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The European Physical Journal Special Topics

, Volume 226, Issue 16–18, pp 3391–3409 | Cite as

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

  • Yong-Kui Chang
  • Rodrigo Ponce
Regular Article
  • 16 Downloads
Part of the following topical collections:
  1. Fractional Dynamical Systems - Recent Trends in Theory and Applications

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.

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References

  1. 1.
    K. Deimling, Multivalued differential equations (Walter de Gruyter, Berlin, 1992) Google Scholar
  2. 2.
    A.A. Tolstonogov, Differential inclusions in a Banach space (Kluwer Academic, Dordrecht, 2000) Google Scholar
  3. 3.
    N. Abada, M. Benchohra, H. Hammouche, J. Differ. Equ. 246, 3834 (2009) ADSCrossRefGoogle Scholar
  4. 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 Google Scholar
  5. 5.
    M. Benchohra, J. Henderson, S.K. Ntouyas, Impulsive differential equations and inclusions (Hindawi Publishing Corporation, New York, 2006) Google Scholar
  6. 6.
    S. Djebali, L. Górniewicz, A. Ouahab, Solutions set for differential equations and inclusions (De Gruyter, Berlin, 2013) Google Scholar
  7. 7.
    K. Ezzinbi, M.S. Lalaoui, Appl. Math. 60, 321 (2015) MathSciNetCrossRefGoogle Scholar
  8. 8.
    J.R. Graef, J. Henderson, A. Ouahab, Impulsive differential inclusions: a fixed point approach (Walter de Gruyter, Berlin, 2013) Google Scholar
  9. 9.
    S. Hu, N.S. Papageorgiou, Handbook of multivalued analysis (Kluwer Academic, Dordrecht, 1997) Google Scholar
  10. 10.
    C. Lizama, A. Pereira, R. Ponce, Semigroup Forum 93, 363 (2016) MathSciNetCrossRefGoogle Scholar
  11. 11.
    S. Abbas, Semigroup Forum 81, 393 (2010) MathSciNetCrossRefGoogle Scholar
  12. 12.
    S. Abbas, M. Banerjee, S. Momani, Comput. Math. Appl. 62, 1098 (2011) MathSciNetCrossRefGoogle Scholar
  13. 13.
    S. Abbas, M. Benchohra, G.M. N’Guérékata, Topics in fractional differential equations (Springer, New York, 2012) Google Scholar
  14. 14.
    K. Diethelm, The analysis of fractional differential equations (Springer, Berlin, 2010) Google Scholar
  15. 15.
    A. Kilbas, H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations (Elsevier Science B.V., Amsterdam, 2006) Google Scholar
  16. 16.
    S. Kumar, N. Sukavanam, J. Differ. Equ. 252, 6163 (2012) ADSCrossRefGoogle Scholar
  17. 17.
    X.B. Shu, Q. Wang, Appl. Math. Comput. 273, 465 (2016) MathSciNetGoogle Scholar
  18. 18.
    G. Wang, D. Baleanu, L. Zhang, Fract. Calc. Appl. Anal. 15, 244 (2011) Google Scholar
  19. 19.
    R.N. Wang, D.H. Chen, T.J. Xiao, J. Differ. Equ. 252, 202 (2012) ADSCrossRefGoogle Scholar
  20. 20.
    R.N. Wang, Q.H. Ma, Appl. Math. Comput. 257, 285 (2015) MathSciNetGoogle Scholar
  21. 21.
    Y. Zhou, Fractional evolution equations and inclusions: analysis and control (Elsevier, New York, 2016) Google Scholar
  22. 22.
    A. Cernea, Math. Rep. 15, 34 (2013) Google Scholar
  23. 23.
    A. Cernea, Electron. J. Qual. Theory Differ. Equ. 2014, 25 (2014) CrossRefGoogle Scholar
  24. 24.
    R.N. Wang, P.X. Zhu, Q.H. Ma, Nonlinear Dyn. 80, 1745 (2015) ADSCrossRefGoogle Scholar
  25. 25.
    A. Benchaabane, R. Sakthivel, J. Comput. Appl. Math. 312, 65 (2017) MathSciNetCrossRefGoogle Scholar
  26. 26.
    A. Debbouche, J.J. Nieto, Appl. Math. Comput. 245, 74 (2014) MathSciNetGoogle Scholar
  27. 27.
    A. Debbouche, D.F.M. Torres, Fract. Calc. Appl. Anal. 18, 95 (2015) MathSciNetCrossRefGoogle Scholar
  28. 28.
    M. Fečkan, J. Wang, Y. Zhou, J. Optim. Theory Appl 156, 79 (2013) MathSciNetCrossRefGoogle Scholar
  29. 29.
    F. Li, J. Liang, H.K. Xu, J. Math. Anal. Appl. 391, 510 (2012) CrossRefGoogle Scholar
  30. 30.
    J. Wang, Z. Fan, Y. Zhou, J. Optim. Theory Appl. 154, 292 (2012) MathSciNetCrossRefGoogle Scholar
  31. 31.
    J. Wang, M. Fečkan, Y. Zhou, Dyn. Part. Differ. Equ. 11, 71 (2014) CrossRefGoogle Scholar
  32. 32.
    J. Liang, T.J. Xiao, J. Math. Anal. Appl. 259, 398 (2001) MathSciNetCrossRefGoogle Scholar
  33. 33.
    R.P. Agarwal, M. Benchohra, S. Hamani, Acta Appl. Math. 109, 973 (2010) MathSciNetCrossRefGoogle Scholar
  34. 34.
    B. Ahmad, S.K. Ntouyas, Fract. Calc. Appl. Anal. 15, 362 (2012) MathSciNetGoogle Scholar
  35. 35.
    Y.K. Chang, A. Pereira, R. Ponce, Fract. Calc. Appl. Anal. 20, 963 (2017) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Y. Chalco-Cano, J.J. Nieto, A. Ouahab, H. Romn-Flores, Fract. Calc. Appl. Anal. 16, 682 (2013) MathSciNetCrossRefGoogle Scholar
  37. 37.
    Z. Fan, Appl. Math. Comput. 232, 60 (2014) MathSciNetGoogle Scholar
  38. 38.
    R. Kamocki, C. Obczynski, J. Math. Phys. 55, 022902 (2014) ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    C. Lizama, G. N’Guérékata, Appl. Anal. 92, 1731 (2013) MathSciNetCrossRefGoogle Scholar
  40. 40.
    R. Ponce, J. Differ. Equ. 255, 3284 (2013) ADSCrossRefGoogle Scholar
  41. 41.
    R. Ponce, Abstr. Appl. Anal. 2016, 4567092 (2016) CrossRefGoogle Scholar
  42. 42.
    R. Sakthivel, Y. Ren, A. Debbouche, N.I. Mahmudov, Appl. Anal. 95, 2361 (2016) MathSciNetCrossRefGoogle Scholar
  43. 43.
    J. Wang, A. Ibrahim, M. Fečkan, Commun. Nonlinear Sci. Numer. Simul. 27, 281 (2015) ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    C. Castaing, M. Valadier, Convex analysis and measurable multifunctions (Springer, Berlin, 1977) Google Scholar
  45. 45.
    N.S. Papageorgiou, S.T. Kyritsi-Yiallourou, Handbook of applied analysis (Springer, New York, 2009) Google Scholar
  46. 46.
    K. Yosida, Functional analysis (Springer, Berlin, 1980) Google Scholar
  47. 47.
    J. Lightbourne, S. Rankin, J. Math. Anal. Appl. 93, 328 (1983) MathSciNetCrossRefGoogle Scholar
  48. 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 Google Scholar

Copyright information

© EDP Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, Xidian UniversityXi’anP.R. China
  2. 2.Universidad de Talca, Instituto de Matemática y FísicaTalcaChile

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