Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
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 $X$ 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.


References [Enhancements On Off] (What's this?)


Similar Articles

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

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


Additional Information

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