PROPER SOLUTIONS AND LIMIT BOUNDARY VALUE PROBLEMS OF NONLINEAR SECOND-ORDER SYSTEMS OF DIFFERENTIAL EQUATIONS

For the system of differential equations(?)=r(t)v_1 (?)=-a(t)f(x)g(v)where a(t)>0,r(t)>0 for t≥t_0;f(x)>0 and is decreasing forx>0,g(y)>0,we give necessary and sufficient condition of a propersolution.a bounded proper solution or sotutions of two kinds of boundary value problems onan infinite interval c,∞),c≥t_0.Several examples are given to illustrate the conditionsof these results.