Abstract
Let k, l be two integers with k >= 0 and l >= 2, c a real number greater than or equal to 1, and f a multivariable function satisfying f (w(1), w(2), w(3), ..., w(l)) >= 0 when w(1), w(2) >= 0. We consider an arbitrary order nonlinear difference equation with the indicated function f : z(n+1) = c(z(n)+z(n-k))+(c-1)z(n)z(n-k) +cf (z(n),z(n-k),w(3),...,w(l))/z(n)z(n-k) + f (z(n),z(n-k),w(3),...,w(l))+c, n >= 0, where initial values z(-k), z(-k+1), ..., z(0) are positive and w(i), i >= 3, are arbitrary functions of z(j), n - k <= j <= n. We classify its solutions into three types with different asymptotic behaviors, and verify the global asymptotic stability of its positive equilibrium solution (z) over bar = c.