Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations

The paper deals with a quasi-linear ordinarydifferential equation when the nonlinearity is not necessarily monotone in its second argument. We find necessary and sufficient conditions for the existence of unbounded non-oscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. In some special cases we are able to show the exact asymptotic growth of these solutions. We apply previous analysis for studying the non-oscillatory problem. Several examples are included.