一类多点边值问题的可解性

Solvability of Multi-Point Boundary Value Problem

中文关键词:  多点边值问题  共振  度理论.

英文关键词:$p$-Laplace  multi-point boundary value problem  resonance  degree theory.

中文摘要:

      本文利用度理论研究了~$p$-Laplace 方程多点边值问题\begin{eqnarray*}\left\{\begin{array}{ll}(\phi_p(u'))'=f(t,u,u'),\ \ t \in (0,1),\\ u'(0)=0,\ \ u(1)=\sum_{i=1}^{n-2}a_iu(\eta_i)\end{array}\right.\end{eqnarray*}解的存在性，其中~$\phi_p(s)=|s|^{p-2}s,\ p>1$，$0<\eta_1<\eta_2<\cdots<\eta_{n-2}<1$,$a_i(i=1,2,\cdots,n-2)$ 是非负常数，且~$\\sum_{i=1}^{n-2}a_i=1$.在符号条件和增长阶条件下,改进了一些已有结果.

英文摘要:

      This paper deals with the existence of solutions for the problem $$\left\{\begin{array}{l}(\phi_p(u'))'=f(t,u,u'),\ \ t \in (0,1),\\ u'(0)=0,\ \ u(1)=\sum_{i=1}^{n-2}a_iu(\eta_i),\end{array}\right.$$ where $\phi_p(s)=|s|^{p-2}s,\ p>1$. $0<\eta_1<\eta_2<\cdots<\eta_{n-2}<1,a_i~(i=1,2,\ldots,n-2)$ are non-negative constants and $\sum_{i=1}^{n-2}a_i=1$. Some known results are improved under some sign and growth conditions. The proof is based on the Brouwer degree theory.