k-summability of formal solutions for certain partial differential equations and the method of successive approximation (Algebraic analytic methods in complex partial differential equations)

Abstract

We study the k-summability of divergent formal solutions to the Cauchy problem of certain linear partial differential operators of the first order with respect to t whose coefficients are polynomial in t. In order to prove the k-summability of divergent solutions, we employ the method of successive approximation for a construction of divergent solutions and the analysis of convolution equations associated with divergent solutions. We give a proof of the existence and uniqueness of local holomorphic solutions for the convolution equations.