Slowly varying solutions of a class of first order systems of nonlinear differential equations (Q2850099)

The authors analyse asymptotic properties of positive solutions of the first-order system of nonlinear differential equations NEWLINE\[NEWLINE x' \pm p(t)y^{\alpha} = 0,\,\,\, y'\pm q(t)x^{\beta} = 0, NEWLINE\]NEWLINE where \(\alpha \) and \(\beta \) are positive constants, \(\alpha \beta < 1\), \(p\) and \(q\) are positive continuous functions on \([a,\infty)\). Results about existence and asymptotic behavior (for \(t\to \infty \)) of decaying slowly varying solutions (in the ``\(+\)'' case) and growing slowly varying solutions (in the ``\(-\)'' case) are derived and illustrated by examples. The theory of regularly varying functions is used and the existence of the desired solutions and their asymptotic properties are deduced from equivalent integral equations using the Schauder-Tychonoff fixed point theorem and an asymptotic technique. An application to generalized Thomas-Fermi equations is performed as well. The authors also underline that the developed approach gives new possibilities to investigate the existence and asymptotic behavior of positive solutions of general multi-dimensional systems of differential equations based on the use of the theory of regular variation combined with fixed point techniques.