Oscillation criteria for three dimensional linear difference systems

Let’s consider the 3-dimensional difference system (1) X(n+ 1) = A(n)X(n), where X(n) = [x1(n), x2(n), x3(n)] T , A(n) = [aij(n)] is a given 3 × 3 matrix, xi : N→ R, aij : N→ R for i, j ∈ {1, 2, 3}. Here N = {0, 1, 2, . . . }, Nn0 = {n0, n0 + 1, n0 + 2, . . . }, n0 ∈ N and R denotes the set of real numbers. If aij(n) ≡ aij ∈ R for any i, j ∈ {1, 2, 3}, then equation (1) is equivalent to (2) X(n+ 1) = AX(n), where X(n) = [x(n), y(n), z(n)] and A(n) = [aij ]3×3. The characteristic equation of (2) is given by det(λI −A) = 0, that is, (3) λ − (trA)λ +mλ− detA = 0, ∗Corresponding Author. Arun Kumar Tripathy 2010 Mathematics Subject Classification: 34K11, 34C10, 39A13.