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)=a₁(n)y(n-1)+···+a_m(n)y(n-m) for ||a(n)||₁<1 was addressed, whereas the case ||a(n)||₁=1 has been left as an open question. Here, we impose the condition of convexity on the set C₀ of the initial values y(n)=[y(n-1),...,y(n-m)]^T εR^m and on the set AεR^m of all allowable values of a(n)=[a₁(n),...,a_m(n)]^T, and derive the results from [1] for a_i≥0, i=1,...,n, as a pure consequence of convexity of the sets C₀ and A. Based upon convexity and the fixed-point iteration (FPI) technique, further GAS results for both ||a(n)||_i<1, and ||a(n)||₁=1 are derived. The issues of convergence in norm, and geometric convergence are tackled.