Abstract: The existence of multiple positive solutions to a class of （k‚n-k） conjugate boundary value problem of the n-th order differential equation with deviating arguments is investigated．By a transform‚the discussed problem is firstly changed into a fixed-point problem of a completely continuous operator‚and by using the fixed-point index theorem in cones and the evaluation of Green functions‚two group of sufficient conditions are then obtained to guarantee the exis-tence of two positive solutions to the problem．The results are the extension of those of the corresponding boundary val -ue problem for the ordinary differential equation without deviating argument．The applicability of the obtained theorems is proved by an examlpe at last．

Key words:  n-th order differential equation, conjugate boundary value problem, fixed-point index theorem, positive solution