On positive solutions of a (k+1)(k+1)th order difference equation

In this work we show that the following difference equation: xn+1=xn−k1+xn+⋯+xn−k+1,n=0,1,…, where k∈Nk∈N is fixed, has a positive solution which converges to zero. This result solves Open Problem 11.4.10 (a) in [M.R.S. Kulenović, G. Ladas, Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures, Chapman and Hall/CRC, 2001].