A Non-Resonant Generalized Multi-Point Boundary Value Problem of Dirichlet-Neumann Type involving a p-Laplacian type operator

Abstract

Let $\phi $, $\theta $ be odd increasing homeomorphisms from $\mathbb{R}$ onto $\mathbb{R}$ satisfying $\phi (0)=\theta (0)=0$, $f:[0,1]\times \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R}$ be a function satisfying Carathéodory conditions and $e:[0,1]\rightarrow \mathbb{R}$ be a function in $L^{1}[0,1]$. Let $\xi _{i}$,$\tau _{j}\in (0,1)$, $a_{i}$, $ b_{j}\in \mathbb{R}$, $i=1$, $2$, $\cdot \cdot \cdot $, $m-2$, $j$ $=$ $1$, $ 2$, $\cdot $ $\cdot $ $\cdot $, $n-2$, $0<\xi _{1}<\xi _{2}<\cdot \cdot \cdot <\xi _{m-2}<1$, $0<\tau _{1}<\tau _{2}<\cdot \cdot \cdot <\tau_{n-2}<1 $ be given. We study the problem of existence of solutions for the generalized multi-point boundary value problem \begin{gather} (\phi (x^{\prime }))^{\prime }=f(t,x,x^{\prime })+e\text{, }0<t<1, \notag \\ x(0)=\sum_{i=1}^{m-2}a_{i}x(\xi _{i}), \theta (x^{\prime }(1))=\sum_{j=1}^{n-2}b_{j}\theta (x^{\prime }(\tau _{j})), \end{gather} in the non-resonance case. We say that this problem is non-resonant if the associated problem: \begin{gather} (\phi (x^{\prime }))^{\prime }=0\text{, }0<t<1, \notag \\ x(0)=\sum_{i=1}^{m-2}a_{i}x(\xi _{i}), \theta (x^{\prime }(1))=\sum_{j=1}^{n-2}b_{j}\theta (x^{\prime }(\tau _{j})), \end{gather} has the trivial solution as its only solution. This is the case if \begin{equation*} (1-\sum_{j=1}^{n-2}b_{j})(1-\sum_{i=1}^{m-2}a_{i})\neq 0. \end{equation*} Our methods consist in using topological degree and some a priori estimates.