Solutions to third-order multi-point boundary-value problems at resonance with three dimensional kernels
In this article, we consider the boundary-value problem $$\displaylines{ x'''(t)=f(t, x(t), x'(t),x''(t)), \quad t\in (0,1),\cr x''(0)=\sum_{i=1}^{m}\alpha_i x''(\xi_i), \quad x'(0)=\sum_{k=1}^{l}\gamma_k x'(\sigma_{k}),\quad x(1)=\sum...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2014-03-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2014/61/abstr.html |
| Summary: | In this article, we consider the boundary-value problem
$$\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. |
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| ISSN: | 1072-6691 |