Extremal solutions to a system of n nonlinear differential equations and regularly varying functions

The precise asymptotic behavior of strongly increasing and strongly decreasing solutions to a system of n nonlinear coupled first order equations is here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We show that, under quite natural conditions, such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish precise asymptotic formulas. The results here presented improve and extend some known results in various aspects. They are new even in some well studied special settings.