On stability of relaxive systems described by polynomials with time-variant coefficients

The problem of global asymptotic stability (GAS) of a time-variant m-th order difference equation y(n) = a(T)(n)y(n - 1) = a(1)(n)y(n - 1) +... + a(m)(n)y(n - m) for \ parallel toa(n)parallel to (1), < 1 was addressed in [1], whereas the case <parallel>a(n)parallel to (1) = 1 has been left as an open question. Here, we impose the condition of convexity on the set C-0 of the initial values y(n) = [y(n - 1),..., y(n - m)](T) is an element of IRm and on the set A is an element of IRm of an allowable values of a(n) = [a(1)(n),..., a(m)(n)](T), and derive the results from [1] for a(i) greater than or equal to 0, i = 1,..., n as a pure consequence of convexity of the sets C-0 and A. Based upon convexity and the fixed-point iteration (FPI) technique, further GAS results for both parallel toa(n)parallel to (1) < 1, and <parallel>a(n)parallel to (1) = 1 are derived, The issues of convergence in norm, and geometric convergence are tackled.