Opuscula Math. 36, no. 1 (2016), 123-137

Opuscula Mathematica

Fractional evolution equation nonlocal problems with noncompact semigroups

Xuping Zhang
Pengyu Chen

Abstract. This paper is concerned with the existence results of mild solutions to the nonlocal problem of fractional semilinear integro-differential evolution equations. New existence theorems are obtained by means of the fixed point theorem for condensing maps. The results extend and improve some related results in this direction.

Keywords: fractional evolution equation, mild solution, nonlocal condition, \(C_0\)-semigroup, condensing maps, measure of noncompactness.

Mathematics Subject Classification: 34A12, 35F25, 35R11.

Full text (pdf)

  1. R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solutions for fractional differential equations with uncertainly, Nonlinear Anal. 72 (2010), 2859-2862.
  2. K. Balachandran, J.J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Anal. 72 (2010), 4587-4593.
  3. K. Balachandran, S. Kiruthika, J.J. Trujillo, On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces, Comput. Math. Appl. 62 (2011), 1157-1165.
  4. J. Banas, K. Goebel, Measure of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., vol. 60, Marcel Dekker, New York, 1980.
  5. M. Benchohra, G.M. N'Guérékata, D. Seba, Measure of noncompactness and nondensly definded semilinear functional differential equations with fractional order, Cubo 12 (2010), 35-48.
  6. L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494-505.
  7. L. Byszewski, Application of properties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems, Nonlinear Anal. 33 (1998), 2413-2426.
  8. L. Byszewski, V. Lakshmikantham, Theorem about the existence and uniqueness of solutions of a nonlocal Cauchy problem in a Banach space, Appl. Anal. 40 (1990), 11-19.
  9. P. Chen, Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys. 65 (2014), 711-728.
  10. P. Chen, Y. Li, Nonlocal problem for fractional evolution equations of mixed type with the measure of noncompactness, Abst. Appl. Anal. 2013, Article ID 784816, 12.
  11. P. Chen, Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math. 63 (2013), 731-744.
  12. P. Chen, Y. Li, Q. Li, Existence of mild solutions for fractional evolution equations with nonlocal initial conditions, Ann. Polon. Math. 110 (2014), 13-24.
  13. P. Chen, Y. Li, H. Yang, Perturbation method for nonlocal impulsive evolution equations, Nonlinear Anal. Hybrid Syst. 8 (2013), 22-30.
  14. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.
  15. M.M. EI-Borai, Some probability densities and funddamental solutions of fractional evolution equations, Chaos Solitons Fractals 14 (2002), 433-440.
  16. M.M. EI-Borai, On some stochastic fractional integro-differential equations, Adv. Dyn. Syst. Appl. 1 (2006), 49-57.
  17. K. Ezzinbi, X. Fu, K. Hilal, Existence and regularity in the \(\alpha\)-norm for some neutral partial differential equations with nonlocal conditions, Nonlinear Anal. 67 (2007), 1613-1622.
  18. L. Fang, G.M. N'Guérékata, An existence result for neutral delay integrodifferentual equations with fractional order nonlocal conditions, Abst. Appl. Anal. 2011, Article ID 952782, 20 pp.
  19. H.P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of rector-valued functions, Nonlinear Anal. 7 (1983), 1351-1371.
  20. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, [in:] North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.
  21. F. Li, J. Liang, H.K. Xu, Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl. 391 (2012), 510-525.
  22. J. Liang, J.H. Liu, T.J. Xiao, Nonlocal impulsive problems for integrodifferential equations, Math. Comput. Modelling 49 (2009), 798-804.
  23. B. Lundstrom, M. Higgs, W. Spain, A. Fairhall, Fractional by neocortial pyramidal neurons, Nat Neurosci. 11 (2008), 1335-1342.
  24. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
  25. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  26. T. Poinot, J.C. Trigeassou, Identification of fractional systems using an output-error technique, Nonlinear Dynam. 38 (2004), 133-154.
  27. Y. Rossikhin, M. Shitikova, Application of fractional derivatives to the analysis of damped vibrationsof viscoelaticsingle mass system, Acta Mech. 120 (1997), 109-125.
  28. N. Tatar, Existence results for an evolution problem with fractional nonlocal conditions, Comput. Math. Appl. 60 (2010), 2971-2982.
  29. J. Wang, Y. Zhou, W. Wei, H. Xu, Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls, Comput. Math. Appl. 62 (2011), 1427-1441.
  30. J. Wang, Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl. 12 (2011), 263-272.
  31. R. Wang, T.J. Xiao, J. Liang, A note on the fractional Cauchy problems with nonlocal conditions, Appl. Math. Lett. 24 (2011), 1435-1442.
  32. Y. Zhou, F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. Real World Appl. 11 (2010), 4465-4475.
  • Xuping Zhang
  • Northwest Normal University, Department of Mathematics, Lanzhou 730 070, P.R. China
  • Zhixing College of Northwest Normal University, Department of Mathematics, Lanzhou 730 070, P.R. China
  • Pengyu Chen
  • Northwest Normal University, Department of Mathematics, Lanzhou 730 070, P.R. China
  • Communicated by Alexander Gomilko.
  • Received: 2014-01-23.
  • Revised: 2015-04-23.
  • Accepted: 2015-05-07.
  • Published online: 2015-09-19.
Opuscula Mathematica - cover

Cite this article as:
Xuping Zhang, Pengyu Chen, Fractional evolution equation nonlocal problems with noncompact semigroups, Opuscula Math. 36, no. 1 (2016), 123-137, http://dx.doi.org/10.7494/OpMath.2016.36.1.123

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.