### A non-resonant multi-point boundary-value problem for a $p$-Laplacian type operator.

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This paper is devoted to the problem of existence of a solution for a non-resonant, non-linear generalized multi-point boundary value problem on the interval $[0,1]$. The existence of a solution is obtained using topological degree and some a priori estimates for functions satisfying the boundary conditions specified in the problem.

Let $g$: $\mathbf{R}\to \mathbf{R}$ be a continuous function, $e$: $[0,1]\to \mathbf{R}$ a function in ${L}^{2}[0,1]$ and let $c\in \mathbf{R}$, $c\ne 0$ be given. It is proved that Duffing’s equation ${u}^{\text{'}\text{'}}+c{u}^{\text{'}}+g\left(u\right)=e\left(x\right)$, $0<x<1$, $u\left(0\right)=u\left(1\right)$, ${u}^{\text{'}}\left(0\right)={u}^{\text{'}}\left(1\right)$ in the presence of the damping term has at least one solution provided there exists an $\mathbf{R}>0$ such that $g\left(u\right)u\ge 0$ for $\left|u\right|\ge \mathbf{R}$ and ${\int}_{0}^{1}e\left(x\right)dx=0$. It is further proved that if $g$ is strictly increasing on $\mathbf{R}$ with ${lim}_{u\to -\infty}g\left(u\right)=-\infty $, ${lim}_{u\to \infty}g\left(u\right)=\infty $ and it Lipschitz continuous with Lipschitz constant $\alpha <4{\pi}^{2}+{c}^{2}$, then Duffing’s equation given above has exactly one solution for every $e\in {L}^{2}[0,1]$.

Let $f:[0,1]\times {\mathbb{R}}^{2}\to \mathbb{R}$ be a function satisfying Caratheodory’s conditions and let $e\left(t\right)\in {L}^{1}[0,1]$. Let ${\xi}_{i},{\tau}_{j}\in (0,1)$, ${c}_{i},{a}_{j}\in \mathbb{R}$, all of the ${c}_{i}$’s, (respectively, ${a}_{j}$’s) having the same sign, $i=1,2,...,m-2$, $j=1,2,...,n-2$, $0<{\xi}_{1}<{\xi}_{2}<...<{\xi}_{m-2}<1$, $0<{\tau}_{1}<{\tau}_{2}<...<{\tau}_{n-2}<1$ be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary value problems $$\begin{array}{c}\hfill {x}^{\text{'}\text{'}}\left(t\right)=f(t,x\left(t\right),{x}^{\text{'}}\left(t\right))+e\left(t\right),\phantom{\rule{2.0em}{0ex}}t\in (0,1)E\\ \hfill x\left(0\right)=\sum _{i=1}^{m-2}{c}_{i}{x}^{\text{'}}\left({\xi}_{i}\right),\phantom{\rule{2.0em}{0ex}}x\left(1\right)=\sum _{j=1}^{n-2}{a}_{j}x\left({\tau}_{j}\right)B{C}_{mn}\end{array}$$ and $$\begin{array}{c}\hfill {x}^{\text{'}\text{'}}\left(t\right)=f(t,x\left(t\right),{x}^{\text{'}}\left(t\right))+e\left(t\right),\phantom{\rule{2.0em}{0ex}}t\in (0,1)E\\ \hfill x\left(0\right)=\sum _{i=1}^{m-2}{c}_{i}{x}^{\text{'}}\left({\xi}_{i}\right),\phantom{\rule{2.0em}{0ex}}{x}^{\text{'}}\left(1\right)=\sum _{j=1}^{n-2}{a}_{j}{x}^{\text{'}}\left({\tau}_{j}\right),B{C}_{mn}^{\text{'}}\end{array}$$ Conditions for the existence of a solution for the above boundary value problems are given using Leray-Schauder Continuation theorem.

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