*(English)*Zbl 1002.60058

Let $H$ be a real separable Hilbert space, $A$ an infinitesimal generator of a ${C}_{0}$-semigroup on $H$, $W$ a cylindrical $Q$-Wiener process on $H$. A stochastic evolution equation

is studied. First, it is shown that existence and uniqueness of a mild solution to (1) may be established by the method of successive approximations under assumptions upon $f$ and $g$ which are weaker than Lipschitz continuity and were proposed, in the finite-dimensional case, by *T. Taniguchi* [J. Differ. Equations 96, No. 1, 152-169 (1992; Zbl 0744.34052)]. In the second part of the paper, it is supposed that the semigroup generated by $A$ is compact, but the functions $f$ and $g$ are merely continuous, of a linear growth, and such that pathwise uniqueness holds for (1). It is proven that Euler and Lie-Trotter approximations converge to the solution of (1) in the ${L}^{p}$-norm and that the solution depends continuously on data and on the coefficients. Finally, equations with nonlinear terms $f$, $g$ defined only on an open subset of $H$ are considered and existence of a solution to (1) is shown under a hypothesis that a suitable sequence of Lyapunov functions exists.

##### MSC:

60H15 | Stochastic partial differential equations |