Stability for non-hyperbolic fixed points of scalar difference equations

We give some new criteria to determine the stability of a non-hyperbolic fixed point of the scalar difference equation xn+1 = f (xn) (n= 0, 1, 2, . . .), where xn ∈ R and f is a sufficient smooth function. Our results are based on higher order derivative f (k)( ¯x) at a fixed point of ¯x.  2005 Published by Elsevier Inc.