\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 112, pp. 1--7.\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 2003 Texas State University-San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/112\hfil Multi-point boundary-value problems] {Multi-point boundary-value problems for the p-Laplacian at resonance} \author[Xiaohong Ni \& Weigao Ge\hfil EJDE--2003/112\hfilneg] {Xiaohong Ni \& Weigao Ge} \address{Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China} \email[Xiaohong Ni]{nixiaohong1126@hotmail.com} \date{} \thanks{Submitted July 8, 2003. Published November 5, 2003.} \thanks{Research supported by National Natural Science Foundation of China} \subjclass[2000]{34B10, 34B15} \keywords{Resonance, multi-point boundary value problem, p-Laplacian} \begin{abstract} In this paper, we consider multi-point boundary-value problems for the p-laplacian at resonance. Applying an extension of Mawhin's continuation theorem, we show the existence of at least one solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} Let $\ f:[0,1]\times \mathbb{R}^2\to \mathbb{R}$ be a continuous function and $0<\xi_1<\xi_2<\dots<\xi_{n-2}<1$, $0<\eta_1<\eta_2<\dots<\eta_{m-2}<1$, $\alpha_j\in \mathbb{R}$ ($1\leq j\leq n-2$), $\beta_i\in \mathbb{R}$ ($1\leq i\leq m-2$) be given. In this paper, we study the differential equation $$(\phi_p(x'))'=f(t,x(t),x'(t)) \eqno(1.1)$$ subject to one of the following boundary-value conditions: $$x'(0)=0,\quad \phi_p(x'(1))=\sum_{i=1}^{m-2}\beta_i\phi_p(x'(\eta_i))\eqno(1.2)$$ or $$\phi_p(x'(0))=\sum_{j=1}^{n-2}\alpha_j\phi_p(x'(\xi_j)),\quad x'(1)=0\eqno(1.3)$$ where $\phi_p(s)=|s|^{p-2}s$, $p>1$, $\sum_{i=1}^{m-2}\beta_i=\sum_{j=1}^{n-2}\alpha_j=1$. Mawhin \cite{g1} proved a continuation theorem for the abstract equation $Lx=Nx$ with $L$ being a non-invertible linear operator. Since then, there has been much attention on the study of boundary-value problems at resonance (cf \cite{b2,m1,p1}). Recently, Liu \cite{l1,l2,l3} considered differential equation (1.1) with $p=2$ and obtained some results. However, the p-laplacian operator is not linear. Our approach is different and based on an extension of Mawhin's continuation theorem in \cite{g2}. We provide sufficient conditions to guarantee the existence of solution for (1.1), (1.2) and for (1.1), (1.3). Moreover, some examples are given to demonstrate our results. For the convenience of the reader, we first recall some notation. Let $X$ and $Z$ be Banach spaces with norms $\|\cdot\|_X$ and $\|\cdot\|_Z$, respectively. A continuous operator $M:X\cap \mathop{\rm dom} M\to Z$ is said to be quasi-linear if \begin{itemize} \item[(i)] $\mathop{\rm Im}M=M(X\cap \mathop{\rm dom}M)$ is a closed subset of $Z$; \item[(ii)] $\mathop{\rm Ker}M=\{x\in X\cap \mathop{\rm dom}M: Mx=0\}$ is linearly homeomorphic to $\mathbb{R}^n$, $n<\infty$. \end{itemize} Let $P:X\to X_1$ and $Q:Z\to Z_1$ be two projectors such that $\mathop{\rm Im}P=\mathop{\rm Ker}M$, $\mathop{\rm Ker}Q=\mathop{\rm Im}M$ and $X=X_1\oplus X_2$, $Z=Z_1\oplus Z_2$, where $X_1=\mathop{\rm Ker}M$, $Z_2=\mathop{\rm Im}M$ and $X_2$, $Z_1$ are respectively the complement space of $X_1$ in $X$, $Z_2$ in $Z$. If $\Omega$ is an open and bounded subset of $X$ such that $\mathop{\rm dom}M\cap \Omega\neq \Phi$, the continuous operator $N_{\lambda}:\overline{\Omega}\to Z, \lambda\in[0,1]$ will be called admissibly compact in $\overline \Omega$ with respect to $M$ if \begin{itemize} \item[(iii)] There is a subset $Z_1$ of $Z$ with $\dim Z_1=\dim X_1$ and an operator $K:\mathop{\rm Im}M\to X_2$ with $K\theta=\theta$ such that for $\lambda\in[0,1]$, $$(I-Q)N_{\lambda}(\bar{\Omega})\subset \mathop{\rm Im}M\subset(I-Q)Z,\eqno(1.4)$$ $$(I-Q)N_0 \mbox{ is a zero operator, and } QN_{\lambda}x=0\Leftrightarrow QNx=0, \lambda\in(0,1), \eqno(1.5)$$ $$KM=I-P,\ K(I-Q)N_{\lambda}:\overline{\Omega}\to X_2\subset X \ {\hbox{is compact}}, \eqno(1.6)$$ $$M[P+K(I-Q)N_{\lambda}]=(I-Q)N_{\lambda}. \eqno(1.7)$$ \end{itemize} \begin{theorem}[\cite{g2}] \label{thm1.1} Let $X$ and $Z$ be two Banach spaces with the norms $\|\cdot\|_X$ and $\|\cdot\|_Z$, respectively, and $\Omega\subset X$ an open and bounded set. Suppose $M: X\cap \mathop{\rm dom}M\to Z$ is a quasi-linear operator and $N_{\lambda}: \overline{\Omega}\to Z$ is admissibly compact with respect to $M$. In addition, if \begin{itemize} \item[(C1)] $Mx\neq N_{\lambda}x, \lambda\in(0,1)$, $x\in\partial\Omega$ \item[(C2)] $\deg\{JQN,\Omega\cap \mathop{\rm Ker}M, 0\}\neq0$, where $N=N_1$ and $J:Z_1\to X_1$ is a homeomorphism with $J(0)=0$; \end{itemize} then the abstract equation $Mx=Nx$ has at least one solution in $\overline{\Omega}$. \end{theorem} \section{General Results} \subsection{Results for boundary-value problem (1.1), (1.2)} Let $X=\{x\in C[0,1]:\ x'(0)=0$, $\phi_p(x'(1))=\sum_{i=1}^{m-2}\beta_i\phi_p(x'(\eta_i))\}$ with norm $\|x\|_1=\max\{\|x\|,\|x'\|\}$ and $Z=C[0,1]$ with norm $\|w\|=\max_{t\in[0,1]}|w(t)|$. Let $M$ be the operator from $\mathop{\rm dom}M\subset X$ to $Z$ with $$\mathop{\rm dom}M=\{x\in C^1[0,1]: \phi_p(x')\in C^1[0,1]\}$$ and $Mx=(\phi_p(x'))'$. For any open and bounded $\Omega\subset X$, we define $N_{\lambda}:\overline{\Omega}\to Z$ by $$(N_{\lambda}x)=\lambda f(t,x(t),x'(t)),\quad t\in[0,1]$$ Then (1.1), (1.2) can be written as $$Mx=Nx,\quad \mbox{where } N=N_1.$$ \begin{lemma} \label{lm2.1} If $0<\eta_1<\eta_2<\dots<\eta_{m-2}<1$, then there exists integer $k\in\{1,2,\dots,m-1\}$ such that $\sum_{i=1}^{m-2}\beta_i\eta^k_i\neq1$. \end{lemma} \begin{proof} If this were not the case, we have $$\begin{pmatrix} \eta_1&\eta_2&\dots&\eta_{m-2}\\ \eta^2_1&\eta^2_2&\dots&\eta^2_{m-2}\\ \vdots&\vdots&\ddots&\vdots\\ \eta^{m-1}_1&\eta^{m-1}_2&\dots&\eta^{m-1}_{m-2} \end{pmatrix} \begin{pmatrix} \beta_1\\ \beta_2\\ \vdots\\ \beta_{m-2} \end{pmatrix}= \begin{pmatrix} 1\\ 1\\ \vdots\\ 1 \end{pmatrix}.$$ This implies $$\begin{pmatrix} \eta_1&\eta_2&\dots&\eta_{m-2}&1\\ \eta^2_1&\eta^2_2&\dots&\eta^2_{m-2}&1\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ \eta^{m-1}_1&\eta^{m-1}_2&\dots&\eta^{m-1}_{m-2}&1 \end{pmatrix} \begin{pmatrix} \beta_1\\ \beta_2\\ \vdots\\ \beta_{m-2}\\ -1 \end{pmatrix}= \begin{pmatrix} 0\\ 0\\ \vdots\\ 0\\ 0 \end{pmatrix}.$$ Putting $$A=\begin{pmatrix} \eta_1&\eta_2&\dots&\eta_{m-2}&1\\ \eta^2_1&\eta^2_2&\dots&\eta^2_{m-2}&1\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ \eta^{m-1}_1&\eta^{m-1}_2&\dots&\eta^{m-1}_{m-2}&1 \end{pmatrix}$$ we have $$\det A=\eta_1\eta_2\eta_{m-2}\left|\begin{matrix} 1&1&\dots&1&1\\ \eta_1&\eta_2&\dots&\eta_{m-2}&1\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ \eta^{m-2}_1&\eta^{m-2}_2&\dots&\eta^{m-2}_{m-2}&1 \end{matrix}\right|\neq 0.$$ Hence, $(\beta_1,\beta_2,\ldots,\beta_{m-2},-1)^T=(0,0,\ldots,0,0)^T$, which leads to a contradiction, $-1=0$. \end{proof} \begin{lemma} \label{lm2.2} If $\sum_{i=1}^{m-2}\beta_i=1$, then the operator $M: \mathop{\rm dom}M\cap X\to Z$ is a quasi-linear and $N_{\lambda}:\overline{\Omega}\to Z$ is admissibly compact in $\overline{\Omega}$ with respect to $M$. \end{lemma} \begin{proof} Obviously, $X_1=\mathop{\rm Ker}M=\{\ x=A,\; A\in \mathbb{R}\}$ and $$Z_2=\mathop{\rm Im}M=\big\{y\in Z,\ \sum_{i=1}^{m-2}\int_{\eta_i}^1 \beta_i y(s)ds=0\big\}.$$ We have $\dim \mathop{\rm Ker}M=1<\infty$ and $M(X\cap \mathop{\rm dom}M)\subset Z$ closed. Hence, $M$ is a quasi-linear operator. Define projectors $P: X\to X_1$ as $Px=x(0)$ for all $x\in X$ and $Q: Z\to Z_1$ as $$Qy=\Big[\frac{k}{1-\sum_{i=1}^{m-2}\beta_i\eta^k_i}\sum_{i=1}^{m-2}\beta_i \int_{\eta_i}^1 y(s)ds\Big]t^{k-1},\quad \forall y\in Z.$$ where $Z_1$ is the complement space of $Z_2$ in $Z$. (Indeed, $Z_1\cong \mathbb{R})$. Thus $\dim X_1=\dim Z_1=1$. Taking $K: \mathop{\rm Im} M\to X_2$ as follows: $$K(I-Q)h(t)=\int_0^t\phi_p^{-1}\Big(\int_0^s((I-Q)h(\tau)d\tau\Big)ds,\quad \forall h(t)\in Z.$$ here $X_2$ is the complement space of $X_1$ in $X$. It is clear that $K\theta=\theta$ and (1.4), (1.5) hold. For for all $x\in X\cap \mathop{\rm dom}M$, we have $$M[P+K(I-Q)N_\lambda]x=(I-Q)N_\lambda x$$ and $$KMx=K(I-Q)Mx=\int_0^t\phi_p^{-1}(\int_0^s(\phi_p(x'(\tau)))'d\tau)ds=x(t)-x(0) =(I-P)x.$$ For any open and bounded set $\Omega\subset X, \lambda\in[0,1]$, by using the Ascoli-Arzela Theorem, we can prove that $[K(I-Q)N_{\lambda}](\overline{\Omega})$ is relatively compact and continuous. This implies that (1.6), (1.7) are valid. Hence, $N_{\lambda}$ is admissibly compact in $\overline{\Omega}$ with respect to $M$. \end{proof} For the next theorem we have the assumptions: \begin{itemize} \item[(H1)] There exist functions $a,b,r\in C([0,1],[0,+\infty))$ such that for all $(x,y)\in \mathbb{R}^2$, $t\in[0,1]$, $$|f(t,x,y)|\leq\phi_p[a(t)|x|+b(t)|y|+r(t)].$$ \item[(H2)] There exists a constant $C>0$ such that for any $x\in X$, if $|x(t)|>c$, for all $t\in[0,1]$, one has $$\mathop{\rm sgn}(x(t))\sum_{i=2}^{m-2}\beta_i\int_{\eta_i}^1f(t,x(t),x'(t))dt<0 \eqno(2.1)$$ or $$\mathop{\rm sgn}(x(t))\sum_{i=2}^{m-2}\beta_i\int_{\eta_i}^1f(t,x(t),x'(t))dt>0 \eqno(2.2)$$ \end{itemize} \begin{theorem} \label{thm2.1} Suppose $f:[0,1]\times \mathbb{R}^2\to \mathbb{R}$ is a continuous function and $0<\eta_1<\eta_2<\dots<\eta_{m-2}<1$, $\beta_i\in \mathbb{R}$ ($1\leq i\leq m-2$) are given. Under Assumptions (H1) and (H2), the boundary-value problem (1.1), (1.2) with $\sum_{i=1}^{m-2}\beta_i=1$ has at least one solution provided that $\|a\|+\|b\|<1$. \end{theorem} \begin{proof} Set $\Omega_1=\{x\in X\cap \mathop{\rm dom}M,\; Mx=N_{\lambda}x,\ \lambda\in(0,1)\}$, then for $x\in \Omega_1$, $Mx=N_{\lambda}x$. Thus, we have $Nx\in \mathop{\rm Im}M=\mathop{\rm Ker}Q$ and $$\sum_{i=2}^{m-2}\beta_i\int_{\eta_i}^1f(t,x(t),x'(t))dt=0.$$ By (H2), there exists $t_0\in[0,1]$ such that $|x(t_0)|\leq C$. Hence, we have $\|x\|\leq C+\|x'\|$, in view of the relation $x(t)=x(t_0)+\int_{t_0}^tx'(s)ds$. At the same time, $$|x'(t)|\leq\phi_p^{-1}(\|Nx\|)\leq\|a\|\|x\|+\|b\|\|x'\|+\|r\|.$$ Noticing $\ \|a\|+\|b\|<1,\$ we know the set $\Omega_1$\ is bounded. Let $\|x\|_1\leq C^*$ for all $x\in \Omega_1$. If (2.1) holds, taking $H(x,\lambda)=-\lambda Ix+(1-\lambda)JQNx$, $\lambda\in[0,1]$ and $$\Omega_2=\{x\in \mathop{\rm Ker}M,\; H(x,\lambda)=0\},$$ where $J: \mathop{\rm Im}Q\to \mathop{\rm Ker}M$ is a homomorphism such that $J(At^{k-1})=A$, for all $A\in \mathbb{R}$. We claim that for any $x\in \Omega_2$, $\|x\|_1\leq C$. In fact, if this were not the case, there exist $\lambda_0\in[0,1]$ and $|x_0|>C$ such that $H(x_0,\lambda_0)=0$. Without loss of generality, suppose $x_0>C$ and $\sum_{i=1}^{m-2}\beta_i\eta^k_i<1$. If $\lambda_0=1$, then $x_0=0$ otherwise, it implies that $$0\leq\lambda_0x_0=(1-\lambda_0)\frac{k}{1-\sum_{i=1}^{m-2}\beta_i\eta^k_i} \sum_{i=2}^{m-2}\beta_i\int_{\eta_i}^1f(t,x_0,0)dt<0,$$ which leads to a contradiction, $0<0$. When (2.2) holds, putting $H(x,\lambda)=\lambda Ix+(1-\lambda)JQNx$. By a similar argument, we can obtain the same conclusion. Next, we shall prove that all conditions of Theorem \ref{thm1.1} are satisfied. Let $\Omega$ be an open bounded subset of $X$ such that $\bigcup\limits_{i=1}^2\overline{\Omega}_i\subset\Omega$. Clearly, we have $$Mx\neq N_{\lambda}x,\quad \lambda\in(0,1),\; x\in \partial\Omega$$ and $$H(x,\lambda)\neq0, \quad \lambda\in[0,1],\; x\in \partial\Omega\cap \mathop{\rm Ker}M.$$ In view of the homotopy property of degree, $$\deg\{JQN,\Omega\cap\mathop{\rm Ker}M,0\} =\deg\{-I,\Omega\cap \mathop{\rm Ker}M,0\}=-1.$$ Theorem \ref{thm1.1} can be generalized to obtain the existence of at least one solution for (1.1), (1.2) with $\sum_{i=2}^{m-2}\beta_i=1$. \end{proof} \subsection{Results for Problem (1.1), (1.3)} Let $X=\{x\in C[0,1]:\phi_p(x'(0))=\sum_{j=1}^{n-2}\alpha_j\phi_p(x'(\xi_j)),\ x'(1)=0\}$. $Z, M, \mathop{\rm dom}M\$and $\ N_{\lambda}\$ be defined as above. \begin{lemma} \label{lm2.3} If $\sum_{j=1}^{n-2}\alpha_j=1$ and $\sum_{j=1}^{n-2}\alpha_j\xi_j\neq 0$ then operator $M: \mathop{\rm dom}M\cap X\to Z$ is a quasi-linear and $N_{\lambda}:\overline{\Omega}\to Z$ is admissibly compact in $\overline{\Omega}$ with respect to $M$. \end{lemma} Note that $\mathop{\rm Ker}M=\{ x=A, A\in \mathbb{R}\}$ and $\mathop{\rm Im}M=\{y\in Z,\;\sum_{j=1}^{n-2}\alpha_j\int_0^{\xi_j}y(s)ds=0\}$. We define $P:X\to X_1$, and $Q: Z\to Z_1$ as follows: $$(Px)(t)=x(0),\quad (Qy)(t)=\frac{1}{\sum_{j=1}^{n-2}\alpha_j\xi_j}\sum_{j=1}^{n-2}\alpha_j \int_0^{\xi_j}y(s)ds.$$ Let $K(I-Q)h(t)=-\int_0^t\phi_p^{-1}\big[\int_s^1((I-Q)h(\tau)d\tau\big]ds$, for $h(t)\in Z$. Using the method in the proof of Lemma \ref{lm2.2}, we can show the theorem bellow. However, we need the assumption: \begin{itemize} \item[(H3)] There exists a constant $D>0$ such that for all $x\in X$, if $\ |x(t)|>D$ for some $t\in[0,1]$, one has $$\mathop{\rm sgn}(x(t))\sum_{j=1}^{n-2}\alpha_j\int_0^{\xi_j}f(t,x(t),x'(t))dt<0 \eqno(2.3)$$ or $$\mathop{\rm sgn}(x(t))\sum_{j=1}^{n-2}\alpha_j\int_0^{\xi_j}f(t,x(t),x'(t))dt>0 \eqno(2.4)$$ \end{itemize} \begin{theorem} \label{thm2.2} Suppose that $f:[0,1]\times \mathbb{R}^2\to \mathbb{R}$ is continuous, $0<\xi_1<\xi_2<\dots<\xi_{n-2}<1$, and $\alpha_j\in \mathbb{R}$ ($1\leq j\leq n-2$) are given. Under Assumptions (H1), (H3), the boundary-value problem (1.1), (1.3) with $\sum_{j=1}^{n-2}\alpha_j=1$, $\sum_{j=1}^{n-2}\alpha_j\xi_j\neq 0$ has at least one solution provided that $\|a\|+\|b\|<1$. \end{theorem} The proof of this theorem is similar to that of Theorem \ref{thm2.1}; therefore, we omit it here. \section{Examples} \subsection*{Example 3.1} Consider the boundary-value problem, at resonance, $\begin{gathered} (\phi_p(x'))'=\frac13x+\frac{(\cos(x'))^2}{2(t+1)}-3t^2,\quad t\in[0,1]\\ x'(0)=0,\quad \phi_p(x'(1))=\sum_{i=1}^{m-2}\beta_i\phi_p(x'(\eta_i)) \end{gathered}\eqno(3.1)$ where $p>2$, $0<\eta_1<\eta_2<\dots<\eta_{m-2}<1$, $\sum_{i=1}^{m-2}\beta_i=1$, $\sum_{i=1}^{m-2}\beta_i\eta_i\neq1$ and $\beta_i(1\leq i\leq m-2)\geq 0$. Let $f(t,x,y)=\frac13x+\frac{(\cos(x'))^2}{2(t+1)}-3t^2$, then $|f(t,x,y)|\leq\frac13|x|+\frac12|y|+3t^2\leq\phi_p(\frac13|x|+\frac12|y|+3)$. Clearly, there exists a constant $C>0$ such that if $|x(t)|>C$, for any $t\in [0,1]$, one has $$\mathop{\rm sgn(x(t))}\sum_{i=2}^{m-2}\beta_i\int_{\eta_i}^1f(t,x(t),x'(t))dt>0.$$ Applying Theorem \ref{thm2.1}, we obtain that (3.1) has at least one solution. \subsection*{Example 3.2} Consider the boundary-value problem $\begin{gathered} (\phi_p(x'))'=\frac14x+\frac1{12}x\sin(x')+t\\ \phi_p(x'(0))=3\phi_p(x'(\frac14))-2\phi_p(x'(\frac12)),\ x'(1)=0, \end{gathered}\eqno(3.2)$ where $p>2$, $\xi_1=\frac14$, $\xi_2=\frac12$, $\alpha_1=3$, $\alpha_2=-2$. Taking $f(t,x,y)=\frac14x+\frac1{12}x\sin(x')+t$, then $|f(t,x,y)|\leq\frac1{12}|x|+\frac14|x|+1\leq\phi_p(\frac13|x|+1)$ and for any $A\in \mathbb{R}$, $$3\int_0^{\frac14}f(t,A,0)dt-2\int_0^{\frac12}f(t,A,0)dt=-\frac{A^3}4-\frac5{32}.$$ Hence, there exists a constant $D>0$ such that for $t\in[0,1]$ $$\sum_{j=1}^{n-2}\alpha_j\int_0^{\xi_j}f(t,D,0)dt<0<\sum_{j=1}^{n-2}\alpha_j \int_0^{\xi_j}f(t,-D,0)dt.$$ From Theorem \ref{thm2.2}, Problem (3.2) has at least one solution. \begin{thebibliography}{00} \bibitem{b2} J. Bouchala, \emph{Resonance problems for p-laplacian}, Math. Compu. Simulation, 61(2003), 599-604. \bibitem{g1} R. Gaines, J. Mawhin; \emph{Coincide degree and nonlinear differential equation}, Springer-Verlag, Berlin, 1977. \bibitem{g2} W. G. Ge, J. L. Ren; \emph{An extension of Mawhin's continuation theorem and its application to boundary value problems with a p-laplacian}, (preprint). \bibitem{l1} B. Liu, J. S. 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