\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Sixth Mississippi State Conference on Differential Equations and Computational Simulations, {\em Electronic Journal of Differential Equations}, Conference 15 (2007), pp. 127--139.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{127} \title[\hfilneg EJDE-2006/Conf/15\hfil A multi-point boundary-value problem] {A non-resonant generalized multi-point boundary-value problem of Dirichelet type involving a p-laplacian type operator} \author[C. P. Gupta\hfil EJDE/Conf/15 \hfilneg] {Chaitan P. Gupta} \address{Chaitan P. Gupta \newline Department of Mathematics, 084\\ University of Nevada, Reno\\ Reno, NV 89557, USA} \email{gupta@unr.edu} \thanks{Published February 28, 2007.} \subjclass[2000]{34B10, 34B15, 34L30, 34L90} \keywords{Generalized multi-point boundary value problems; non-resonance; \hfill\break\indent p-Laplace type operator; a priori estimates; topological degree} \begin{abstract} We study the existence of solutions for the generalized multi-point boundary-value problem \begin{gather*} (\phi (x'))'=f(t,x,x')+e\quad 00$, $$\limsup_{z\to \infty }\frac{\phi (Mz)}{\phi (z)}\equiv \alpha (M)<\infty . \label{cond1}$$ (b) For any$\sigma $,$0\leq \sigma <1$, $$\widetilde{\alpha }(\sigma )\equiv \limsup_{z\to \infty }\frac{ \phi (\sigma z)}{\phi (z)}<1. \label{cond2}$$ \begin{proposition} \label{Prop0} The boundary-value problem \eqref{1hmbp} has only the trivial solution if and only if $$\Big(\sum_{i=1}^{m-2}a_i\xi _i\Big)\Big(1-\sum_{j=1}^{n-2}b_j\Big)\neq \Big(1-\sum_{i=1}^{m-2}a_i\Big)\Big(\sum_{j=1}^{n-2}b_j\tau _j-1\Big). \label{NRcond}$$ \end{proposition} \begin{proof} It is obvious that$x(t)=At+B$,$t\in [0,1]$,$A$,$B\in \mathbb{R}$, is a general solution for the differential equation \begin{equation*} (\phi (x'))'=0, \quad 00$ be such that $\widetilde{\alpha }(\sigma ^{\ast })+\varepsilon <1$ and the constant $C_{\varepsilon }$ be such that $$\phi (\sigma ^{\ast }z)\leq (\widetilde{\alpha }(\sigma ^{\ast })+\varepsilon )\phi (z)+C_{\varepsilon },\quad \text{for every }z\in \mathbb{R}. \label{eq1}$$ \begin{proposition}\label{Prop2} Let $\xi _i$,$\tau _j\in (0,1)$, $a_i$, $b_j\in \mathbb{R}$, $i=1, 2, \dots , m-2$, $j=1, 2, \dots, n-2$, $0<\xi _{1}<\xi _{2}<\dots <\xi _{m-2}<1$, $0<\tau _{1}<\tau _{2}<\dots <\tau _{n-2}<1$, with $(\sum_{i=1}^{m-2}a_i\xi _i)(1-\sum_{j=1}^{n-2}b_j)\neq (1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j\tau _j-1)$ be given. Also let the function $x(t)$ be such that $x(t)$, $x'(t)$ be absolutely continuous on $[0,1]$ with $(\phi (x'))'\in L^{1}(0,1)$ and $x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i)$, $x(1)=\sum_{j=1}^{n-2}b_jx( \tau _j)$. Then $$\| \phi (x')\| _{\infty }\leq \frac{1}{1-\widetilde{ \alpha }(\sigma ^{\ast })-\varepsilon }\| (\phi (x'))'\| _{L^{1}(0,1)}+\frac{C_{\varepsilon }}{1-\widetilde{ \alpha }(\sigma ^{\ast })-\varepsilon }, \label{est17}$$ where $\varepsilon$ and $C_{\varepsilon }$ are as in \eqref{eq1}. \end{proposition} \begin{proof} For $i=1,2,\dots,m-2$ we see using mean value theorem that there exist $\chi _i$ in $[0,1]$ such that \begin{equation*} x(\xi _i)-x(0)=\xi _ix'(\chi _i). \end{equation*} It then follows using $x(0)=\sum_{i=1}^{m-2}a_ix(\xi _i)$ that $$(1-\sum_{i=1}^{m-2}a_i)x(0)=\sum_{i=1}^{m-2}a_i\xi _ix'(\chi _i). \label{eq2}$$ Again, for $j=1,2,\dots,n-2$ we see using mean value theorem that there exist $\lambda _j$ in $[0,1]$ such that \begin{equation*} x(1)-x(\tau _j)=(1-\tau _j)x'(\lambda _j), \end{equation*} and we see using $x(1)=\sum_{j=1}^{n-2}b_jx(\tau _j)$ that $$(\sum_{j=1}^{n-2}b_j-1)x(1)=\sum_{j=1}^{n-2}b_j(1-\tau _j)x'(\lambda _j). \label{eq3}$$ Also, we see that there exists a $\lambda \in [0,1]$ such that $$x(1)-x(0)=x'(\lambda ). \label{eq4}$$ Now, we see from equations (\ref{eq2}), (\ref{eq3}), (\ref{eq4}) that \begin{align*} &(1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j-1)x'(\lambda ) \\ &=(1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j-1)(x(1)-x(0)) \\ &=(1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j(1-\tau _j) x'(\lambda _j))-(\sum_{j=1}^{n-2}b_j-1)(\sum_{i=1}^{m-2} a_i\xi_ix'(\chi _i)). \end{align*} It follows that \begin{align*} &(1-\sum_{i=1}^{m-2}a_i)(1-\sum_{j=1}^{n-2}b_j)x'(\lambda )+\sum_{j=1}^{n-2}b_j(1-\tau _j)(1-\sum_{i=1}^{m-2}a_i)x'(\lambda _j) \\ &+\sum_{i=1}^{m-2}a_i\xi _i(1-\sum_{j=1}^{n-2}b_j)x'(\chi_i)=0. \end{align*} Using, next, the intermediate value theorem we see that there exist $\upsilon _{1}$, $\upsilon _{2}$ in $[0,1]$ such that $$Ax'(\upsilon _{1})-Bx'(\upsilon _{2})=0, \label{eq15}$$ where $A$, $B$ are as defined in (\ref{defA}), (\ref{defB}). Suppose, now, one of $x'(\upsilon _{1})$, $x'(\upsilon _{2})$ is zero. We then see from one of the following equations $$\phi (x'(t))=\phi (x'(\upsilon _{k}))+\int_{\upsilon _{k}}^{t}(\phi (x'))'(s)ds,\quad k=1,2;\; t\in [0,1] \label{eq5}$$ that $$\| \phi (x')\| _{\infty }\leq \| (\phi (x'))'\| _{L^{1}(0,1)}. \label{est7}$$ Let us, next, suppose that both $x'(\upsilon _{1})$, $x'(\upsilon _{2})$ are non-zero. Since, now, $A\neq B$, in view of Lemma \ref{lemma1} we see from equation (\ref{eq15}) that $x'(\upsilon _{1})\neq x'(\upsilon _{2})$. We now use the equations \begin{gather*} \phi (x'(t)) =\phi (x'(\upsilon _{1}))+\int_{\upsilon _{k}}^{t}(\phi (x'))'(s)ds =\phi (\frac{B}{A}x'(\upsilon _{2}))+\int_{\upsilon _{k}}^{t}(\phi (x'))'(s)ds, \\ \phi (x'(t)) =\phi (x'(\upsilon _{2}))+\int_{\upsilon _{k}}^{t}(\phi (x'))'(s)ds =\phi (\frac{A}{B}x'(\upsilon _{1}))+\int_{\upsilon _{k}}^{t}(\phi (x'))'(s)ds, \end{gather*} along with the definition of $\sigma ^{\ast }$, as given in \eqref{eq0}, (\ref{eq1}) and the estimate (\ref{est7}) to obtain the estimate (\ref{est17}). This completes the proof of the proposition. \end{proof} \section{Existence Theorem} Let $\phi$ be an odd increasing homeomorphism from $\mathbb{R}$ onto $\mathbb{R}$ satisfying $\phi (0)=0$, $f:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be a function satisfying Carath\'{e}odory conditions and $e:[0,1]\to \mathbb{R}$ be a function in $L^{1}[0,1]$. Let $\xi _i$, $\tau _j\in (0,1)$, $a_i$, $b_j\in \mathbb{R}$, $i=1, 2,\dots,m-2$, $j=1,2,\dots,n-2$, $0<\xi _{1}<\xi _{2}<\dots <\xi _{m-2}<1$, $0<\tau _{1}<\tau _{2}<\dots <\tau _{n-2}<1$ with $(\sum_{i=1}^{m-2}a_i\xi _i)(1-\sum_{j=1}^{n-2}b_j)\neq (1-\sum_{i=1}^{m-2}a_i)(\sum_{j=1}^{n-2}b_j\tau _j-1)$. \begin{theorem} \label{Thm1} Let $f:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be a function satisfying Carath\'{e}odory's conditions such that there exist non-negative functions $d_{1}(t)$, $d_{2}(t)$, and $r(t)$ in $L^{1}(0,1)$ such that \begin{equation*} |f(t,u,v)|\leq d_{1}(t)\phi (|u|)+d_{2}(t)\phi (|v|)+r(t), \end{equation*} for a. e. $t\in [0,1]$ and all $u$, $v\in \mathbb{R}$. Suppose, further, $$\alpha (M)\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)}<1-\widetilde{\alpha } (\sigma ^{\ast }) \label{existcond}$$ where $M$ is as defined in Proposition \ref{Prop1}, $\alpha (M)$ is as defined in \eqref{cond1}, $\sigma ^{\ast }$ and $\widetilde{\alpha }(\sigma ^{\ast })$ are as defined in \eqref{eq0}, \eqref{eq1}. Then, for every given function $e(t)\in L^{1}[0,1]$, the boundary value problem \eqref{1mbp} has at least one solution $x(t)\in$ $C^{1}[0,1]$. \end{theorem} \begin{proof} We consider the family of boundary-value problems \begin{gathered} (\phi (x'))'=\lambda f(t,x,x')+\lambda e, 00$, independent of$\lambda \in [0,1]$, such that if$x(t)\in C^{1}[0,1]$is a solution to (\ref {eq6}), equivalently to the boundary value problems (\ref{f1mbp}), for some$\lambda \in [0,1]$then$\|x\|_{C^{1}[0,1]}0$such that$\widetilde{\alpha }(\sigma ^{\ast })+\varepsilon <1$and $$(\alpha (M)+\varepsilon )\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)}<1- \widetilde{\alpha }(\sigma ^{\ast })-\varepsilon , \label{eq9}$$ which is possible to do, in view of our assumption (\ref{existcond}). Here$M $is as defined in Propostion \ref{Prop1} and$\alpha (M)$is as defined in \eqref{cond1} so that for the$\varepsilon >0$, chosen above, there exists a constant$C_{\varepsilon }^{1}>0$such that $$\phi (Mz)\leq (\alpha (M)+\varepsilon )\phi (z)+C_{\varepsilon }^{1},\quad \text{for every }z\in \mathbb{R}. \label{eq10}$$ Also, from Proposition \ref{Prop2} we see that there is a constant$C_{\varepsilon }^{2}>0$, for the chosen$\varepsilon >0$, such that $$\phi (\|x'\|_{\infty })\leq \frac{1}{1-\widetilde{\alpha }(\sigma ^{\ast })-\varepsilon }\|(\phi (x'))'\|_{L^{1}(0,1)}+C_{\varepsilon }^{2}. \label{eq11}$$ We, now, see from the equation in (\ref{f1mbp}), using our assumptions on the function$f, Proposition \ref{Prop1}, and estimates (\ref{eq10}), (\ref{eq11}) that \begin{align*} &\|(\phi (x'))'\|_{L^{1}(0,1)} \\ &\leq \phi (\|x\|_{\infty })\|d_{1}\|_{L^{1}(0,1)}+\phi (\|x'\|_{\infty })\|d_{2}\|_{L^{1}(0,1)}+\|r\|_{L^{1}(0,1)}+\|e\|_{L^{1}(0,1)} \\ &\leq \phi (M\|x'\|_{\infty })\|d_{1}\|_{L^{1}(0,1)}+\phi (\|x'\|_{\infty})\|d_{2}\|_{L^{1}(0,1)}+\|r\|_{L^{1}(0,1)}+\|e\|_{L^{1}(0,1)} \\ &\leq ((\alpha (M)+\varepsilon)\|d_{1}\|_{L^{1}(0,1)} +\|d_{2}\|_{L^{1}(0,1)})\phi (\|x'\|_{\infty})+\|r\|_{L^{1}(0,1)} +\|e\|_{L^{1}(0,1)} \\ &\quad +C_{\varepsilon }^{1}\|d_{1}\|_{L^{1}(0,1)} \\ &\leq \frac{(\alpha (M)+\varepsilon )\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)}}{1-\widetilde{\alpha }(\sigma ^{\ast })-\varepsilon }\|(\phi (x'))'\|_{L^{1}(0,1)}+C\varepsilon , \end{align*} whereC\varepsilon =\|r\|_{L^{1}(0,1)}+\|e\|_{L^{1}(0,1)}+C_{\varepsilon }^{1}\|d_{1}\|_{L^{1}(0,1)}+C_{\varepsilon }^{2}[(\alpha (M)+\varepsilon )\|d_{1}\|_{L^{1}(0,1)}+\|d_{2}\|_{L^{1}(0,1)}]$. It, now, follows from (\ref{eq9}) that there exists a constant$R_{0}$, independent of$\lambda \in [0,1]$, such that if$x(t)\in C^{1}[0,1]$is a solution to the boundary value problems (\ref{f1mbp}) for some$\lambda \in [0,1]$then \begin{equation*} \|(\phi (x'))'\|_{L^{1}(0,1)}\leq R_{0}. \end{equation*} This combined with (\ref{eq11}) and (\ref{est0}) give that there exists a constant$R>0$such that \begin{equation*} \|x\|_{C^{1}[0,1]}0 \\ -1, &\text{if }\det \mathbb{A}<0. \end{cases} \end{equation*} Accordingly, we see from the non-resonance assumption \eqref{NRcond} i.e. $\det \mathbb{A=(}1-\sum_{i=1}^{m-2}a_i)(1-\sum_{j=1}^{n-2}b_j\tau _j)+(\sum_{i=1}^{m-2}a_i\xi _i)(1-\sum_{j=1}^{n-2}b_j)\neq 0$ that$\deg _{LS}(I-\Psi (\cdot ,1),B(0,R),0)\neq 0$and there is$x(t)\in B(0,R)\subset C^{1}[0,1]$that satisfies \begin{equation*} x=\Psi (x,1), \end{equation*} equivalently$x(t)$is a solution to the boundary value \eqref{1mbp}. 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