Parametric asymptotic expansions and confluence for Banach valued solutions to some singularly perturbed nonlinear q-difference-differential Cauchy problem

We investigate a singularly perturbed q-difference differential Cauchy problem with polynomial coefficients in complex time t and space z and with quadratic nonlinearity. We construct local holomorphic solutions on sectors in the complex plane with respect to the perturbation parameter ϵ with values in some Banach space of formal power series in z with analytic coefficients on shrinking domains in t. Two aspects of these solutions are addressed. One feature concerns asymptotic expansions in ϵ for which a Gevrey type structure is unveiled. The other fact deals with confluence properties as q > 1 tends to 1. In particular the built up Banach valued solutions are shown to merge in norm to a fully bounded holomorphic map in all the variables t,z and ϵ that solves a nonlinear partial differential Cauchy problem.