Abstract
The purpose of this chapter is to state some basic concepts and to introduce well known results in the nonlinear operator theory and the theory of nonlinear evolutions in real Banach spaces that will play very important roles in the subsequent development of this book. In Section 1.1 we consider properties of accretive operators with one-sided directional derivatives and duality mappings. Section 1.2 contains the generation of nonlinear semigroups and solutions of autonomous homogeneous nonlinear evolutions. Section 1.3 deals with the existence for strong solutions, integral solutions, and limit solutions of autonomous non-homogeneous nonlinear evolutions. Section 1.4 is devoted to definitions and properties of generalized domains. Section 1.5 treats properties of evolution operators and their generation. In Section 1.6, Section 1.7, and Section 1.8 we discuss how to solve non-autonomous nonlinear evolutions by Crandall-Liggett, Crandall-Pazy, and Evans, respectively, whilst Section 1.9 deals with them by Pavel’s method. Comments and notes for references are found in Section 1.10.
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Notes for References
V. Barbu (1976).Nonlinear Semigroups and Differential Equations in Banach Spaces, (Noordhoff International Publishing: Leyden, The Netherlands).
Ph. B–nilan (1972).Equations d––volution dans un espace de Ba-nach quelconque et application, (Th–se de Doctorat d–Etat), Publications Math–matiques d–Orsay 25, Universit– de Paris XI.
Ph. Bénilan (1972). Solutions integrales d’équations d’évolution dans un espace de Banach,C. R. Acad. Sci. Paris 274, 47 – 50.
H. Brézis (1973).Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, ( North-Holland Publishing Company: Amsterdam).
H. Brézis (1974).New results concerning monotone operators and nonlinear semigroups, Anal. Nonlin. Probl., RIMS, 2 – 27
H. Brézis and A. Pazy (1970). Accretive sets and differential equations in Banach spaces,Israel J. Math. 8, 367 – 383.
M. G. Crandall and T. M. Liggett (1971). Generation of semi-groups of nonlinear transformations on general Banach spaces,Amer. J. Math. 93, 265 – 298.
K. Deimling (1985).Nonlinear Functional Analysis, (Springer-Verlag: Berlin, Germany).
K. S. Ha (1987). Theory of Nonlinear Partial Differential Equations with Nonlinear Semigroups (in Korean), (Mineumsa: Seoul).
V. Lakshmikantham and S. Leela (1981).Nonlinear Differntial Equations in Abstract Spaces, ( Pergamon Press: New York).
G. Lumer and R. S. Phillips (1961). Dissipative operators in a Banach space,Pacific J. Math. 11, 679 – 698.
R. H. Martin Jr. (1976).Nonlinear Operators and Differential Equations in Banach Spaces, ( John Wiley and Sons: New York).
I. Miyadera (1977).Nonlinear Semigroups(in Japanese), ( Kinokuniya: Tokyo).
G. Morsanu (1988).Nonlinear Evolution Equations and Applications, (D. Rei-del Publishing Company: Dordrecht).
R. Nagel (1986).One-Parameter Semigroups of Positive Operators, Lecture Notes on Mathematics 1184, ( Springer—Verlag: Berlin).
D. Pascali and S. Sburlan (1978).Nonlinear Mappings of Monotone Type, ( Sijthoff and Noordhoff International Publishers: Burcharest).
N. H. Pavel (1987).Nonlinear Evolution Operators and Semigroups, Lecture Notes in Mathematics 1260, ( Springer—Verlag: Berlin).
A. Pazy (1983).Semigroups of Linear Operators and Applications to Partial Differential Equations, ( Springer—Verlag: Berlin).
I. I. Vrabie (1987).Compactness Methods for Nonlinear Evolutions, Pitman Monographs and Surveys in Pure and Applied Mathematics 32, ( Longman Scientific and Technical: London).
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Ha, K.S. (2003). Nonlinear Evolutions. In: Nonlinear Functional Evolutions in Banach Spaces. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0365-9_1
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DOI: https://doi.org/10.1007/978-94-017-0365-9_1
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