Geodesic
Boundary value problems with compatible boundary conditions
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 581-592.

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If $Y$ is a subset of the space $\mathbb{R}^{n}\times {\mathbb{R}^{n}}$, we call a pair of continuous functions $U$, $V$ $Y$-compatible, if they map the space $\mathbb{R}^{n}$ into itself and satisfy $Ux\cdot Vy\ge 0$, for all $(x,y)\in Y$ with $x\cdot y\ge {0}$. (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential $n$-dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its domain, Brouwer’s fixed point theorem is applied to the composition of the consequent mapping with some projections and a one-parameter family of fixed points $P_{\delta }$ is obtained. Then passing to the limits as $\delta $ tends to zero the so-obtained accumulation points are solutions of the problem.
Classification : 34B15, 34C30, 34C99
Keywords: differential equations of second order; two-point boundary value problems 
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Karakostas, G. L.; Palamides, P. K. Boundary value problems with compatible boundary conditions. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 3, pp. 581-592. http://geodesic.mathdoc.fr/item/CMJ_2005__55_3_a1/