Weakly continuous nonlinear accretive operators in reflexive Banach spaces
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- by W. E. Fitzgibbon PDF
- Proc. Amer. Math. Soc. 41 (1973), 229-236 Request permission
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 \[ (1.1)\quad u’(t) + Au(t) = 0,\quad u(0) = X.\] 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), \[ (1.2)\quad u(t) = T(t)x = \lim \limits _{n \to \infty } (I + (t/n)A){x^n}\quad {\text {for }}t \in [0,\infty ).\] We then investigate properties of semigroups which have weakly continuous infinitesimal generators.References
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Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 41 (1973), 229-236
- MSC: Primary 47H15; Secondary 34G05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0324496-2
- MathSciNet review: 0324496