Uniqueness and parameter dependence of solutions of second-order boundary value problems

We consider the boundary value problem with nonhomogeneous multi-point boundary condition u ″ + a ( t ) f ( u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = ∑ i = 1 m a i u ( t i ) + λ , u ( 1 ) = ∑ i = 1 m b i u ( t i ) + μ . A sufficient condition is obtained for the existence and uniqueness of a positive solution. The dependence of the solution on the parameters λ and μ is also studied. Our work complements some results in the literature, especially those in our earlier papers [L. Kong, Q. Kong, Second-order boundary value problems with nonhomogeneous boundary conditions I, Math. Nachr. 278 (2005) 173–193; L. Kong, Q. Kong, Second-order boundary value problems with nonhomogeneous boundary conditions II, J. Math. Anal. Appl. 330 (2007) 1393–1411].