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quasi-linear second order equations; unbounded; oscillatory and non-oscillatory solutions; fixed-point techniques
The paper deals with the quasi-linear ordinary differential equation $(r(t)\varphi (u^{\prime }))^{\prime }+g(t,u)=0$ with $t \in [0, \infty )$. We treat the case when $g$ is not necessarily monotone in its second argument and assume usual conditions on $r(t)$ and $\varphi (u)$. 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 Math. Hung. 1990]. 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 associated to the equation when $\varphi (u)=u$. Several examples are included.
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