Nonoscillation of second-order linear difference systems with varying coefficients

This paper deals with nonoscillation problem about the non-autonomous linear difference system
xn = Anxn−1, n = 1,2,...,
where An is a 2×2 variable matrix that is nonsingular for n ∈ N. In the special case that A is a constant matrix, it is well-known that all non-trivial solutions are nonoscillatory if and only if all eigenvalues of A are positive real numbers; namely, detA > 0, trA > 0 and detA/(trA) 2 ≤ 1/4. The well-known result can be said to be an analogy of ordinary differential equations. The results obtained in this paper extend this analogy result. In other words, this paper clarifies the distinction between difference equations and ordinary differential equations. Our results are explained with some specific examples. In addition, figures are attached to facilitate understanding of those examples.