Higher order boundary value problems with nonhomogeneous boundary conditions

We study the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition u(2n)=g(t)f(t,u),t∈(0,1),{u(2i)(0)−∑j=1maju(2i)(tj)=(−1)iλ2i,u(2i)(1)−∑j=1mbju(2i)(tj)=(−1)iλ2i+1,i=0,…,n−1. Sufficient conditions are obtained for the problem to have at least one positive solution and to have no solution, respectively. Moreover, under certain conditions, we prove that there exists a bounded and continuous surface ΓΓ separating R+2n∖{0,…,0} into two disjoint subsets ΛEΛE and ΛNΛN with Γ⊆ΛEΓ⊆ΛE such that the problem has at least two positive solutions for each (λ0,…,λ2n−1)∈ΛE∖Γ(λ0,…,λ2n−1)∈ΛE∖Γ, one positive solution for each (λ0,…,λ2n−1)∈Γ(λ0,…,λ2n−1)∈Γ, and no positive solution for any (λ0,…,λ2n−1)∈ΛN(λ0,…,λ2n−1)∈ΛN.