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 conditionu″+f(t,u,u′)=0,t∈(0,1),u(0)−∑i=1maiu(ti)=λ1,u(1)−∑i=1mbiu(ti)=λ2. We prove that the whole plane R2R2 is divided by a “continuous decreasing curve” Γ   into two disjoint connected regions ΛEΛE and ΛNΛN such that the above problem has at least one solution for (λ1,λ2)∈Γ(λ1,λ2)∈Γ, has at least two solutions for (λ1,λ2)∈ΛE∖Γ(λ1,λ2)∈ΛE∖Γ, and has no solution for (λ1,λ2)∈ΛN(λ1,λ2)∈ΛN. We also find explicit   subregions of ΛEΛE where the above problem has at least two solutions and two positive solutions, respectively.