a discrete-time system conjecture

x(k)=ax(k-1)+b(k)x(k-n)+u(k)-------------------------(1), where x(k), a>0, b(k) ,u(k) are all scalar, n is positive integer, and b(k) is time varying, u(k) is control input. Denote b_0=sup{b(k)}. I conjecture that if a+|b(k)|n

|x(k-1)-x(k-n)|