Abstract
In this article, we investigate the SturmLiouville boundary value problems of fractional differential equations with pLaplacian
where , , , are the standard Caputo fractional derivatives, , , , , , , and is continuous. By means of the properties of the Green’s function, LeggettWilliams fixedpoint theorems, and fixedpoint index theory, several new sufficient conditions for the existence of at least two or at least three positive solutions are obtained. As an application, an example is given to demonstrate the main result.
MSC: 34A08, 34B18, 35J05.
Keywords:
SturmLiouville boundary value problem; positive solution of fractional differential equation; LeggettWilliams fixedpoint theorem; fixedpoint index theory; pLaplacian operator1 Introduction
During the past decades, much attention has been focused on the study of equations with pLaplacian differential operator. The motivation for those works stems from the applications in the modeling of different physical and natural phenomena: nonNewtonian mechanics [1], system of MongeKantorovich partial differential equations [2], population biology [3], nonlinear flow laws [4], combustion theory [5]. There exist a very large number of papers devoted to the existence of solutions for the equation with pLaplacian operator.
The ordinary differential equation with pLaplacian operator
subject to various boundary conditions, has been studied by many authors, see [6,7] and the references therein.
The existence of positive solutions of the differential equation with pLaplacian operator
satisfying different boundary conditions have been established by using fixedpoint theorems and monotone iterative technique, see [8,9] and the references therein.
In [10], Hai considered the existence of positive solutions for the boundary value problem
where , , and λ is a positive parameter, f is psuperlinear or psublinear at ∞ and maybe singular at .
However, few papers can be found in the literature on the existence of multiple positive solutions for the thirdorder SturmLiouville boundary value problem with pLaplacian.
In [11], Zhai and Guo studied the thirdorder SturmLiouville boundary value problem with pLaplacian
where , , , , , . By means of the LeggettWilliams fixedpoint theorems, some existence and multiplicity results of positive solutions are obtained. In later work, Yang and Yan [12] also studied the above problem by means of the fixedpoint index method.
Recently, fractional differential equations have been of great interest. The motivation for those works stems from both the intensive development of the theory of fractional calculus itself and the applications such as economics, engineering and other fields [1317]. Much attention has been focused on the study of the existence and multiplicity of solutions or positive solutions for boundary value problems of fractional differential equations by the use of techniques of nonlinear analysis (fixedpoint theorems [1825], upper and lower solutions method [26], fixedpoint index theory [27,28], coincidence theory [29], etc.).
Although the boundary value problems of fractional differential equation with pLaplacian have been studied in many literature, only few papers can be found in the literature on the existence of multiple positive solutions for the SturmLiouville boundary value problems of fractional differential equations with pLaplacian. As the extension and supplement of some results in [11,12], in this article, we investigate the SturmLiouville boundary value problems of fractional differential equations with pLaplacian subject Robin boundary value conditions
where , , , are the standard Caputo fractional derivatives, , , , , , , and is continuous. By means of the properties of the Green’s function, LeggettWilliams fixedpoint theorems and fixedpoint index theory, we establish the existence of at least two or at least three positive solutions for the SturmLiouville boundary value problem (1.1). As an application, an example is given to demonstrate the main result.
The rest of this paper is organized as follows. In Section 2, we shall introduce some definitions and lemmas to prove our main results. In Section 3, we state our main results. We prove our main results by LeggettWilliams fixedpoint theorems and fixedpoint index theory in Section 4. As an application, an example is presented to illustrate our main result in Section 5.
2 Preliminaries and lemmas
For the convenience of the reader, we give some background materials from fractional calculus theory to facilitate analysis of problem (1.1). These materials can be found in the recent literature, see [14,17,18,3133].
Definition 2.1 ([17])
The RiemannLiouville fractional integral of order of a function is given by
provided the right side is pointwise defined on .
Definition 2.2 ([17])
The Caputo fractional derivative of order of a continuous function is given by
where n is the smallest integer greater than or equal to α, provided that the right side is pointwise defined on .
Remark 2.1 ([14])
By Definition 2.2, under natural conditions on the function , as the Caputo derivative becomes a conventional nth derivative of the function .
Remark 2.2 ([18])
As a basic example, we have
given in particular that , , where is the Caputo fractional derivative, and n is the smallest integer greater than or equal to α.
From the definition of the Caputo derivative and Remark 2.2, we can obtain the following statement.
Lemma 2.1 ([17])
Let. Then the fractional differential equation
has
as the unique solution, wherenis the smallest integer greater than or equal toα.
Lemma 2.2 ([17])
Let. Assume that. Then the following equality holds:
for some, , wherenis the smallest integer greater than or equal to α.
Lemma 2.3Letand. Then the boundary value problem of the fractional differential equation
has a unique solution,
where
Proof By the Lemma 2.2, we can reduce the equation of problem (2.1) to an equivalent integral equation
for some constants . Moreover, we have
From the boundary conditions , , we have
So,
Hence, the unique solution of (2.1) is
which completes the proof. □
Lemma 2.4Let, . Then the boundary value problem of the fractional differential equation
has a unique solution,
Proof From Lemma 2.2 and the boundary value problem (2.3), we have
that is
By , we have . So, . Thus, the boundary value problem (2.3) is equivalent to the following problem:
Lemma 2.3 implies that boundary value problem (2.3) has a unique solution,
which completes the proof. □
Lemma 2.5The Green’s functiondefined by (2.2) is continuous on.
Assume, thenalso has the following properties:
(3) there exists a positive numberλsuch that, for, where.
The method of proof is similar to Lemma 3.2 in [30], and we omit it here.
Definition 2.3 ([31])
Let E be a real Banach space and P be a nonempty, convex closed set in E. We say that P is a cone if it satisfies the following properties:
(ii) implies , where θ denotes the null element of E.
If is a cone, we denote the order induced by P on E by ≤. For , we write if .
Definition 2.4 ([31])
The map φ is said to be a nonnegative continuous concave functional on P of a real Banach space E provided that is continuous and
Definition 2.5 ([31])
Let be given and let φ be a nonnegative continuous concave functional on the cone P. Define the convex sets , and by , , .
Lemma 2.6 (LeggettWilliams [32])
Letbe a completely continuous operator and letφbe a nonnegative continuous concave functional onPsuch thatfor all. Suppose that there existsuch that
ThenThas at least three fixed points, , andinsatisfying, , , and.
Lemma 2.7 ([32])
Letbe a completely continuous operator and letφbe a nonnegative continuous concave functional onPsuch thatfor all. Suppose that there existsuch that
ThenThas at least two fixed pointsandinsatisfying, and.
Lemma 2.8 ([31])
LetPbe a closed convex set in a Banach spaceEand let Ω be a bounded open set such that. Letbe a compact map. Suppose thatfor all.
(C_{1}) (Existence) If, then T has a fixed point in.
(C_{2}) (Normalization) If, then, wherefor.
(C_{3}) (Homotopy) Letbe a compact map such thatforand. Then.
(C_{4}) (Additivity) If, are disjoint relatively open subsets ofsuch thatfor, then, where ().
Lemma 2.9 ([33])
LetPbe a cone in a Banach spaceE. For, define. Assume thatis a compact map such thatfor. Thus, one has the following conclusions:
3 Main theorems
In this section, let be the Banach space of continuous functions endowed with , and the ordering if for all . Define the cone by
where λ is given as in Lemma 2.5.
For convenience of the reader, we denote
Lemma 3.1Letbe the operator defined by
Proof By Lemma 2.5, we have
Thus, . In view of nonnegativity and continuity of , and , we find that is continuous.
Let be bounded, i.e., there exists a positive constant such that , for all . Let , then, for , we have
Hence, is uniformly bounded. Further for any and , we have
Hence, . For any and , we have
That is to say, is equicontinuous. By the ArzelaAscoli theorem, we see that is completely continuous. The proof is completed. □
We are now ready to prove our main results.
Theorem 3.1Letbe nonnegative continuous on. Assume that there exist constantsa, bwithsuch that
Then the boundary value problem (1.1) has at least two positive solutionsandsatisfying, and, whereλis given as in Lemma 2.5.
Proof Let be the nonnegative continuous concave functional defined by
Evidently, for each , we have .
It’s easy to see that is completely continuous and . We choose , then
So . Hence, if , then for . Thus for , from assumption (H_{1}), we have
Consequently,
That is,
Therefore, condition (B_{1}) of Lemma 2.7 is satisfied. Now if , then . By assumption (H_{2}), we have
which shows that , that is, for . This shows that condition (B_{2}) of Lemma 2.7 is satisfied. Finally, we show that (B_{3}) of Lemma 2.7 also holds. Assume that with , then by the definition of cone P, we have
So condition (B_{3}) of Lemma 2.7 is satisfied. Thus using Lemma 2.7, T has at least two fixed points. Consequently, the boundary value problem (1.1) has at least two positive solutions and in satisfying , and . The proof is completed. □
Theorem 3.2Letbe nonnegative continuous on. Assume that there exist constantsa, b, cwithsuch that
Then the boundary value problem (1.1) has at least three positive solutions, andwith, , and, whereλis given as in Lemma 2.5.
Proof If , then . By assumption (H_{5}), we have
This shows that . Using the same arguments as in the proof of Lemma 3.1, we can show that is a completely continuous operator. It follows from the conditions (H_{3}) and (H_{4}) in Theorem 3.2 that . Similarly with the proof of Theorem 3.1, we have and
So all the conditions of Lemma 2.6 are satisfied. Thus using Lemma 2.6, T has at least three fixed points. So, the boundary value problem (1.1) has at least three positive solutions , and with , , and . The proof is completed. □
Theorem 3.3Letbe nonnegative continuous on. If the following assumptions are satisfied:
(H_{7}) there exists a constantsuch that
then the boundary value problem (1.1) has at least two positive solutionsandsuch that.
Proof From Lemma 3.1, we obtain is completely continuous. In view of , there exists such that
which implies for . Hence, Lemma 2.9 implies
On the other hand, since , there exists such that
Let and . Then , for any . By using the method to get (3.1), we obtain
which implies for . Thus, from Lemma 2.9, we have
Finally, let . Then, for any , by (H_{7}), we then get
which implies for . Using Lemma 2.9 again, we get
Note that , by the additivity of fixedpoint index and (3.1)(3.3), we obtain
and
Hence, T has a fixed point in , and has a fixed point in . Clearly, and are positive solutions of the boundary value problem (1.1) and . The proof is completed. □
Theorem 3.4Letbe nonnegative continuous on. If the following assumptions are satisfied:
(H_{9}) there exists a constantsuch that
Then the boundary value problem (1.1) has at least two positive solutionsandsuch that.
Proof From Lemma 3.1, we obtain is completely continuous. In view of , there exists such that
which implies for . Hence, Lemma 2.9 implies
Next, since , there exists such that
where . We consider two cases.
Case 1: Suppose that f is bounded, which implies that there exists such that for all and .
Take . Then, for with , we get
Case 2: Suppose that f is unbounded. In view of being continuous, there exist and such that
So, in either case, if we always choose , then we have
Thus, from Lemma 2.9, we have
Finally, Let . Then, for any , , by (H_{9}), and we then obtain
which implies for . An application of Lemma 2.9 again shows that
Note that ; by the additivity of fixedpoint index and (3.4)(3.6), we obtain
and
Hence, T has a fixed point in , and it has a fixed point in . Consequently, and are positive solutions of the boundary value problem (1.1) and . The proof is completed. □
4 Example
In this section, we present an example to illustrate the main result.
Example 4.1 We consider the boundary value problem of the fractional differential equation
where
Let . We note that , , , . By a simple calculation, we obtain , and
Consequently, all the conditions of Theorem 3.2 are satisfied. With the use of Theorem 3.2, the boundary value problem (4.1) has at least three positive solutions , , and with
5 Conclusion
In this paper, the SturmLiouville boundary value problems of fractional differential equations with pLaplacian are investigated, the existence of at least two or at least three positive solutions for the fractional differential equations with Robin boundary conditions are given by using LeggettWilliams fixedpoint theorems and the fixedpoint index theory, respectively.
It is worth emphasizing that our work presented in this article has the following features: Firstly, the boundary conditions in (1.1) are important Robin boundary conditions. Secondly, our results improve and extend the main results of [11,12] for the SturmLiouville boundary value problems of integerorder differential equations with pLaplacian. For example, if , , then the problem (1.1) reduces to
which is studied in [11,12]. Furthermore, if we take , problem (1.1) is the usual form of thirdorder SturmLiouville boundary value problem
The method can be applied on the SturmLiouville boundary value problems of higherorder fractional differential equations with pLaplacian and boundary conditions involving fractional derivatives
Based on this paper, one can consider boundary value problems of fractional differential equations with parameters, and also one can do further research on eigenvalue problems of fractional differential equations with pLaplacian.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143, 61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009).
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