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

Abstract

The two-dimensional systems of first order nonlinear differential equations
$$({\rm S}_1)\quad x'=p(t) y^\a,\quad y'=q(t)x^\b \quad \textrm{and}\quad ({\rm S}_2)\quad x'+p(t) y^\a=0,\quad y'+q(t)x^\b=0$$
are analyzed using the theory of rapid variation. This approach allows us to prove that all strongly increasing solutions of system $({\rm S}_1)$ (and, respectively, all strongly decreasing solutions of system $({\rm S}_2)$) are rapidly varying functions under the assumption that $p$ and $q$ are rapidly varying. Also, the asymptotic equivalence relations for these solutions are given.