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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-1989-0978369-9
MathSciNet review: 978369
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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 $ X$ is introduced by means of a lower semicontinuous convex functional $ [{\text{unk}}]:X \to [0,\infty ]$ 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 $ A + B$ of the class to be the infinitesimal generator of a nonlinear semigroup $ \{ S(t):t \geqslant 0\} $ on the domain $ C$ of $ B$ such that for $ x \in C$ the $ C$-valued function $ S( \cdot )x$ on $ [0,\infty )$ provides a unique mild solution of the semilinear evolution equation $ u'(t) = (A + B)u(t)$ satisfying a growth condition for the function $ [{\text{unk]}}(u( \cdot ))$. 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.


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  • [1] J. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977), 370-373. MR 0442748 (56:1128)
  • [2] M. Crandall and T. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265-298. MR 0287357 (44:4563)
  • [3] E. Hille and R. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR 0089373 (19:664d)
  • [4] T. Iwamiya, Global existence of mild solutions to semilinear differential equations in Banach spaces, Hiroshima Math. J. 16 (1986), 499-530. MR 867577 (88a:34098)
  • [5] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • [6] Y. Kobayashi, Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. Math. Soc. Japan 27 (1975), 640-665. MR 0399974 (53:3812)
  • [7] K. Kobayasi, Y. Kobayashi and S. Oharu, Nonlinear evolution operators in Banach spaces, Osaka J. Math. 21 (1984), 281-310. MR 752464 (85h:47073)
  • [8] V. Lakshmikantham and S. Leela, Differential and integral inequalities, Academic Press, New York, 1969.
  • [9] R. Martin, Jr., Invariant sets for perturbed semigroups of linear operators, Ann. Mat. Pura Appl. 150 (1975), 221-239. MR 0390414 (52:11240)
  • [10] -, Nonlinear operators and differential equations in Banach spaces, Wiley-Interscience, New York, 1976. MR 0492671 (58:11753)
  • [11] S. Oharu and T. Takahashi, Locally Lipschitz continuous perturbations of linear dissipative operators and nonlinear semigroups, Proc. Amer. Math. Soc. 97 (1987), 139-145. MR 883426 (88d:47076)
  • [12] N. Pavel, Nonlinear evolution equations governed by $ f$-quasi-dissipative operators, Nonlinear Anal. 5 (1981), 449-468. MR 613054 (82i:34073)
  • [13] -, Semilinear equations with dissipative $ t$-dependent domain perturbations, Israel J. Math. 46 (1983), 103-122. MR 727025 (85b:47067)
  • [14] -, Differential equations, flow invariance and applications, Pitman, London, 1984.
  • [15] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci., vol. 44, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
  • [16] G. Webb, Continuous nonlinear perturbations of linear accretive operators in Banach spaces, J. Funct. Anal. 10 (1972), 191-203. MR 0361965 (50:14407)
  • [17] K. Yosida, Functional analysis, Springer-Verlag, New York, 1968. MR 0239384 (39:741)

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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

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