Geodesic
Singular nonlinear problem for ordinary differential equation of the second order
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 46 (2007) no. 1, pp. 75-84.

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The paper deals with the singular nonlinear problem \[ u^{\prime \prime }(t) + f(t,u(t),u^{\prime }(t)) = 0,\quad u(0) = 0,\quad u^{\prime }(T) = \psi (u(T)), \] where $f \in \mathop {\mathit{Car}}((0,T)\times D)$, $D = (0,\infty )\times $. We prove the existence of a solution to this problem which is positive on $(0,T]$ under the assumption that the function $f(t,x,y)$ is nonnegative and can have time singularities at $t = 0$, $t = T$ and space singularity at $x = 0$. The proof is based on the Schauder fixed point theorem and on the method of a priori estimates.
Classification : 34B15, 34B16, 34B18
Keywords: singular ordinary differential equation of the second order; lower and upper functions; nonlinear boundary conditions; time singularities; phase singularity 
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Rachůnková, Irena; Tomeček, Jan. Singular nonlinear problem for ordinary differential equation of the second order. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 46 (2007) no. 1, pp. 75-84. http://geodesic.mathdoc.fr/item/AUPO_2007__46_1_a7/