Solutions to m-point boundary value problems of third order ordinary differential equations at resonance

Abstract

In this paper, we study the third order ordinary differential equation:

$$x'''(t) = f(t,x(t),x'(t),x''(t)), t \in (0,1)$$

subject to the boundary value conditions:

$$x'(0) = x'(\xi ), x'(1) = \sum\limits_{i - 1}^{m - 3} {\beta _i x'(\eta _i )} , x''(1) = 0.$$

Hereβ _i ∈R,$\sum\limits_{i = 1}^{m - 3} {\beta _i = 1, 0< \eta _1< \eta _2< \ldots< \eta _{m - 3}< 1, 0< \xi< 1}$. This is the case dimKerL=2. When theβ _i have different signs, we prove some existence results for the m-point boundary value problem at resonance by use of the coincidence degree theory of Mawhin [12, 13]. Since all the existence results obtained in previous papers are for the case dimKerL=1, our work is new.