Ahmed, N. U. Nonlinear evolution equations on Banach space. (English) Zbl 0744.34057 J. Appl. Math. Stochastic Anal. 4, No. 3, 187-202 (1991). 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. Reviewer: D.Nualart (Barcelona) Cited in 1 Document 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 Keywords:existence and uniqueness of a mild solution; semilinear evolution equation on a Banach space; integral equation; stochastic semilinear and quasilinear evolution equations; semigroup theory PDFBibTeX XMLCite \textit{N. U. Ahmed}, J. Appl. Math. Stochastic Anal. 4, No. 3, 187--202 (1991; Zbl 0744.34057) Full Text: DOI EuDML