The author investigates a method of determining the basin of attraction of an asymptotically stable fixed point of a discrete time autonomous dynamical system (where , using Lyapunov functions constructed by approximating the solution of the equation . The author makes reference to P. Giesl [J. Difference Equ. Appl. 13, No. 6, 523–546 (2007; Zbl 1120.39018)] for a constructive existence theorem for a smooth solution of the above difference equation.
However, he seems unaware of the result of St. Balint, E. Kaslik, A. M. Balint and A. Grigis [Adv. Difference Equ., Article ID23939 (2006; Zbl 1134.39013)] (and the references within) which addresses a similar problem. Considering the solution of the above difference equation and its Taylor polynomial like functions , the function is constructed and its properties are given. Approximations of and are constructed, using radial basis functions, and local and global error estimates are provided. It is shown that the function defined by is a local and global Lyapunov function. Two examples confirming the effectiveness of proposed method are presented.