Nonlinear evolution equations and product stable operators on Banach spaces

Webb, G. F.

doi:10.1090/S0002-9947-1971-0276842-9

Nonlinear evolution equations and product stable operators on Banach spaces
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Trans. Amer. Math. Soc. 155 (1971), 409-426 Request permission

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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