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Proceedings of the American Mathematical Society

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Locally Lipschitz continuous perturbations of linear dissipative operators and nonlinear semigroups


Authors: Shinnosuke Oharu and Tadayasu Takahashi
Journal: Proc. Amer. Math. Soc. 100 (1987), 187-194
MSC: Primary 47H20; Secondary 34G20
DOI: https://doi.org/10.1090/S0002-9939-1987-0883426-5
MathSciNet review: 883426
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1987-0883426-5
Keywords: Nonlinear perturbations of linear dissipative operators, semilinear evolution equations, mild solutions, nonlinear semigroups, measure of noncompactness
Article copyright: © Copyright 1987 American Mathematical Society