Jump to ContentJump to Main Navigation
Show Summary Details

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board Member: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

8 Issues per year


IMPACT FACTOR 2015: 0.687

SCImago Journal Rank (SJR) 2015: 0.298
Source Normalized Impact per Paper (SNIP) 2015: 0.476
Impact per Publication (IPP) 2015: 0.677

Mathematical Citation Quotient (MCQ) 2015: 0.04

Online
ISSN
2191-0294
See all formats and pricing
Volume 18, Issue 1 (Feb 2017)

Issues

Nontrivial Solutions of Higher-Order Nonlinear Singular Fractional Differential Equations with Fractional Multi-point Boundary Conditions

Shengli Xie
  • Corresponding author
  • Shengli Xie, School of Mathematics and Physics, Urban Construction College, Anhui Jianzhu University, Hefei, Anhui 230601, P. R. China,
  • Email:
/ Yiming Xie
  • Yiming Xie: School of Civil Engineering, Anhui Jianzhu University, Hefei, Anhui 230601, P. R. China
Published Online: 2016-12-20 | DOI: https://doi.org/10.1515/ijnsns-2016-0060

Abstract

This paper deals with the existence and multiplicity of nontrivial solutions of fractional multi-point boundary value problems for higher-order nonlinear singular fractional differential equations with sign-changing nonlinear term. The main tool used in the proof is topological degree theory. Some examples are given to illustrate our main results.

Keywords: singular fractional differential equations; nontrivial solution; topological degree; Riemann–Liouville fractional derivative

MSC 2010: 34B10; 34B16; 34B27; 45G10

References

  • [1] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.

  • [2] A. Kilbas, H. Srivastava and J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.

  • [3] V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of fractional dynamic systems, Cambridge Academic Publishers, Cambridge, 2009.

  • [4] Z. B. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal. 72 (2010), 916–924.

  • [5] J. Henderson and R. Luca, Positive solution for a system of nonlocal fractional boundary value problems, Fract. Calc. Appl. Anal. 16 (2013), 985–1008.

  • [6] J. Henderson and R. Luca, Positive solutions for a system of fractional differential equations with coupled integral boundary conditions, Appl. Math. Comput. 249 (2014), 182–197.

  • [7] W. G. Yang, Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions, Comput. Math. Appl. 63 (2012), 288–297.

  • [8] S. Q. Zhang, Existence results of positive solutions to fractional differential equation with integral boundary conditions, Math. Bohem. 135 (2010), 299–317.

  • [9] C. X. Zhu, X. Z. Zhang and Z.Q. Wu, Solvability for a coupled system of fractional differential equations with nonlocal integral boundary conditions, Taiwanese J. Math. 17 (2013), 2039–2054.

  • [10] X. Z. Zhang, C. X. Zhu and Z. Q. Wu, Solvability for a coupled system of fractional differential equations with impulses at resonance, Bound. Value Probl. 2013 (2013), 80.

  • [11] B. Ahmad and J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 58 (2009), 1838–1843.

  • [12] M. Jia, X. G. Zhang and X. M. Gu, Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions, Bound. Value Probl. 2012 (2012), 70.

  • [13] M. El-Shahed and J. J. Nieto, Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order, Comput. Math. Appl. 59 (2010), 3438–3443.

  • [14] X. W. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22 (2009), 64–69.

  • [15] J. X Sun and X. Y Liu, Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations, J. Math. Anal. Appl. 326 (2008), 927–937.

  • [16] J. X Sun and X. Y Liu, Computation of topological degree for nonlinear operators and applications, Nonlinear Anal. 69 (2008), 4121–4130.

  • [17] T. H. Wu, X. G. Zhang and Yinan Lu, Solutions of sign-changing fractional differential equation with the fractional derivatives, Abstr. Appl. Anal. (2012), Art. ID 797398, 16 pp. [

  • [18] Keyu Zhang and Jiafa Xu, Nontrivial solutions for a fractional boundary value problem, Adv. Difference Equ. 2013 (2013), 171.

  • [19] J. X. Sun and G. W. Zhang, Nontrivial solutions of singular superlinear Sturm-Liouville problems, J. Math. Anal. Appl. 313 (2006), 518–536.

  • [20] J. X. Sun and G. W. Zhang, Nontrivial solutions of singular superlinear Sturm-Liouville problems, J. Math. Anal. Appl. 326 (2007), 242–251.

  • [21] Y. J. Cui and Yumei Zou, Nontrivial solutions of singular superlinear m-point boundary value problems, Appl. Math. Comput. 187 (2007), 1256–1264.

  • [22] C. S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010), 1050–1055.

  • [23] W. H. Jiang, J. Q. Qiu and W. W. Guo The Existence of positive solutions for fractional differential equations with sign changing Nonlinearities, Abstr. Appl. Anal. (2012), Art. ID 180672, 13 pp.

  • [24] C. F. Li, X. N. Luo and Y. Zhou, Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Comput. Math. Appl. 59 (2010), 1363–1375.

  • [25] S. L. Xie, Positive solutions for a system of higher-order singular nonlinear fractional differential equations with nonlocal boundary conditions, Electron. J. Qual. Theory Differ. Equ. 18 (2015), 1–17.

  • [26] X. G. Zhang, Y. H. Wu and L. Caccetta, Nonlocal fractional order differential equations with changing-sign singular perturbation. Appl. Math. Model. 39 (2015), 6543–6552. [

  • [27] X. Q. Zhang and Q. Y. Zhong, Multiple positive solutions for nonlocal boundary value problems of singular fractional differential equations, Bound. Value Probl. 2016 (2016), 65.

  • [28] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Gordon and Breach, Yverdon, 1993.

  • [29] M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice Hall, New York, 1967.

  • [30] D. Guo and V. Laksmikantham, Nonlinear problems in abstract cones, Academic Press, Boston, NY, 1988.

  • [31] K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985.

  • [32] Liu Xiao-Ying and Sun Jing-Xian, Computation of topological degree and applications to superlinear system of equations, J. Sys. Math. Scis. 16 (1996), 51–59 (Chinese).

  • [33] M. Ur Rehman, R. Ali Khan, Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, Appl. Math. Lett. 23 (2010), 1038–1044.

About the article

Received: 2016-04-18

Accepted: 2016-11-27

Published Online: 2016-12-20

Published in Print: 2017-02-01


Funding: The work was supported by Natural Science Foundation of Anhui Province and Anhui Provincial Education Department (1508085MA08, KJ2014A043), P. R. China.


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2016-0060. Export Citation

Comments (0)

Please log in or register to comment.
Log in