Second-order boundary value problems with nonhomogeneous boundary conditions (II)

Sufficient conditions are given for the existence of solutions of the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition u + f (t,u,u ) = 0, t∈ (0, 1), u(0) − m i=1 ai u(ti ) = λ1, u(1) − m i=1 bi u(ti ) = λ2. We prove that the whole plane R2 is divided by a “continuous decreasing curve” Γ into two disjoint connected regions ΛE and ΛN such that the above problem has at least one solution for (λ1,λ2) ∈ Γ , has at least two solutions for (λ1,λ2) ∈ ΛE \ Γ , and has no solution for (λ1,λ2) ∈ ΛN. We also find explicit subregions of ΛE where the above problem has at least two solutions and two positive solutions, respectively. © 2006 Elsevier Inc. All rights reserved.