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Nonlinear evolution equations and product stable operators on Banach spaces


Author: G. F. Webb
Journal: Trans. Amer. Math. Soc. 155 (1971), 409-426
MSC: Primary 47.80
DOI: https://doi.org/10.1090/S0002-9947-1971-0276842-9
MathSciNet review: 0276842
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Abstract: The method of product integration is used to obtain solutions to the time dependent Banach space differential equation $ u'(t) = A(t)(u(t)),t \geqq 0$, where $ A$ is a function from $ [0,\infty )$ to the set of nonlinear operators from the Banach space $ X$ to itself and $ u$ is a function from $ [0,\infty )$ to $ X$. The main requirements placed on $ A$ are that $ A$ is $ m$-dissipative and product stable on its domain. Applications are given to a linear partial differential equation, to nonlinear dissipative operators in Hilbert space, and to continuous, $ m$-dissipative, everywhere defined operators in Banach spaces.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0276842-9
Keywords: Nonlinear evolution equation, product integral, $ m$-dissipative mapping, product stable operator, evolution operator
Article copyright: © Copyright 1971 American Mathematical Society

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