Characterization of nonlinear semigroups associated with semilinear evolution equations

Authors:
Shinnosuke Oharu and Tadayasu Takahashi

Journal:
Trans. Amer. Math. Soc. **311** (1989), 593-619

MSC:
Primary 47H20; Secondary 58D25

MathSciNet review:
978369

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Nonlinear continuous perturbations of linear dissipative operators are considered from the point of view of the nonlinear semigroup theory. A general class of nonlinear perturbations of linear contraction semigroups in a Banach space is introduced by means of a lower semicontinuous convex functional and two notions of semilinear infinitesimal generators of the associated nonlinear semigroups are formulated. Four types of necessary and sufficient conditions are given for a semilinear operator of the class to be the infinitesimal generator of a nonlinear semigroup on the domain of such that for the -valued function on provides a unique mild solution of the semilinear evolution equation satisfying a growth condition for the function . It turns out that various types of characterizations of nonlinear semigroups associated with semilinear evolution equations are obtained and, in particular, a semilinear version of the Hille-Yosida theorem is established in a considerably general form.

**[1]**J. M. Ball,*Strongly continuous semigroups, weak solutions, and the variation of constants formula*, Proc. Amer. Math. Soc.**63**(1977), no. 2, 370–373. MR**0442748**, 10.1090/S0002-9939-1977-0442748-6**[2]**M. G. Crandall and T. M. Liggett,*Generation of semi-groups of nonlinear transformations on general Banach spaces*, Amer. J. Math.**93**(1971), 265–298. MR**0287357****[3]**Einar Hille and Ralph S. Phillips,*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR**0089373****[4]**Toshiyuki Iwamiya,*Global existence of mild solutions to semilinear differential equations in Banach spaces*, Hiroshima Math. J.**16**(1986), no. 3, 499–530. MR**867577****[5]**Tosio Kato,*Perturbation theory for linear operators*, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR**0203473****[6]**Yoshikazu Kobayashi,*Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups*, J. Math. Soc. Japan**27**(1975), no. 4, 640–665. MR**0399974****[7]**Kazuo Kobayasi, Yoshikazu Kobayashi, and Shinnosuke Oharu,*Nonlinear evolution operators in Banach spaces*, Osaka J. Math.**21**(1984), no. 2, 281–310. MR**752464****[8]**V. Lakshmikantham and S. Leela,*Differential and integral inequalities*, Academic Press, New York, 1969.**[9]**Robert H. Martin Jr.,*Invariant sets for perturbed semigroups of linear operators*, Ann. Mat. Pura Appl. (4)**105**(1975), 221–239. MR**0390414****[10]**Robert H. Martin Jr.,*Nonlinear operators and differential equations in Banach spaces*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. Pure and Applied Mathematics. MR**0492671****[11]**Shinnosuke Oharu and Tadayasu Takahashi,*Locally Lipschitz continuous perturbations of linear dissipative operators and nonlinear semigroups*, Proc. Amer. Math. Soc.**100**(1987), no. 1, 187–194. MR**883426**, 10.1090/S0002-9939-1987-0883426-5**[12]**Nicolae H. Pavel,*Nonlinear evolution equations governed by 𝑓-quasidissipative operators*, Nonlinear Anal.**5**(1981), no. 5, 449–468. MR**613054**, 10.1016/0362-546X(81)90094-8**[13]**Nicolae H. Pavel,*Semilinear equations with dissipative time-dependent domain perturbations*, Israel J. Math.**46**(1983), no. 1-2, 103–122. MR**727025**, 10.1007/BF02760625**[14]**-,*Differential equations, flow invariance and applications*, Pitman, London, 1984.**[15]**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486****[16]**G. F. Webb,*Continuous nonlinear perturbations of linear accretive operators in Banach spaces*, J. Functional Analysis**10**(1972), 191–203. MR**0361965****[17]**Kôsaku Yosida,*Functional analysis*, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag New York Inc., New York, 1968. MR**0239384**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
47H20,
58D25

Retrieve articles in all journals with MSC: 47H20, 58D25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0978369-9

Keywords:
Nonlinear perturbations of linear operators,
semilinear evolution equation,
mild solution,
nonlinear semigroup,
full infinitesimal generator,
range condition,
local quasi-dissipativity

Article copyright:
© Copyright 1989
American Mathematical Society