Abstract In this paper, we are concerned with the three-point boundary value problem for second-order differential equations { u ″ ( t ) + w ( t ) f ( u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = β u ′ ( 0 ) , u ( 1 ) = α u ( η ) , where β ≥ 0 , 0 < η < 1 , 0 < α η < 1 and 1 + β − α η − α β > 0 ; w ∈ C ( [ 0 , 1 ] , ( 0 , + ∞ ) ) and f ∈ C ( R + , R + ) , R + = [ 0 , ∞ ) satisfies f ( u ) > 0 for u > 0 . The existence of the continuum of a positive solution is established by utilizing the Leray-Schauder global continuation principle. Furthermore, the interval of α about the nonexistence of a positive solution is also given. MSC:34B10, 34B18, 34G20.