Well-Defined Solutions of a Three-Dimensional System of Difference Equations

Abstract

We show that the three-dimensional system of difference equations𝑥𝑛+1 =𝑎𝑥𝑛𝑧𝑛−1𝑧𝑛−𝛽+ 𝛾, 𝑦𝑛+1 =𝑏𝑦𝑛𝑥𝑛−1𝑥𝑛−𝛾+ 𝛼, 𝑧𝑛+1 =𝑐𝑧𝑛𝑦𝑛−1𝑦𝑛−𝛼+ 𝛽 , 𝑛 ∈ 𝑁0 ,where the parameters 𝑎, 𝑏, 𝑐, 𝛼, 𝛽, 𝛾 and the initial conditions 𝑥−𝑖 , 𝑦−𝑖 , 𝑧−𝑖 , 𝑖 ∈ {0,1}, are nonzeroreal numbers, can be solved. Using the obtained formulas, we determine the asymptoticbehavior of solutions and give conditions for which periodic solutions exist. Some numericalexamples are given to demonstrate the theoretical results.