On the Dynamics of the Recursive Sequence xn+1=α+xn−kpxnq

Abstract EN

We investigate the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation xn+1=α+xn-kp/xnq, n=0,1,…, where k∈{2,3…}, α, p, q∈(0,∞) and the initial conditions x-k,…,x0 are arbitrary positive numbers.

We investigate the boundedness character for p∈(0,∞).

Also, we investigate the existence of a prime two periodic solution for k is odd.

Moreover, when k is even, we prove that there are no prime two periodic solutions of the equation above.