Analytic general solutions of nonlinear second-order $q$-difference equations with a double characteristic value

Abstract

As far as the author knows, it seems that an existence theorem of a solution of a general nonlinear $q$-difference equation is not known. In this paper we will investigate a nonlinear second order $q$-difference equation whose characteristic equation has only one solution and will show analytic general solutions of such an equation. Further, we will show an example.