Geodesic
Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 44 (2005) no. 1, pp. 97-113.

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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.
Classification : 34C10, 34C11, 47N20
Keywords: quasi-linear second order equations; unbounded; oscillatory and non-oscillatory solutions; fixed-point techniques 
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Malaguti, Luisa; Taddei, Valentina. Fixed point analysis for non-oscillatory solutions of quasi linear ordinary differential equations. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 44 (2005) no. 1, pp. 97-113. http://geodesic.mathdoc.fr/item/AUPO_2005__44_1_a9/