# Solutions of a multi-point boundary value problem for higher-order differential equations at resonance. (II)

Archivum Mathematicum (2005)

• Volume: 041, Issue: 2, page 209-227
• ISSN: 0044-8753

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## Abstract

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In this paper, we are concerned with the existence of solutions of the following multi-point boundary value problem consisting of the higher-order differential equation ${x}^{\left(n\right)}\left(t\right)=f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right),\cdots ,{x}^{\left(n-1\right)}\left(t\right)\right)+e\left(t\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}0 and the following multi-point boundary value conditions $\begin{array}{cc}\hfill 1*-1{x}^{\left(i\right)}\left(0\right)& =0\phantom{\rule{1.0em}{0ex}}for\phantom{\rule{1.0em}{0ex}}i=0,1,\cdots ,n-3\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {x}^{\left(n-1\right)}\left(0\right)& =\alpha {x}^{\left(n-1\right)}\left(\xi \right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}{x}^{\left(n-2\right)}\left(1\right)=\sum _{i=1}^{m}{\beta }_{i}{x}^{\left(n-2\right)}\left({\eta }_{i}\right)\phantom{\rule{0.166667em}{0ex}}.**\hfill \end{array}$ Sufficient conditions for the existence of at least one solution of the BVP $\left(*\right)$ and $\left(**\right)$ at resonance are established. The results obtained generalize and complement those in [13, 14]. This paper is directly motivated by Liu and Yu [J. Pure Appl. Math. 33 (4)(2002), 475–494 and Appl. Math. Comput. 136 (2003), 353–377].

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