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=1maiu(ti)+λ,u(1)=∑i=1mbiu(ti)+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].