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Part of the book series: Developments in Mathematics ((DEVM,volume 39))

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Abstract

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

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References

  1. 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. H. Amann, Linear and Quasilinear Parabolic Problems (Birkhäuser, Berlin, 1995)

    Book  Google Scholar 

  3. 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)

    Article  Google Scholar 

  4. W. Arendt, Vector valued Laplace transforms and Cauchy problems. Israel J. Math. 59, 327–352 (1987)

    Article  Google Scholar 

  5. C. Avramescu, Some remarks on a fixed point theorem of Krasnoselskii. Electron. J. Qual. Differ. Equ. 5, 1–15 (2003)

    Google Scholar 

  6. A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values. Stud. Math. 90, 70–85 (1988)

    Google Scholar 

  7. T.A. Burton, A fixed-point theorem of Krasnoselskii. Appl. Math. Lett. 11(1), 85–88 (1998)

    Article  Google Scholar 

  8. T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii type. Math. Nachr. 189, 23–31 (1998)

    Article  Google Scholar 

  9. C. Corduneanu, Integral Equations and Stability of Feedback Systems (Acedemic, New York, 1973)

    Google Scholar 

  10. G. Da Prato, E. Grisvard, On extrapolation spaces. Rend. Accad. Naz. Lincei. 72, 330–332 (1982)

    Google Scholar 

  11. K. Deimling, Multivalued Differential Equations (Walter de Gruyter, Berlin/New York, 1992)

    Book  Google Scholar 

  12. B.C. Dhage, Fixed-point theorems for discontinuous multivalued operators on ordered spaces with applications. Comput. Math. Appl. 51, 589–604 (2006)

    Article  Google Scholar 

  13. S. Djebali, L. Gorniewicz, A. Ouahab, Solution Sets for Differential Equations and Inclusions (Walter de Gruyter, Berlin, 2013)

    Book  Google Scholar 

  14. J. Dugundji, A. Granas, Fixed point Theory (Springer, New York 2003)

    Google Scholar 

  15. K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer, New York, 2000)

    Google Scholar 

  16. H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces. North Holland, Mathematical Studies (North Holland, Amsterdam, 1985)

    Google Scholar 

  17. M. Frigon, Fixed point results for generalized contractions in gauge spaces and applications. Proc. Am. Math. Soc. 128(10), 2957–2965 (2000)

    Article  Google Scholar 

  18. 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. 175–181

    Google Scholar 

  19. 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. 89–114

    Google Scholar 

  20. 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. J.A. Goldstein, Semigroups of Linear operators and Applications (Oxford University Press, New York, 1985)

    Google Scholar 

  22. L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings. Mathematics and its Applications, vol. 495 (Kluwer Academic, Dordrecht, 1999)

    Google Scholar 

  23. 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. J. Hale, J. Kato, Phase space for retarded equations with infinite delay. Funkcial. Ekvac. 21, 11–41 (1978)

    Google Scholar 

  25. S. Heikkila, V. Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations (Marcel Dekker Inc., New York, 1994)

    Google Scholar 

  26. Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Unbounded Delay (Springer, Berlin, 1991)

    Google Scholar 

  27. Sh. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, vol. I (Kluwer, Dordrecht/Boston/London, 1997)

    Book  Google Scholar 

  28. 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)

    Book  Google Scholar 

  29. F. Kappel, W. Schappacher, Some considerations to the fundamental theory of infinite delay equations. J. Differ. Equ. 37, 141–183 (1980)

    Article  Google Scholar 

  30. H. Kellermann, M. Hieber, Integrated semigroup. J. Funct. Anal. 84, 160–180 (1989)

    Article  Google Scholar 

  31. M. Kisielewicz, Differential Inclusions and Optimal Control (Kluwer, Dordrecht, 1991)

    Google Scholar 

  32. 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. L. Maniar, A. Rhandi, Inhomogeneous retarded equation in infinite dimentional space via extrapolation spaces. Rend. Circ. Mat. Palermo 47, 331–346 (1998)

    Article  Google Scholar 

  34. H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, 985–999 (1980)

    Article  Google Scholar 

  35. 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. 51–70

    Google Scholar 

  36. J. Neerven, The Adjoint of a Semigroup of Linear Operators. Lecture Notes in Math., vol. 1529 (Springer, New York, 1992)

    Google Scholar 

  37. 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. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, New York, 1983)

    Book  Google Scholar 

  39. K. Schumacher, Existence and continuous dependence for differential equations with unbounded delay. Arch. Ration. Mech. Anal. 64, 315–335 (1978)

    Google Scholar 

  40. C.C. Travis, G.F. Webb, Existence and stability for partial functional differential equations. Trans. Am. Math. Sci. 200, 395–418 (1974)

    Article  Google Scholar 

  41. 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. A.A. Tolstonogov, Differential Inclusions in a Banach Space (Kluwer Academic, Dordrecht, 2000)

    Book  Google Scholar 

  43. K. Yosida, Functional Analysis, 6th edn. (Springer, Berlin, 1980)

    Book  Google Scholar 

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Abbas, S., Benchohra, M. (2015). Preliminary Background. In: Advanced Functional Evolution Equations and Inclusions. Developments in Mathematics, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-17768-7_1

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