• Views 160
• Citations 0
• ePub 0
• PDF 54
`Journal of Function SpacesVolume 2018, Article ID 8346398, 6 pageshttps://doi.org/10.1155/2018/8346398`
Research Article

## The Exact Iterative Solution of Fractional Differential Equation with Nonlocal Boundary Value Conditions

1School of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China
2Department of Mathematics, Jining University, Qufu, Shandong 273155, China

Correspondence should be addressed to Jinxiu Mao; moc.361@2891uixnijoam

Received 3 May 2018; Accepted 26 July 2018; Published 6 August 2018

Copyright © 2018 Jinxiu Mao 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.

#### Abstract

We deal with a singular nonlocal fractional differential equation with Riemann-Stieltjes integral conditions. The exact iterative solution is established under the iterative technique. The iterative sequences have been proved to converge uniformly to the exact solution, and estimation of the approximation error and the convergence rate have been derived. An example is also given to demonstrate the results.

#### 1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines; see [15]. Much attention has been paid to study fractional differential equations both with initial and boundary conditions; see, for example, [6, 7]. In [8, 9], they focused on sign-changing solution for some fractional differential equations. In [10], they get the existence of solutions for impulsive fractional differential equations. In [1113], they get the existence and multiplicity of nontrivial solutions for a class of fractional differential equations. The mainly techniques authors need are fixed point theory, variational method, and global bifurcation techniques.

Also, ordinary differential equations and partial differential equations involving nonlocal boundary conditions have been studied extensively in recent years, see [1422], including integral boundary conditions and multipoint boundary conditions.

In [23], authors obtained results on the uniqueness of positive solution for problem where is a real number. Under the assumption that where , and is the first eigenvalue of the corresponding linear operator.

Motivated by the above works, we study the following nonlocal boundary value problems:where denotes the left-handed Riemann-Liouville derivative of order q and is a real number. denotes a Stieltjes integral with a suitable function of bounded variation. Different from [23] and other works, we only use the iterative methods to obtain the existence and uniqueness of positive solution. Moreover, the estimation of the approximation error and the convergence rate have also been derived.

For clarity in presentation, we also list below some assumptions to be used later in the paper.

is continuous, and for , is increasing with respect to and there exists a constant such that, for ,It is easy to see that if , then

,

#### 2. Preliminaries

For the convenience of the reader, we present here some necessary definitions from fractional calculus theory. These definitions and properties can be found in the recent monograph [23].

Definition 1. The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined on

Definition 2. The Riemann-Liouville fractional derivative of order of a continuous function is given by where , , provided that the right-hand side is pointwise defined on In particular,

Lemma 3 (see [13]). Assume that hold. Let Then boundary value problem has the unique solution given by the following formula: where

One can prove that have the following properties.

Lemma 4. Note that is the Green function of problem (8).

Lemma 5 (see [12]). For , one has where

Lemma 6. where is a constant and is nonnegative for any

Proof. We have the estimation where and Thus, (13) holds.

#### 3. The Main Results

Throughout this paper, we will work in the space , which is a Banach space if it is endowed with the norm for any

Define the set in as follows:

there exists positive constants such that

And define the operator .Evidently Therefore, is not empty.

Theorem 7. Assume that - hold. And Then BVP (3) has at least one positive solution , and there exist constants satisfying

Proof. It is clear that is a solution of (3) if and only if is a fixed point of .
Claim 1. The operator is nondecreasing.
In fact, for , it is obvious that , , and for For any , we have that, for , and where and are positive constants satisfying Thus, it follows that there are constants such that, for , Therefore, for any , , i.e., is the operator From (16), it is easy to see that is nondecreasing for Hence, Claim 1 holds.
Claim 2. We take . Let and be fixed numbers satisfyingand assume thatThenand there exists such thatuniformly on
In fact, since Therefore, From (24), we have and
On the other hand, and since and is nondecreasing, by induction, (26) holds.
Let , and then It follows from (4) that And for any natural number , Thus, for any natural number and , we havewhich implies that there exists such that (27) holds and Claim 2 holds.
Letting in and noting the fact that is continuous, we obtain , which is a positive solution of BVP (3). The proof of Theorem 7 is now complete.

Theorem 8. Assume that - hold. Then
(i) BVP (3) has unique positive solution , and there exist constants with such that (ii) For any initial value , there exists a sequence that uniformly converges to the unique positive solution , and one has the error estimationwhere is a constant with and determined by

Proof. Let be defined in (24) and (25).
(i) It follows from Theorem 7 that BVP (3) has a positive solution , which implies that there exist constants and with such that satisfies (18). Let be another positive solution of BVP (3); then from Theorem 7 we have that there exist constants and with such that Let defined in (23) be small enough such that and defined in (23) be large enough such that Then Note that and is nondecreasing; we haveLetting in (36), we obtain that Hence, the positive solution of BVP (3) is unique.
(ii) From (i), we know that the positive solution to BVP (3) is unique. For any , there exist constants and with such that Similar to (i), we can let and defined by (23) satisfy and Then Let Note that is nondecreasing; we haveLetting in (39), it follows that uniformly converges to the unique positive solution for BVP (3), where At the same time, (33) follows from (31). Thus, the proof of the theorem is complete.

#### 4. An Example

where Analysis 1. Let and then for any , we take and have Then holds.

In addition, we have Then and hold.

And

Hence all conditions of Theorem 7 are satisfied, and consequently we have the following corollary.

Corollary 9. Problem (41) has unique positive solution . For any initial value , the successive iterative sequence generated by uniformly converges to the unique positive solution on . One has the error estimation where is a constant with and determined by the initial value And there are constants with such that

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work is supported by the Natural Science Foundation of China (11571197), the Science Foundation of Qufu Normal University of China (XJ201112), Jining University Youth Research Foundation (2015QNK102), a project of Shandong Province Higher Educational Science and Technology Program (J17KB143), Shandong Education Science 13th Five-Year Plan Project (BYK2017003).

#### References

1. E. Ahmed and H. A. El-Saka, “On fractional order models for Hepatitis C,” Nonlinear Biomedical Physics, vol. 4, 2010.
2. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, NY, USA, 1999.
3. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
4. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, B.V, Netherlands, 2006.
5. K. B. Borberg, Cracks and Fracture, Academic Press, San Diego, SD, USA, 1999.
6. X. G. Zhang, L. S. Liu, Y. H. Wu, and B. Wiwatanapataphee, “The spectral analysis for a singular fractional differential equation with a signed measure,” Applied Mathematics and Computation, vol. 257, pp. 252–263, 2015.
7. X. A. Hao, “Positive solution for singular fractional differential equations involving derivatives,” Advances in Difference Equations, vol. 139, pp. 1–12, 2016.
8. K. M. Zhang, “On sign-changing solution for some fractional differential equations,” Boundary Value Problems, vol. 59, 2017.
9. Y. L. Guan, Z. Q. Zhao, and X. L. Lin, “On the existence of positive solutions and negative solutions of singular fractional differential equations via global bifurcation techniques,” Boundary Value Problems, vol. 141, pp. 1–18, 2016.
10. Y. L. Guan, Z. Q. Zhao, and X. L. Lin, “On the Existence of Solutions for Impulsive Fractional Differential Equations,” Advances in Mathematical Physics, Article ID 1207456, 2017.
11. X. S. Du and A. M. Mao, “Existence and multiplicity of nontrivial solutions for a class of semilinear fractional Schrdinger equations,” Journal of Function Spaces, vol. 7, Article ID 3793872, 2017.
12. X. G. Zhang, L. S. Liu, and Y. H. Wu, “Variational structure and multiple solutions for a fractional advection-dispersion equation,” Computers & Mathematics with Applications, vol. 68, pp. 1794–1805, 2014.
13. X. G. Zhang, L. S. Liu, Y. H. Wu, and B. Wiwatanapataphee, “Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion,” Applied Mathematics Letters, vol. 66, pp. 1–8, 2017.
14. C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,” Computers & Mathematics with Applications, vol. 61, no. 2, pp. 191–202, 2011.
15. J. R. Webb and G. Infante, “Positive solutions of nonlocal boundary value problems involving integral conditions,” Nonlinear Differential Equations and Applications, vol. 15, pp. 45–67, 2008.
16. J. X. Mao, Z. Q. Zhao, and N. W. Xu, “The existence and uniqueness of positive solutions for integral boundary value problems,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 34, no. 1, pp. 153–164, 2011.
17. J. X. Mao, Z. Q. Zhao, and N. W. Xu, “On existence and uniqueness of positive solutions for integral boundary boundary value problems,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 16, 2010.
18. J. R. L. Webb and G. Infante, “Positive solutions of nonlocal boundary value problems: a unied approach,” Journal of the London Mathematical Society, vol. 74, pp. 673–693, 2006.
19. A. M. Mao, R. N. Jing, S. X. Luan, J. L. Chu, and Y. C. Kong, “Some nonlocal elliptic problem involving positive parameter,” Topological Methods in Nonlinear Analysis, vol. 42, pp. 207–220, 2013.
20. Y. Q. Wang and L. S. Liu, “Positive solutions for a class of fractional 3-point boundary value problems at resonance,” Advances in Difference Equations, vol. 13, 2017.
21. X. L. Lin and Z. Q. Zhao, “Iterative technique for a third-order differential equation with three-point nonlinear boundary value conditions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 12, pp. 1–10, 2016.
22. M. Rehman and R. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 23, pp. 1038–1044, 2010.
23. Y. J. Cui, “Uniqueness of solution for boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 51, pp. 48–54, 2016.