On difference Riccati equation and continued fractions

Abstract

We study a difference Riccati equation $\hat{I}¦(x)+\ddot{I}(x)/\hat{I}¦(x-\ddot{I}‰)=v(x)$ with $1-$periodic continuos coefficients. Using continued fraction theory we investigate a problem of existence of continuos solutions for this equation. It is shown that convergence of a continued fraction representing a solution of the Riccati equation can be expressed in terms of hyperbolicity of a cocycle naturally associated to this continued fraction. We apply the critical set method to establish the uniform hyperbolicity of the cocycle and to obtain sufficient conditions for the convergence of a continued fraction giving a representation for a solution of the Riccati equation.