Solvability of a higher-order multi-point boundary value problem at resonance

Lin, Xiaojie, Zhang, Qin, and Du, Zengji. "Solvability of a higher-order multi-point boundary value problem at resonance." Applications of Mathematics 56.6 (2011): 557-575. <http://eudml.org/doc/196674>.

@article{Lin2011,
abstract = {Based on the coincidence degree theory of Mawhin, we get a new general existence result for the following higher-order multi-point boundary value problem at resonance \[ \{ x^\{(n)\}(t)=f(t, x(t), x^\{\prime \}(t),\cdots , x^\{(n-1)\}(t)),\quad t\in (0,1),\cr x(0)=\sum \_\{i=1\}^\{m\}\alpha \_\{i\}x(\xi \_\{i\}),\quad x^\{\prime \}(0)=\cdots =x^\{(n-2)\}(0)=0,\quad x^\{(n-1)\}(1)=\sum \_\{j=1\}^\{l\}\beta \_\{j\}x^\{(n-1)\}(\eta \_\{j\}),\cr \} \] where $f\colon [0, 1]\times \mathbb \{R\}^n\rightarrow \mathbb \{R\}$ is a Carathéodory function, $0<\xi _\{1\}<\xi _\{2\}<\cdots <\xi _\{m\}<1$, $\alpha _\{i\}\in \mathbb \{R\}$, $i=1,2,\cdots , m$, $m\ge 2$ and $0<\eta _\{1\}<\cdots <\eta _\{l\}<1$, $\beta _\{j\}\in \mathbb \{R\}$, $j=1,\cdots , l$, $l\ge 1$. In this paper, two of the boundary value conditions are responsible for resonance.},
author = {Lin, Xiaojie, Zhang, Qin, Du, Zengji},
journal = {Applications of Mathematics},
keywords = {multi-point boundary value problem; coincidence degree theory; resonance; higher-order ODE; degree arguments; multi-point BVP; higher-order ODE; resonance; degree arguments},
language = {eng},
number = {6},
pages = {557-575},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Solvability of a higher-order multi-point boundary value problem at resonance},
url = {http://eudml.org/doc/196674},
volume = {56},
year = {2011},
}

TY - JOUR
AU - Lin, Xiaojie
AU - Zhang, Qin
AU - Du, Zengji
TI - Solvability of a higher-order multi-point boundary value problem at resonance
JO - Applications of Mathematics
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 6
SP - 557
EP - 575
AB - Based on the coincidence degree theory of Mawhin, we get a new general existence result for the following higher-order multi-point boundary value problem at resonance \[ { x^{(n)}(t)=f(t, x(t), x^{\prime }(t),\cdots , x^{(n-1)}(t)),\quad t\in (0,1),\cr x(0)=\sum _{i=1}^{m}\alpha _{i}x(\xi _{i}),\quad x^{\prime }(0)=\cdots =x^{(n-2)}(0)=0,\quad x^{(n-1)}(1)=\sum _{j=1}^{l}\beta _{j}x^{(n-1)}(\eta _{j}),\cr } \] where $f\colon [0, 1]\times \mathbb {R}^n\rightarrow \mathbb {R}$ is a Carathéodory function, $0<\xi _{1}<\xi _{2}<\cdots <\xi _{m}<1$, $\alpha _{i}\in \mathbb {R}$, $i=1,2,\cdots , m$, $m\ge 2$ and $0<\eta _{1}<\cdots <\eta _{l}<1$, $\beta _{j}\in \mathbb {R}$, $j=1,\cdots , l$, $l\ge 1$. In this paper, two of the boundary value conditions are responsible for resonance.
LA - eng
KW - multi-point boundary value problem; coincidence degree theory; resonance; higher-order ODE; degree arguments; multi-point BVP; higher-order ODE; resonance; degree arguments
UR - http://eudml.org/doc/196674
ER -