Strongly monotone solutions of systems of nonlinear differential equations with rapidly varying coefficients

The two-dimensional systems of first order nonlinear differential equations (S1) x? = p(t)y?, y? q(t)x? and (S2) + p(t)y? 0, 0 are analyzed using the theory rapid variation. This approach allows us to prove that all strongly increasing solutions system (and, respectively, decreasing ) rapidly varying functions under assumption p q varying. Also, asymptotic equivalence relations for these given.