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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Weakly continuous nonlinear accretive operators in reflexive Banach spaces

Author: W. E. Fitzgibbon
Journal: Proc. Amer. Math. Soc. 41 (1973), 229-236
MSC: Primary 47H15; Secondary 34G05
MathSciNet review: 0324496
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Abstract: Let $ X$ be a reflexive space and $ A$ be a weakly continuous (possibly nonlinear) operator which maps $ X$ to $ X$. We are concerned with the autonomous differential equation

$\displaystyle (1.1)\quad u'(t) + Au(t) = 0,\quad u(0) = X.$

We first provide a local solution to (1.1) and then we apply the additional hypothesis that $ A$ is accretive to extend the local solution of (1.1) to a global solution. If $ A$ is weakly continuous and accretive then $ A$ is shown to be $ m$-accretive, i.e. $ R(I + \lambda A) = X$ for all $ \lambda \geqq 0$. The $ m$-accretiveness of $ A$ enables us to provide a semigroup representation of solutions to (1.1),

$\displaystyle (1.2)\quad u(t) = T(t)x = \mathop {\lim }\limits_{n \to \infty } (I + (t/n)A){x^n}\quad {\text{for }}t \in [0,\infty ).$

We then investigate properties of semigroups which have weakly continuous infinitesimal generators.

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Keywords: Accretive, weakly continuous, semigroup of nonexpansive nonlinear operators
Article copyright: © Copyright 1973 American Mathematical Society