The authors investigate the -point boundary value problem consisting of the differential equation
and the boundary conditions of the form
or of the form
where is a continuous function, belongs to , the numbers , have the same sign, and the numbers satisfy . In the case the conditions (2) and (3) simplify to the conditions
respectively, where and .
Using the coincidence degree theory developed by J. Mawhin, the authors show that the question of existence and uniqueness of solutions to the problems (1), (2) and (1), (3) can be reduced to the investigation of the simpler problems (1), (4) and (1), (5) respectively.