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On a Fractional Differential Inclusion in Banach Space Under Weak Compactness Condition

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Advances in Mathematical Economics Volume 20

Part of the book series: Advances in Mathematical Economics ((MATHECON,volume 20))

Abstract

We consider a class of boundary value problem in a separable Banach space governed by a fractional differential inclusion with integral boundary conditions

$$\displaystyle{\left \{\begin{array}{lll} w\text{-}D^{\alpha }u(t) \in F(t,u(t),w\text{-}D^{\alpha -1}u(t)), t \in [0,1] \\ I^{\beta }u(t)\vert _{t=0} = 0, u(1) =\int _{ 0}^{1}u(t)dt\end{array} \right.}$$

where α ∈ ]1, 2], \(\beta \in ]0,\infty [\) are given constant and w-D γ is the fractional w-R.L derivative of order γ ∈ {α − 1, α}, F is a convex weakly compact valued mapping. Topological properties of the solutions set are presented. Applications to control problems and further variants are provided.

JEL Classification: C61, C73

Mathematics Subject Classification (2010): 34A60, 34B15, 47H10

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Notes

  1. 1.

    Since E is a separable Banach space, \(\mathcal{B}(E_{\sigma }) = \mathcal{B}(E).\)

References

  1. Agarwal RP, Arshad S, O’ Reagan D, Lupulescu V (2012) Fuzzy fractional integral equations under compactness type condition. Frac Calc App Anal 15(4):572–590

    Google Scholar 

  2. Agarwal RP, Ahmad B, Alsaedi A, Shahzad N (2013) Dimension of the solution set for fractional differential inclusion. J Nonlinear Convex Anal 14(2):314–323

    MathSciNet  MATH  Google Scholar 

  3. Ahmad B, Nieto J (2012) Riemann-Liouville fractional differential equations with fractional boundary conditions. Fixed Point Theory 13(2):329–336

    MathSciNet  MATH  Google Scholar 

  4. Amrani A, Castaing C, Valadier M (1998) Convergence in Pettis norm under extreme point condition. Vietnam J Math (4):323–335

    MathSciNet  MATH  Google Scholar 

  5. Angosto C, Cascales B (2009) Measures of weak noncompactness in Banach spaces. Topol Appl 156:1412–1421

    Article  MathSciNet  MATH  Google Scholar 

  6. Attouch H, Cabot A, Redont P (2002) The dynamics of elastic shocks via epigraphical regularization of a differential inclusion. Barrier and penalty approximations. Adv Math Sci Appl 12(1):273–306. Gakkotosho, Tokyo

    Google Scholar 

  7. Azzam DL, Castaing C, Thibault L (2001) Three boundary value problems for second order differential inclusions in Banach spaces. Control Cybern 31:659–693

    MathSciNet  MATH  Google Scholar 

  8. Balder EJ (2000) Lectures on Young measure theory and its applications in economics. Rend Istit Mat Univ Trieste 31:1–69. Workshop di Teoria della Misura et Analisi Reale Grado, 1997 (Italia)

    Google Scholar 

  9. Benchohra M, Henderson J, Ntouyas SK, Ouahab A (2008) Existence results for fractional functional differential inclusions with infinite delay and applications to control theory. Frac Cal Appl Anal 11:35–56

    MathSciNet  MATH  Google Scholar 

  10. Benchohra M, Graef J, Mostefai F-Z (2011) Weak solutions for boundary-value problems with nonlinear fractional differential inclusions. Nonlinear Dyn Syst Theory 11(3):227–237

    MathSciNet  MATH  Google Scholar 

  11. Castaing C (1972) Quelques résultats de compacité liés a l’intégration. CR Acd Sc Paris 270:1732–1735 (1970) et Bulletin Soc Math France 31:73–81

    Google Scholar 

  12. Castaing C (1980) Topologie de la convergence uniforme sur les parties uniformément intégrables de L E 1 et théorèmes de compacité faible dans certains espaces du type Köthe-Orlicz. Sém Anal Convexe 10:5.1–5.27

    Google Scholar 

  13. Castaing C (1996) Weak compactness and convergences in Bochner and Pettis integration. Vietnam J Math 24(3):241–286

    MathSciNet  Google Scholar 

  14. Castaing C, Godet-Thobie C (2014) Le Xuan Truong and Satco B, Optimal control problems governed by a second order ordinary differential equation with m-point boundary condition. Adv Math Econ 18:1–56

    Article  MathSciNet  MATH  Google Scholar 

  15. Castaing C, Truong LX (2012) Some topological properties of solutions set in a second order inclusion with m-point boundary condition. Set-Valued Var Anal 20:249–277

    Article  MathSciNet  MATH  Google Scholar 

  16. Castaing C, Truong LX (2013) Bolza, relaxation and viscosity problems governed by a second order differential equation. J Nonlinear Convex Anal 14(2):451–482

    MathSciNet  MATH  Google Scholar 

  17. Castaing C, Valadier M (1977) Convex analysis and measurable multifunctions. Lecture notes in mathematics, vol 580. Springer, Berlin/Heidelberg/New York

    Google Scholar 

  18. Castaing C, Raynaud de Fitte P, Valadier M (2004) Young measures on topological spaces. With applications in control theory and probability theory. Kluwer Academic, Dordrecht

    MATH  Google Scholar 

  19. Castaing C, Ibrahim AG, Yarou M (2008) Existence problems in second order evolution inclusion: discretization and variational approach. Taiwan J Math 12(6):1435–1477

    MathSciNet  MATH  Google Scholar 

  20. Castaing C, Godet-Thobie C, Hoang TD, Raynaud de Fitte P (2015) On the integration of fuzzy level sets. Adv Math Econ 19:1–32

    Article  MathSciNet  MATH  Google Scholar 

  21. Castaing C, Phung PD, Truong LX (submitted) On a fractional differential inclusion with integral boundary condition in Banach spaces

    Google Scholar 

  22. Cernea A (2010) On a fractional differential inclusion with boundary condition. Studia Universitatis Babes-Bolyai Mathematica LV:105–113

    Google Scholar 

  23. Cichon M, Salem HAH (2014) Second order Three-point boundary value problems in abstract spaces. Acta Mathematicae Applicatae Sinica 30(4):1131–1152

    Article  MathSciNet  MATH  Google Scholar 

  24. De Blasi FS (1977) On a property of the unit sphere in a Banach space. Bull Math de la Soc Sci Math de la R.S de la Roumanie. Tome 21(64):3–4

    Google Scholar 

  25. El-Sayed AMA (1998) Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal 33:181–186

    Article  MathSciNet  MATH  Google Scholar 

  26. El-Sayed AMA, Abd El-Salam ShA (2009) Nonlocal boundary value problem of a fractional-order functional differential equation. Int J Nonlinear Sci 7:436–442

    MathSciNet  MATH  Google Scholar 

  27. El-Sayed AMA, Ibrahim AG (2001) Set-valued integral equations of arbitrary (fractional) order. Appl Math Comput 118:113–121

    MathSciNet  MATH  Google Scholar 

  28. Florescu LC, Godet-Thobie C (2012) Young measures and compactness in measure spaces. De Gruyter, Berlin

    Book  MATH  Google Scholar 

  29. Grothendieck A (1964) Espaces vectoriels topologiques. Publicacão da Sociedade de Matemática de Sao Paulo

    MATH  Google Scholar 

  30. Ibrahim AG, El-Sayed MA (1997) Definite integral of fractional order for Set-Valued function. Fractional Calc Appl Anal 11(3):81–87

    MathSciNet  MATH  Google Scholar 

  31. Kilbas AA, Marichev OI, Samko SG (1993) Fractional integrals and derivatives: theory and applications. Gorden and Breach, New York

    MATH  Google Scholar 

  32. Laksmikantham F, Mohapatra RN (2003) Theory of fuzzy differential equations. Taylor and Francis, London/New York

    Book  Google Scholar 

  33. Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, NewYork

    MATH  Google Scholar 

  34. O’Regan D (2000) Fixed point theorem for weakly sequentially closed maps. Archivum Mathematicum (BRNO) Tomus 36:61–70

    MATH  Google Scholar 

  35. Ouahab A (2008) Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal 69:3877–3896

    Article  MathSciNet  MATH  Google Scholar 

  36. Phung PH, Truong LX (2013) On a fractional differential inclusion with integral boundary conditions in Banach space. Fractional Calc Appl Anal 16(3):538–558

    MathSciNet  MATH  Google Scholar 

  37. Podlubny I (1999) Fractional differential equation. Academic, New York

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Département de Mathématiques, Université Montpellier II, Case Courrier 051, 34095, Montpellier Cedex 5, France

    C. Castaing

  2. Laboratoire de Mathématiques de Bretagne Atlantique, Université de Bretagne Occidentale, CNRS UMR 6205, 6, avenue Victor Le Gorgeu, CS 9387, F-29238, Brest Cedex 3, France

    C. Godet-Thobie

  3. Department of Mathematics and Statistics, University of Economics of HoChiMinh City, 59C Nguyen Dinh Chieu Str. Dist. 3, HoChiMinh City, Vietnam

    L. X. Truong

  4. Département de Mathématiques, Université de Saida, BP 138, Cite Ennasr, 20000, Saida, Algérie

    F. Z. Mostefai

Authors
  1. C. Castaing
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  2. C. Godet-Thobie
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  3. L. X. Truong
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  4. F. Z. Mostefai
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Correspondence to C. Castaing .

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Editors and Affiliations

  1. Professor Emeritus, The University of Tokyo, Tokyo, Tokyo, Japan

    Shigeo Kusuoka

  2. Keio University, Minato-ku, Japan

    Toru Maruyama

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Castaing, C., Godet-Thobie, C., Truong, L.X., Mostefai, F.Z. (2016). On a Fractional Differential Inclusion in Banach Space Under Weak Compactness Condition. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics Volume 20. Advances in Mathematical Economics, vol 20. Springer, Singapore. https://doi.org/10.1007/978-981-10-0476-6_2

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  • DOI: https://doi.org/10.1007/978-981-10-0476-6_2

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