Abstract
Let k >= 0 and l >= 2 be integers, c a nonnegative number and f an arbitrary multivariate function such that f (x(1), x(2), x(3),.., x(l)) >= x(1) + x(2) for x(1), x(2) >= 0. This work deals with the higher-order nonlinear difference equation
z(n+1) = (c + 1)z(n)z(n-k) + c [f (z(n), z(n-k), w(3),..., w(l))) - z(n) - z(n-k)] + 2c(2)/z(n)z(n-k) + f (z(n), z(n-k), w(3),..., w(l))) + c, n >= 0,
where z(-k), z(-k+1),..., z(0) are positive initial values and w(i), 3 <= i <= l, arbitrary functions of variables z(n-k), z(n-k+1),..., z(n). All solutions of this equation are classified into three groups, according to their asymptotic behavior, and a decreasing and increasing characteristic of oscillatory solutions is also explored. Finally, the global asymptotic stability of the positive equilibrium solution (z) over bar = c is exhibited by establishing a strong negative feedback property.