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Journal of Function Spaces
Volume 2018, Article ID 6461930, 6 pages
https://doi.org/10.1155/2018/6461930
Research Article

Positive Solutions for Boundary Value Problems of Fractional Differential Equation with Integral Boundary Conditions

1Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, China
2Department of Mathematics and Physics, Nantong Normal College, Nantong 226010, China
3State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Yujun Cui; moc.361@102027jyc

Received 29 November 2017; Accepted 22 February 2018; Published 22 April 2018

Academic Editor: Dhananjay Gopal

Copyright © 2018 Qiao Sun 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

By using two fixed-point theorems on cone, we discuss the existence results of positive solutions for the following boundary value problem of fractional differential equation with integral boundary conditions: , , , and .

1. Introduction

Boundary value problem for fractional differential equation has aroused much attention in the past few years; many professors devoted themselves to the solvability of fractional differential equations, especially to the study of the existence of solutions for boundary value problems of fractional differential equation (see [128]). For example, Wang et al. [19] studied the existence of positive solutions for the following problem: with the boundary conditions where is the Riemann-Liouville differential operator of , is a positive parameter, and may be singular at and may change sign. And Ma [14] discussed the positive solutions of where is integer and , .

There have already been lots of books and papers involving the positive solutions for boundary value problems of fractional differential equation; however, only a few papers cover that for fractional differential equation boundary value problems with integral boundary conditions. Motivated by [14], we shall investigate the positive solutions of the following boundary value problem:where is the Riemann-Liouville differential operator of , is right continuous on , left continuous at , and nondecreasing on with , and denotes the Riemann-Stieltjes integrals of with respect to . And , satisfies the following conditions:(H1) is nonnegative and not identically zero on any compact subset of , .(H2) is continuous.

This paper consists of four sections. After the introduction, we recall some definitions, lemmas, and theorems in Section 2. And the main results of this paper are stated in Section 3. In the last section, we give two examples of the main results.

2. Preliminaries

Firstly, for convenience we recall some definitions, lemmas, and theorems.

Definition 1 (see [29, 30]). Let define the Riemann-Liouville fractional integral of order for as where is Euler gamma function.

Definition 2 (see [29, 30]). Define the Riemann-Liouville fractional derivative of order for as where has absolutely continuous derivatives up to order on .

Lemma 3. Let , ; then the boundary value problem has the unique solution , where

Proof. The boundary value problem can be converted to an equivalent integral equation: Then the solution is It follows from the boundary conditions that and Thus we get Then we can obtain which means So

Lemma 4 (see [20]). defined in (9) has the following properties:(i), .(ii), .(iii), where

Lemma 5. If (H2) is satisfied, then defined in (8) has the following properties: where .

Proof. The proof can be easily accomplished by Lemma 4, so we omitted it.

Theorem 6 (see [31]). Let be a Banach space and be a cone in . Suppose that are two bounded open sets of with . Assume that is a completely continuous operator such that either(i) for any and for any or (ii) for any and for any . Then has a fixed point in .

Theorem 7 (see [32]). Let be a cone of a Banach space . . is a nonnegative continuous concave function on , such that, for any , , and . Assume that is completely continuous, and there exist constants such that(c1) and for ;(c2) for ;(c3) for any with . Then has at least three fixed points , , with , , , and .

Let be a Banach space with the maximum norm ; define the cone as

Define a continuous operator as

Lemma 8. Assume that (H1) and (H2) hold; then is a completely continuous operator.

Proof. The lemma can be easily proven, so we omitted it.

3. Main Results

We define the following notation: given , takeNow we can obtain the following theorems.

Theorem 9. Suppose that (H1) and (H2) are satisfied; there exist two positive constants such that(H3), ;(H4), . Then boundary value problem (4) has at least one positive solution such that .

Proof. The solution of boundary value problem (4) is equivalent to the fixed point of operator . Let ; when , for any we have . By Lemma 5 and (H3) we get which means when , .
Let ; when , for any we have . By Lemma 5 and (H4) we get which means when , .
It follows from Theorem 6 that we know that has at least one fixed point in , which means that the boundary value problem (4) has at least one solution.

Theorem 10. Suppose that (H1) and (H2) are satisfied; there exist four positive constants , , , with , such that(H5), ;(H6), ;(H7), . Then boundary value problem (4) has at least three positive solutions , , , such that

Proof. Define a nonnegative continuous concave function on as If , then ; it follows from (H7) that ; hence Thus, . It follows from Lemma 8 that is completely continuous. In the same way, let ; it follows from (H5) that for any , which shows that condition (c2) of Theorem 7 is fulfilled.
Let ; it is easy to know that and . If , we have for any . We know for by (H6). So we get So condition (c1) of Theorem 7 holds.
When with , noting that thus so we obtainThat is to say, (c3) is satisfied.
All conditions of Theorem 7 are satisfied, so has at least three fixed points , , , which means that the boundary value problem (4) has at least three positive solutions , , , such that The proof of this theorem is finished.

4. Some Examples

Now we present two examples to illustrate our main results.

Example 1. Let us see the following problem:

Choose ; we obtain that , , , and Then for any , we have , and for any , we get

Then the boundary value problem has at least one positive solution such that .

Example 2. We now study the following problem: where

Choose ; we have , , , and . Let , , , and ; then for any , we have , for any , we have , and for any , we have . Then by Theorem 10, we conclude that this boundary value problem has at least three positive solutions , , , such that

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project is supported by NNSF of China (11371221, 11571207), SDNSF (ZR2018MA011), and the Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.

References

  1. Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 916–924, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. Y. Cui, “Uniqueness of solution for boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 51, pp. 48–54, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  3. Y. Cui, “Existence results for singular boundary value problem of nonlinear fractional differential equation,” Abstract and Applied Analysis, vol. 2011, Article ID 605614, 2011. View at Publisher · View at Google Scholar · View at Scopus
  4. Y. Cui, Q. Sun, and X. Su, “Monotone iterative technique for nonlinear boundary value problems of fractional order p (2 , 3 ],” Advances in Difference Equations, vol. 2017, no. 1, article no. 248, 2017. View at Publisher · View at Google Scholar · View at Scopus
  5. Y. Cu, W. Ma, Q. Sun, and X. Su, “New uniqueness results for boundary value problem of fractional differential equation,” Nonlinear Analysis: Modelling and Control, pp. 31–39, 2018. View at Publisher · View at Google Scholar
  6. H. H. Dong, B. Y. Guo, and B. S. Yin, “Generalized fractional supertrace identity for Hamiltonian structure of NLS–MKdV hierarchy with self-consistent sources,” Analysis and Mathematical Physics, vol. 6, no. 2, pp. 199–209, 2016. View at Publisher · View at Google Scholar · View at Scopus
  7. A. M. El-Sayed, “Nonlinear functional-differential equations of arbitrary orders,” Nonlinear Analysis: Theory, Methods & Applications, vol. 33, no. 2, pp. 181–186, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. R. Graef and L. Kong, “Positive solutions for a class of higher order boundary value problems with fractional q-derivatives,” Applied Mathematics and Computation, vol. 218, no. 19, pp. 9682–9689, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  9. L. Guo, L. Liu, and Y. Wu, “Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions,” Lithuanian Association of Nonlinear Analysts. Nonlinear Analysis: Modelling and Control, vol. 21, no. 5, pp. 635–650, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. H. Maagli, “Existence of positive solutions for a nonlinear fractional differential equation,” Electronic Journal of Differential Equations, vol. 2013, 2013. View at Google Scholar · View at Scopus
  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, vol. 72, no. 2, pp. 710–719, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. L. Liu, H. Li, C. Liu, and Y. Wu, “Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary conditions,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 10, no. 1, pp. 243–262, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  13. D. Liu, Z. Ouyang, and H. Wang, “Positive solutions for class of state dependent boundary value problems with fractional order differential operators,” Abstract and Applied Analysis, vol. 2015, Article ID 263748, 2015. View at Publisher · View at Google Scholar · View at Scopus
  14. D.-x. Ma, “Positive solutions of multi-point boundary value problem of fractional differential equation,” Arab Journal of Mathematical Sciences, vol. 21, no. 2, pp. 225–236, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  15. N. Nyamoradi, “Existence of solutions for multi point boundary value problems for fractional differential equations,” Arab Journal of Mathematical Sciences, vol. 18, no. 2, pp. 165–175, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  16. T. Qi, Y. Liu, and Y. Cui, “Existence of Solutions for a Class of Coupled Fractional Differential Systems with Nonlocal Boundary Conditions,” Journal of Function Spaces, vol. 2017, Article ID 6703860, 2017. View at Publisher · View at Google Scholar · View at Scopus
  17. T. Qi, Y. Liu, and Y. Zou, “Existence result for a class of coupled fractional differential systems with integral boundary value conditions,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 10, no. 7, pp. 4034–4045, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  18. Y. Wang, L. Liu, X. Zhang, and Y. Wu, “Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of {HIV} infection,” Applied Mathematics and Computation, vol. 258, pp. 312–324, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. 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, vol. 74, no. 17, pp. 6434–6441, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. 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 MathSciNet · View at Scopus
  21. X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition,” Applied Mathematics and Computation, vol. 235, pp. 412–422, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  22. X. Zhang, L. Liu, and Y. Wu, “Variational structure and multiple solutions for a fractional advection-dispersion equation,” Computers & Mathematics with Applications. An International Journal, vol. 68, no. 12, part A, pp. 1794–1805, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  23. D. Zhao and Y. Liu, “Positive solutions for a class of fractional differential coupled system with integral boundary value conditions,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 9, no. 5, pp. 2922–2942, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  24. Y. Zhao, S. Sun, Z. Han, and Q. 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 · View at Google Scholar · View at MathSciNet · View at Scopus
  25. B. Zhu, L. Liu, and Y. Wu, “Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay,” Applied Mathematics Letters, vol. 61, pp. 73–79, 2016. View at Publisher · View at Google Scholar · View at Scopus
  26. Y. Zou and G. He, “On the uniqueness of solutions for a class of fractional differential equations,” Applied Mathematics Letters, vol. 74, pp. 68–73, 2017. View at Publisher · View at Google Scholar · View at Scopus
  27. Y. Zou and G. He, “The existence of solutions to integral boundary value problems of fractional differential equations at resonance,” Journal of Function Spaces, Art. ID 2785937, 7 pages, 2017. View at Google Scholar · View at MathSciNet
  28. Y. Zou and Y. Cui, “Existence results for a functional boundary value problem of fractional differential equations,” Advances in Difference Equations, vol. 2013, article no. 233, 2013. View at Publisher · View at Google Scholar · View at Scopus
  29. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, New York, NY, USA, Elsevier, 2006. View at MathSciNet
  30. 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. View at MathSciNet
  31. M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, The Netherlands, 1964. View at MathSciNet
  32. R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,” Indiana University Mathematics Journal, vol. 28, no. 4, pp. 673–688, 1979. View at Publisher · View at Google Scholar · View at MathSciNet