Skip to main content
SpringerLink
Log in

Properties of solution sets for Sobolev type fractional differential inclusions via resolvent family of operators

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

In this manuscript, by properties on some corresponding resolvent operators and techniques in multivalued analysis, we establish some results for solution sets of Sobolev type fractional differential inclusions in the Caputo and Riemann-Liouville fractional derivatives with order 1 < α < 2, respectively. We show that the solution sets are nonempty, compact, contractible and thus arcwise connected under some suitable conditions. We remark that our results are directly established through resolvent operators instead of subordination formulas usually applied, and the existence and compactness of E−1 is not necessarily needed. Some applications are also given in the final.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K. Deimling, Multivalued differential equations (Walter de Gruyter, Berlin, 1992)

  2. A.A. Tolstonogov, Differential inclusions in a Banach space (Kluwer Academic, Dordrecht, 2000)

  3. N. Abada, M. Benchohra, H. Hammouche, J. Differ. Equ. 246, 3834 (2009)

    Article  ADS  Google Scholar 

  4. J. Barrios, A. Piétrus, A. Marrero, H. Arazoza, HIV model described by differential inclusions, in IWANN 2009, Part I, LNCS 5517, edited by J. Cabestany et al. (Springer-Verlag, Berlin, Heidelberg, 2009), p. 909

  5. M. Benchohra, J. Henderson, S.K. Ntouyas, Impulsive differential equations and inclusions (Hindawi Publishing Corporation, New York, 2006)

  6. S. Djebali, L. Górniewicz, A. Ouahab, Solutions set for differential equations and inclusions (De Gruyter, Berlin, 2013)

  7. K. Ezzinbi, M.S. Lalaoui, Appl. Math. 60, 321 (2015)

    Article  MathSciNet  Google Scholar 

  8. J.R. Graef, J. Henderson, A. Ouahab, Impulsive differential inclusions: a fixed point approach (Walter de Gruyter, Berlin, 2013)

  9. S. Hu, N.S. Papageorgiou, Handbook of multivalued analysis (Kluwer Academic, Dordrecht, 1997)

  10. C. Lizama, A. Pereira, R. Ponce, Semigroup Forum 93, 363 (2016)

    Article  MathSciNet  Google Scholar 

  11. S. Abbas, Semigroup Forum 81, 393 (2010)

    Article  MathSciNet  Google Scholar 

  12. S. Abbas, M. Banerjee, S. Momani, Comput. Math. Appl. 62, 1098 (2011)

    Article  MathSciNet  Google Scholar 

  13. S. Abbas, M. Benchohra, G.M. N’Guérékata, Topics in fractional differential equations (Springer, New York, 2012)

  14. K. Diethelm, The analysis of fractional differential equations (Springer, Berlin, 2010)

  15. A. Kilbas, H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations (Elsevier Science B.V., Amsterdam, 2006)

  16. S. Kumar, N. Sukavanam, J. Differ. Equ. 252, 6163 (2012)

    Article  ADS  Google Scholar 

  17. X.B. Shu, Q. Wang, Appl. Math. Comput. 273, 465 (2016)

    MathSciNet  Google Scholar 

  18. G. Wang, D. Baleanu, L. Zhang, Fract. Calc. Appl. Anal. 15, 244 (2011)

    Google Scholar 

  19. R.N. Wang, D.H. Chen, T.J. Xiao, J. Differ. Equ. 252, 202 (2012)

    Article  ADS  Google Scholar 

  20. R.N. Wang, Q.H. Ma, Appl. Math. Comput. 257, 285 (2015)

    MathSciNet  Google Scholar 

  21. Y. Zhou, Fractional evolution equations and inclusions: analysis and control (Elsevier, New York, 2016)

  22. A. Cernea, Math. Rep. 15, 34 (2013)

    Google Scholar 

  23. A. Cernea, Electron. J. Qual. Theory Differ. Equ. 2014, 25 (2014)

    Article  Google Scholar 

  24. R.N. Wang, P.X. Zhu, Q.H. Ma, Nonlinear Dyn. 80, 1745 (2015)

    Article  ADS  Google Scholar 

  25. A. Benchaabane, R. Sakthivel, J. Comput. Appl. Math. 312, 65 (2017)

    Article  MathSciNet  Google Scholar 

  26. A. Debbouche, J.J. Nieto, Appl. Math. Comput. 245, 74 (2014)

    MathSciNet  Google Scholar 

  27. A. Debbouche, D.F.M. Torres, Fract. Calc. Appl. Anal. 18, 95 (2015)

    Article  MathSciNet  Google Scholar 

  28. M. Fečkan, J. Wang, Y. Zhou, J. Optim. Theory Appl 156, 79 (2013)

    Article  MathSciNet  Google Scholar 

  29. F. Li, J. Liang, H.K. Xu, J. Math. Anal. Appl. 391, 510 (2012)

    Article  Google Scholar 

  30. J. Wang, Z. Fan, Y. Zhou, J. Optim. Theory Appl. 154, 292 (2012)

    Article  MathSciNet  Google Scholar 

  31. J. Wang, M. Fečkan, Y. Zhou, Dyn. Part. Differ. Equ. 11, 71 (2014)

    Article  Google Scholar 

  32. J. Liang, T.J. Xiao, J. Math. Anal. Appl. 259, 398 (2001)

    Article  MathSciNet  Google Scholar 

  33. R.P. Agarwal, M. Benchohra, S. Hamani, Acta Appl. Math. 109, 973 (2010)

    Article  MathSciNet  Google Scholar 

  34. B. Ahmad, S.K. Ntouyas, Fract. Calc. Appl. Anal. 15, 362 (2012)

    MathSciNet  Google Scholar 

  35. Y.K. Chang, A. Pereira, R. Ponce, Fract. Calc. Appl. Anal. 20, 963 (2017)

    Article  MathSciNet  Google Scholar 

  36. Y. Chalco-Cano, J.J. Nieto, A. Ouahab, H. Romn-Flores, Fract. Calc. Appl. Anal. 16, 682 (2013)

    Article  MathSciNet  Google Scholar 

  37. Z. Fan, Appl. Math. Comput. 232, 60 (2014)

    MathSciNet  Google Scholar 

  38. R. Kamocki, C. Obczynski, J. Math. Phys. 55, 022902 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  39. C. Lizama, G. N’Guérékata, Appl. Anal. 92, 1731 (2013)

    Article  MathSciNet  Google Scholar 

  40. R. Ponce, J. Differ. Equ. 255, 3284 (2013)

    Article  ADS  Google Scholar 

  41. R. Ponce, Abstr. Appl. Anal. 2016, 4567092 (2016)

    Article  Google Scholar 

  42. R. Sakthivel, Y. Ren, A. Debbouche, N.I. Mahmudov, Appl. Anal. 95, 2361 (2016)

    Article  MathSciNet  Google Scholar 

  43. J. Wang, A. Ibrahim, M. Fečkan, Commun. Nonlinear Sci. Numer. Simul. 27, 281 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  44. C. Castaing, M. Valadier, Convex analysis and measurable multifunctions (Springer, Berlin, 1977)

  45. N.S. Papageorgiou, S.T. Kyritsi-Yiallourou, Handbook of applied analysis (Springer, New York, 2009)

  46. K. Yosida, Functional analysis (Springer, Berlin, 1980)

  47. J. Lightbourne, S. Rankin, J. Math. Anal. Appl. 93, 328 (1983)

    Article  MathSciNet  Google Scholar 

  48. E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, in Proceedings of the 6th AIMS International Conference, Dynamical Systems and Differential Equations (Discrete Contin. Dyn. Syst.) (2007), p. 277

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, Shaanxi, P.R. China

    Yong-Kui Chang

  2. Universidad de Talca, Instituto de Matemática y Física, Casilla 747, Talca, Chile

    Rodrigo Ponce

Authors
  1. Yong-Kui Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rodrigo Ponce
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong-Kui Chang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chang, YK., Ponce, R. Properties of solution sets for Sobolev type fractional differential inclusions via resolvent family of operators. Eur. Phys. J. Spec. Top. 226, 3391–3409 (2017). https://doi.org/10.1140/epjst/e2018-00015-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2018-00015-y

Search

Navigation