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Nonlinear evolution equations on Banach space. (English) Zbl 0744.34057

In the first part of this paper the author proves the existence and uniqueness of a mild solution for a semilinear evolution equation on a Banach space of the form \(dx(t)+A(t)x(t)dt=f(t,x(t))dt, x(0)=x_ 0\). The family of operators \(\{A(t), t\in[0,a]\}\) is supposed to generate an evolution operator \(U(t,s)\) which allows to write down the corresponding integral equation \[ x(t)=U(t,0)x_ 0+\int^ t_ 0U(t,x)f(s,x(s))ds. \] Two different types of conditions are imposed on the nonlinear term f: A Carathéodory type property and a Lipschitz and linear growth condition. The second part deals with stochastic semilinear and quasilinear evolution equations perturbed by a Hilbert-valued Brownian motion. The existence of a unique solution is proved using again the semigroup theory and fixed point arguments.

MSC:

34G20 Nonlinear differential equations in abstract spaces
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
60H20 Stochastic integral equations
93C25 Control/observation systems in abstract spaces
34F05 Ordinary differential equations and systems with randomness
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