Another understanding of fourth-order four-point boundary-value problems

Abstract

In this article we investigate the existence of positive and/or negative solutions of a classes of four-point boundary-value problems for fourth-order ordinary differential equations. The assumptions in this article are more relaxed than the known assumptions. Our technique relies on the continuum property (connectedness and compactness) of the solutions funnel (Knesser's Theorem), combined with the corresponding vector field's ones. This approach permits the extension of results (getting positive solutions) to nonlinear boundary conditions, whenever the corresponding Green's kernel is not of definite sign or there does not exist (see the last Corollary).

Keywords

• Multipoint boundary value problem
• positive solution
• vector field
• third order differential equation
• Green function
• Krasnoselskii's fixed point theorem