Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 61, pp. 1-14.

Title: Solutions to third-order multi-point boundary-value problems 
       at resonance with three dimensional kernels

Authors: Shuang Li (China Univ. of Mining and Tech., Xuzhou, Jiangsu, China)
         Jian Yin  (Jiangsu Normal Univ., Xuzhou, Jiangsu, China)
         Zengji Du (Jiangsu Normal Univ., Xuzhou, Jiangsu, China)

Abstract:
  In this article, we consider the boundary-value problem
 $$\displaylines{
 x'''(t)=f(t, x(t), x'(t),x''(t)), \quad t\in (0,1),\cr
 x''(0)=\sum_{i=1}^{m}\alpha_i x''(\xi_i), \quad
 x'(0)=\sum_{k=1}^{l}\gamma_k x'(\sigma_{k}),\quad
 x(1)=\sum_{j=1}^{n}\beta_jx(\eta_j),
 }$$
 where $f: [0, 1]\times \mathbb{R}^3\to \mathbb{R}$ is a Caratheodory
 function, and the kernel to the linear operator has dimension three.
 Under some resonance conditions, by using the coincidence
 degree theorem, we show the existence of solutions. An example
 is given to illustrate our results.

Submitted October 13, 2013. Published March 05, 2014.
Math Subject Classifications: 34B15.
Key Words: Multi-point boundary-value problem; coincidence degree theory; 
           resonance.