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Preliminary Background

  • Saïd Abbas
  • Mouffak Benchohra
Chapter
  • 590 Downloads
Part of the Developments in Mathematics book series (DEVM, volume 39)

Abstract

In this chapter, we introduce notations, definitions, and preliminary facts which are used throughout this book.

Keywords

Banach Space Phase Space Continuous Semigroup Order Interval Contraction Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 16.
    N.U. Ahmed, Semigroup Theory with Applications to Systems and Control. Pitman Research Notes in Mathematics Series, vol. 246 (Longman Scientific & Technical, Harlow; Wiley, New York, 1991)Google Scholar
  2. 23.
    H. Amann, Linear and Quasilinear Parabolic Problems (Birkhäuser, Berlin, 1995)CrossRefGoogle Scholar
  3. 25.
    B. Amir, L. Maniar, Composition of pseudo almost periodic functions and Cauchy problems with operator of nondense domain. Ann. Math. Blaise Pascal 6, 1–11 (1999)CrossRefGoogle Scholar
  4. 29.
    W. Arendt, Vector valued Laplace transforms and Cauchy problems. Israel J. Math. 59, 327–352 (1987)CrossRefGoogle Scholar
  5. 32.
    C. Avramescu, Some remarks on a fixed point theorem of Krasnoselskii. Electron. J. Qual. Differ. Equ. 5, 1–15 (2003)Google Scholar
  6. 86.
    A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values. Stud. Math. 90, 70–85 (1988)Google Scholar
  7. 87.
    T.A. Burton, A fixed-point theorem of Krasnoselskii. Appl. Math. Lett. 11(1), 85–88 (1998)CrossRefGoogle Scholar
  8. 88.
    T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii type. Math. Nachr. 189, 23–31 (1998)CrossRefGoogle Scholar
  9. 96.
    C. Corduneanu, Integral Equations and Stability of Feedback Systems (Acedemic, New York, 1973)Google Scholar
  10. 99.
    G. Da Prato, E. Grisvard, On extrapolation spaces. Rend. Accad. Naz. Lincei. 72, 330–332 (1982)Google Scholar
  11. 101.
    K. Deimling, Multivalued Differential Equations (Walter de Gruyter, Berlin/New York, 1992)CrossRefGoogle Scholar
  12. 102.
    B.C. Dhage, Fixed-point theorems for discontinuous multivalued operators on ordered spaces with applications. Comput. Math. Appl. 51, 589–604 (2006)CrossRefGoogle Scholar
  13. 104.
    S. Djebali, L. Gorniewicz, A. Ouahab, Solution Sets for Differential Equations and Inclusions (Walter de Gruyter, Berlin, 2013)CrossRefGoogle Scholar
  14. 105.
    J. Dugundji, A. Granas, Fixed point Theory (Springer, New York 2003)Google Scholar
  15. 106.
    K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer, New York, 2000)Google Scholar
  16. 111.
    H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces. North Holland, Mathematical Studies (North Holland, Amsterdam, 1985)Google Scholar
  17. 113.
    M. Frigon, Fixed point results for generalized contractions in gauge spaces and applications. Proc. Am. Math. Soc. 128(10), 2957–2965 (2000)CrossRefGoogle Scholar
  18. 114.
    M. Frigon, Fixed Point Results for Multivalued Contractions on Gauge Spaces. Set Valued Mappings with Applications in Nonlinear Analysis. Ser. Math. Anal. Appl., vol. 4 (Taylor & Francis, London, 2002), pp. 175–181Google Scholar
  19. 115.
    M. Frigon, Fixed Point and Continuation Results for Contractions in Metric and Gauge Spaces. Fixed Point Theory and Its Applications, vol. 77 (Banach Center/Polish Academic Science, Warsaw, 2007), pp. 89–114Google Scholar
  20. 116.
    M. Frigon, A. Granas, Résultats de type Leray-Schauder pour des contractions sur des espaces de Fréchet. Ann. Sci. Math. Québec 22(2), 161–168 (1998)Google Scholar
  21. 122.
    J.A. Goldstein, Semigroups of Linear operators and Applications (Oxford University Press, New York, 1985)Google Scholar
  22. 123.
    L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings. Mathematics and its Applications, vol. 495 (Kluwer Academic, Dordrecht, 1999)Google Scholar
  23. 130.
    G. Guhring, F. Rabiger, W. Ruess, Linearized stability for semilinear non-autonomous evolution equations to retarded differential equations. Differ. Integr. Equ. 13, 503–527 (2000)Google Scholar
  24. 132.
    J. Hale, J. Kato, Phase space for retarded equations with infinite delay. Funkcial. Ekvac. 21, 11–41 (1978)Google Scholar
  25. 134.
    S. Heikkila, V. Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations (Marcel Dekker Inc., New York, 1994)Google Scholar
  26. 142.
    Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Unbounded Delay (Springer, Berlin, 1991)Google Scholar
  27. 143.
    Sh. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, vol. I (Kluwer, Dordrecht/Boston/London, 1997)CrossRefGoogle Scholar
  28. 144.
    M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces (Walter de Gruyter Series in Nonlinear Analysis and Applications, Berlin, 2001)CrossRefGoogle Scholar
  29. 145.
    F. Kappel, W. Schappacher, Some considerations to the fundamental theory of infinite delay equations. J. Differ. Equ. 37, 141–183 (1980)CrossRefGoogle Scholar
  30. 146.
    H. Kellermann, M. Hieber, Integrated semigroup. J. Funct. Anal. 84, 160–180 (1989)CrossRefGoogle Scholar
  31. 147.
    M. Kisielewicz, Differential Inclusions and Optimal Control (Kluwer, Dordrecht, 1991)Google Scholar
  32. 154.
    A. Lasota, Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13, 781–786 (1965)Google Scholar
  33. 160.
    L. Maniar, A. Rhandi, Inhomogeneous retarded equation in infinite dimentional space via extrapolation spaces. Rend. Circ. Mat. Palermo 47, 331–346 (1998)CrossRefGoogle Scholar
  34. 162.
    H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, 985–999 (1980)CrossRefGoogle Scholar
  35. 163.
    R. Nagel, E. Sinestrari, Inhomogeneous Volterra Integrodifferential Equations for Hille-Yosida operators, In Functional Analysis, ed. by K.D. Bierstedt, A. Pietsch, W.M. Ruess, D. Voigt (Marcel Dekker, New York, 1998) pp. 51–70Google Scholar
  36. 164.
    J. Neerven, The Adjoint of a Semigroup of Linear Operators. Lecture Notes in Math., vol. 1529 (Springer, New York, 1992)Google Scholar
  37. 167.
    V. Obukhovskii, Semilinear functional-differential inclusions in a Banach space and controlled parabolic systems. Soviet J. Autom. Inf. Sci. 24, 71–79 (1991)Google Scholar
  38. 168.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, New York, 1983)CrossRefGoogle Scholar
  39. 173.
    K. Schumacher, Existence and continuous dependence for differential equations with unbounded delay. Arch. Ration. Mech. Anal. 64, 315–335 (1978)Google Scholar
  40. 179.
    C.C. Travis, G.F. Webb, Existence and stability for partial functional differential equations. Trans. Am. Math. Sci. 200, 395–418 (1974)CrossRefGoogle Scholar
  41. 180.
    C.C. Travis, G.F. Webb, Existence, stability and compactness in the α−norm for partial functionaldifferential equations. Trans. Am. Math. Sci. 240, 129–143 (1978)Google Scholar
  42. 181.
    A.A. Tolstonogov, Differential Inclusions in a Banach Space (Kluwer Academic, Dordrecht, 2000)CrossRefGoogle Scholar
  43. 185.
    K. Yosida, Functional Analysis, 6th edn. (Springer, Berlin, 1980)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Saïd Abbas
    • 1
  • Mouffak Benchohra
    • 2
  1. 1.Laboratoire de MathématiquesUniversité de SaïdaSaïdaAlgeria
  2. 2.Department of MathematicsUniversity of Sidi Bel AbbesSidi Bel AbbesAlgeria

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