Geodesic
On a two-point boundary value problem for second order singular equations
Czechoslovak Mathematical Journal, Tome 53 (2003) no. 1, pp. 19-43.

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The problem on the existence of a positive in the interval $\mathopen ]a,b\mathclose [$ solution of the boundary value problem \[ u^{\prime \prime }=f(t,u)+g(t,u)u^{\prime };\quad u(a+)=0, \quad u(b-)=0 \] is considered, where the functions $f$ and $g\:\mathopen ]a,b\mathclose [\times \mathopen ]0,+\infty \mathclose [ \rightarrow \mathbb R$ satisfy the local Carathéodory conditions. The possibility for the functions $f$ and $g$ to have singularities in the first argument (for $t=a$ and $t=b$) and in the phase variable (for $u=0$) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.
Classification : 34B10, 34B16, 34B18
Keywords: second order singular equation; two-point boundary value problem; solvability 
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Lomtatidze, A.; Torres, P. On a two-point boundary value problem for second order singular equations. Czechoslovak Mathematical Journal, Tome 53 (2003) no. 1, pp. 19-43. http://geodesic.mathdoc.fr/item/CMJ_2003__53_1_a2/