Applications of Mathematics

, Volume 56, Issue 2, pp 227–250 | Cite as

Impulsive semilinear neutral functional differential inclusions with multivalued jumps

  • Nadjet Abada
  • Ravi P. Agarwal
  • Mouffak Benchohra
  • Hadda Hammouche
Article
First Online:
Received:

Abstract

In this paper we establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some densely defined impulsive semilinear neutral functional differential inclusions in separable Banach spaces. We rely on a fixed point theorem for the sum of completely continuous and contraction operators.

Keywords

impulsive semilinear neutral functional differential equation densely defined operator infinite delay phase space fixed point, mild solutions extremal mild solution 

MSC 2010

34A37 34G25 34K30 34K35 34K45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. Abada, M. Benchohra, H. Hammouche: Existence and controllability results for impulsive partial functional differential inclusions. Nonlinear Anal., Theory Methods Appl. 69 (2008), 2892–2909.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    N. Abada, M. Benchohra, H. Hammouche, A. Ouahab: Controllability of impulsive semilinear functional differential inclusions with finite delay in Fréchet spaces. Discuss. Math., Differ. Incl. Control Optim. 27 (2007), 329–347.MathSciNetMATHGoogle Scholar
  3. [3]
    N.U. Ahmed: Semigroup Theory with Applications to Systems and Control. Pitman Research Notes in Mathematics Series, 246. Longman Scientific & Technical/John Wiley & Sons, Harlow/New York, 1991.MATHGoogle Scholar
  4. [4]
    N.U. Ahmed: Dynamic Systems and Control with Applications. World Scientific Publishing, Hackensack, 2006.MATHGoogle Scholar
  5. [5]
    N.U. Ahmed: Systems governed by impulsive differential inclusions on Hilbert spaces. Nonlinear Anal., Theory Methods Appl. 45 (2001), 693–706.MATHCrossRefGoogle Scholar
  6. [6]
    N.U. Ahmed: Optimal impulse control for impulsive systems in Banach spaces. Int. J. Differ. Equ. Appl. 1 (2000), 37–52.MathSciNetMATHGoogle Scholar
  7. [7]
    D.D. Bajnov, P. S. Simeonov: Systems with Impulse Effect. Stability, Theory and Applications. Ellis Horwood, Chichester, 1989.MATHGoogle Scholar
  8. [8]
    M. Belmekki, M. Benchohra, K. Ezzinbi, S.K. Ntouyas: Existence results for some partial functional differential equations with infinite delay. Nonlinear Stud. 15 (2008), 373–385.MathSciNetMATHGoogle Scholar
  9. [9]
    M. Benchohra, L. Górniewicz, S.K. Ntouyas: Controllability of Some Nonlinear Systems in Banach Spaces (The Fixed Point Theory Approach). Pawel Wlodkowic University College, Wydawnictwo Naukowe NOVUM, Plock, 2003.MATHGoogle Scholar
  10. [10]
    M. Benchohra, J. Henderson, S.K. Ntouyas: Impulsive Differential Equations and Inclusions, Vol. 2. Hindawi Publishing Corporation, New York, 2006.CrossRefGoogle Scholar
  11. [11]
    I. Benedetti: An existence result for impulsive functional differential inclusions in Banach spaces. Discuss. Math., Differ. Incl. Control Optim. 24 (2004), 13–30.MathSciNetMATHGoogle Scholar
  12. [12]
    C. Corduneanu, V. Lakshmikantham: Equations with unbounded delay. A survey. Nonlinear Anal., Theory Methods Appl. 4 (1980), 831–877.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    G. Da Prato, E. Grisvard: On extrapolation spaces. Atti. Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 72 (1982), 330–332.MATHGoogle Scholar
  14. [14]
    B.C. Dhage: Multivalued maping and fixed point. Nonlinear Funct. Anal. Appl. 10 (2005), 359–378.MathSciNetMATHGoogle Scholar
  15. [15]
    B.C. Dhage: A fixed point theorem for multi-valued mappings on ordered Banach spaces with applications II. Panam. Math. J. 15 (2005), 15–34.MathSciNetMATHGoogle Scholar
  16. [16]
    B.C. Dhage, E. Gatsori, S.K. Ntouyas: Existence theory for perturbed functional differential inclusions. Commun. Appl. Nonlinear Anal. 13 (2006), 15–26.MathSciNetMATHGoogle Scholar
  17. [17]
    K. Deimling: Multivalued Differential Equations. Walter De Gruyter, Berlin, 1992.MATHCrossRefGoogle Scholar
  18. [18]
    K. J. Engel, R. Nagel: One-Parameter Semigroups for Linear Evolution Equations. Springer, Berlin, 2000.MATHGoogle Scholar
  19. [19]
    D. Guo, V. Lakshmikantham: Nonlinear Problems in Abstract Cones. Academic Press, Boston, 1988.MATHGoogle Scholar
  20. [20]
    L. Górniewicz: Topological Fixed Point Theory of Multivalued Mappings. Mathematics and Its Applications, 495. Kluwer Academic Publishers, Dordrecht, 1999.MATHGoogle Scholar
  21. [21]
    J.K. Hale: Theory of Functional Differential Equations. Springer, New York, 1977.MATHGoogle Scholar
  22. [22]
    J.K. Hale, J. Kato: Phase space for retarded equations with infinite delay. Funkc. Ekvacioj, Ser. Int. 21 (1978), 11–41.MathSciNetMATHGoogle Scholar
  23. [23]
    J.K. Hale, S.H. Verduyn Lunel: Introduction to Functional Differential Equations. Applied Mathematical Sciences 99. Springer, New York, 1993.MATHGoogle Scholar
  24. [24]
    S. Heikkila, V. Lakshmikantham: Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations. Marcel Dekker Inc., New York, 1994.Google Scholar
  25. [25]
    E. Hernández, H.R. Henríquez: Existence results for partial neutral functional differential equations with unbounded delay. J. Math. Anal. Appl. 221 (1998), 452–475.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    H.R. Henríquez: Existence of periodic solutions of partial neutral functional differential equations with unbounded delay. J. Math. Anal. Appl. 221 (1998), 499–522.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    Y. Hino, S. Murakami, T. Naito: Functional Differential Equations with Infinite Delay. Lecture Notes Math. Vol. 1473. Springer, Berlin, 1991.MATHGoogle Scholar
  28. [28]
    Sh. Hu, N. S. Papageorgiou: Handbook of Multivalued Analysis. Volume I: Theory. Kluwer Academic Publishers, Dordrecht, 1997.MATHGoogle Scholar
  29. [29]
    M. Kamenskii, V. Obukhovskii, P. Zecca: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. De Gruyter Series in Nonlinear Analysis and Applications. De Gruyter, Berlin, 2001.Google Scholar
  30. [30]
    F. Kappel, W. Schappacher: Some considerations to the fundamental theory of infinite delay equations. J. Differ. Equations 37 (1980), 141–183.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    M. Kisielewicz: Differential Inclusions and Optimal Control. Kluwer Academic Publishers, Dordrecht, 1990.MATHGoogle Scholar
  32. [32]
    V. Kolmanovskii, A. Myshkis: Introduction to the Theory and Applications of Functional-Differential Equations. Mathematics and Its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999.MATHGoogle Scholar
  33. [33]
    Y. Kuang: Delay Differential Equations: with Applications in Population Dynamics. Academic Press, Boston, 1993.MATHGoogle Scholar
  34. [34]
    V. Lakshmikantham, D.D. Bainov, P. S. Simeonov: Theory of Impulsive Differential Equations. World Scientific, Singapore, 1989.MATHGoogle Scholar
  35. [35]
    V. Lakshmikantham, L. Wen, B. Zhang: Theory of Differential Equations with Unbounded Delay. Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht, 1994.Google Scholar
  36. [36]
    A. Lasota, Z. Opial: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astronom. Phys. 13 (1965), 781–786.MathSciNetMATHGoogle Scholar
  37. [37]
    J.H. Liu: Nonlinear impulsive evolution equations. Dyn. Contin. Discrete Impulsive Syst. 6 (1999), 77–85.MATHGoogle Scholar
  38. [38]
    S. Migorski, A. Ochal: Nonlinear impulsive evolution inclusions of second order. Dyn. Syst. Appl. 16 (2007), 155–173.MathSciNetMATHGoogle Scholar
  39. [39]
    A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983.MATHGoogle Scholar
  40. [40]
    Y.V. Rogovchenko: Impulsive evolution systems: Main results and new trends. Dyn. Contin. Discrete Impulsive Syst. 3 (1997), 57–88.MathSciNetMATHGoogle Scholar
  41. [41]
    Y.V. Rogovchenko: Nonlinear impulsive evolution systems and applications to population models. J. Math. Anal. Appl. 207 (1997), 300–315.MathSciNetCrossRefGoogle Scholar
  42. [42]
    A.M. Samoilenko, N.A. Perestyuk: Impulsive Differential Equations. World Scientific, Singapore, 1995.MATHCrossRefGoogle Scholar
  43. [43]
    K. Schumacher: Existence and continuous dependence for functional-differential equations with unbounded delay. Arch. Ration. Mech. Anal. 67 (1978), 315–335.MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    J. S. Shin: An existence of functional differential equations. Arch. Ration. Mech. Anal. 30 (1987), 19–29.MATHGoogle Scholar
  45. [45]
    J. Wu: Theory and Applications of Partial Functional Differential Equations. Applied Mathematical Sciences 119. Springer, New York, 1996.MATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2011

Authors and Affiliations

  • Nadjet Abada
    • 1
    • 2
  • Ravi P. Agarwal
    • 3
  • Mouffak Benchohra
    • 4
  • Hadda Hammouche
    • 5
  1. 1.École Normale Supérieure, Département Sciences ExactesPlateau MansourahConstantineAlgérie
  2. 2.Laboratoire Modelisation Mathématiques et SimulationsUniversité MentouriConstantineAlgérie
  3. 3.Department of Mathematical SciencesFlorida Institute of TechnologyMelbourneUSA
  4. 4.Laboratoire de MathématiquesUniversité de Sidi Bel AbbèsSidi Bel AbbèsAlgérie
  5. 5.Département de MathématiquesUniversité Kasdi Merbah de OuarglaRoute de Ghardaia, OuarglaAlgérie

Personalised recommendations