Abstract
In this article, we investigate the Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian
where , , , are the standard Caputo fractional derivatives, , , , , , , and is continuous. By means of the properties of the Green’s function, Leggett-Williams fixed-point theorems, and fixed-point 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.
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1 Introduction
During the past decades, much attention has been focused on the study of equations with p-Laplacian differential operator. The motivation for those works stems from the applications in the modeling of different physical and natural phenomena: non-Newtonian mechanics [1], system of Monge-Kantorovich 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 p-Laplacian operator.
The ordinary differential equation with p-Laplacian 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 p-Laplacian operator
satisfying different boundary conditions have been established by using fixed-point 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 p-superlinear or p-sublinear at ∞ and maybe singular at .
However, few papers can be found in the literature on the existence of multiple positive solutions for the third-order Sturm-Liouville boundary value problem with p-Laplacian.
In [11], Zhai and Guo studied the third-order Sturm-Liouville boundary value problem with p-Laplacian
where , , , , , . By means of the Leggett-Williams fixed-point 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 fixed-point 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 [13–17]. 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 (fixed-point theorems [18–25], upper and lower solutions method [26], fixed-point index theory [27, 28], coincidence theory [29], etc.).
Although the boundary value problems of fractional differential equation with p-Laplacian have been studied in many literature, only few papers can be found in the literature on the existence of multiple positive solutions for the Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian. As the extension and supplement of some results in [11, 12], in this article, we investigate the Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian 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, Leggett-Williams fixed-point theorems and fixed-point index theory, we establish the existence of at least two or at least three positive solutions for the Sturm-Liouville 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 Leggett-Williams fixed-point theorems and fixed-point 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, 31–33].
Definition 2.1 ([17])
The Riemann-Liouville 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 n th 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, where n is the smallest integer greater than or equal to α.
Lemma 2.2 ([17])
Let . Assume that . Then the following equality holds:
for some , , where n is the smallest integer greater than or equal to α.
Lemma 2.3 Let and . 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 2.4 Let , . Then the boundary value problem of the fractional differential equation
has a unique solution,
where is defined as (2.2).
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 2.5 The Green’s function defined by (2.2) is continuous on .
Assume , then also has the following properties:
-
(1)
, for ;
-
(2)
, for ;
-
(3)
there exists a positive number λ such 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:
-
(i)
for , ;
-
(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
for all 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 (Leggett-Williams [32])
Let be a completely continuous operator and let φ be a nonnegative continuous concave functional on P such that for all . Suppose that there exist such that
(A1) and for ;
(A2) for ;
(A3) for with .
Then T has at least three fixed points , , and in satisfying , , , and .
Lemma 2.7 ([32])
Let be a completely continuous operator and let φ be a nonnegative continuous concave functional on P such that for all . Suppose that there exist such that
(B1) , and for ;
(B2) for ;
(B3) for with .
Then T has at least two fixed points and in satisfying , and .
Lemma 2.8 ([31])
Let P be a closed convex set in a Banach space E and let Ω be a bounded open set such that . Let be a compact map. Suppose that for all .
(C1) (Existence) If , then T has a fixed point in .
(C2) (Normalization) If , then , where for .
(C3) (Homotopy) Let be a compact map such that for and . Then .
(C4) (Additivity) If , are disjoint relatively open subsets of such that for , then , where ().
Lemma 2.9 ([33])
Let P be a cone in a Banach space E. For , define . Assume that is a compact map such that for . Thus, one has the following conclusions:
(D1) if for , then ;
(D2) if for , then .
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 3.1 Let be the operator defined by
Then is completely continuous.
Proof By Lemma 2.5, we have
Thus, . In view of non-negativity 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 Arzela-Ascoli theorem, we see that is completely continuous. The proof is completed. □
We are now ready to prove our main results.
Theorem 3.1 Let be nonnegative continuous on . Assume that there exist constants a, b with such that
(H1) , for ;
(H2) , for .
Then the boundary value problem (1.1) has at least two positive solutions and satisfying , and , where λ is 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 (H1), we have
Consequently,
That is,
Therefore, condition (B1) of Lemma 2.7 is satisfied. Now if , then . By assumption (H2), we have
which shows that , that is, for . This shows that condition (B2) of Lemma 2.7 is satisfied. Finally, we show that (B3) of Lemma 2.7 also holds. Assume that with , then by the definition of cone P, we have
So condition (B3) 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 3.2 Let be nonnegative continuous on . Assume that there exist constants a, b, c with such that
(H3) , for ;
(H4) , for ;
(H5) , for .
Then the boundary value problem (1.1) has at least three positive solutions , and with , , and , where λ is given as in Lemma 2.5.
Proof If , then . By assumption (H5), 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 (H3) and (H4) in Theorem 3.2 that . Similarly with the proof of Theorem 3.1, we have and
Moreover, for and , we have
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 3.3 Let be nonnegative continuous on . If the following assumptions are satisfied:
(H6) ;
(H7) there exists a constant such that
then the boundary value problem (1.1) has at least two positive solutions and such that .
Proof From Lemma 3.1, we obtain is completely continuous. In view of , there exists such that
where .
Let . Then, for any , we have
which implies for . Hence, Lemma 2.9 implies
On the other hand, since , there exists such that
where .
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 (H7), we then get
which implies for . Using Lemma 2.9 again, we get
Note that , by the additivity of fixed-point 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 3.4 Let be nonnegative continuous on . If the following assumptions are satisfied:
(H8) ;
(H9) there exists a constant such that
Then the boundary value problem (1.1) has at least two positive solutions and such that .
Proof From Lemma 3.1, we obtain is completely continuous. In view of , there exists such that
where .
Let . Then, for any , we have
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
Then, for with , we obtain
So, in either case, if we always choose , then we have
Thus, from Lemma 2.9, we have
Finally, Let . Then, for any , , by (H9), and we then obtain
which implies for . An application of Lemma 2.9 again shows that
Note that ; by the additivity of fixed-point 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
Choosing , , , evidently, 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 Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian 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 Leggett-Williams fixed-point theorems and the fixed-point 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 Sturm-Liouville boundary value problems of integer-order differential equations with p-Laplacian. 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 third-order Sturm-Liouville boundary value problem
The method can be applied on the Sturm-Liouville boundary value problems of higher-order fractional differential equations with p-Laplacian 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 p-Laplacian.
References
Diaz JI, De Thelin F: On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 1994, 25(4):1085-1111. 10.1137/S0036141091217731
Evans LC, Gangbo W: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Am. Math. Soc. 1999, 137(653):1-75.
Oruganti S, Shi J, Shivaji R: Logistic equation with p -Laplacian and constant yield harvesting. Abstr. Appl. Anal. 2004, 2004(9):723-727. 10.1155/S1085337504311097
Ly I, Seck D: Isoperimetric inequality for an interior free boundary problem with p -Laplacian operator. Electron. J. Differ. Equ. 2004, 2004(109):1-12.
Ramaswamy M, Shivaji R: Multiple positive solutions for classes of p -Laplacian equations. Differ. Integral Equ. 2004, 17(11-12):1255-1261.
Wang Y, Hou C: Existence of multiple positive solutions for one-dimensional p -Laplacian. J. Math. Anal. Appl. 2006, 315(1):144-153. 10.1016/j.jmaa.2005.09.085
Lian H, Ge W: Positive solutions for a four-point boundary value problem with the p -Laplacian. Nonlinear Anal. 2008, 68(11):3493-3503. 10.1016/j.na.2007.03.042
Sun B, Qu Y, Ge W: Existence and iteration of positive solutions for a multipoint one-dimensional p -Laplacian boundary value problem. Appl. Math. Comput. 2008, 197(1):389-398. 10.1016/j.amc.2007.07.071
Zhao X, Ge W: Successive iteration and positive symmetric solution for a Sturm-Liouville-like four-point boundary value problem with a p -Laplacian operator. Nonlinear Anal. 2009, 71(11):5531-5544. 10.1016/j.na.2009.04.060
Hai DD: On singular Sturm-Liouville boundary-value problems. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 2010, 140(1):49-63.
Zhai C, Guo C: Positive solutions for third-order Sturm-Liouville boundary-value problems with p -Laplacian. Electron. J. Differ. Equ. 2009, 2009(154):1-9.
Yang C, Yan J: Positive solutions for third-order Sturm-Liouville boundary value problems with p -Laplacian. Comput. Math. Appl. 2010, 59(6):2059-2066. 10.1016/j.camwa.2009.12.011
Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.
Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, New York; 1999.
Meral FC, Royston TJ, Magin R: Fractional calculus in viscoelasticity: an experimental study. Commun. Nonlinear Sci. Numer. Simul. 2010, 15(4):939-945. 10.1016/j.cnsns.2009.05.004
Tenreiro Machado J, Kiryakova V, Mainardi F: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(3):1140-1153. 10.1016/j.cnsns.2010.05.027
Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Zhao Y, Sun S, Han Z, Zhang M: Positive solutions for boundary value problems of nonlinear fractional differential equations. Appl. Math. Comput. 2011, 217(16):6950-6958. 10.1016/j.amc.2011.01.103
Feng W, Sun S, Han Z, Zhao Y: Existence of solutions for a singular system of nonlinear fractional differential equations. Comput. Math. Appl. 2011, 62(3):1370-1378. 10.1016/j.camwa.2011.03.076
Zhao Y, Sun S, Han Z, Li Q: The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(4):2086-2097. 10.1016/j.cnsns.2010.08.017
Nyamoradi N, Bashiri T: Multiple positive solutions for nonlinear fractional differential systems. Fract. Differ. Calc. 2012, 2(2):119-128.
Ahmad B, Nieto JJ, Alsaedi A, El-Shahed M: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl. 2012, 13(2):599-606. 10.1016/j.nonrwa.2011.07.052
Han Z, Lu H, Sun S, Yang D: Positive solutions to boundary value problems of p -Laplacian fractional differential equations with a parameter in the boundary conditions. Electron. J. Differ. Equ. 2012, 2012(213):1-14.
Chen T, Liu W: An anti-periodic boundary value problem for fractional differential equation with p -Laplacian operator. Appl. Math. Lett. 2012, 25(11):1671-1675. 10.1016/j.aml.2012.01.035
Chai G: Positive solutions for boundary value problem of fractional differential equation with p -Laplacian operator. Bound. Value Probl. 2012, 2012: 1-18. 10.1186/1687-2770-2012-1
Wang J, Xiang H: Upper and lower solutions method for a class of singular fractional boundary value problems with p -Laplacian operator. Abstr. Appl. Anal. 2010., 2010: Article ID 971824 10.1155/2010/971824
Dix JG, Karakostas GL: A fixed-point theorem for S -type operators on Banach spaces and its applications to boundary-value problems. Nonlinear Anal. 2009, 71(9):3872-3880. 10.1016/j.na.2009.02.057
Xu J, Wei Z, Dong W: Uniqueness of positive solutions for a class of fractional boundary value problems. Appl. Math. Lett. 2012, 25(3):590-593. 10.1016/j.aml.2011.09.065
Chen T, Liu W, Hu Z: A boundary value problem for fractional differential equation with p -Laplacian operator at resonance. Nonlinear Anal. 2012, 75(6):3210-3217. 10.1016/j.na.2011.12.020
Zhao Y, Chen H, Huang L: Existence of positive solutions for nonlinear fractional functional differential equation. Comput. Math. Appl. 2012, 64(10):3456-3467. 10.1016/j.camwa.2012.01.081
Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, Orlando; 1988.
Leggett RW, Williams LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 1979, 28(4):673-688. 10.1512/iumj.1979.28.28046
Deimling K: Nonlinear Functional Analysis. Springer, New York; 1985.
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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Lu, H., Han, Z. & Sun, S. Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian. Bound Value Probl 2014, 26 (2014). https://doi.org/10.1186/1687-2770-2014-26
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DOI: https://doi.org/10.1186/1687-2770-2014-26