Fractional Calculus and Applied Analysis

, Volume 16, Issue 3, pp 538–558 | Cite as

On a fractional differential inclusion with integral boundary conditions in Banach space

  • Phan Dinh Phung
  • Le Xuan Truong
Research Paper


We consider a class of boundary value problem in a separable Banach space E, involving a nonlinear differential inclusion of fractional order with integral boundary conditions, of the form
$$\left\{ \begin{gathered} D^\alpha u(t) \in F(t,u(t),D^{\alpha - 1} u(t)),a.e.,t \in [0,1], \hfill \\ I^\beta u(t)|_{t = 0} = 0,u(1) = \int\limits_0^1 {u(t)dt,} \hfill \\ \end{gathered} \right. $$
where D α is the standard Riemann-Liouville fractional derivative, F is a closed valued mapping. Under suitable conditions we prove that the solutions set of (*) is nonempty and is a retract in W E α,1 (I). An application in control theory is also provided by using the Young measures.

Key Words and Phrases

fractional differential inclusion boundary value problem Green’s function contractive set valued-map retract Young measures 

MSC 2010

26A33 34A60 34B10 34A08 47N70 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    B. Ahmad, S. Ntouyas, Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 15, No 3 (2012), 362–382; DOI: 10.2478/s13540-012-0027-y, at Scholar
  2. [2]
    D.L. Azzam, C. Castaing and L. Thibault, Three boundary value problems for second order differential inclusions in Banach spaces. Control Cybernet. 31 (2001), 659–693.MathSciNetGoogle Scholar
  3. [3]
    M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, Existence results for fractional functional differential inclusions with infinite delay and applications to control theory. Frac. Calc. Applied Anal. 11, No 1 (2008), 35–56; at Scholar
  4. [4]
    Z. Bai, H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311 (2005), 495–505.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    A. Bressan, A. Cellina and A. Fryszkowski, A class of absolute retracts in spaces of integrable functions. Proc. Amer. Math. Soc. 112 (1991), 413–418.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    C. Castaing, L.X. Truong, Second order differential inclusions with mpoint boundary conditions. J. Nonlinear Convex Anal. 12, No 2 (2011), 199–224.MathSciNetzbMATHGoogle Scholar
  7. [7]
    C. Castaing, P. Raynaud de Fitte and M. Valadier, Young Measures on Topological Spaces. With Applications in Control Theory and Probability Theory. Kluwer Academic Publishers, Dordrecht (2004).zbMATHGoogle Scholar
  8. [8]
    C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin, Heidelberg, New York (1977).zbMATHCrossRefGoogle Scholar
  9. [9]
    A. Cernea, On a fractional differential inclusion with boundary condition. Studia Univ. Babes-Bolyai Mathematica. LV (2010), 105–113.MathSciNetGoogle Scholar
  10. [10]
    A. Cernea, A note on the existence of solutions for some boundary value problems of fractional differential inclusions. Fract. Calc. Appl. Anal. 15, No 2 (2012), 183–194; DOI: 10.2478/s13540-012-0013-4; at Scholar
  11. [11]
    H. Covitz, S.B. Nadler, Multivalued contraction mappings in generalized metric spaces. Israel J. Math. 8 (1970), 5–11.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal. 33 (1998), 181–186.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    A.M.A. El-Sayed, Sh.A. Abd El-Salam, Nonlocal boundary value problem of a fractional-order functional differential equation. International J. Nonlinear. Science 7 (2009), 436–442.Google Scholar
  14. [14]
    A.M.A. El-Sayed, A.G. Ibrahim, Set-valued integral equations of arbitrary (fractional) order. Appl. Math. Comput. 118 (2001), 113–121.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    A.M. Gomaa, On the solution sets of the three-points boundary value problems for nonconvex differential inclusions. J. Egypt. Math. Soc. 12 (2004), 97–107.MathSciNetzbMATHGoogle Scholar
  16. [16]
    E. Hille and R.S. Phillips, Functional Analysis and Semi-Groups. Amer. Math. Soc. Colloq. Publ., Vol. 31 (1957).Google Scholar
  17. [17]
    S. Liang, J. Zhang, Existence and uniqueness of positive solutions to m-points boundary value problem for nonlinear fractional differential equation. J. Appl. Math. Comput. 38, No 1–2 (2012), 225–241; DOI 10.1007/s12190-011-0475-2.MathSciNetCrossRefGoogle Scholar
  18. [18]
    K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Willey, New York (1993).zbMATHGoogle Scholar
  19. [19]
    A. Ouahab, Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal. 69 (2008), 3877–3896.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    I. Podlubny, Fractional Differential Equation. Academic Press, New York (1999).Google Scholar
  21. [21]
    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993).zbMATHGoogle Scholar
  22. [22]
    H.A.H. Salem, A.M.A. El-Sayed, O.L. Moustafa, A note on the fractional calculus in Banach spaces. Studia Sci. Math. Hungar. 42 (2005), 115–130.MathSciNetzbMATHGoogle Scholar

Copyright information

© Versita Warsaw and Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Nguyen Tat Thanh UniversityHoChiMinh CityVietnam
  2. 2.Department of Mathematics and StatisticsUniversity of Economics HoChiMinh CityHoChiMinh CityVietnam

Personalised recommendations