ON LINEAR RECURRENCES WITH POSITIVE VARIABLE COEFFICIENTS IN BANACH SPACES

The purpose of the paper is to study the convergence of a linear recurrence with positive variable coefficients with elements from a Banach space and to estimate the speed of the convergence. The conditions depend of the sum of the coefficients and the minimum of the coefficients. In this paper we study the case when the sequence of the sum of the coefficients is convergent to I. The results are first proved for recurrences of real numbers. These results are extended later for Banach spaces.