described as follows.
Definition 1 (see [1, Page 111]). The th Hadamard fractional order derivative of a function is defined by where , , and denotes the largest integer which is less than or equal to . Correspondingly, the th Hadamard fractional order integral of is defined by where is the gamma function.
To the best of our knowledge, there are few contributions to the Hadamard type with -Laplacian operator; we fill the gap in this paper. In fact, we will discuss the existence and the uniqueness of the positive solutions of (5). The structure of this paper goes on as follows. In Section 2, we will introduce some basic lemmas that will be used. In Section 3, we first give some existence results including Theorems 10 and 11, Corollary 12, and Theorem 13. Then, we will prove Theorems 14 and 15 which reveal the uniqueness of the solution. In Section 4, we give two examples to illustrate our results.
2. Preliminary Results
In this section, we will first recall the following preliminary facts that will be used in our main results.
Lemma 2 (see [1, Theorem 2.3]). Let , ; then where , , are some constants in .
The following lemma is the Schauder fixed point theorem which is well known; see Theorem 2.10 in .
Lemma 3. If is a nonempty closed, bounded, and convex subset of a Banach space and is completely continuous, then has a fixed point in .
Lemma 4 (see [24, Lemma 2.7]). Let be a Banach space, let be a cone, and let , be two bounded open balls of centered at the origin with . Suppose that is a completely continuous operator such that either(i), and , , or(ii), and , ,holds. Then has a fixed point in .
The following conclusion is the nonlinear alternative of Leray-Schauder type; see Lemma 2.6 in .
Lemma 5. Let be a Banach space with being closed and convex. Assume that is a relatively open subset of with and is a continuous, compact map. Then either (1) has a fixed point in , or (2)there exists and , with .
Next, we give several lemmas which will be applied in the proofs of our main results.
Lemma 6. Let be the solution of the problem (5); then it can be described as below: where
Proof. Putting , we have . By Lemma 2 and the fact that ,
The boundary value hypotheses give . So we can get that
Notice the fact that , ; we have Putting , it follows from Lemma 2 that This, combined with the fact that , yields Thus, where
This completes the proof of Lemma 6.
Lemma 7. Suppose that , . Then the functions and defined in (10) have the following properties: (1), are continuous on ; (2)for any , , ; (3)for any , , ; (4)there exist two positive functions such that
Proof. (1) and (2) are evident from the expression of and . Since, for any fixed number , is an increasing function on and is a decreasing function on and increasing on , we get (3). To prove (4), suppose that
The monotonicity of gives
Which implies that (19) holds.
Similarly, by writing and applying the monotonicity of , we have where is the unique solution of the equation Hence, setting we obtain (20). This completes the proof of Lemma 7.
Let , . we define the cone and the bounded closed set .
The operator is defined as the following form: Evidently, the solutions of boundary value problem (5) are the corresponding fixed points of the operator .
Lemma 8. Suppose that is an operator as above; then is completely continuous.
Proof. It is easy to see that is continuous. Let be a bounded set; then there is a positive constant such that for any . Write . For any , we have
which shows that is uniformly bounded.
Next, the continuity of implies that, for any , there exists a constant such that, for any , if , then Therefore, for any , That is, is equicontinuity. By the means of Arzela-Ascoli theorem , we have that is completely continuous. This completes the proof of Lemma 8.
In the final part of this section, we list the following basic properties of the -Laplacian operator.
(1) If , , and , then
(2) If , then
3. Proofs of the Main Results
In this section, first, we consider the existence of the solutions of problem (5).
Theorem 10. If , then the boundary value problem (5) has at least one positive solution.
Proof. For any , by the assumption as above and the nonnegativeness of , , and , we have
Therefore, is a mapping from to . This, combined with the continuity of , , and , implies that is continuous.
Let be a bounded set; then there exists a positive constant such that for any . So we have, for any , Therefore, is uniformly bounded.
Since is continuous, for any , there exists a constant satisfying that, for any and , Then, for any , which shows that is equicontinuous. By Arzela-Ascoli theorem , is a completely continuous operator. It follows from Lemma 3 that has a fixed point in . That is, problem (5) has at least one positive solution. This completes the proof of Theorem 10.
Let us denote
Theorem 11. Suppose that is a continuous function and there exist two constants satisfying that (i), for ; (ii), for . Then the boundary value problem (5) has at least one positive solution which satisfies that .
Proof. Let . For any , we have for . By the assumption (i), for any , Hence, Similarly, let . For any , we get , . It follows from (ii) that for any , Therefore, By Lemmas 4 and 8, has a fixed point in . Therefore, the boundary value problem (5) has one positive solution in . This completes the proof of Theorem 11.
Corollary 12. Suppose that is a continuous function and there exist two constants satisfying that (i), for ; (ii), for . Then the boundary value problem (5) has at least one positive solution which satisfies that .
Theorem 13. Suppose that is a positive continuous function and there exists a constant such that Then the boundary value problem (5) has at least one positive solution.
Proof. Let From Lemma 8, we know is completely continuous. Assume that there exist , such that . Then we have Thus, By (43), we can imply that , which means that . That is to say, there is no such that for some . Therefore, by Lemma 5, we conclude that the problem (5) has at least one positive solution. This completes the proof of Theorem 13.
Now we turn to the uniqueness of solution for boundary value problem (5).
Theorem 14. Suppose that . If there exists a nonnegative function satisfying that (1)for any , ; (2); (3)for any , , , where then the boundary value problem (5) has a unique solution.
Proof. Assume that are two positive solutions of problem (5). It is easy to see that then by the fact (i.e., its dual exponent ) and Lemma 9, we have where . So we can get By the third hypothesis, , which implies that . And this completes the proof of Theorem 14.
By using the same way, we can prove the last one of our main uniqueness results.
Theorem 15. Suppose that . If there exists a nonnegative function satisfying that (1)for any , ; (2); (3)for any , , , where then the boundary value problem (5) has a unique solution.
In this section we give several examples to illustrate our main results.
Example 16. Consider the boundary value problem: Then the boundary value problem has a unique positive solution.
Proof. Since , , a straightforward calculation gives
Taking ,, and (its dual exponent ), we have By Theorem 10, the boundary value problem (52) has at least one positive solution.
Choosing the nonnegative function , for any , we gain that . Then