Geodesic
The problem of the solvability of nonlinear two-point boundary-value problems
Matematičeskie zametki, Tome 16 (1974) no. 3, pp. 479-490.

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We establish sufficient conditions for the solvability of boundary-value problems of the form \begin{gather*} u''=f(t,u,u'); \\ (u(0),u'(0))\in S_0,\quad(u(1),u'(1))\in S_1. \end{gather*} 
@article{MZM_1974_16_3_a16,
     author = {I. T. Kiguradze and N. R. Lezhava},
     title = {The problem of the solvability of nonlinear two-point boundary-value problems},
     journal = {Matemati\v{c}eskie zametki},
     pages = {479--490},
     publisher = {mathdoc},
     volume = {16},
     number = {3},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_3_a16/}
}
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I. T. Kiguradze; N. R. Lezhava. The problem of the solvability of nonlinear two-point boundary-value problems. Matematičeskie zametki, Tome 16 (1974) no. 3, pp. 479-490. http://geodesic.mathdoc.fr/item/MZM_1974_16_3_a16/