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

Abstract

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