Asymptotic Behavior of Positive Solutions of a Class of Systems of Second Order Nonlinear Differential Equations

The two-dimensional system of nonlinear differential equations (A) x′′ = p(t)y, y′′ = q(t)x , with positive exponents α and β satisfying αβ < 1 is analyzed in the framework of regular variation. Under the assumption that p(t) and q(t) are nearly regularly varying it is shown that system (A) may possess three types of positive solutions (x(t), y(t)) which are strongly monotone in the sense that (i) both components are strongly decreasing, (ii) both components are strongly increasing, and (iii) one of the components is strongly decreasing, while the other is strongly increasing. The solutions in question are sought in the three classes of nearly regularly varying functions of positive or negative indices. It is also shown that if we make a stronger assumption that p(t) and q(t) are regularly varying, then the solutions from the above three classes are fully regularly varying functions, too.