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Two-point boundary value problems for m-dimensional second order systems are considered. The method of upper and lower functions is applied to problems of the Dirichlet type and problems with nonlinear boundary conditions. The conditions on upper and lower functions are substantially relaxed comparing with the classical C 2 -class and properties of them are studied for systems with monotone in x right sides. Consequences for even order differential equations with mixed monotonicities are given.doi:10.4171/zaa/1042 fatcat:7jspapgzhvcwplyqvdkg3jyida