THE ASYMPTOTIC-BEHAVIOR OF CERTAIN DIFFERENCE-EQUATIONS WITH PROPORTIONAL DELAYS

This paper is concerned with the nonstationary linear difference equat ion y(n) = lambday(n-1) + mu(y[n/2] + y[(n-1)/2]), y0 = 1 We demonstra te by Fourier techniques that the sequence is asymptotically stable wh en \lambda\ < 1 and \mu\ < (1/2)[1-lambda\, but our main effort is dev oted to the marginal case \lambda\ = 1. We derive the solution explici tly as a power series in mu for lambda = -1, thereby demonstrating tha t it is uniformly bounded and that, for mu not-equal 0, its attractor contains a countable subset of distinct points. Finally, we consider a somewhat more general difference equation and prove that, for specifi c choices of parameters therein, the attractor is a probabilistic mixt ure of Julia sets.