Research Article | Open Access

# Iterative Approximation of the Minimal and Maximal Positive Solutions for Multipoint Fractional Boundary Value Problem on an Unbounded Domain

**Academic Editor:**Aurelian Gheondea

#### Abstract

By employing the monotone iterative method, this paper not only establishes the existence of the minimal and maximal positive solutions for multipoint fractional boundary value problem on an unbounded domain, but also develops two computable explicit monotone iterative sequences for approximating the two positive solutions. An example is given for the illustration of the main result.

#### 1. Introduction

The fractional calculus has been recognized as an effective modeling methodology for describing hereditary properties of various materials and processes widely. For a lot of applications, we refer the reader to the books [1–5]. For some new development on the topic, see [6–17] and the references therein.

Recently, there has been a significant development on boundary value problems for fractional differential equations on *infinite intervals*; see papers [18–26], in which authors are devoted to investigating the existence of solutions and positive solutions by employing some fixed point theorems, Leray-Schauder nonlinear alternative theorem, or fixed point index theory.

By using Schauder’s fixed point theorem combined with the diagonalization method, Arara et al. [18] studied the existence of the bounded solution of the following problem on infinite intervals: where , , , and is the Caputo fractional derivative of order .

In [19], Zhao and Ge investigated the existence of positive solutions for the following fractional boundary value problem by employing the Leray-Schauder nonlinear alternative theorem: where , , , , and is the standard Riemann-Liouville fractional derivative.

Liang and Zhang [20] were concerned with the following nonlinear fractional differential equations with multipoint fractional boundary conditions on an unbounded domain: where , , denotes the Riemann-Liouville fractional derivative, , and , , satisfy . By using the fixed point index theory, authors gave sufficient conditions for the existence of multiple positive solutions to the above multi-point fractional boundary value problem.

However, very interesting and important question is “If we know the existence of the solution, how can we find it?” This question motivates us to reconsider problem (3). In this paper, we not only establish the existence of two positive solutions for problem (3), but also develop two computable explicit monotone iterative sequences for approximating the minimal and maximal positive solutions of (3), which is indeed an important and useful contribution to the existing literature on the topic. In addition, to start our work, we employ the monotone iterative method, which is different from the ones used in [18–26]. Let us state that this method was widely used for nonlinear problem; see, for instance, [27–38].

#### 2. Preliminaries and Several Lemmas

In this section, we present some useful definitions and related theorems.

*Definition 1 (see [2]). *The Riemann-Liouville fractional derivative of order for a continuous function is defined by
provided the right-hand side is pointwise defined on and is the integer part of .

*Definition 2 (see [2]). *The Riemann-Liouville fractional integral of order for a function is defined as
provided that such integral exists.

Lemma 3 (see [20]). *Let . For , the fractional boundary value problem
**
has a unique solution
**
where
**
with
*

Lemma 4 (see [20]). *For , then Green’s function has the following properties:*(1)*(2)**where
*

For the forthcoming analysis, we will use a Banach space: equipped with the norm

Define a cone by and an operator as follows:

Observe that multi-point fractional boundary value problem (3) has a solution if and only if the integral operator has a fixed point.

#### 3. Main Results

In this section, we shall construct two explicit monotone iterative sequences which converge to the minimal and maximal positive solutions of (3).

Theorem 5. *Assume that the following conditions hold:*(H_{1})*, on any subinterval of , and when is bounded, is bounded on ;*(H_{2})* does not identically vanish on any subinterval of and ;*(H_{3})* is nondecreasing for any , and there exists a constant , such that for .**Then the multi-point fractional boundary value problem (3) has the minimal and maximal positive solutions , in , which can be obtained by the following two explicit monotone iterative sequences:
**
Moreover,
*

*Proof. *By a similar process used in [20], it is easy to show that is completely continuous.

Now denote ; then we have . In fact, let ; then by and (12), we have
That is, .

Denote that , and , for all . Since and , then and . So, we have
By condition , for and , we have
This proves that is a nondecreasing operator.

So, we have

By the induction, define , . Then the sequence and satisfies the following relation:

In view of the complete continuity of the operator and , then is relative compact. That is, has a convergent subsequence and there exists a such that as . This, together with (23), holds .

Since is continuous and , then we have . That is, is a fixed point of the operator .

Denote that , , and , for all . Since and , then and . By , we have

Since is nondecreasing, then we have

By the induction, define , . Then the sequence and satisfies the following relation:

With an analysis exactly parallel to the proving process of , we have that there exists a such that .

Since is continuous and , we have . That is, is a fixed point of the operator .

Now, we are in a position to show that and are the maximal and minimal positive solutions of (3) in .

Let be any solution of (3). That is . Noting that is nondecreasing and , then we have , for all .

Similarly, we can obtain

Since and , it follows from (23)(27) that

Since , for all , then is not a solution of problem (3). Thus, by (28), we know that and are the maximal and minimal positive solutions of (3) in , which can be obtained by the corresponding iterative sequences in (17).

This completes the proof.

#### 4. Example

*Example 1. *Take , , , , and . Consider the following boundary value problem:
where and

Now, we show that is bounded on when is bounded. Since
Then we have . So condition holds.

In view of , condition holds.

By a simple computation, we have that and . Taking , it follows that

Hence, condition holds. Thus all conditions of Theorem 5 are satisfied. Therefore, the fractional boundary value problem (29) has the minimal and maximal positive solutions in , which can be obtained by two explicit monotone iterative sequences.

#### Conflict of Interests

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

#### Acknowledgments

The authors would like to express their gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which improved the quality of the original paper. This work is supported by the NNSF of China (no. 61373174) and the Natural Science Foundation for Young Scientists of Shanxi Province, China (no. 2012021002-3).

#### References

- I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calf, USA, 1999. View at: MathSciNet - 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 B.V., Amsterdam, The Netherlands, 2006. View at: MathSciNet - V. Lakshmikantham, S. Leela, and J. V. Devi,
*Theory of Fractional Dynamic Systems*, Cambridge Scientific, Cambridge, UK, 2009. - J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds.,
*Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering*, Springer, Dordrecht, The Netherlands, 2007. - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, vol. 3 of*Series on Complexity, Nonlinearity and Chaos*, World Scientific, Hackensack, NJ, USA, 2012. View at: Publisher Site | MathSciNet - F. Jarad, T. Abdeljawad, and D. Baleanu, “Stability of $q$-fractional non-autonomous systems,”
*Nonlinear Analysis: Real World Applications*, vol. 14, no. 1, pp. 780–784, 2013. View at: Publisher Site | Google Scholar | MathSciNet - G.-C. Wu and D. Baleanu, “Variational iteration method for the Burgers' flow with fractional derivatives–-new Lagrange multipliers,”
*Applied Mathematical Modelling*, vol. 37, no. 9, pp. 6183–6190, 2013. View at: Publisher Site | Google Scholar | MathSciNet - E. Hernández, D. O'Regan, and K. Balachandran, “On recent developments in the theory of abstract differential equations with fractional derivatives,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 73, no. 10, pp. 3462–3471, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Cabada and G. Wang, “Positive solutions of nonlinear fractional differential equations with integral boundary value conditions,”
*Journal of Mathematical Analysis and Applications*, vol. 389, no. 1, pp. 403–411, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Wang, A. Cabada, and L. Zhang, “Integral boundary value problem for nonlinear differential equations of fractional order on an unbounded domain,” to appear in
*Journal of Integral Equations and Applications*. View at: Google Scholar - G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 3, pp. 792–804, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Dabas and A. Chauhan, “Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay,”
*Mathematical and Computer Modelling*, vol. 57, no. 3-4, pp. 754–763, 2013. View at: Publisher Site | Google Scholar | MathSciNet - R. Sakthivel, N. I. Mahmudov, and J. J. Nieto, “Controllability for a class of fractional-order neutral evolution control systems,”
*Applied Mathematics and Computation*, vol. 218, no. 20, pp. 10334–10340, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Ahmad, J. J. Nieto, A. Alsaedi, and M. El-Shahed, “A study of nonlinear Langevin equation involving two fractional orders in different intervals,”
*Nonlinear Analysis: Real World Applications*, vol. 13, no. 2, pp. 599–606, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Ahmad and S. K. Ntouyas, “Existence results for nonlocal boundary value problems of fractional differential equations and inclusions with strip conditions,”
*Boundary Value Problems*, vol. 2012, article 55, 21 pages, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. P. Agarwal, D. O'Regan, and S. Staněk, “Positive solutions for mixed problems of singular fractional differential equations,”
*Mathematische Nachrichten*, vol. 285, no. 1, pp. 27–41, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an unbounded domain,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 2, pp. 580–586, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Zhao and W. Ge, “Unbounded solutions for a fractional boundary value problems on the infinite interval,”
*Acta Applicandae Mathematicae*, vol. 109, no. 2, pp. 495–505, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Liang and J. Zhang, “Existence of multiple positive solutions for $m$-point fractional boundary value problems on an infinite interval,”
*Mathematical and Computer Modelling*, vol. 54, no. 5-6, pp. 1334–1346, 2011. View at: Publisher Site | Google Scholar | MathSciNet - S. Liang and J. Zhang, “Existence of three positive solutions of $m$-point boundary value problems for some nonlinear fractional differential equations on an infinite interval,”
*Computers & Mathematics with Applications*, vol. 61, no. 11, pp. 3343–3354, 2011. View at: Publisher Site | Google Scholar | MathSciNet - X. Su, “Solutions to boundary value problem of fractional order on unbounded domains in a Banach space,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 8, pp. 2844–2852, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Su and S. Zhang, “Unbounded solutions to a boundary value problem of fractional order on the half-line,”
*Computers & Mathematics with Applications*, vol. 61, no. 4, pp. 1079–1087, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. P. Agarwal, M. Benchohra, S. Hamani, and S. Pinelas, “Boundary value problems for differential equations involving Riemann-Liouville fractional derivative on the half-line,”
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A*, vol. 18, no. 2, pp. 235–244, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet - L. Zhang, B. Ahmad, G. Wang, R. P. Agarwal, M. Al-Yami, and W. Shammakh, “Nonlocal integrodifferential boundary value problem for nonlinear fractional differential equations on an unbounded domain,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 813903, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - G. Wang, B. Ahmad, and L. Zhang, “A coupled system of nonlinear fractional differential equations with multipoint fractional boundary conditions on an unbounded domain,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 248709, 11 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala,
*Monotone Iterative Techniques for Nonlinear Differential Equations*, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics, 27, Pitman, Boston, Mass, USA, 1985. View at: MathSciNet - J. J. Nieto, “An abstract monotone iterative technique,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 28, no. 12, pp. 1923–1933, 1997. View at: Google Scholar - J. D. Ramírez and A. S. Vatsala, “Monotone iterative technique for fractional differential equations with periodic boundary conditions,”
*Opuscula Mathematica*, vol. 29, no. 3, pp. 289–304, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Wang, “Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments,”
*Journal of Computational and Applied Mathematics*, vol. 236, no. 9, pp. 2425–2430, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Wang, R. P. Agarwal, and A. Cabada, “Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations,”
*Applied Mathematics Letters*, vol. 25, no. 6, pp. 1019–1024, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Wang, D. Baleanu, and L. Zhang, “Monotone iterative method for a class of nonlinear fractional differential equations,”
*Fractional Calculus and Applied Analysis*, vol. 15, no. 2, pp. 244–252, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Zhang, B. Ahmad, G. Wang, and R. P. Agarwal, “Nonlinear fractional integro-differential equations on unbounded domains in a Banach space,”
*Journal of Computational and Applied Mathematics*, vol. 249, pp. 51–56, 2013. View at: Publisher Site | Google Scholar | MathSciNet - T. Jankowski, “Fractional equations of Volterra type involving a Riemann-Liouville derivative,”
*Applied Mathematics Letters*, vol. 26, no. 3, pp. 344–350, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. A. McRae, “Monotone iterative technique and existence results for fractional differential equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 12, pp. 6093–6096, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Wei, Q. Li, and J. Che, “Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative,”
*Journal of Mathematical Analysis and Applications*, vol. 367, no. 1, pp. 260–272, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Liu, J. Sun, and I. Szántó, “Monotone iterative technique for Riemann-Liouville fractional integro-differential equations with advanced arguments,”
*Results in Mathematics*, vol. 63, no. 3-4, pp. 1277–1287, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Zhang, L. Liu, Y. Wu, and Y. Lu, “The iterative solutions of nonlinear fractional differential equations,”
*Applied Mathematics and Computation*, vol. 219, no. 9, pp. 4680–4691, 2013. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Guotao Wang et al. 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.