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

Lin, Xiaojie ; Zhang, Qin ; Du, Zengji

Applications of Mathematics, Tome 56 (2011) no. 6, pp. 557-575

Résumé

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 \[ \displaylines { x^{(n)}(t)=f(t, x(t), x'(t),\cdots , x^{(n-1)}(t)),\quad t\in (0,1),\cr x(0)=\sum _{i=1}^{m}\alpha _{i}x(\xi _{i}),\quad x'(0)=\cdots =x^{(n-2)}(0)=0,\quad x^{(n-1)}(1)=\sum _{j=1}^{l}\beta _{j}x^{(n-1)}(\eta _{j}),\cr } \] where $f\colon [0, 1]\times \mathbb R^n\rightarrow \mathbb R$ is a Carathéodory function, $0\xi _{1}\xi _{2}\cdots \xi _{m}1$, $\alpha _{i}\in \mathbb R$, $i=1,2,\cdots , m$, $m\geq 2$ and $0\eta _{1}\cdots \eta _{l}1$, $\beta _{j}\in \mathbb R$, $j=1,\cdots , l$, $l\geq 1$. In this paper, two of the boundary value conditions are responsible for resonance.

Détail

DOI : 10.1007/s10492-011-0033-0

Classification : 34B10, 34B15, 47N20
Keywords: multi-point boundary value problem; coincidence degree theory; resonance; higher-order ODE; degree arguments