Nonlinear Dynamics

, Volume 71, Issue 4, pp 685–700 | Cite as

Abstract Cauchy problem for fractional differential equations

  • JinRong Wang
  • Yong Zhou
  • Michal Fec̆kan
Original Paper


In this paper, a generalized Darbo’s fixed-point theorem associated with Hausdorff measure of noncompactness is established. Then we apply this new variant fixed-point theorem to study some fractional differential equations in Banach spaces via the technique of measure of noncompactness. Many novel existence and uniqueness results for solutions are obtained under the more general conditions.


Fractional differential equations Cauchy problems Impulsive problems Nonlocal problems Hausdorff measure of noncompactness 



The first author acknowledges the support by National Natural Science Foundation of China (11201091) and Key Projects of Science and Technology Research in the Chinese Ministry of Education (211169); the second author acknowledges the support by National Natural Science Foundation of China (11271309), Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001) and Key Projects of Hunan Provincial Natural Science Foundation of China (12JJ2001) and the third author acknowledges the support by Grants VEGA-MS 1/0507/11, VEGA-SAV 2/0124/10 and APVV-0414-07.


  1. 1.
    Liu, L., Guo, F., Wu, C., Wu, Y.: Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces. J. Math. Anal. Appl. 309, 638–649 (2005) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Baleanu, D., Machado, J.A.T., Luo, A.C.-J.: Fractional Dynamics and Control. Springer, Berlin (2012) MATHCrossRefGoogle Scholar
  3. 3.
    Diethelm, K.: The analysis of fractional differential equations. In: Lecture Notes in Mathematics. Springer, Berlin (2010) Google Scholar
  4. 4.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science, Amsterdam (2006) MATHCrossRefGoogle Scholar
  5. 5.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
  6. 6.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
  7. 7.
    Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. HEP. Springer, Berlin (2010) Google Scholar
  8. 8.
    Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ahmad, B., Nieto, J.J.: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray–Schauder degree theory. Topol. Methods Nonlinear Anal. 35, 295–304 (2010) MathSciNetMATHGoogle Scholar
  10. 10.
    Bai, C.: Impulsive periodic boundary value problems for fractional differential equation involving Riemann–Liouville sequential fractional derivative. J. Math. Anal. Appl. 384, 211–231 (2011) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340–1350 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Chen, F., Nieto, J.J., Zhou, Y.: Global attractivity for nonlinear fractional differential equations. Nonlinear Anal., Real World Appl. 13, 287–298 (2012) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Chang, Y.K., Kavitha, V., Arjunan, M.M.: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. Nonlinear Anal. TMA 71, 5551–5559 (2009) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Henderson, J., Ouahab, A.: Fractional functional differential inclusions with finite delay. Nonlinear Anal. TMA 70, 2091–2105 (2009) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12, 262–272 (2011) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Wang, J., Zhou, Y.: Analysis of nonlinear fractional control systems in Banach spaces. Nonlinear Anal. TMA 74, 5929–5942 (2011) MATHCrossRefGoogle Scholar
  17. 17.
    Wang, J., Zhou, Y.: Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal., Real World Appl. 12, 3642–3653 (2011) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Wang, J., Zhou, Y., Medved̆, M.: On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay. J. Optim. Theory Appl. 389, 261–274 (2012) MathSciNetMATHGoogle Scholar
  19. 19.
    Wang, R., Chen, D., Xiao, T.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252, 202–235 (2012) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Zhang, S.: Existence of positive solution for some class of nonlinear fractional differential equations. J. Math. Anal. Appl. 278, 136–148 (2003) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Zhou, Y., Jiao, F., Li, J.: Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal. TMA 71, 3249–3256 (2009) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl. 11, 4465–4475 (2010) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Pazy, A.: Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) CrossRefGoogle Scholar
  24. 24.
    Banas̀, J., Goebel, K.: Measure of Noncompactness in Banach Spaces. Marcel Dekker, New York (1980) Google Scholar
  25. 25.
    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985) MATHCrossRefGoogle Scholar
  26. 26.
    Heinz, H.-P.: On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Anal. TMA 7, 1351–1371 (1983) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Lakshmikantham, V., Leela, S.: Nonlinear Differential Equations in Abstract Spaces. Pergamon Press, New York (1969) Google Scholar
  28. 28.
    Fec̆kan, M., Zhou, Y., Wang, J.: On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 3050–3060 (2012). MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Department of MathematicsGuizhou UniversityGuiyangP.R. China
  2. 2.Department of MathematicsXiangtan UniversityXiangtanP.R. China
  3. 3.Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and InformaticsComenius UniversityBratislavaSlovakia
  4. 4.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia

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