# 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].

## How to cite

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Liu, Yuji, and Ge, Weigao. "Solutions of a multi-point boundary value problem for higher-order differential equations at resonance. (II)." Archivum Mathematicum 041.2 (2005): 209-227. <http://eudml.org/doc/249511>.

@article{Liu2005,
abstract = {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 $x^\{(n)\}(t)=f(t,x(t),x^\{\prime \}(t),\dots ,x^\{(n-1)\}(t))+e(t)\,,\quad 0<t<1\,,\qquad \mathrm \{\{(\ast )\}\}$ and the following multi-point boundary value conditions \begin\{align*\}\{1\}\{*\}\{-1\} x^\{(i)\}(0)&=0\quad \mbox \{for\}\quad i=0,1,\dots ,n-3\,,\\ x^\{(n-1)\}(0)&=\alpha x^\{(n-1)\}(\xi )\,,\quad x^\{(n-2)\}(1)=\sum \_\{i=1\}^m\beta \_ix^\{(n-2)\}(\eta \_i)\,. **\end\{align*\} Sufficient conditions for the existence of at least one solution of the BVP $(\ast )$ and $(\ast \ast )$ 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].},
author = {Liu, Yuji, Ge, Weigao},
journal = {Archivum Mathematicum},
keywords = {solution; resonance; multi-point boundary value problem; higher order differential equation},
language = {eng},
number = {2},
pages = {209-227},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Solutions of a multi-point boundary value problem for higher-order differential equations at resonance. (II)},
url = {http://eudml.org/doc/249511},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Liu, Yuji
AU - Ge, Weigao
TI - Solutions of a multi-point boundary value problem for higher-order differential equations at resonance. (II)
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 2
SP - 209
EP - 227
AB - 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 $x^{(n)}(t)=f(t,x(t),x^{\prime }(t),\dots ,x^{(n-1)}(t))+e(t)\,,\quad 0<t<1\,,\qquad \mathrm {{(\ast )}}$ and the following multi-point boundary value conditions \begin{align*}{1}{*}{-1} x^{(i)}(0)&=0\quad \mbox {for}\quad i=0,1,\dots ,n-3\,,\\ x^{(n-1)}(0)&=\alpha x^{(n-1)}(\xi )\,,\quad x^{(n-2)}(1)=\sum _{i=1}^m\beta _ix^{(n-2)}(\eta _i)\,. **\end{align*} Sufficient conditions for the existence of at least one solution of the BVP $(\ast )$ and $(\ast \ast )$ 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].
LA - eng
KW - solution; resonance; multi-point boundary value problem; higher order differential equation
UR - http://eudml.org/doc/249511
ER -

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