The paper deals with the quasi-linear ordinary differential equation ${\left(r\left(t\right)\varphi \left(u^{\text{'}}\right)\right)}^{\text{'}}+g(t,u)=0$ with $t\in [0,\infty )$. We treat the case when $g$ is not necessarily monotone in its second argument and assume usual conditions on $r\left(t\right)$ and $\varphi \left(u\right)$. 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. The results generalize previous ones due to Elbert–Kusano, [Acta...