On product formulas for nonlinear semigroups

We study a number of sufficient conditions which guarantee the convergence of semigroup product formulas of the type

$\text{H}{}_{\mathrm{t}}\text{=}\text{lim}\text{n→∞}\text{(F}{}_{\mathrm{<mtext>t</mtext><mtext>n</mtext>}}\text{°G}{}_{\mathrm{<mtext>t</mtext><mtext>n</mtext>}}\text{)}{}^{\mathrm{n}}$

and its generalizations. Our hypotheses differ from those of other authors in that we do not assume in advance that the limit operator is a generator. Rather this is a consequence and hence the above formula yields an existence theorem (local in time) for nonlinear semigroups. A number of smoothness properties are studied as well. The results may be applied to and are motivated by the Navier-Stokes equations.