Positive solutions for the nonhomogeneous three-point boundary value problem of second-order differential equations

This paper is concerned  with the existence of positive solutions to the nonhomogeneous three-point boundary value problem  of the second-order ordinary differential equation u ″ ( t ) + a ( t ) u ′ ( t ) + b ( t ) u ( t ) + h ( t ) f ( u ) = 0 u ( 0 ) = 0 , u ( 1 ) − α u ( η ) = b where 0 < η < 1 , 0 < α η < 1 and b ∈ [ 0 , ∞ ) are given. a ( t ) ∈ C [ 0 , 1 ] , b ( t ) ∈ C ( [ 0 , 1 ] , ( − ∞ , 0 ] ) and h ( t ) ∈ C ( [ 0 , 1 ] , [ 0 , ∞ ) ) satisfying that there exists x 0 ∈ [ 0 , 1 ] such that h ( x 0 ) > 0 , and f ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) . By applying Krasnosel’skii’s fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution if f is either superlinear or sublinear are established for the above boundary value problem. The results obtained extend and complement some known results.