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

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.

Keywords

• Multi-point boundary-value problem
• coincidence degree theory
• resonance