On the Well-Defined Solutions of a Nonlinear Difference Equation with Variable Coefficients

Obtaining analytical solutions of nonlinear difference equations is of great importance for a full understanding of the long-term behavior of dynamical systems. Such equations are widely used in mathematical modeling of complex systems, especially population dynamics, the spread of infectious diseases, economic models, and biological processes. Therefore, obtaining analytical solutions of nonlinear difference equations is indispensable not only for theoretical purposes but also for making reliable predictions in applied sciences. Based on this undeniable fact, in this paper, we study the following nonlinear difference equation: xi(n+1) = [xi(n) xi(n−1)] / [a(n) xi(n) − b(n) xi(n−2)], n in N0, where (a(n)) and (b(n)) are periodic sequences of positive real numbers with prime period two and with real initial values xi(−2), xi(−1), xi(0). We introduce the well-defined solutions and study their global behavior.