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Journal of Function Spaces
Volume 2018, Article ID 1547293, 7 pages
https://doi.org/10.1155/2018/1547293
Research Article

Existence of Uniqueness and Nonexistence Results of Positive Solution for Fractional Differential Equations Integral Boundary Value Problems

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

Correspondence should be addressed to Yongqing Wang; moc.361@1089gniqyw

Received 14 April 2018; Revised 20 July 2018; Accepted 17 October 2018; Published 4 December 2018

Academic Editor: Yong H. Wu

Copyright © 2018 Yongqing Wang. 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

In this paper, we consider a class of fractional differential equations with conjugate type integral conditions. Both the existence of uniqueness and nonexistence of positive solution are obtained by means of the iterative technique. The interesting point lies in that the assumption on nonlinearity is closely associated with the spectral radius corresponding to the relevant linear operator.

1. Introduction

In this paper, we consider the existence of uniqueness and nonexistence of positive solution for the following fractional differential equations:where , , , , , is the standard Riemann-Liouville derivative, is continuous, and is nonnegative.

In the recent years, many results were obtained to deal with the existence of solutions for nonlinear differential equations by using nonlinear analysis methods; see [116] and references therein. The fractional nonlocal boundary value problems have particularly attracted a great deal of attention (see [1727]). While there are a lot of works dealing with the existence and multiplicity of solutions for nonlinear fractional differential equations, the results dealing with the uniqueness of solution are relatively scarce (see [2835]). The main tool used in most of the papers dealing with the uniqueness of solution is the Banach contraction map principle provided that the nonlinearity is a Lipschitz continuous function. When , and is continuous on , Zhang and Zhong [34] established the uniqueness results of solution to problem (1) by using the Banach contraction map principle. It is worth mentioning that only positive solutions are meaningful in most practical problems. As far as we know, the nonexistence of positive solution has seldom been considered up to now.

Motivated by the above work, the aim of this paper is to establish the existence of uniqueness and nonexistence of positive solution to problem (1). Our analysis relies on the iterative technique on the cone derived from the properties of the Green function. This article provides some new insights. Firstly, the uniqueness results are obtained under some conditions concerning the spectral radius with respect to the relevant linear operator. In addition, the error estimation of the iterative sequences is given. Secondly, we impose weaker positivity conditions on ; that is, the Lipschitz constant is generalized to a function and may be singular at . Finally, the nonexistence results of positive solution are obtained under conditions concerning the spectral radius of the relevant linear operator.

2. Preliminaries

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory and lemmas.

Definition 1. The fractional integral of order of a function is given byprovided that the right-hand side is point-wise defined on .

Definition 2. The Riemann-Liouville fractional derivative of order of a function is given by where ; denotes the integer part of number , provided that the right-hand side is point-wise defined on .

For convenience, we here list the assumptions to be used throughout the paper.

   is nonnegative, and

   is continuous.

Lemma 3 ([34]). For any , the unique solution of the boundary value problemiswhere

Lemma 4. The function has the following properties:(1);(2);(3).

Proof. It is obvious that (1) and (2) hold. In the following, we will prove (3).
Case (i) (). Noticing , we have which impliesNoticing , we have On the other hand, it follows fromthatTherefore,Case (ii) (). It is easy to see that On the other hand, we have It follows from (10), (13), (14), and (15) that (3) holds.

Lemma 5. The function has the following properties:(1);(2);(3), where

Proof. It can be directly deduced from Lemma 4 and the definition of , so we omit the proof.

Let be endowed with the maximum norm , . Define cones , byIt is clear that is nonempty set since .

For convenience, we list here one more assumption to be used later:

There exists such thatMoreover,

Define operators and as follows:

Lemma 6. Assume that holds; then .

Proof. It is clear that ,  . For any , we have Then is well-defined on . It follows from Lemma 5 that .

By virtue of the Krein-Rutmann theorem and Lemma 5, we have the following lemma.

Lemma 7. Assume that holds. Then is a completely continuous linear operator. Moreover, the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue ; that is, .

3. Main Results

Theorem 8. Assume that holds. Then (1) has a unique positive solution if the spectral radius .

Proof. It follows from that is not a fixed point of . Then we only need to prove that has a unique fixed point in .
Firstly, we will prove has a fixed point in .
For any , let Then By Lemma 7, we have that . It is easy to see that For any , set We may suppose that (otherwise, the proof is finished). It follows from (23) and (24) that Then, we have By induction, we can get Then, for any , one has It follows from thatwhich implies is a Cauchy sequence. Therefore, there exists such that converges to . Clearly is a fixed point of .
In the following, we will prove the fixed point of is unique.
Suppose is a positive fixed point of . Then there exists such that Therefore,By induction, we can get It follows from that which implies the positive fixed point of is unique.

Remark 9. The unique positive solution of (1) can be approximated by the iterative schemes: for any , let and then . Furthermore, we have error estimationand with the rate of convergence

Remark 10. The spectral radius satisfies and . Particularly,

Theorem 11. Assume that the following condition holds:
There exists satisfying such thatThen (1) has no positive solution if the spectral radius , where

Proof. We only need to prove that has no fixed point in . Otherwise, there exists , such that .
By Lemma 7, we have that the spectral radius and has a positive eigenfunction satisfying It is clear that . Therefore, there exists such that It follows from that . It is obvious that is increasing on . By induction, we can get , . Thus, Noticing , we have , which contradicts with .

Theorem 12. Assume that the following condition holds:
There exists satisfying such thatThen (1) has no positive solution if the spectral radius , where

Proof. Suppose that there exists , such that .
By Lemma 7, we have that the spectral radius and has a positive eigenfunction satisfying It is clear that . Therefore, there exists such that It follows from that . Noticing that is increasing on , by induction, we have , . Thus, It follows from that , which contradicts with .

4. Example

Example 1. Consider the following integral boundary value problem: withwhere

By direct calculations, we have It is clear that and hold. Clearly, we have Let ; then we have It is easy to get that holds.

Denote By direct calculations, we have , we have

Therefore,

It is obvious that andwhich implies that Lemma 7 and Remark 10 can guarantee that So all of the assumptions of Theorem 8 are satisfied. As a result, BVP (51) has a unique positive solution.

Example 2. Consider BVP (51) with

Clearly, we haveIt is not difficult to check that holds.

It follows from Example 1 that So all of the assumptions of Theorem 11 are satisfied. As a result, BVP (51) has no positive solution.

5. Conclusions

In this paper, we consider the existence of positive solution for fractional differential equations with conjugate type integral conditions. Both the existence of uniqueness and nonexistence of positive solution are established under conditions closely associated with the spectral radius with respect to the relevant linear operator. In addition, the unique positive solution can be approximated by an iterative scheme, and the error estimation of the iterative sequences is also given.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of Shandong Province of China (ZR2017MA036) and the National Natural Science Foundation of China (11571296).

References

  1. 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 ID 248, 2017. View at Publisher · View at Google Scholar · View at Scopus
  2. X. Du and A. Mao, “Existence and Multiplicity of Nontrivial Solutions for a Class of Semilinear Fractional Schrödinger Equations,” Journal of Function Spaces, vol. 2017, Article ID 3793872, 7 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  3. Y. Guan, Z. Zhao, and X. Lin, “On the existence of positive solutions and negative solutions of singular fractional differential equations via global bifurcation techniques,” Boundary Value Problems, vol. 2016, no. 1, Article ID 141, 2016. View at Publisher · View at Google Scholar · View at Scopus
  4. X. Hao, H. Wang, L. Liu, and Y. Cui, “Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator,” Boundary Value Problems, vol. 2017, no. 1, Article ID 182, 2017. View at Publisher · View at Google Scholar · View at Scopus
  5. J. Jiang and L. Liu, “Existence of solutions for a sequential fractional differential system with coupled boundary conditions,” Boundary Value Problems, vol. 2016, Article ID 159, 2016. View at Google Scholar · View at MathSciNet
  6. Tingting Qi, Yansheng Liu, and Yujun 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, 9 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  7. Yanbin Sang, Xiaorong Luo, and Yongqing Wang, “The Choquard Equation with Weighted Terms and Sobolev-Hardy Exponent,” Journal of Function Spaces, vol. 2018, Article ID 7210680, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  8. 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
  9. Y. Wang and L. Liu, “Necessary and sufficient condition for the existence of positive solution to singular fractional differential equations,” Advances in Difference Equations, vol. 2015, Article ID 207, pp. 1–14, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  10. F. Yan, M. Zuo, and X. Hao, “Positive solution for a fractional singular boundary value problem with p-Laplacian operator,” Boundary Value Problems, Article ID 51, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  11. K. M. Zhang, “On a sign-changing solution for some fractional differential equations,” Boundary Value Problems, vol. 2017, Article ID 59, 2017. View at Google Scholar · View at MathSciNet
  12. 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 Scopus
  13. X. Zhang, L. Liu, Y. Wu, and Y. Cui, “Entire blow-up solutions for a quasilinear p-Laplacian Schrodinger equation with a non-square diffusion term,” Applied Mathematics Letters, vol. 74, pp. 85–93, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  14. Xinguang Zhang, Lishan Liu, Yonghong Wu, and Yujun Cui, “New Result on the Critical Exponent for Solution of an Ordinary Fractional Differential Problem,” Journal of Function Spaces, vol. 2017, Article ID 3976469, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  15. X. Zhang, Z. Shao, and Q. Zhong, “Positive solutions for semipositone (k, n-k) conjugate boundary value problems with singularities on space variables,” Applied Mathematics Letters, vol. 72, pp. 50–57, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  16. M. Zuo, X. Hao, L. Liu, and Y. Cui, “Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions,” Boundary Value Problems, vol. 2017, article no. 161, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  17. 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
  18. Z. Bai and Y. Zhang, “Solvability of fractional three-point boundary value problems with nonlinear growth,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1719–1725, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. Z. Bai, Y. Chen, H. Lian, and S. Sun, “On the existence of blow up solutions for a class of fractional differential equations,” Fractional Calculus and Applied Analysis, vol. 17, no. 4, pp. 1175–1187, 2014. View at Publisher · View at Google Scholar · View at Scopus
  20. L. Guo, L. Liu, and Y. Wu, “Existence of positive solutions for singular higher-order fractional differential equations with infinite-point boundary conditions,” Boundary Value Problems, vol. 2016, Article ID 114, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  21. Q. Song, X. Dong, Z. Bai, and B. Chen, “Existence for fractional Dirichlet boundary value problem under barrier strip conditions,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 10, no. 7, pp. 3592–3598, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  22. Q. Song and Z. Bai, “Positive solutions of fractional differential equations involving the Riemann-Stieltjes integral boundary condition,” Advances in Difference Equations, vol. 2018, Article ID 183, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  23. Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a nonlocal fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 11, pp. 3599–3605, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. Y. Wang and L. Liu, “Positive solutions for a class of fractional 3-point boundary value problems at resonance,” Advances in Difference Equations, vol. 2017, Article ID 7, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  25. C. Zhai and L. Xu, “Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 8, pp. 2820–2827, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. X. Zhang and Q. Zhong, “Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables,” Applied Mathematics Letters, vol. 80, pp. 12–19, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  27. Y. Zhang, Z. Bai, and T. Feng, “Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1032–1047, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. 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
  29. 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, vol. 23, no. 1, pp. 31–39, 2018. View at Publisher · View at Google Scholar
  30. Y. Wang, L. Liu, and Y. Wu, “Existence and uniqueness of a positive solution to singular fractional differential equations,” Boundary Value Problems, vol. 2012, Article ID 81, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  31. J. Wu, X. Zhang, L. Liu, Y. Wu, and Y. Cui, “The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity,” Boundary Value Problems, vol. 2018, Article ID 82, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  32. X. Zhang, L. Liu, and Y. Wu, “The uniqueness of positive solution for a singular fractional differential system involving derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 6, pp. 1400–1409, 2013. View at Publisher · View at Google Scholar · View at Scopus
  33. X. Zhang, L. Liu, and Y. Wu, “The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium,” Applied Mathematics Letters, vol. 37, pp. 26–33, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  34. X. Zhang and Q. Zhong, “Uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions,” Fractional Calculus and Applied Analysis, vol. 20, no. 6, pp. 1471–1484, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  35. 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 MathSciNet