Electron. J. Diff. Equ., Vol. 2014 (2014), No. 61, pp. 1-14.

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

Shuang Li, Jian Yin, Zengji Du

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, 2014. Published March 5, 2014.
Math Subject Classifications: 34B15.
Key Words: Multi-point boundary-value problem; coincidence degree theory; resonance.

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Shuang Li
School of Management
China University of Mining and Technology
Xuzhou, Jiangsu 221116, China
email: lishuangchina@163.com
Jian Yin
School of Mathematics and Statistics
Jiangsu Normal University
Xuzhou, Jiangsu 221116, China
email: yinjian719@163.com
Zengji Du
School of Mathematics and Statistics
Jiangsu Normal University
Xuzhou, Jiangsu 221116, China
email: duzengji@163.com, Fax 0086-516-83403299

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