A holomorphic Lie-Trotter product formula for nonlinear local semigroup

We prove a new, holomorphic version of the Lie-Trotter product formula for nonlinear local semigroups with a parameter. We show that the product converges to the solutions of a nonlinear evolutions equation with a parameter. The solution is holomorphic in time on a centered at zero conic set in the complex plane and depends continuously on the parameter, and analytically on the initial data.