Multiple Positive Solutions to Nonlinear Boundary Value Problems of a System for Fractional Differential Equations

Zhai, Chengbo; Hao, Mengru

doi:https://doi.org/10.1155/2014/817542

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Research Article | Open Access

Volume 2014 | Article ID 817542 | https://doi.org/10.1155/2014/817542

Multiple Positive Solutions to Nonlinear Boundary Value Problems of a System for Fractional Differential Equations

Chengbo Zhai¹and Mengru Hao¹

Academic Editor: L. Acedo, N. Hussain

Received05 Aug 2013

Accepted03 Nov 2013

Published23 Jan 2014

Abstract

By using Krasnoselskii's fixed point theorem, we study the existence of at least one or two positive solutions to a system of fractional boundary value problems given by where is the standard Riemann-Liouville fractional derivative, for and , subject to the boundary conditions , for , and , for , or , for , and , , for , Our results are new and complement previously known results. As an application, we also give an example to demonstrate our result.

1. Introduction

The purpose of this paper is to consider the existence of multiple positive solutions for the following system of nonlinear fractional differential equations: where , for and , and , subject to a couple of boundary conditions. In particular, we first consider (1) subject to where , and . We then consider the case in which the boundary conditions are changed to where .

Fractional differential equations arise in many fields, such as physics, mechanics, chemistry, economics, and engineering and biological sciences; see [1–11] for example. In recent years, the study of positive solutions for fractional differential equation boundary value problems has attracted considerable attention, and fruits from research into it emerge continuously. For a small sample of such work, we refer the reader to [12–20] and the references therein. The situation of at least one positive solution has been studied in many excellent monograph; see [12–19, 21] and other references therein. In [22], by means of Schauder fixed point theorem, Su investigated the existence of one positive solution to the following boundary value problem for a coupled system of nonlinear fractional differential equations: where .

In [21], Goodrich established the existence of one positive solution to problems (1)-(2) and (1), (3) by using Krasnoselskii's fixed point theorem. Different from the above works mentioned, in this paper we will present the existence of at least two positive solutions to problems (1)-(2) and (1), (3) by using the similar method presented in [21]. Moreover, under different conditions, we also present the existence of at least one positive solution to problems (1)-(2) and (1), (3) with .

2. Preliminaries

For the convenience of the reader, we present here some definitions, lemmas, and basic results that will be used in the proofs of our theorems.

Definition 1 (see [23]). Let with . Suppose that . Then the th Riemann-Liouville fractional integral is defined to be whenever the right-hand side is defined. Similarly, with and , we define the th Riemann-Liouville fractional derivative to be where is the unique positive integer satisfying and .

Lemma 2 (see [24]). Let be given. Then the unique solution to problem together with the boundary conditions , where and , is where is the Green function for this problem.

Lemma 3 (see [24]). Let be as given in the statement of Lemma 2. Then one finds that(i) is a continuous function on the unit square ;(ii) for each ;(iii), for each .

Lemma 4 (see [24]). Let be as given in the statement of Lemma 2. Then there exists a constant such that To prove our results, we need the following Krasnoselskii's fixed point theorem which can be seen in Guo and Lakshmikantham [25].

Lemma 5 (see [25]). Let be a Banach space, and let be a cone. Assume that are open bounded subsets of with , , and let be a completely continuous operator such that(i), and ; or(ii), and .

Then has a fixed point in .

3. Main Results

In this section, we apply Lemma 5 to study problems (1)-(2) and (1), (3), and we obtain some new results on the existence of multiple positive solutions.

3.1. Problem (1)-(2) in the General Case

In our considerations, let represent the Banach space of when equipped with the usual supremum norm, . Then put , where is equipped with the norm for . Observe that is also a Banach space (see [26]). In addition, we define two operators by where is the Green function of Lemma 2 with replaced by and, likewise, is the Green function of Lemma 2 with replaced by . Now, we define an operator by We claim that whenever is a fixed point of the operator defined in (11), it follows that and solve problems (1)-(2). That is, a pair of functions is a solution of problems (1)-(2) if and only if is a fixed point of the operator defined in (11) (see [26]).

In the following, we will look for fixed points of the operator , because these fixed points coincide with solutions of problems (1)-(2). For use in the sequel, let and be the constants given by Lemma 4 associated, respectively, with the Green functions and , and define by , and notice that .

For the sake of convenience, we set

Now we list some assumptions:();();()there are numbers , where such that , .

Next, we define the cone by

Lemma 6 (see [21]). Let be the operator defined by (11). Then .

Lemma 7. is a completely continuous operator.

Proof. The operator is continuous in view of nonnegativeness and continuity of , and .
Let be bounded; that is, there exists a positive constant such that , for all . Let ; then, for , we have Hence, is bounded.
On the other hand, given , setting , then, for each , , , and , one has . That is to say, is equicontinuity. In fact,
In the following, we divide the proof into two cases.
Case 1. If , then we have where .
Case 2. If , then we have By the means of the Arzela-Ascoli theorem, we have that is completely continuous. Similarly, is completely continuous. Consequently, is a completely continuous operator. This completes the proof.

In [21], Goodrich established the following result.

Theorem 8 (see Theorem 3.3 in [21]). Suppose that are satisfied. Then problem (1)-(2) has at least one positive solution.

From Theorem 8, the following problem is natural: whether we can obtain some conclusions or not, if or In the rest of this paper, we give some answers to this problem.

For the sake of convenience, we make some assumptions:there exist constants , such that there exist constants , such that there are numbers , where such that ;there are numbers , where such that .

Theorem 9. Suppose that and are satisfied. Then problem (1)-(2) has at least two positive solutions , , such that .

Proof. From Lemma 7, is a completely continuous operator. At first, in view of , we have , for ; , for , where satisfies . Set . So we define by . Then for each , we find that So for .
Similarly, we find that for . Consequently, whenever . Thus, is cone expansion on .
Next, since , we have for ; for , where satisfies . Set . Let and . Then implies So we obtain So for .
Similarly, we find that for .
Consequently, , whenever . Thus, is cone expansion on .
Finally, let . For , from , , we have Similarly, we find that for .
Consequently, , whenever . Thus, is cone compression on .
So, from Lemma 5, has a fixed point and a fixed point . Both are positive solutions of BVP (1)-(2) with The proof is complete.

Theorem 10. Suppose that and , are satisfied. Then problem (1)-(2) has at least two positive solutions , , such that .

Proof. At first, in view of , we have , , for , where satisfies . Let .
Then for each , we find that Like Theorem 9, we get for .
Next, in view of , we have , , for , where satisfies . We consider two cases.
Case 1. Suppose that is unbounded; there exists such that Since , one has for . Then, for and , we obtain
Case 2. Suppose that is bounded; there exists such that for all . Taking , for and , we obtain Hence, in either case, we always may set such that for . Like Theorem 9, we get , for .
Finally, set . Then implies Hence we have Consequently, for . Like Theorem 9, we get for .
So, from Lemma 5, has a fixed point and a fixed point . Both are positive solutions of BVP (1)-(2) with which complete the proof.

3.2. Problem (1)(3) in Case

In the following, for the sake of convenience, set Assume that there exist two positive constants such that, for ;, for .

Theorem 11. Suppose that and are satisfied. Then problem (1)-(2), in the case where , has at least one positive solution such that between and .

Proof. With loss of generality, we may assume that .
Let . For , one has Like Theorem 9, we get for .
Now, set . Then for , one has Thus, we get Like Theorem 9, we get for . Hence, from Lemma 5, we complete the proof.

Remark 12. In [21], problem (1)-(2) with is not considered.

3.3. Problem (1), (3) in the General Case

Consider the following.

Lemma 13 (see [21]). A pair of functions is a solution of (1), (3) if and only if is a fixed point of the operator defined by where are defined by

Lemma 14 (see [21]). Each of and is strictly increasing in t and satisfies and . Moreover, there exist constants and satisfying such that and .
Let one define a new cone by where . It is obvious that .

Lemma 15 (see [21]). is a completely continuous operator.
Now, one assumes, for each ;There are numbers , where such that .There are numbers , where such that .

Theorem 16. Suppose that and , , are satisfied. Then problem (1), (3) has at least two positive solutions ,, such that

Proof. At first, in view of , we have , , for , where satisfies .
Let . Then for each , we find that So for .
Similarly, we find that for . Consequently, whenever . Thus, is cone expansion on .
Next, since , we get , , for , where satisfies . Let and ; then, implies So for , we obtain That is, for .
Similarly, we find that for . Consequently, , whenever . Thus, is cone expansion on .
Finally, let . For , from , and , we have Similarly, we find that for . Consequently, , whenever . Thus, is cone compression on .
So, from Lemma 5, has a fixed point and a fixed point . Both are positive solutions of BVP (1), (3) with The proof is complete.

Theorem 17. Suppose that and , , are satisfied. Then problem (1), (3) has at least two positive solutions , , such that

Proof. At first, in view of , we have , for , where satisfies . Let . Then for each , we find that Like Theorem 16, we get for .
Next, in view of , we have , , for , where satisfies . We consider two cases.
Case 1. Suppose that is unbounded and there exists such that Since , one has for .
Then, for and , we obtain
Case 2. Suppose that is bounded; there, exists such that for all . Taking , for and , we obtain Hence, in either case, we always may set such that for .
Like Theorem 16, we get , for .
Finally, set . Then implies Hence we have So, for .
Like Theorem 16, we get for . So, from Lemma 5, has a fixed point and a fixed point . Both are positive solutions of BVP (1), (3) with , which complete the proof.

3.4. Problem (1), (3) in Case

In [21], the author obtained that problem (1), (3) with having at least one positive solution. In the following, we also establish the existence of one positive solution to problem (1), (3) with under different conditions.

For the sake of convenience, set Assume that there exist two positive constants such that for ; for .

Theorem 18. Suppose that ,, and are satisfied. Then problem (1), (3), in the case where , has at least one positive solution such that between and .

Proof. With loss of generality, we may assume that . Let . For , from , , one has Like Theorem 16, we get for .
Now, set . For , one has Thus, from , we get Like Theorem 16, we get for . Hence, from Lemma 5, we complete the proof.

4. An Example

To illustrate how our main results can be used in practice, we present one example.

Example 1. Consider the following BVP, for : subject to the boundary conditions Obviously, problem (63)–(65) fits the framework of problem (1)-(2) with In addition, we have set We can see that and are continuous. The functions and are obviously nonnegative.
Now, observe that holds. Again set , because , is monotone increasing function for , taking ; then, when , we get which implies that holds.
On the other hand, to calculate the admissible range of the eigenvalues , , as given by condition , observe by numerical approximation, we find that Thus, for any , satisfying , , condition will be satisfied.
Consequently, by Theorem 9, problem (63)–(65) has at least two positive solutions.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgments

The first author was supported financially by the Youth Science Foundations of China (11201272) and Shanxi Province (2010021002-1).

References

K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, London, UK, 1974.
L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, no. 2, pp. 81–88, 1991.
View at: Google Scholar
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Geneva, Switzerland, 1993.
R. Metzler, W. Schick, H. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” The Journal of Chemical Physics, vol. 103, no. 16, pp. 7180–7186, 1995.
View at: Google Scholar
W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46–53, 1995.
View at: Google Scholar
I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006.
View at: Google Scholar
E. M. Rabei, K. I. Nawafleh, R. S. Hijjawi, S. I. Muslih, and D. Baleanu, “The Hamilton formalism with fractional derivatives,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 891–897, 2007.
View at: Publisher Site | Google Scholar
V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 69, no. 10, pp. 3337–3343, 2008.
View at: Publisher Site | Google Scholar
N. Kosmatov, “A singular boundary value problem for nonlinear differential equations of fractional order,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 125–135, 2009.
View at: Publisher Site | Google Scholar
Z. B. Bai and H. S. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.
View at: Publisher Site | Google Scholar
E. R. Kaufmann and E. Mboumi, “Positive solutions of a boundary value problem for a nonlinear fractional differential equation,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 3, pp. 1–11, 2008.
View at: Google Scholar
S. H. Liang and J. H. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 11, pp. 5545–5550, 2009.
View at: Publisher Site | Google Scholar
K. Sadarangani, J. Caballero Mena, and J. Harjani, “Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems,” Boundary Value Problems, vol. 2009, Article ID 421310, 10 pages, 2009.
View at: Publisher Site | Google Scholar
S. Q. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1300–1309, 2010.
View at: Publisher Site | Google Scholar
L. Yang and H. Chen, “Unique positive solutions for fractional differential equation boundary value problems,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1095–1098, 2010.
View at: Publisher Site | Google Scholar
M. El-Shahed, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Abstract and Applied Analysis, vol. 2007, Article ID 10368, 8 pages, 2007.
View at: Publisher Site | Google Scholar
S. H. Liang and J. H. Zhang, “Existence and uniqueness of strictly nondecreasing and positive solution for a fractional three-point boundary value problem,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1333–1340, 2011.
View at: Publisher Site | Google Scholar
Y. G. Zhao, S. R. Sun, Z. L. Han, and Q. P. Li, “The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 2086–2097, 2011.
View at: Publisher Site | Google Scholar
C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1251–1268, 2011.
View at: Publisher Site | Google Scholar
X. W. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 22, no. 1, pp. 64–69, 2009.
View at: Publisher Site | Google Scholar
I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
C. S. Goodrich, “Existence of a positive solution to a class of fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1050–1055, 2010.
View at: Publisher Site | Google Scholar
D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, NY, USA, 1988.
D. R. Dunninger and H. Wang, “Existence and multiplicity of positive solutions for elliptic systems,” Nonlinear Analysis, Theory, Methods and Applications, vol. 29, no. 9, pp. 1051–1060, 1997.
View at: Google Scholar

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Copyright © 2014 Chengbo Zhai and Mengru Hao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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