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Solvability for Some Higher Order Multi-Point Boundary Value Problems at Resonance

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Abstract

In this paper, by using the Mawhin’s continuation theorem, we obtain an existence theorem for some higher order multi-point boundary value problems at resonance in the following form:

$$\begin{array}{lll}x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))+e(t),\ t\in(0,1),\\x^{(i)}(0) = 0, i=0,1,\ldots,n-1,\ i\neq p, \\x^{(k)}(1) = \sum\limits_{j=1}^{m-2}{\beta_j}x^{(k)}(\eta_j),\end{array}$$

where \({f:[0,1]\times \mathbb{R}^n \to \mathbb{R}=(-\infty,+\infty)}\) is a continuous function, \({e(t)\in L^1[0,1], p, k\in\{0,1,\ldots,n-1\}}\) are fixed, m ≥ 3 for p ≤ k (m ≥ 4 for p > k), \({\beta_j \in \mathbb{R}, j=1,2,\ldots,m-2, 0 < \eta_1 < \eta_2 < \cdots < \eta_{m-2} <1 }\) . We give an example to demonstrate our results.

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Authors and Affiliations

  1. Department of Mathematics, Yeungnam University, Kyongsan, 712-749, Republic of Korea

    Sung Kag Chang

  2. Department of Mathematics, Beihua University, Jilin, 132013, People’s Republic of China

    Minghe Pei

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  1. Sung Kag Chang
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  2. Minghe Pei
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Correspondence to Minghe Pei.

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This research was supported by Yeungnam University Research grants 2010.

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Chang, S.K., Pei, M. Solvability for Some Higher Order Multi-Point Boundary Value Problems at Resonance. Results. Math. 63, 763–777 (2013). https://doi.org/10.1007/s00025-012-0232-3

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  • DOI: https://doi.org/10.1007/s00025-012-0232-3

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