Multi-point boundary value problems of second-order differential equations (I) Original Research Article

We study the nonlinear boundary value problem with nonhomogeneous multi-point boundary condition u″+f(t,u,u′)=0,t∈(0,1), View the MathML source Sufficient conditions are found for the existence of solutions of the problem based on the existence of lower and upper solutions with certain relations. Using one of the results, under some assumptions, we obtain explicit ranges of values of λ1 and λ2 with which the problem has a solution, has a positive solution, and has no solution, respectively. Furthermore, we prove that the whole plane for λ1 and λ2 can be divided into two disjoint connected regions ΛE and ΛN such that the problem has a solution for (λ1,λ2)∈ΛE and has no solution for (λ1,λ2)∈ΛN.