Solvability of a three-dimensional system of nonlinear difference equations

In this paper, we solve the following three-dimensional system of difference equationsxn=yn−4zn−5yn−1(an+bnzn−2xn−3yn−4zn−5),yn=zn−4xn−5zn−1(αn+βnxn−2yn−3zn−4xn−5),zn=xn−4yn−5xn−1(An+Bnyn−2zn−3xn−4yn−5), n∈N0,xn=yn−4zn−5yn−1(an+bnzn−2xn−3yn−4zn−5),yn=zn−4xn−5zn−1(αn+βnxn−2yn−3zn−4xn−5),zn=xn−4yn−5xn−1(An+Bnyn−2zn−3xn−4yn−5), n∈N0,where the sequences (an)n∈N0(an)n∈N0, (bn)n∈N0(bn)n∈N0, (αn)n∈N0(αn)n∈N0, (βn)n∈N0(βn)n∈N0, (An)n∈N0(An)n∈N0, (Bn)n∈N0(Bn)n∈N0 and the initial values x−j,y−jx−j,y−j, j= ̄ ̄ ̄ ̄ ̄ ̄ ̄ ̄1,5j=1,5 ̄, are real numbers. In addition, the constant coefficient of the mentioned system is solved in closed form. Finally, we also describe the forbidden set of solutions of the system of difference equations.