\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 65, 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} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/65\hfil Constant sign solutions] {Constant sign solutions for second-order $m$-point boundary-value problems} \author[J. Yang \hfil EJDE-2013/65\hfilneg] {Jingping Yang} % in alphabetical order \address{Jingping Yang \newline Gansu Institute of Political Science and Law \\ Lanzhou, 730070, China} \email{fuj09@lzu.cn} \thanks{Submitted November 8, 2012. Published March 5, 2013.} \subjclass[2000]{34B18, 34C25} \keywords{Constant sign solutions; eigenvalue; bifurcation methods} \begin{abstract} We will study the existence of constant sign solutions for the second-order $m$-point boundary-value problem \begin{gather*} u''(t)+f(t,u(t))=0,\quad t\in(0,1),\\ u(0)=0, \quad u(1)=\sum^{m-2}_{i=1}\alpha_i u(\eta_i), \end{gather*} where $m\geq3$, $\eta_i\in(0,1)$ and $\alpha_i>0$ for $i=1,\dots,m-2$, with $\sum^{m-2}_{i=1}\alpha_i<1$, we obtain that there exist at least a positive and a negative solution for the above problem. Our approach is based on unilateral global bifurcation theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In recent years, there has been considerable interests in the existence of nodal solutions of second-order $m$-point boundary value problems (BVPs) of the form $$\begin{gathered} u''(t)+ f(u(t))=0,\quad t\in(0,1),\\ u(0)=0, \quad u(1)=\sum^{m-2}_{i=1}\alpha_i u(\eta_i), \end{gathered}\label{e1.1}$$ see \cite{a1,a2,m1,r2,x1} and the references therein. Ma and O'Regan \cite{m1} considered \eqref{e1.1} under the assumption $f\in C^1(\mathbb{R},\mathbb{R} )$ with $sf(s)>0$ for $s\neq0$. They obtained the existence of nodal solutions for $f_0, f_\infty\in (0,\infty)$, where $f_0=\lim_{u\to 0}\frac{f(u)}{u}$, $f_\infty =\lim_{u\to \infty}\frac{f(u)}{u}$. In 2011, An \cite{a2} considered the problem $$\begin{gathered} u''(t)+\lambda f(u(t))=0,\quad t\in(0,1),\\ u(0)=0, \quad u(1)=\sum^{m-2}_{i=1}\alpha_i u(\eta_i) \end{gathered}\label{e1.2}$$ under the assumption $f\in C^1(\mathbb{R}\backslash\{0\}, \mathbb{R})\cap C(\mathbb{R}, \mathbb{R})$ with $sf(s)>0$ for $s\neq0$. She investigated the global structure of nodal solutions of \eqref{e1.2} in the case $f_0=\infty$, $f_{\infty}\in[0,\infty]$, by using Rabinowit's global bifurcation theorem. From above results, we can see that the existence results are largely based on the assumption that $f_0, f_\infty$ are constants and nonlinearity term is autonomous. It is interesting to know what will happen if $f_0, f_\infty$ are functions and the nonlinear term is non-autonomous? The above results rely largely on the direct computation of eigenvalues and eigenfunctions of the linear problem associated with \eqref{e1.2}, hence, it can not be extended to the more general problem. In view of the fact that the principle eigenvalue can be easily obtained by Krein-Rutman Theorem, in this paper, we obtain the existence of constant sign solution for $$\begin{gathered} u''(t)+ f(t,u(t))=0,\quad t\in(0,1),\\ u(0)=0, \quad u(1)=\sum^{m-2}_{i=1}\alpha_i u(\eta_i) \end{gathered}\label{e1.3}$$ by relating it to the principle eigenvalue of the associated linear problem. We make the following assumptions: \begin{itemize} \item[(H1)] $\lambda_1\leq a(t)\equiv\lim_{|s|\to+\infty}\frac{f(t,s)}{s}$ uniformly on $[0,1]$, and the inequality is strict on some subset of positive measure in $(0,1)$; where $\lambda_1$ denotes the principle eigenvalue of $$\begin{gathered} \psi''(t)+\lambda\psi(t)=0,\quad t\in(0,1),\\ \psi(0)=0, \quad \psi(1)=\sum^{m-2}_{i=1}\alpha_i \psi(\eta_i); \end{gathered}\label{e1.4}$$ \item[(H2)] $0\leq\lim_{|s|\to 0} \frac{f(t,s)}{s}\equiv c(t)\leq \lambda_1$ uniformly on $[0,1]$, and all the inequalities are strict on some subset of positive measure in $(0,1)$; \item[(H3)] $f(t,s)s>0$ for all $t\in (0,1)$ and $s\neq 0$. \end{itemize} By applying the bifurcation theorem of L\'{o}pez-G\'{o}mez \cite[Theorem 6.4.3]{l1}, we will establish the following results. \begin{theorem} \label{thm1.1} Suppose that $f(t, u)$ satisfies {\rm(H1)--(H3)}. Then \eqref{e1.3} possesses at least one positive and one negative solution. \end{theorem} Similar result is obtained under the following assumptions. \begin{itemize} \item[(H1')] $\lambda_1\geq a(t)\equiv\lim_{| s|\to +\infty} \frac{f(t,s)}{s}\geq0$ uniformly on $[0,1]$, and all the inequalities are strict on some subset of positive measure in $(0,1)$, where $\lambda_1$ denotes the principle eigenvalue of \eqref{e1.4}; \item[(H2')] $\lim_{| s|\to0} \frac{f(t,s)}{s}\equiv c(t)\geq \lambda_1$ uniformly on $[0,1]$, and the inequality is strict on some subset of positive measure in $(0,1)$. \end{itemize} \begin{theorem} \label{thm1.2} Suppose that $f(t, u)$ satisfies {\rm (H1'), (H2'), (H3)}. Then \eqref{e1.3} possesses at least one positive and one negative solution. \end{theorem} The existence of constant sign solutions of \eqref{e1.3} is related to the eigenvalue problem $$\begin{gathered} u''(t)+ \mu f(t,u(t))=0,\quad t\in(0,1),\\ u(0)=0, \quad u(1)=\sum^{m-2}_{i=1}\alpha_i u(\eta_i), \end{gathered}\label{e1.5}$$ where $\mu>0$ is a parameter. Therefore, we will study the bifurcation phenomena for \eqref{e1.5} with crossing nonlinearity. Moreover, the bifurcation point of \eqref{e1.5} is related to the principle eigenvalues of the problem $$\begin{gathered} u''(t)+ \mu c(t)u(t)=0,\quad t\in(0,1),\\ u(0)=0, \quad u(1)=\sum^{m-2}_{i=1}\alpha_i u(\eta_i), \end{gathered}\label{e1.6}$$ it is well-known that there exists a principle eigenvalue $\mu_1(c(t))$ of \eqref{e1.6} (see \cite{z1}). The rest of the paper is organized as follows: in Section 2, we state some notations and preliminary results. In Section 3, we prove the main results. \section{Notation and preliminary results} To show the constant sign solutions of \eqref{e1.5}, we consider the operator equation $$u=\mu Tu.\label{e2.1}$$ This equations are usually called nonlinear eigenvalue problems. L\'{o}pez-G\'{o}mez \cite{l1} studied a nonlinear eigenvalue problem of the form $$u=\mu Tu+H(\mu,u),\label{e2.2}$$ where $H(\mu,u)=o(\|u\|)$ as $\|u\|\to 0$ uniformly for $\mu$ on a bounded interval, and $T$ is a linear completely continuous operator on a Banach space $X$. A solution of \eqref{e2.2} is a pair $(\mu,u)\in \mathbb{R}\times X$, which satisfies \eqref{e2.2}. The closure of the set nontrivial solutions of \eqref{e2.2} is denoted by $\mathcal{C}$. Let $\Sigma(T)$ denote the set of eigenvalues of linear operator $T$. L\'{o}pez-G\'{o}mez \cite{l1} established the following results. \begin{lemma}[{\cite[Theorem 6.4.3]{l1}}] \label{lem2.1} Assume $\Sigma(T)$ is discrete. Let $\mu_0\in\Sigma(T)$ such that $ind (I-\mu T,\theta)$ changes sign as $\mu$ crosses $\mu_0$, then each of the components $\mathcal{C}$ (denote the components of $S$ emanating of $(\mu, \theta)$ at $(\mu_0, \theta)$), satisfies $(\mu_0,\theta)\in\mathcal{C}$, and either \begin{itemize} \item[(i)] $\mathcal{C}$ is unbounded in $\mathbb{R}\times X$; \item[(ii)] there exist $\lambda_1\in\Sigma(T)\setminus\{\lambda_0\}$ such that $(\lambda_1,\theta)\in\mathcal{C}$; or \item[(iii)] $\mathcal{C}$ contains a point $$(\iota,y)\in\mathbb{R}\times (V\backslash\{\theta\}),$$ where $V$ is the complement of $\operatorname{span}\{\varphi_{\mu_0}\}$, $\varphi_{\mu_0}$ denotes the eigenfunction corresponding to eigenvalue $\mu_0$. \end{itemize} \end{lemma} \begin{lemma}[{\cite[Theorem 6.5.1]{l1}}] \label{lem2.2} Under the assumptions: \begin{itemize} \item[(A)] $X$ is an ordered Banach space, whose positive cone, denoted by $P$, is normal and has a nonempty interior; \item[(B)] The family $\Upsilon(\mu)$ has the special form $$\Upsilon(\mu)=I_X-\mu T,$$ where $T$ is a compact strongly positive operator, i.e., $T(P\backslash\{\theta\})\subset$int P; \item[(C)] The solutions of $u=\mu Tu+H(\mu,u)$ satisfy the strong maximum principle. \end{itemize} Then the following assertions are true: \begin{itemize} \item[(1)] $\operatorname{Spr}(T)$ is a simple eigenvalue of $T$, having a positive eigenfunction denoted by $\psi_0>0$, i.e., $\psi_0\in\operatorname{int} P$, and there is no other eigenvalue of $T$ with a positive eigenfunction; \item[(2)] For every $y\in$int P, the equation $$u-\mu Tu=y$$ has exactly one positive solution if $\mu<\frac{1}{\operatorname{Spr} (T)}$, whereas it does not admit a positive solution if $\mu\geq\frac{1}{\operatorname{Spr} (T)}$. \end{itemize} \end{lemma} \begin{lemma}[{cite[Theorem 2.5]{b1}}] \label{lem2.3} Assume $T: X\to X$ is a linear completely continuous operator, and 1 is not an eigenvalue of $T$, then $$\operatorname{ind}(I-T,\theta)=(-1)^\beta,$$ where $\beta$ is the sum of the algebraic multiplicities of the eigenvalues of $T$ large than 1, and $\beta=0$ if $T$ has no eigenvalue of this kind. \end{lemma} Let $Y$ be the space $C[0,1]$ with the norm $\|u\|_{\infty}=\max_{t\in[0,1]}|u(t)|$. Let $$E=\{u\in C^1[0,1]: u(0)=0, \; u(1)=\sum^{m-2}_{i=1}\alpha_i u(\eta_i)\}$$ with the norm $$\|u\|=\max_{t\in[0,1]}|u(t)|+\max_{t\in[0,1]}|u'(t)|.$$ Define $L:D(L)\to Y$ by setting $$Lu(t):=-u''(t),\quad t\in [0,1],\; u\in D(L),$$ where $$D(L)=\{u\in C^2[0,1]: u(0)=0, \; u(1)=\sum^{m-2}_{i=1}\alpha_i u(\eta_i) \}.$$ Then $L^{-1}: Y\to E$ is compact. Let $\mathbb{E}=\mathbb{R}\times E$ under the product topology. As in \cite{r1}, we add the point $\{(\mu,\infty)|\ \mu\in \mathbb{R}\}$ to our space $\mathbb{E}$. For any $u\in C^1[0,1]$, if $u(x_0)=0$, then $x_0$ is a simple zero of $u$ if $u'(x_0)\neq 0$. For $\nu\in\{+,-\}$, define: \begin{itemize} \item $S_1^\nu$ is the set of functions such that \begin{itemize} \item[(i)] $u(0) = 0$, $\nu u'(0)>0;$ \item[(ii)] $u$ has constant sign in $(0,1)$. \end{itemize} \item $T_1^\nu$ is the set of functions such that \begin{itemize} \item[(i)] $u(0) = 0$, $\nu u'(0)>0$ and $u'(1)\neq0$; \item[(ii)] $u'$ has exactly one simple zero point in $(0,1)$; \item[(iii)] $u$ has a zero strictly between each two consecutive zeros of $u'$. \end{itemize} \end{itemize} Obviously, if $u\in T_1^\nu$, then $u\in S_1^\nu$. The sets $T_1^\nu$ are disjoint and open in $E$, (see \cite[Remark 2.2]{r2}). Finally, let $\phi_1^\nu=\mathbb{R}\times T_1^\nu$. Furthermore, let $\zeta\in C([0,1]\times\mathbb{R})$ be such that $f(t,u)=c(t)u+\zeta(t,u)$ with $$\lim_{| u|\to0}\frac{\zeta(t,u)}{u}=0\quad \text{uniformly on } [0,1].$$ Let $$\bar{\zeta}(t,u)=\max_{0\leq | s|\leq u}| \zeta(t,s)|\quad \text{for } t\in[0,1].$$ Then $\bar{\zeta}$ is nondecreasing with respect to $u$ and $$\lim_{ u\to 0^+}\frac{\bar{\zeta}(t,u)}{u}=0.\label{e2.3}$$ From this equality, it follows that $$\frac{{\zeta}(t,u)}{\| u\|} \leq\frac{ \bar{\zeta}(t,| u|)}{\| u\|} \leq \frac{ \bar{\zeta}(t,\| u\|_\infty)}{\| u\|} \leq \frac{ \bar{\zeta}(t,\| u\|)}{\| u\|}\to0,\,\,\ \ \text{as}\,\, \| u\|\to 0$$ uniformly for $t\in[0,1]$. Let us study $$Lu-\mu c(t) u=\mu \zeta(t,u)\label{e2.4}$$ as a bifurcation problem from the trivial solution $u\equiv 0$. Equation \eqref{e2.4} can be converted to the equivalent equation \begin{align*} u(t)&= \int_0^1G(t,s)[\mu c(s) u(s)+\mu \zeta(s, u(s))]ds\\ &:=\mu L^{-1}[c(t) u(t)]+ \mu L^{-1}[\zeta(t, u(t))], \end{align*} where $G(t,s)$ denotes the Green's function of $Lu=0$. We note that $\|L^{-1}[ \zeta(t, u(t))]\|=o(\|u\|)$ for $u$ near $0$ in $E$. Since \begin{align*} \|L^{-1}[\zeta(t, u(t))]\| &= \max_{t\in[0,1]}|\int_0^1G(t,s) \zeta(s, u(s))ds| +\max_{t\in[0,1]}|\int_0^1G_t(t,s) \zeta(s,u(s))ds|\\ &\leq C \|\zeta(t, u(t))\|_{\infty}. \end{align*} \begin{lemma}[{\cite[Proposition 4.1]{r2}}] \label{lem2.4} If $(\mu,u)\in\mathbb{E}$ is a non-trivial solution of \eqref{e2.4}, then $u\in T_1^\nu$ for $\nu\in\{+,-\}$. \end{lemma} \begin{lemma} \label{lem2.5} For $\nu\in\{+,-\}$, there exists a continuum $\mathcal{C}_1^\nu\subset\mathbb{E}$ of solutions of \eqref{e2.4} with the properties: \begin{itemize} \item[(i)] $(\mu_1(c(t)),\theta)\in \mathcal{C}_1^\nu;$ \item[(ii)] $\mathcal{C}_1^\nu\backslash\{(\mu_1(c(t)),\theta)\}\subset \mathbb{R}\times T_1^\nu$; \item[(iii)] $\mathcal{C}_1^\nu$ is unbounded in $\mathbb{E}$, where $\mu_1(c(t))$ denotes the principle eigenvalue of \eqref{e1.6}. \end{itemize} \end{lemma} \begin{proof} From above, we know that problem \eqref{e2.4} is of the form considered in \cite{l1}, and satisfies the general hypotheses imposed in that paper. From \cite{z1}, we know that the principle eigenvalues of \eqref{e1.6} is simple. So for $\nu\in\{+,-\}$, combining Lemma 2.1 with Lemma 2.3, we know that there exists a continuum, $\mathcal{C}_1^\nu\subset\mathbb{E}$, of solutions of \eqref{e2.4} such that: \begin{itemize} \item[(a)] $\mathcal{C}_1^\nu$ is unbounded and $(\mu_1(c(t)),\theta)\in \mathcal{C}_1^\nu, \mathcal{C}_1^\nu\backslash\{(\mu_1(c(t)),\theta)\}\subset\mathbb{E}$, or \item[(b)] $(\mu_j(c(t)),\theta)\in \mathcal{C}_1^\nu$, where $j\in\mathbb{N},\mu_j(c(t))$ is another eigenvalue of \eqref{e1.6} if possible, or \item[(c)] $\mathcal{C}_1^\nu$ contains a point $$(\iota,y)\in\mathbb{R}\times (V\backslash\{\theta\}),$$ where $V$ is the complement of $\operatorname{span}\{\varphi_{1}\}$, $\varphi_{1}$ denotes the eigenfunction corresponding to principle eigenvalue $\mu_1(c(t))$. \end{itemize} We finally prove that the first choice (a) is the only possibility. In fact, all functions belong to the continuum set $\mathcal{C}_1^\nu$ are constant sign, this implies that it is impossible to exist $(\mu_j(c(t)),\theta) \in \mathcal{C}_1^\nu, j\in\mathbb{N},j\neq 1$, where $\mu_j(c(t))$ is another eigenvalue of \eqref{e1.6} if possible. If this happened, it will be contracted with the definition of $S_1^\nu$. Next, we will prove (c) is impossible, suppose (c) occurs, without loss of generality, suppose there exists a point $(\iota,y)\in\mathbb{R}\times (V\backslash\{\theta\})\cap \mathcal{C}_1^+$. Define $$P=\{u\in C^1[0,1]: u(t)\geq 0, \; t\in [0,1]\},$$ then $P$ is a normal cone and has a nonempty interior, and $\mathcal{C}_1^+\backslash\{(\mu_1(c(t)),\theta)\}\subset \operatorname{int} P$. Note that as the complement $V$ of Span$\{\varphi_1\}$ in $E$, we can take $$V:=R[c(t)I_E-\frac{1}{\mu_1(c(t))}L].$$ Thus, for this choice of $V$, if the component $\mathcal{C}_1^+$ contains a point $$(\iota,y)\in\mathbb{R}\times (V\backslash\{\theta\})\cap \mathcal{C}_1^+.$$ Then there exists $u\in E$ for which $$c(t)u-\frac{1}{\mu_1(c(t))}Lu=y>0,\quad \text{in } (0,1).$$ Thus, for each sufficiently large $\eta>0$, we have that $c(t)u+\eta \varphi_1(t)>0$ in $(0,1)$ and $$c(t)u+\eta c(t)\varphi_1(t)-\frac{1}{\mu_1(c(t))} L(u+\eta \varphi_1)=y>0\quad \text{in } (0,1).$$ Hence, by Lemma 2.2, we have $$\operatorname{Spr}(\frac{1}{\mu_1(c(t))} L)<1,$$ which is impossible. since $Spr(L)=\mu_1(c(t))$. \end{proof} \section{Proof of main results} \begin{proof}[Proof of Theorem 1.1] Theorem 1.2 is proved in similar manner. It is clear that any solution of \eqref{e2.4} of the form $(1,u)$ yields a solution $u$ of \eqref{e1.3}. We will show $\mathcal{C}_1^\nu$ crosses the hyperplane $\{1\}\times E$ in $\mathbb{R}\times E$. By $\mu_1(c(t))$ being strict decreasing with respect to $c(t)$ (see \cite{l2}), where $\mu_1(c(t))$ is the principle eigenvalue of \eqref{e1.6}, we have $\mu_1(c(t)) > \mu_1(\lambda_1) = 1$. Let $(\mu_n, u_n) \in \mathcal{C}_{1}^\nu$ with $u_n\not\equiv 0$ satisfies $\mu_n+\| u_n\|\to+\infty.$ We note that $\mu_n >0$ for all $n \in \mathbb{N}$, since $(0,\theta)$ is the only solution of \eqref{e2.4} for $\mu = 0$ and $\mathcal{C}_{1}^\nu\cap(\{0\}\times E)=\emptyset$. \textbf{Step 1:} We show that if there exists a constant $M>0$, such that $\mu_n\subset(0,M]$ for $n\in\mathbb{N}$ large enough, then $\mathcal{C}_1^\nu$ crosses the hyperplane $\{1\}\times E$ in $\mathbb{R}\times E$. In this case it follows that $\|u_n\|\to\infty$. Let $\xi\in C([0,1]\times\mathbb{R})$ be such that $f(t,u)=a(t)u+\xi(t,u)$ with $$\label{e3.1} \lim_{| u|\to+\infty}\frac{\xi(t,u)}{u}=0\quad \text{uniformly on } [0,1].$$ We divide the equation $$Lu_n-\mu_n a(t) u_n=\mu_n \xi(t,u_n)\label{e3.2}$$ by $\|u_n\|$ and set $\bar{u}_n=\frac{u_n}{\|u_n\|}$. Since $\bar{u}_n$ is bounded in $C^2[0,1]$, after taking a subsequence if necessary, we have that $\bar{u}_n\to\bar{u}$ for some $\bar{u}\in E$ with $\|\bar{u}\|=1$. By \eqref{e3.1}, using the similar proof of \eqref{e2.3}, we have that $\lim_{n\to+\infty}\frac{ \xi(t,u_n(t))}{\| u_n\|}=0\quad \text{in} Y.$ Thus, we obtain $-\bar{u}''-\overline{\mu}(a(t))a(t) \bar{u}=0,$ where $\overline{\mu}(a(t))=\underset{n\to+\infty}\lim \mu_n$. It is clear that $\overline{u}\in \overline{\mathcal{C}_{1}^\nu}\subseteq \mathcal{C}_{1}^\nu$, since $\mathcal{C}_{1}^\nu$ is closed in $\mathbb{R}\times E$. Therefore, $\overline{\mu}(a(t))$ is the principle eigenvalue of \eqref{e1.6} corresponding to weight function $a(t)$. By the strict decreasing of $\overline{\mu}(a(t))$ with respect to $a(t)$ (see \cite{l2}), we have $\overline{\mu}(a(t)) < \overline{\mu}(\lambda_1) = 1$. Therefore, $\mathcal{C}_{1}^\nu$ crosses the hyperplane $\{1\}\times E$ in $\mathbb{R}\times E$. \textbf{Step 2:} We show that there exists a constant $M$ such that $\mu_n \in(0,M]$ for $n\in \mathbb{N}$ large enough. On the contrary, we suppose that $\lim_{n\to +\infty}\mu_n=+\infty$. On the other hand, we note that $-u_n''=\mu_n\frac{f(t,u_n)}{u_n}u_n.$ We have $\mu_n\frac{f(t,u_n)}{u_n}>\lambda_1$ for $n$ large enough and all $t\in (0,1]$. We get $u_n$ must change its sign in $(0,1)$ for $n$ large enough, which contradicts the fact that $u_n\in T_1^\nu$. Therefore, $\mu_n\leq M$ for some constant positive $M$ and $n\in \mathbb{N}$ sufficiently large. \end{proof} \begin{thebibliography}{00} \bibitem{a1} Y. An, R. 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