Locally Lipschitz continuous perturbations of linear dissipative operators and nonlinear semigroups

Oharu, Shinnosuke; Takahashi, Tadayasu

doi:10.1090/S0002-9939-1987-0883426-5

Locally Lipschitz continuous perturbations of linear dissipative operators and nonlinear semigroups
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by Shinnosuke Oharu and Tadayasu Takahashi PDF

Proc. Amer. Math. Soc. 100 (1987), 187-194 Request permission

Abstract:

Locally Lipschitz continuous perturbations of linear $m$-dissipative operators in Banach spaces are considered from the point of view of the nonlinear semigroup theory. A necessary and sufficient condition is given for a semilinear operator $A + F$ to be the infinitesimal generator of a nonlinear semigroup which provides mild solutions (with exponential growth) of the semilinear evolution equation $u’(t) = (A + F)u(t)$. It turns out that a characterization of Hille-Yosida type for nonlinearly perturbed contraction semigroups is obtained.

References

Jean-Bernard Baillon, Générateurs et semi-groupes dans les espaces de Banach uniformement lisses, J. Functional Analysis 29 (1978), no. 2, 199–213 (French, with English summary). MR 504459, DOI 10.1016/0022-1236(78)90006-X
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 287357, DOI 10.2307/2373376
Klaus Deimling, Ordinary differential equations in Banach spaces, Lecture Notes in Mathematics, Vol. 596, Springer-Verlag, Berlin-New York, 1977. MR 0463601
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 399974, DOI 10.2969/jmsj/02740640
Robert H. Martin Jr., Invariant sets for perturbed semigroups of linear operators, Ann. Mat. Pura Appl. (4) 105 (1975), 221–239. MR 390414, DOI 10.1007/BF02414931
Robert H. Martin Jr., Nonlinear operators and differential equations in Banach spaces, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. MR 0492671
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
G. F. Webb, Continuous nonlinear perturbations of linear accretive operators in Banach spaces, J. Functional Analysis 10 (1972), 191–203. MR 0361965, DOI 10.1016/0022-1236(72)90048-1