On a three-dimensional system of difference equations with variable coefficients

Abstract

Consider the three-dimensional system of difference equations xn+1 = ∏k j=0 zn−3j ∏k j=1 xn−(3j−1) ( an + bn ∏k j=0 zn−3j ), yn+1 = ∏k j=0 xn−3j ∏k j=1 yn−(3j−1) ( cn + dn ∏k j=0 xn−3j ), zn+1 = ∏k j=0 yn−3j ∏k j=1 zn−(3j−1) ( en + fn ∏k j=0 yn−3j ), n ∈ N0, where k ∈ N0, the sequences (an)n∈N0 , (bn)n∈N0 , (cn)n∈N0 , (dn)n∈N0 , (en)n∈N0 , (fn)n∈N0 and the initial values x−3k, x−3k+1, . . . , x0, y−3k, y−3k+1, . . . , y0, z−3k, z−3k+1, . . . , z0 are real numbers. In this work, we give explicit formulas for the well defined solutions of the above system. Also, the forbidden set of solution of the system is found. For the constant case, a result on the existence of periodic solutions is provided and the asymptotic behavior of the solutions is investigated in detail. AMS Mathematics Subject Classification : 39A10, 39A20, 39A23, 40A05. Key words and phrases : Three-dimensional systems of difference equations, explicit formulas, periodicity, asymptotic behavior.