On the Cushing-Henson conjecture, delay difference equations and attenuant cycles

We consider the second part of the Cushing-Henson conjecture (the cycle's average is less than the average of carrying capacities; the first part of the conjecture deals with the existence and global stability of periodic cycles) for a periodic delay difference equation x(n+1) = f(k(n),x(n-h1(n)), ..., x(n-hr(n))). Sufficient conditions on f and h(i) are obtained, when the second part of the conjecture is valid. We demonstrate the sharpness of these conditions by presenting several counterexamples. In addition, sufficient global attractivity conditions are deduced for the Pielou equation.