Fourth-order boundary value problems with multi-point boundary conditions

This paper is Concerned with the existence, uniqueness, and method of Construction of a solution for a class of fourth-order and second-order multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as the standard two-point boundary condition, three or four-point boundary condition, and one-end multi-point boundary condition. The existence problem is based on the method of upper and lower solutions and its associated monotone iterations. Various verifiable sufficient conditions for the existence of maximal and minimal solutions, uniqueness of the solution, and a continuum of infinite number of constant solutions are given. Several examples are presented to illustrate the application of the existence and uniqueness results for both the fourth-order and the second-order problems.