Abstract
In this chapter, we introduce notations, definitions, and preliminary facts which are used throughout this book.
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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)
H. Amann, Linear and Quasilinear Parabolic Problems (Birkhäuser, Berlin, 1995)
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)
W. Arendt, Vector valued Laplace transforms and Cauchy problems. Israel J. Math. 59, 327–352 (1987)
C. Avramescu, Some remarks on a fixed point theorem of Krasnoselskii. Electron. J. Qual. Differ. Equ. 5, 1–15 (2003)
A. Bressan, G. Colombo, Extensions and selections of maps with decomposable values. Stud. Math. 90, 70–85 (1988)
T.A. Burton, A fixed-point theorem of Krasnoselskii. Appl. Math. Lett. 11(1), 85–88 (1998)
T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii type. Math. Nachr. 189, 23–31 (1998)
C. Corduneanu, Integral Equations and Stability of Feedback Systems (Acedemic, New York, 1973)
G. Da Prato, E. Grisvard, On extrapolation spaces. Rend. Accad. Naz. Lincei. 72, 330–332 (1982)
K. Deimling, Multivalued Differential Equations (Walter de Gruyter, Berlin/New York, 1992)
B.C. Dhage, Fixed-point theorems for discontinuous multivalued operators on ordered spaces with applications. Comput. Math. Appl. 51, 589–604 (2006)
S. Djebali, L. Gorniewicz, A. Ouahab, Solution Sets for Differential Equations and Inclusions (Walter de Gruyter, Berlin, 2013)
J. Dugundji, A. Granas, Fixed point Theory (Springer, New York 2003)
K.J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer, New York, 2000)
H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces. North Holland, Mathematical Studies (North Holland, Amsterdam, 1985)
M. Frigon, Fixed point results for generalized contractions in gauge spaces and applications. Proc. Am. Math. Soc. 128(10), 2957–2965 (2000)
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
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
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)
J.A. Goldstein, Semigroups of Linear operators and Applications (Oxford University Press, New York, 1985)
L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings. Mathematics and its Applications, vol. 495 (Kluwer Academic, Dordrecht, 1999)
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)
J. Hale, J. Kato, Phase space for retarded equations with infinite delay. Funkcial. Ekvac. 21, 11–41 (1978)
S. Heikkila, V. Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations (Marcel Dekker Inc., New York, 1994)
Y. Hino, S. Murakami, T. Naito, Functional Differential Equations with Unbounded Delay (Springer, Berlin, 1991)
Sh. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, vol. I (Kluwer, Dordrecht/Boston/London, 1997)
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)
F. Kappel, W. Schappacher, Some considerations to the fundamental theory of infinite delay equations. J. Differ. Equ. 37, 141–183 (1980)
H. Kellermann, M. Hieber, Integrated semigroup. J. Funct. Anal. 84, 160–180 (1989)
M. Kisielewicz, Differential Inclusions and Optimal Control (Kluwer, Dordrecht, 1991)
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)
L. Maniar, A. Rhandi, Inhomogeneous retarded equation in infinite dimentional space via extrapolation spaces. Rend. Circ. Mat. Palermo 47, 331–346 (1998)
H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, 985–999 (1980)
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
J. Neerven, The Adjoint of a Semigroup of Linear Operators. Lecture Notes in Math., vol. 1529 (Springer, New York, 1992)
V. Obukhovskii, Semilinear functional-differential inclusions in a Banach space and controlled parabolic systems. Soviet J. Autom. Inf. Sci. 24, 71–79 (1991)
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, New York, 1983)
K. Schumacher, Existence and continuous dependence for differential equations with unbounded delay. Arch. Ration. Mech. Anal. 64, 315–335 (1978)
C.C. Travis, G.F. Webb, Existence and stability for partial functional differential equations. Trans. Am. Math. Sci. 200, 395–418 (1974)
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)
A.A. Tolstonogov, Differential Inclusions in a Banach Space (Kluwer Academic, Dordrecht, 2000)
K. Yosida, Functional Analysis, 6th edn. (Springer, Berlin, 1980)
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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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