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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 254012, 8 pages
http://dx.doi.org/10.1155/2014/254012
Research Article

Monotone Iterative Methods of Positive Solutions for Fractional Differential Equations Involving Derivatives

College of Electron and Information, Zhejiang University of Media and Communications, Hangzhou, Zhejiang 310018, China

Received 23 September 2013; Accepted 27 December 2013; Published 4 February 2014

Academic Editor: Rosana Rodriguez-Lopez

Copyright © 2014 Xiaoping Zhang and Yongping Sun. 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.

Abstract

This paper studies the existence and computing method of positive solutions for a class of nonlinear fractional differential equations involving derivatives with two-point boundary conditions. By applying monotone iterative methods, the existence results of positive solutions and two iterative schemes approximating the solutions are established. The interesting point of our method is that the iterative scheme starts off with a known simple function or the zero function and the nonlinear term in the fractional differential equation is allowed to depend on the unknown function together with derivative terms. Two explicit numerical examples are given to illustrate the results.

1. Introduction

In this paper, we discuss the existence and computing method of positive solutions of the th-order fractional boundary value problem consisting of the nonlinear fractional differential equation and the two-point boundary conditions where is an integer, is a real number, is the Riemann-Liouville fractional derivative of order , denotes the integer part of real number , and in boundary conditions (2) represents the th (ordinary) derivative of .

Fractional differential equations arise in many fields such as physics, mechanics, chemistry, economics, engineering, and biological sciences. Recently, there have been many papers dealing with the solutions or positive solutions of boundary value problems for nonlinear fractional differential equations. We refer the reader to the papers of Agarwal et al. [1], Ahmad and Sivasundaram [2], Ahmad and Nieto [3], Babakhani and Daftardar-Gejji [4], Bai and Sun [5], Bai et al. [6], Bai and Qiu [7], Caballero et al. [8], Delbosco and Rodino [9], Graef et al. [10], Jiang and Yuan [11], Lakshmikantham and Vatsala [12], Liang and Zhang [13], Qiu and Bai [14], Tian and Liu [15], Wang et al. [16], Yang and Chen [17], Yuan et al. [18], Zhang [19], Zhang and Han [20], and Zhang et al. [21, 22] and the references therein. Nonlinear fractional differential equations with two-point boundary conditions (2) have been studied by several authors. For example, in [23], Goodrich studied a fractional differential equation of the form with boundary conditions (2), where is continuous. The author obtained Green’s function of the problem and proved that Green’s function satisfied a Harnack-like inequality. By using a fixed point theorem due to Krasnosel’skii, the author established the existence results for at least one positive solution. Graef et al. in [24] found sufficient conditions to guarantee that the following fractional differential equation: with boundary conditions (2) has at least one or two positive solutions when is small and large, where is a parameter, , , , and are continuous functions. Zhai and Hao [25] discussed the existence and uniqueness of positive solutions for the following fractional differential equation: with boundary conditions (2), where and are continuous functions and satisfy some monotonicity conditions. The analysis relies on two new fixed point theorems for mixed monotone operators with perturbation. In [26], Su and Feng studied a fractional differential equation with deviating argument of the form with boundary conditions (2), where , and are continuous functions. The author obtained novel sufficient conditions for the existence of at least one or two positive solutions by using Krasnosel’skii’s fixed point theorem, and some other new sufficient conditions for the existence of at least triple positive solutions by using the fixed point theorems developed by Leggett and Williams, and so forth. Yuan [27] gave sufficient conditions for the existence of multiple positive solution for the semipositone -type boundary value problems of nonlinear fractional differential equations where is a parameter, is a real number and , is fixed and integer, and is a sign-changing continuous function. The author derived an interval of such that for any lying in this interval, the semipositone boundary value problem has multiple positive solutions. The analysis relied on nonlinear alternative of Leray-Schauder type and the Krasnosel’skii fixed point theorem.

We notice that the methods used in the above papers are all fixed point theorems and the derivatives of unknown function are not involved in the nonlinear term explicitly. Different from the works mentioned above, motivated by the works [2832], we will use monotone iterative techniques to study the existence and iteration of positive solutions for the problem (1)-(2). We not only obtain the existence of positive solutions, but also give two iterative schemes approximating the solutions. Moreover, this method does not demand the existence of upper-lower solutions. To the best of our knowledge, few authors utilize the monotone methods to study the existence of positive solutions for nonlinear fractional boundary value problems. So, it is worthwhile to investigate the problem (1)-(2) by using monotone iterative techniques.

This paper is organized as follows. In Section 2, we recall some definitions and notations from the theory of fractional calculus and give expression and properties of Green’s function. The main results will be given in Section 3. Finally, in Section 4, some examples are included to demonstrate the applicability of our results.

2. Preliminaries

Here we present some necessary basic knowledge and definitions for fractional calculus theory that can be found in the literature [33, 34].

Definition 1. The Riemann-Liouville fractional derivative of order of a continuous function is defined to be where denotes the Euler gamma function and denotes the integer part of number provided that the right side is pointwise defined on .

Definition 2. The Riemann-Liouville fractional integral of order is defined as Provided that the integral exists.

In [23], the author obtain Green’s function associated with the problem (1)-(2). More precisely, the author proved the following lemma.

Lemma 3 (see [23]). Let , then the differential equation with boundary conditions (2) has a unique solution where

Obviously, for , is continuous on .

The following properties of Green’s function defined by (12) will be used later.

Lemma 4. Green’s function defined by (12) has the following properties: (1) on , for ,(2) on .

Proof. Firstly, we prove that (1) is true. In fact, for all , if , it is obvious that for . If , from (13), we obtain that On the other hand, by (13), we find From (14) and (15) we get part (1).
Next, we show part (2). In fact, on the one hand, from (1), we know that for any . Thus, is increasing in , so On the other hand, if , then from (13), we have If , then from (13), we have Thus, (17) and (18) show From (16) and (19), we get part (2). Then the proof is completed.

3. Main Results

In this section, we discuss the existence and iteration of positive solutions for the problem (1)-(2). In the sequel, the following conditions hold:(H1) is continuous and .(H2) is nonnegative and .For any , we define . Let the Banach space be equipped with the norm We define a cone by and an integral operator by where Obviously, the fixed points of are solutions of the problem (1)-(2).

Lemma 5. is completely continuous and .

Proof. Since , are continuous for and is integrable on , we get that the operator is well defined on . By (13), we get . Let be bounded. Then there exists a positive constant such that . Denote Then for , by Lemma 4(1) and (22), we have Hence, is bounded. For , one has Thus, By means of the Arzela-Ascoli theorem, we claim that is completely continuous.
Now, we conclude that . In fact, for any , it follows from Lemma 4(2) that which implies that On the other hand, which together with (29) implies In addition, it follows from Lemma 4(1) that Therefore, (31) and (32) show that ; that is, . Then the proof is completed.

For notational convenience, we denote By , we know that is well defined.

Theorem 6. Suppose that and hold. In addition, assume that there exists such that (H3) for , , ;(H4).Then, the problem (1)-(2) has two positive solutions and satisfying . Moreover, there exist monotone increasing sequence and monotone decreasing sequence in such that , , and and , where , and .

The iterative schemes in Theorem 6 start off with the zero function and a known simple function, respectively.

Proof. We divide the proof into four steps.
Step 1. Let . Then .
In fact, if , then ; thus, By the conditions and , we have Thus, by the definition of and Lemma 4(2), for , we get Then (36) shows that ; thus, .
Step 2. Let . Then is increasing; there exists such that , and is a positive solution of the problem (1)-(2).
Obviously, . Since , we have . Since is completely continuous, we assert that is a sequentially compact set. Since , we have It follows from that is increasing; then Thus, by the induction, we have Hence, there exists such that . Applying the continuity of and equation , we get . Moreover, because the zero function is not a solution of the problem (1)-(2), thus, . It follows from the definition of the cone that we have . That is, is a positive solution of the problem (1)-(2).
Step 3. Let . Then is decreasing; there exists such that , and is a positive solution of the problem (1)-(2).
Obviously, . Since , we have . Since is completely continuous, we assert that is a sequentially compact set. Since , by Lemma 4(2), , and , for , we have Thus, we obtain that So by , we have By the induction, we have Hence, there exists such that . Applying the continuity of and equation , we get . Thus, is a nonnegative solution of the problem (1)-(2). Moreover, the zero function is not a solution of the problem (1)-(2). Thus, , it follows from the definition of the cone that we have ; that is, is a positive solution of the problem (1)-(2).
Step 4. From , we have By the induction, we have The proof is complete.

Remark 7. Of course, may happen and then the problem (1)-(2) has only one solution in .

Corollary 8. Assume that and hold. Suppose that is increasing in . Moreover, Then the problem (1)-(2) has at least two monotone positive solutions.

4. Examples

To illustrate the usefulness of the results, we provide two examples.

Example 1. Consider the fractional boundary value problem

Obviously, the problem (47) fits the framework of problem (1)-(2) with , . In addition, we have set , . Obviously, and satisfy the conditions and . Moreover, It is easy to see that is increasing in and , and Let ; then for any , we have Then conditions and hold. Consequently, applying Theorem 6, the problem (47) has at least two positive solutions and satisfying .

Moreover, the two iterative schemes are

After direct calculations, we get

Example 2. Consider the problem

In this problem, , , , and . Obviously, and satisfy the conditions and . In addition, is increasing with regard to , and and Let ; then for any , by simple computation, we obtain that Therefore, all assumptions of Theorem 6 are satisfied. Thus, Theorem 6 ensures that the problem (52) has two monotone positive solutions and satisfying and and .

Moreover, the two iterative schemes are

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the referees for their careful reading of the paper and a lot of valuable suggestions and comments, which greatly improved this paper. This work was supported financially by the Natural Science Foundation of Zhejiang Province of China (LY12A01012).

References

  1. R. P. Agarwal, Y. Liu, D. O'Regan, and C. Tian, “Positive solutions of two-point boundary value problems for fractional singular differential equations,” Differential Equations, vol. 48, no. 5, pp. 619–629, 2012. View at Publisher · View at Google Scholar
  2. B. Ahmad and S. Sivasundaram, “Existence and uniqueness results for nonlinear boundary value problems of fractional differential equations with separated boundary conditions,” Communications in Applied Analysis, vol. 13, no. 1, pp. 121–128, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. B. Ahmad and J. J. Nieto, “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2009, Article ID 494720, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Babakhani and V. Daftardar-Gejji, “Existence of positive solutions of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 434–442, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Z. Bai and W. Sun, “Existence and multiplicity of positive solutions for singular fractional boundary value problems,” Computers & Mathematics with Applications, vol. 63, no. 9, pp. 1369–1381, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Z. Bai, W. Sun, and W. Zhang, “Positive solutions for boundary value problems of singular fractional differential equations,” Abstract and Applied Analysis, vol. 2013, Article ID 129640, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  7. Z. Bai and T. Qiu, “Existence of positive solution for singular fractional differential equation,” Applied Mathematics and Computation, vol. 215, no. 7, pp. 2761–2767, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Caballero, J. Harjani, and K. Sadarangani, “Positive and nondecreasing solutions to a singular boundary value problem for nonlinear fractional differential equations,” Communications in Applied Analysis, vol. 15, no. 2-3, pp. 265–272, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. R. Graef, L. Kong, and Q. Kong, “Application of the mixed monotone operator method to fractional boundary value problems,” Fractional Differential Calculus, vol. 2, no. 1, pp. 87–98, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  11. D. Jiang and C. Yuan, “The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 72, no. 2, pp. 710–719, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 69, no. 8, pp. 2677–2682, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 71, no. 11, pp. 5545–5550, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. T. Qiu and Z. Bai, “Existence of positive solutions for singular fractional differential equations,” Electronic Journal of Differential Equations, vol. 2008, article 146, 9 pages, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. C. Tian and Y. Liu, “Multiple positive solutions for a class of fractional singular boundary value problems,” Memoirs on Differential Equations and Mathematical Physics, vol. 56, pp. 115–131, 2012. View at Google Scholar · View at MathSciNet
  16. Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 74, no. 17, pp. 6434–6441, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. C. Yuan, D. Jiang, and X. Xu, “Singular positone and semipositone boundary value problems of nonlinear fractional differential equations,” Mathematical Problems in Engineering, vol. 2009, Article ID 535209, 17 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 804–812, 2000. View at Publisher · View at Google Scholar
  20. X. Zhang and Y. Han, “Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 3, pp. 555–560, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8526–8536, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. J. R. Graef, L. Kong, and B. Yang, “Positive solutions for a semipositone fractional boundary value problem with a forcing term,” Fractional Calculus and Applied Analysis, vol. 15, no. 1, pp. 8–24, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. C. Zhai and M. Hao, “Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems,” Boundary Value Problems, vol. 2013, article 85, 13 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  26. Y. Su and Z. Feng, “Existence theory for an arbitrary order fractional differential equation with deviating argument,” Acta Applicandae Mathematicae, vol. 118, no. 1, pp. 81–105, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. C. Yuan, “Multiple positive solutions for semipositone (n,p)-type boundary value problems of nonlinear fractional differential equations,” Analysis and Applications, vol. 9, no. 1, pp. 97–112, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  28. D.-X. Ma and X.-Z. Yang, “Existence and iteration of positive solutions to third order three-point BVP with increasing homeomorphism and positive homomorphism,” The Rocky Mountain Journal of Mathematics, vol. 43, no. 2, pp. 539–550, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. B. Sun and W. Ge, “Successive iteration and positive pseudo-symmetric solutions for a three-point second-order p-Laplacian boundary value problems,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1772–1779, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  30. Y. Sun, X. Zhang, and M. Zhao, “Successive iteration of positive solutions for fourth-order two-point boundary value problems,” Abstract and Applied Analysis, vol. 2013, Article ID 621315, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  31. Q. Yao, “Successively iterative technique of a classical elastic beam equation with Carathéodory nonlinearity,” Acta Applicandae Mathematicae, vol. 108, no. 2, pp. 385–394, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. X. Zhang, “Existence and iteration of monotone positive solutions for an elastic beam equation with a corner,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2097–2103, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006. View at MathSciNet
  34. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. View at MathSciNet