## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

Chaitan P. Gupta

#### 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.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 15, Number 2 (2008), 237-248.

Dates
First available in Project Euclid: 8 May 2008

https://projecteuclid.org/euclid.bbms/1210254821

Digital Object Identifier
doi:10.36045/bbms/1210254821

Mathematical Reviews number (MathSciNet)
MR2424109

Zentralblatt MATH identifier
1156.34012

#### Citation

Gupta, Chaitan P. A Non-Resonant Generalized Multi-Point Boundary Value Problem of Dirichlet-Neumann Type involving a p-Laplacian type operator. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 2, 237--248. doi:10.36045/bbms/1210254821. https://projecteuclid.org/euclid.bbms/1210254821