In this article, the author studies the boundedness and convergence for the {(x) = a(y) - f(x),(y) = b(y)β(x) - g(x) + e(t),where a(y), b(y), f(x),g(x),β(x) are real continuous functions in y ∈ R or x ∈ R,β(x) ≥ 0 for all x and e(t) is a real continuous function on R+ = {t: t ≥ 0} such that the equation has a unique solution for the initial value problem. The necessary and sufficient conditions are obtained and some of the results in the literatures are improved and extended.