\documentclass[reqno]{amsart} \usepackage{graphicx} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 94, pp. 1--12.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/94\hfil Potential Landesman-Lazer type condition] {Potential Landesman-Lazer type conditions and the Fu\v{c}\'{i}k spectrum} \author[P. Tomiczek\hfil EJDE-2005/94\hfilneg] {Petr Tomiczek} \address{Petr Tomiczek \hfill\break Department of Mathematics, University of West Bohemia\\ Univerzitn\'{\i} 22, 306 14 Plze\v{n}, Czech Republic} \email{tomiczek@kma.zcu.cz} \date{} \thanks{Submitted November 26, 2004. Published August 29, 2005.} \thanks{Partially supported by the Grant Agency of Czech Republic, MSM 4977751301.} \subjclass[2000]{35J70, 58E05, 49B27} \keywords{Resonance; eigenvalue; jumping nonlinearities; Fucik spectrum} \begin{abstract} We prove the existence of solutions to the nonlinear problem $$\displaylines{ u''(x)+\lambda_+ u^+(x)-\lambda_- u^-(x)+g(x,u(x))=f(x)\,,\quad x\in (0,\pi)\,,\cr u(0)=u(\pi)=0 }$$ where the point $[\lambda_+,\lambda_-]$ is a point of the Fu\v{c}\'{\i}k spectrum and the nonlinearity $g(x,u(x))$ satisfies a potential Landesman-Lazer type condition. We use a variational method based on the generalization of the Saddle Point Theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} We investigate the existence of solutions for the nonlinear boundary-value problem $$\label{1.1} \begin{gathered} u''(x)+ \lambda_+ u^+(x)-\lambda_- u^-(x)+g(x,u(x))=f(x) \,,\quad x\in (0,\pi)\,,\\ u(0)=u(\pi)=0\,. \end{gathered}$$ Here $u^{\pm}=\max\{\pm u,0\}$, $\lambda_+,\lambda_- \in \mathbb{R}$, the nonlinearity $g\,\colon( 0,\pi )\times\mathbb{R} \mapsto \mathbb{R}$ is a Caratheodory function and $f\in L^1(0,\pi)$. For $g\equiv 0$ and $f\equiv 0$ problem (\ref{1.1}) becomes $$\label{1.2} \begin{gathered} u''(x)+ \lambda_+ u^+(x)-\lambda_- u^-(x)=0\,,\quad x\in (0,\pi)\,,\\ u(0)=u(\pi)=0\,. \end{gathered}$$ We define $\Sigma=\{[\lambda_+,\lambda_-]\in\mathbb{R}^2 : \mbox{(\ref{1.2}) has a nontrivial solution}\}$. This set is called the Fu\v{c}\'{\i}k spectrum (see \cite{lit2}), and can be expressed as $\Sigma=\bigcup_{j=1}^{\infty}\Sigma_j$ where \begin{gather*} \Sigma_1 = \bigl\{[\lambda_+ ,\lambda_- ] \in \mathbb{R}^2: \; (\lambda_+ -1)(\lambda_- -1) = 0 \bigr\}\,, \\ \Sigma_{2i} = \bigl\{[\lambda_+,\lambda_-] \in \mathbb{R}^2: \; i\, \Bigl({1\over {\sqrt {\lambda_+}}}+{1\over {\sqrt {\lambda_-}}} \Bigr) = 1 \bigr\}\,, \\ \Sigma_{2i+1} = \Sigma_{2i+1,1} \cup \Sigma_{2i+1,2} \quad \mbox{where} \\ \Sigma_{2i+1,1} = \bigl\{[\lambda_+,\lambda_-] \in \mathbb{R}^2: \; i\,\Bigl({1\over {\sqrt {\lambda_+}}}+ {1\over {\sqrt {\lambda_-}}} \Bigr)+ {1\over {\sqrt {\lambda_+}}}= 1\bigr\}\,,\\ \Sigma_{2i+1,2} = \bigl\{[\lambda_+,\lambda_-] \in \mathbb{R}^2: \; i\,\Bigl({1\over {\sqrt {\lambda_+}}}+ {1\over {\sqrt {\lambda_-}}}\Bigr)+ {1\over {\sqrt {\lambda_-}}}= 1 \bigr\} \,. \end{gather*} We suppose that $$\label{1.3} \begin{gathered} \ [\lambda_+,\lambda_-]\in \Sigma_m\,, \mbox{ if m\in\mathbb{N} is even} \\ \ [\lambda_+,\lambda_-]\in \Sigma_{m2}\,, \mbox{ if m\in\mathbb{N} is odd} \\ \mbox{and }\lambda_-<\lambda_+ <(m+1)^2\,. \end{gathered}$$ \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig1} \end{center} %\includegraphics[width=16cm,height=21.5cm]{fucikclanek} \caption{Fu\v{c}\'{i}k spectrum} \end{figure} \begin{remark} \label{rem1} \rm Assuming that $(m+1)^2>\lambda_+>\lambda_-$, if $[\lambda_+,\lambda_-]\in \Sigma_m$, $m\in\mathbb{N}$, then $\lambda_- > (m-1)^2$. \end{remark} We define the potential of the nonlinearity $g$ as $$G(x,s)=\int_0^{s}g(x,t)\,dt$$ and $$G_+(x)=\liminf_{s\to +\infty}{{G(x,s)}\over s}\,,\quad G_-(x)=\limsup_{s\to -\infty}{{G(x,s)}\over s}\,.$$ We denote by $\varphi_m$ a nontrivial solution of (\ref{1.2}) corresponding to $[\lambda_+,\lambda_-]$ (see Remark \ref{rem2}). We assume that for any $\varphi_m$ the following potential Landesman-Lazer type condition holds: $$\label{1.4} \int_0^{\pi}f(x)\varphi_m(x) \,dx < \int_0^{\pi}\bigl[ G_+(x)(\varphi_m(x))^+ - G_-(x)(\varphi_m(x))^-\bigr] \,dx\,.$$ We suppose that the nonlinearity $g$ is bounded, i.e. there exists $p(x)\in L^1(0,\pi)$ such that $$\label{1.5} |g(x,s)|\le p(x)\quad \mbox{for a.e.}\; x\in(0,\pi)\,,\; \forall \,s\in\mathbb{R}$$ and we prove the solvability of (\ref{1.1}) in Theorem (\ref{thm2}) below. This article is inspired by a result in \cite{lit3} where the author studies the case when $g(x,s)/s$ lies (in some sense) between $\Sigma_1$ and $\Sigma_2$ and by a result in \cite{lit1} with the classical Landesman-Lazer type condition \cite[Corollary 2]{lit1}. \begin{remark} \label{rem2} \rm First we note that if $m$ is even then two different functions $\varphi_{m1},\varphi_{m2}$ of norm 1 correspond to the point $[\lambda_+,\lambda_-]\in \Sigma_m$. For example for $m=2$, $\lambda_+>\lambda_-$ we have $$\varphi_{21}(x)=\begin{cases} k_1\sqrt{\lambda_-}\sin(\sqrt{\lambda_+}x), & x\in\langle 0,\pi/\sqrt{\lambda_+}\rangle ,\\ -k_1\sqrt{\lambda_+}\sin(\sqrt{\lambda_-}(x-\pi/\sqrt{\lambda_+})), & x\in\langle\pi/\sqrt{\lambda_+},\pi\rangle , \end{cases}$$ where $k_1>0$, and $$\varphi_{22}(x)=\begin{cases} -k_2\sqrt{\lambda_+}\sin(\sqrt{\lambda_-}x), & x\in\langle 0,\pi/\sqrt{\lambda_-}\rangle ,\\ k_2\sqrt{\lambda_-}\sin(\sqrt{\lambda_+}(x-\pi/\sqrt{\lambda_-})), & x\in\langle\pi/\sqrt{\lambda_-},\pi\rangle , \end{cases}$$ where $k_2>0$. For $\lambda_+=\lambda_-=4$ we set $\varphi_{21}(x)=k_1\sin 2x$ and $\varphi_{22}(x)=-k_2\sin 2x$, where $k_1,k_2>0$. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig2} \end{center} %\includegraphics[width=17cm,height=19cm]{varphiclanek} \caption{Solutions corresponding to $\Sigma_2$} \end{figure} If $m$ is odd, then $\Sigma_m=\Sigma_{m1}\cup \Sigma_{m2}$ and it corresponds only one function $\varphi_{m1}$ od norm 1 to the point $[\lambda_+',\lambda_-'] \in \Sigma_{m1}$,\, one function $\varphi_{m2}$ of norm 1 to the point $[\lambda_+,\lambda_-] \in \Sigma_{m2}$, respectively. For $m=3$, $\lambda_+'>\lambda_-'$, $\lambda_+>\lambda_-$ we have \begin{align*} &\varphi_{31}(x)\\ &=\begin{cases} k_1\sqrt{\lambda_-'}\sin(\sqrt{\lambda_+'}x), & x\in\langle 0,\pi/\sqrt{\lambda_+'}\rangle ,\\ -k_1\sqrt{\lambda_+'}\sin(\sqrt{\lambda_-'}(x-\pi/\sqrt{\lambda_+'})), & x\in\langle\pi/\sqrt{\lambda_+'},\pi/\sqrt{\lambda_+'}+\pi/\sqrt{\lambda_-'}\rangle ,\\ k_1\sqrt{\lambda_-'}\sin(\sqrt{\lambda_+'}(x-\pi/\sqrt{\lambda_+'}-\pi/\sqrt{\lambda_-'})), & x\in\langle\pi/\sqrt{\lambda_+'}+\pi/\sqrt{\lambda_-'},\pi\rangle , \end{cases} \end{align*} where $k_1>0$. \begin{align*} &\varphi_{32}(x)\\ &= \begin{cases} -k_2\sqrt{\lambda_+}\sin(\sqrt{\lambda_-}x), & x\in\langle 0,\pi/\sqrt{\lambda_-}\rangle ,\\ k_2\sqrt{\lambda_-}\sin(\sqrt{\lambda_+}(x-\pi/\sqrt{\lambda_-})), & x\in\langle\pi/\sqrt{\lambda_-},\pi/\sqrt{\lambda_-}+\pi/\sqrt{\lambda_+}\rangle ,\\ -k_2\sqrt{\lambda_+}\sin(\sqrt{\lambda_-}(x-\pi/\sqrt{\lambda_-}-\pi/\sqrt{\lambda_+})), & x\in\langle\pi/\sqrt{\lambda_-}+\pi/\sqrt{\lambda_+},\pi\rangle, \end{cases} \end{align*} where $k_2>0$. \begin{figure}[ht] \begin{center} \includegraphics[width=0.7\textwidth]{fig3} \end{center} %\includegraphics[width=17cm,height=19cm]{varphiclanek3} \caption{Solutions corresponding to $\Sigma_3$} \end{figure} For $\lambda_+=\lambda_-=m^2$ we set $\varphi_{m1}(x)=k_1\sin mx$, and $\varphi_{m2}(x)=-k_2\sin mx$, where $k_1,k_2>0$ and from the condition (\ref{1.4}) we obtain $$\int_0^{\pi}f(x)\sin mx \,dx < \int_0^{\pi}\bigl[ G_+(x)(\sin mx)^+ - G_-(x)(\sin mx)^-\bigr] \,dx$$ and $$\int_0^{\pi}f(x)(-\sin mx) \,dx < \int_0^{\pi}\bigl[ G_+(x)(-\sin mx)^+ - G_-(x)(-\sin mx)^-\bigr] \,dx\,.$$ Hence it follows \begin{aligned} &\int_0^{\pi}\bigl[ G_-(x)(\sin mx)^+ - G_+(x)(\sin mx)^-\bigr] \,dx \\ &<\int_0^{\pi}f(x)\sin mx \,dx < \int_0^{\pi}\bigl[ G_+(x)(\sin mx)^+ - G_-(x)(\sin mx)^-\bigr] \,dx\,. \end{aligned} We obtained the potential Landesman-Lazer type condition (see \cite{lit6}). \end{remark} \begin{remark} \label{rem3} \rm We have $$\langle v, \sin mx\rangle=\int_0^{\pi} v'(x)(\sin mx)'\,dx= m^2\int_0^{\pi} v(x)\sin mx\,dx \quad \forall\, v\in H$$ ($H$ is a Sobolev space defined below). Since and from the definition of the functions $\varphi_{m1},\varphi_{m2}$ (see remark \ref{rem2}) it follows $$\label{1.6} \langle \varphi_{m1}, \sin mx\rangle>0\,\quad \mbox{and}\quad \, \langle \varphi_{m2}, \sin mx\rangle<0\,.$$ \end{remark} \section{Preliminaries} \subsection*{Notation} We shall use the classical spaces ${C}(0,\pi)$, $L^p(0,\pi)$ of continuous and measurable real-valued functions whose $p$-th power of the absolute value is Lebesgue integrable, respectively. $\!H$ is the Sobolev space of absolutely continuous functions $u \colon (0,\pi)\to \mathbb{R}$ such that $u'\!\in\!L^2(0,\pi)$ and $u(0)\!=\!u(\pi)\!=\!0.$ We denote by the symbols $\| \cdot \|$, and $\| \cdot \|_2$ the norm in $H$, and in $L^2(0,\pi)$, respectively. We denote $\langle\,\cdot,\cdot\,\rangle$ the pairing in the space $H$. By a solution of (\ref{1.1}) we mean a function $u\in {C}^1(0,\pi)$ such that $u'$ is absolutely continuous, $u$ satisfies the boundary conditions and the equations (\ref{1.1}) holds a.e. in $(0,\pi)$. Let $I\,\colon\,H\to \mathbb{R}$ be a functional such that $I\in {C}^1(H,\mathbb{R})$ (continuously differentiable). We say that $u$ is a critical point of $I$, if $$\langle I'(u), v\rangle = 0 \quad \mbox{for all } v\in H\,.$$ We say that $\gamma$ is a critical value of $I$, if there is $u_0\in H$ such that $I(u_0)=\gamma$ and $I'(u_0)=0$. We say that $I$ satisfies Palais-Smale condition (PS) if every sequence $(u_n)$ for which $I(u_n)$ is bounded in $H$ and $I'(u_n)\to 0$ (as $n\to \infty)$ possesses a~convergent subsequence. We study (\ref{1.1}) by the use of a variational method. More precisely, we look for critical points of the functional $I:H\to \mathbb{R}$, which is defined by $$\label{2.1} I(u)={1\over 2}\int_0^{\pi }\bigl[(u')^2-\lambda_+(u^+)^2- \lambda_-(u^-)^2 \bigr]\,dx - \int_0^{\pi }\bigl[ G(x,u)-fu\bigr] \,dx\,.$$ Every critical point $u\in H$ of the functional $I$ satisfies $$\int_0^{\pi}\bigl[ u'v'-(\lambda_+u^+ - \lambda_-u^-)v \bigr] \,dx - \int_0^{\pi }\bigl[g(x,u)v-fv\bigr] \,dx=0\quad \mbox{for all } v\in H\,.$$ Then $u$ is also a weak solution of (\ref{1.1}) and vice versa. The usual regularity argument for ODE yields immediately (see Fu\v{c}\'{\i}k \cite{lit2}) that any weak solution of (\ref{1.1}) is also a solution in the sense mentioned above. We will use the following variant of the Saddle Point Theorem (see \cite{lit4}) which is proved in Struwe \cite[Theorem 8.4]{lit5}. \begin{theorem} \label{thm1} Let $S$ be a closed subset in $H$ and $Q$ a bounded subset in $H$ with boundary $\partial Q$. Set $\Gamma = \{h : \,h\in {\bf C}(H,H), \,h(u)=u\,\mbox{ on } \partial Q\}$. Suppose $I\in {C}^1(H,\mathbb{R})$ and \begin{itemize} \item[$(i)$] $S \cap \partial Q = \emptyset$, \item[$(ii)$] $S \cap h(Q) \not= \emptyset$, for every $h\in\Gamma$, \item[$(iii)$] there are constants $\mu,\nu$ such that\, $\mu=\inf_{u\in S} I(u) > \sup_{u\in \partial Q} I(u)=\nu$, \item[$(iv)$] $I$ satisfies Palais-Smale condition. \end{itemize} Then the number $$\gamma=\inf_{h\in \Gamma}\sup_{u\in Q} I(h(u))$$ defines a critical value $\gamma>\nu$ of $I$. \end{theorem} We say that $S$ and $\partial Q$ {\it link} if they satisfy conditions (i), (ii) of the theorem above. We denote the first integral in the functional $I$ by $$J(u)=\int_0^\pi \bigl[(u')^2-\lambda_+(u^+)^2- \lambda_-(u^-)^2\bigr] \,dx\,.$$ Now we present a few results needed later. \begin{lemma} \label{l1} Let $\varphi$ be a solution of (\ref{1.2}) with $[\lambda_+,\lambda_-]\, \in \Sigma$, $\lambda_+\ge\lambda_-$. We put $u=a\varphi+ w$, $a\ge 0$, $w\in H$. Then the following relation holds $$\label{2.2} \int_0^{\pi }\bigl[(w')^2-\lambda_+ w^2 \bigr] \,dx \le J(u)\le \int_0^{\pi }\bigl[(w')^2-\lambda_- w^2 \bigr] \,dx\,.$$ \end{lemma} \begin{proof} We prove only the right inequality in (\ref{2.2}), the proof of the left inequality is similar. Since $\varphi$ is a solution of (\ref{1.2}) we have $$\label{2.3} \int_0^\pi \varphi'w' \,dx=\int_0^\pi \bigl[\lambda_+ \varphi^+ w -\lambda_-\varphi^- w \bigr] \,dx\quad \mbox{for } w\in H$$ and $$\label{2.4} \int_0^\pi (\varphi')^2 \,dx=\int_0^\pi \bigl[\lambda_+(\varphi^+)^2+\lambda_-(\varphi^-)^2\bigr] \,dx\,.$$ By (\ref{2.3}) and (\ref{2.4}), we obtain \label{2.5} \begin{aligned} J(u) &= \int_0^{\pi} \Bigl[((a\varphi+w)')^2 -\lambda_+ ((a\varphi+w)^+)^2-\lambda_-((a\varphi+w)^-)^2\Bigr]\,dx \\ &= \int_0^{\pi} \Bigl[(a\varphi')^2 +2a\varphi'w'+(w')^2- (\lambda_+-\lambda_-)((a\varphi+w)^+)^2 \\ &\quad -\lambda_-(a\varphi+w)^2\Bigr]\,dx \\ &= \int_0^{\pi} \Bigl[(\lambda_+-\lambda_-)(a\varphi^+)^2 +\lambda_-(a\varphi)^2+2a((\lambda_+-\lambda_-)\varphi^++\lambda_-\varphi)w \\ &\quad +(w')^2 -(\lambda_+-\lambda_-)((a\varphi+w)^+)^2-\lambda_-((a\varphi)^2 +2a\varphi w+w^2)\Bigr]\,dx \\ &= \int_0^{\pi} \Bigl\{(\lambda_+-\lambda_-)\bigl[(a\varphi^+)^2 +2a\varphi^+w-((a\varphi+w)^+)^2\bigr] \\ & \quad +(w')^2-\lambda_-w^2\Bigr\}\,dx\,. \end{aligned} For the function $(a\varphi^+)^2+2a\varphi^+w-((a\varphi+w)^+)^2$ in the last integral in (\ref{2.5}) we have \begin{align*} &(a\varphi^+)^2+2a\varphi^+w-((a\varphi+w)^+)^2\\ &=\begin{cases} -((a\varphi+w)^+)^2\le 0 & \varphi<0 \\ -w^2\le 0 & \varphi\ge0, a\varphi+w\ge 0 \\ a\varphi^+(a\varphi^++w+w)\le 0 & \varphi\ge0, a\varphi+w< 0\,. \end{cases} \end{align*} By the assumption $\lambda_+\ge\lambda_-$, we obtain the assertion of the Lemma~\ref{l1}. \end{proof} \begin{remark}\label{rem4} \rm It follows from the previous proof that we obtain the equality $$J(u)=\int_0^\pi \bigl[(w')^2-\lambda_-w^2\bigr] \,dx$$ in (\ref{2.2}) if $a\varphi+w\le 0$ when $\varphi<0$, and $w= 0$ when $\varphi\ge0$. Consequently, if the equality holds and if $w$ in span$\{\sin x, \dots, \sin kx\},\, k\in\mathbb{N}$, then $w=0$. \end{remark} \section{Main result} \begin{theorem} \label{thm2} Under the assumptions (\ref{1.3}), (\ref{1.4}), and (\ref{1.5}), Problem (\ref{1.1}) has at least one solution in $H$. \end{theorem} \begin{proof} First we suppose that $m$ is even. We shall prove that the functional $I$ defined by (\ref{2.1}) satisfies the assumptions in Theorem~\ref{thm1}. Let $\varphi_{m1}, \varphi_{m2}$ be the normalized solutions of (\ref{1.2}) described above (see Remark \ref{rem2})\,. Let $H^-$ be the subspace of $H$ spanned by functions $\sin x, \dots, \sin (m-1)x$. We define $V\equiv V_1\cup V_2$ where \begin{gather*} V_1=\{u\in H : u=a_1\varphi_{m1}+w,\; 0\le a_1, \; w\in H^-\}, \\ V_2=\{u\in H : u=a_2\varphi_{m2}+w,\; 0\le a_2, \; w\in H^-\}. \end{gather*} Let $K>0$, $L>0$ then we define $Q\equiv Q_1\cup Q_2$ where \begin{gather*} Q_1=\{u\in V_1 : 0\le a_1\le K,\; \|w\|\le L\},\\ Q_2=\{u\in V_2 : 0\le a_2\le K,\; \|w\|\le L\}. \end{gather*} Let $S$ be the subspace of $H$ spanned by functions $\sin(m+1)x, \sin(m+2)x, \dots$. Next, we verify the assumptions of Theorem \ref{thm1}. We see that $S$ is a closed subset in $H$ and $Q$ is a bounded subset in $H$. \noindent(i) Firstly we note that for $z\in H^-\oplus S$ we have $\langle z, \sin mx\rangle=0$. We suppose for contradiction that there is $u\in \partial Q\cap S$. Then $$0\stackrel{u\in S}=\langle u, \sin mx\rangle \stackrel{u\in \partial Q}{=} \langle a_i\varphi_{mi}+w,\sin mx\rangle \stackrel{w\in H^-}{=} a_i\langle \varphi_{mi},\sin mx\rangle\,$$ $i=1,2$. From previous equalities and inequalities (\ref{1.6}) it follows that $a_i=0$, $i=1,2$ and $u=w$. For $u=w\in\partial Q$ we have $\|u\|=L>0$ and we obtain a contradiction with $u\in H^-\cap S =\{o\}$. \noindent(ii) We prove that $H=V\oplus S$. We can write a function $h\in H$ in the form \begin{align*} h&=\sum_{i=1}^{m-1} b_i \sin ix + b_m\sin mx +\sum_{i=m+1}^{\infty}\! b_i \sin ix\\ &= \overline{h} + b_m\sin mx + \widetilde{h},\, b_i\in\mathbb{R}, \end{align*} $i\in \mathbb{N}$. The inequalities (\ref{1.6}) yield that there are constants $b_{m1}, b_{m2}>0$ such that $\sin mx=b_{m1}(\varphi_{m1}-\overline{\varphi}_{m1}-\widetilde{\varphi}_{m1})$ and $-\sin mx=b_{m2}(\varphi_{m2}-\overline{\varphi}_{m2}-\widetilde{\varphi}_{m2})$. Hence we have for $b_m\ge 0$, \begin{align*} h&= \overline{h} + b_mb_{m1}(\varphi_{m1}-\overline{\varphi}_{m1} -\widetilde{\varphi}_{m1}) + \widetilde{h}\\ &= \underbrace{(\overbrace{\overline{h} -b_mb_{m1}\overline{\varphi}_{m1}}^{\in H^-} + \overbrace{b_mb_{m1}}^{\ge 0} \varphi_{m1})}_{\in V} +\underbrace{(\widetilde{h}-b_mb_{m1}\widetilde{\varphi}_{m1})}_{\in S}\,. \end{align*} Similarly for $b_m\le 0$, \begin{align*} h&=\overline{h} + |b_m|b_{m2}(\varphi_{m2}-\overline{\varphi}_{m2} -\widetilde{\varphi}_{m2}) + \widetilde{h}\\ &= \underbrace{(\overbrace{\overline{h}-|b_m|b_{m2} \overline{\varphi}_{m2}}^{\in H^-} + \overbrace{|b_m|b_{m2}}^{\ge 0}\varphi_{m2})}_{\in V} +\underbrace{(\widetilde{h}-|b_m|b_{m2}\widetilde{\varphi}_{m2})}_{\in S}\,. \end{align*} We proved that $H$ is spanned by $V$ and $S$. The proof of the assumption $S\cap h(Q)\not=\emptyset\quad \forall h\in\Gamma$ is similar to the proof in \cite[example 8.2]{lit5}. Let $\pi \colon H\to V$ be the continuous projection of $H$ onto $V$. We have to show that $0\in \pi(h(Q))$. For $t\in[0,1]$, $u\in Q$ we define $$h_t(u)=t \pi(h(u))+(1-t)u\,.$$ The function $h_t$ defines a homotopy of $h_0= id$ with $h_1=\pi\circ h$. Moreover, $h_t|\partial Q=\mathop{\rm id}$ for all $t\in[0,1]$. Hence the topological degree $\deg(h_t,Q,0)$ is well-defined and by homotopy invariance we have $$\deg(\pi\circ h,Q,0)=\deg(\mathrm{id},Q,0)=1\,.$$ Hence $0\in \pi(h(Q))$, as needed. \noindent (iii) Firstly, we note that by assumption (\ref{1.5}), one has $$\label{3.1} \lim_{\|u\|\to \infty} \int_0^{\pi } \frac{G(x,u) - fu}{\|u\|^2} \,dx = 0\,.$$ First we show that the infimum of functional $I$ on the set $S$ is a real number. We prove for this that $$\label{3.2} \lim_{\|u\|\to \infty} I(u)= \infty\quad\mbox{for all } u\in S$$ and $I$ is bounded on bounded sets. Because of the compact imbedding of $H$ into ${C}(0,\pi)$ $(\| u \|_{{C}(0,\pi)}\leq c_1\| u \|)$, and of $H$ into L$^2(0,\pi)$ $(\| u \|_2\leq c_2\| u \|)$, and the assumption (\ref{1.5}) one has \begin{align*} I(u)&=\frac{1}{2}\int_0^{\pi }\bigl[(u')^2- \lambda_+(u^+)^2-\lambda_-(u^-)^2\bigr]\,dx -\int_0^{\pi } \bigl[G(x,u) - fu \bigr] \,dx \\ &\le \frac{1}{2}\left( \|u\|^2+\lambda_+ \|u^+\|_2^2+ \lambda_- \|u^-\|_2^2 \right) + \int_0^{\pi } \bigl[\,(|p|+|f|) |u|\bigr] \,dx \\ &\le \frac{1}{2}\left( \|u\|^2+\lambda_+ c_2\|u^+\|^2+ \lambda_- c_2 \|u^-\|^2 \right) + \left( \|p\|_1+\|f\|_1 \right) c_1\|u\| \,. \nonumber \end{align*} Hence $I$ is bounded on bounded subsets of S. To prove (\ref{3.2}), we argue by contradiction. We suppose that there is a sequence $(u_n)\subset S$ such that $\|u_n\|\to \infty$ and a constant $c_3$ satisfying $$\label{3.3} \liminf_{n\to \infty} I(u_n)\le c_3\,.$$ For $u\in S$ the following relation holds $$\label{3.4} \| u\|^2=\int_0^{\pi } (u')^2 \,dx \ge (m+1)^2\int_0^{\pi } u^2 \,dx= (m+1)^2 \| u\|_2^2\,.$$ The definition of $I$, (\ref{3.1}), (\ref{3.3}) and (\ref{3.4}) yield $$\label{3.5} 0\ge \liminf_{n\to \infty} \frac{I(u_n)}{\|u_n\|^2}\ge \liminf_{n\to \infty} \frac{((m+1)^2-\lambda_+)\|u_n^+\|_2^2+((m+1)^2-\lambda_-)\|u_n^-\|_2^2}{2\|u_n\|^2}\,.$$ It follows from (\ref{3.5}) and (\ref{1.3}) that $\| u_n\|_2^2/ \| u_n \|^2 \to 0$ and from the definition of $I$ and (\ref{3.1}) we have $$\liminf_{\| u_n \|\to \infty} \frac{I(u_n)}{\| u_n \|^2} = \frac{1}{2}$$ a contradiction to (\ref{3.5}). We proved that there is $\mu\in\mathbb{R}$ such that $\inf_{u\in S} I(u) =\mu$. Second we estimate the value $I(u)$ for $u\in\partial Q$. We remark that $u\in\partial Q$ can be either of the form $K\varphi_m+w$, with $\|w\|\le L$ or of the form $a_i\varphi_{mi}$, with $0\le a_i\le K$, $\|w\|=L$ ($i=1,2$). We prove that $$\label{3.6} \sup_{(K+L) \to \infty}I(K\varphi_m+w) =\sup_{\| u \| \to \infty}I(u)=-\infty\, \quad\mbox{for}\quad u\in\partial Q\,.$$ For (\ref{3.6}), we argue by contradiction. Suppose that (\ref{3.6}) is not true then there are a sequence $(u_n)\subset \partial Q$ such that $\| u_n \| \to \infty$ and a constant $c_4$ satisfying $$\label{3.7} \limsup_{n\to \infty} I(u_n)\ge c_4\,.$$ Hence, it follows $$\label{3.8} \limsup_{n\to \infty} \Bigl[ \frac{1}{2}\int_0^{\pi} \frac{(u_n')^2- \lambda_+(u_n^+)^2-\lambda_-(u_n^-)^2}{\|u_n\|^2}\,dx -\int_0^{\pi } \frac{G(x,u_n) - fu_n}{\|u_n\|^2} \,dx \Bigr]\ge 0\,.$$ Set $v_n=u_n/\| u_n \|$. Since $\dim\partial Q<\infty$ there is $v_0\in \partial Q$ such that $v_n\to v_0$ strongly in $H$ (also strongly in $L^2(0,\pi))$. Then (\ref{3.8}) and (\ref{3.1}) yield $$\label{3.9} \frac{1}{2}\int_0^{\pi} \bigl[ (v_0')^2- \lambda_+(v_0^+)^2-\lambda_-(v_0^-)^2 \bigr] \,dx \ge 0\,.$$ Let $v_0=a_0\varphi_m + w_0$, $a_0\in\mathbb{R}_0^+, w_0\in H^-$. It follows from Lemma \ref{l1} that $$\label{3.10} \int_0^{\pi} \bigl[ (v_0')^2- \lambda_+(v_0^+)^2-\lambda_-(v_0^-)^2 \bigr] \,dx \le \int_0^{\pi} \bigl[ (w_0')^2- \lambda_-(w_0)^2 \bigr] \,dx\,.$$ For $w_0\in H^-$ we have $$\label{3.11} \int_0^{\pi} \bigl[ (w_0')^2- \lambda_- w_0^2 \bigr] \,dx\le \int_0^{\pi} \bigl[ ((m-1)^2- \lambda_-)\, w_0^2 \bigr] \,dx \,.$$ Since $(m-1)^2<\lambda_-$ (see Remark \ref{rem1}) then (\ref{3.9}), (\ref{3.10}) and (\ref{3.11}) yield $$\int_0^{\pi} \bigl[ (v_0')^2- \lambda_+(v_0^+)^2-\lambda_-(v_0^-)^2\bigr] \,dx = ((m-1)^2- \lambda_-) \|w_0\|_2^2 = 0\,.$$ Hence we obtain $w_0=0$ and $v_0=a_0 \varphi_m$, $\|v_0 \|=1$. Now we divide (\ref{3.7}) by $\|u_n\|$ then $$\label{3.12} \limsup_{n\to \infty} \Bigl[ \frac{1}{2}\int_0^{\pi} \frac{(u_n')^2- \lambda_+(u_n^+)^2-\lambda_-(u_n^-)^2}{\|u_n\|}\,dx -\int_0^{\pi } \frac{G(x,u_n) - fu_n}{\|u_n\|} \,dx \Bigr]\ge 0\,.$$ By Lemma \ref{l1} the first integral in (\ref{3.12}) is less then or equal to 0. Hence it follows $$\label{3.13} \limsup_{n\to \infty}\int_0^{\pi } \frac{-G(x,u_n) + fu_n}{\|u_n\|} \,dx = \limsup_{n\to \infty} \int_0^{\pi } \Bigl[ \frac{-G(x,u_n)}{u_n} v_n + f v_n \Bigr] \,dx\ge 0\,.$$ Because of the compact imbedding $H^-\subset {C}(0,\pi)$, we have $v_n\to a_0 \varphi_m$ in ${C}(0,\pi)$ and we get $$\lim_{n\to \infty}u_n(x) = \begin{cases} +\infty &\mbox{for x\in(0,\pi) such that  \varphi_m(x)>0}\,,\\ -\infty& \mbox{for x\in(0,\pi) such that \varphi_m(x)<0}\,. \end{cases}$$ We note that from (\ref{1.5}) it follows that $-|p(x)|\le G_+(x)$, $G_-(x)\le |p(x)|$ for a.e. $x\in(0,\pi)$. We obtain from Fatou's lemma and (\ref{3.13}) $$\int_0^{\pi} f(x)\varphi_m(x) \,dx \ge \int_0^{\pi} \bigl[ G_+(x) (\varphi_m(x))^+ -G_-(x) (\varphi_m(x))^-\bigr] \,dx\,,$$ a contradiction to (\ref{1.4}). We proved that by choosing $K,L$ sufficiently large there is $\nu\in\mathbb{R}$ such that $\sup_{u\in \partial Q} I(u) =\nu<\mu$. Then Assumption (iii) of Theorem \ref{thm2} is verified. \noindent (iv) Now we show that $I$ satisfies the Palais-Smale condition. First, we suppose that the sequence $(u_n)$ is unbounded and there exists a constant $c_5$ such that $$\label{3.14} \Bigl| {1\over 2}\int_0^{\pi} \bigl[(u'_n)^2-\lambda_+ (u_n^+)^2-\lambda_- (u_n^-)^2 \bigr] \,dx -\int_0^{\pi} \bigl[ G(x,u_n)- f u_n \bigr] \,dx \Bigr| \le c_5$$ and $$\label{3.15} \lim_{n\to \infty}\| I'(u_n)\|=0\,.$$ Let $(w_k)$ be an arbitrary sequence bounded in $H$. It follows from (\ref{3.15}) and the Schwarz inequality that \label{3.16} \begin{aligned} &\Bigl| \lim_{{n\to \infty}\atop{k\to \infty}} \int_0^{\pi}\bigl[ u_n'w_k'-(\lambda_+ u_n^+ -\lambda_- u_n^-)\,w_k\bigr] \,dx -\int_0^{\pi} \bigl[ g(x,u_n)w_k - fw_k \bigr] \,dx \,\Bigr| \\ & = \bigl| \lim_{{n\to \infty}\atop{k\to \infty}}\langle I'(u_n), w_k\rangle \bigr|\\ & \le \lim_{{n\to \infty}\atop{k\to \infty}} \| I'(u_n)\|\cdot\| w_k\| =0\,. \end{aligned} Put $v_n=u_n/ \| u_n \|$. Due to compact imbedding $H\subset L^2(0,\pi)$ there is $v_0\in H$ such that (up to subsequence) $v_n\rightharpoonup v_0$ weakly in H, $v_n\to v_0$ strongly in $L^2(0,\pi)$. We divide (\ref{3.16}) by $\|u_n\|$ and we obtain $$\label{3.17} \lim_{n,k\, \to \infty} \int_0^{\pi}\bigl[ v_n'w_k'-(\lambda_+ v_n^+ -\lambda_- v_n^-)\,w_k \bigr] \,dx=0$$ and $$\label{3.18} \lim_{i, k\,\to \infty} \int_0^{\pi}\bigl[ v_i'w_k'-(\lambda_+ v_i^+ -\lambda_- v_i^-)\, w_k \bigr] \,dx=0\,.$$ We subtract equalities (\ref{3.17}) and (\ref{3.18}) we have $$\label{3.19} \lim_{n,i,k\, \to \infty} \int_0^{\pi}\bigl[ (v_n'-v_i')w_k'-(\lambda_+ (v_n^+-v_i^+) -\lambda_- (v_n^--v_i^-)\,)\, w_k \bigr] \,dx=0\,.$$ Because $(w_k)$ is a arbitrary bounded sequence we can set $w_k=v_n-v_i$ in (\ref{3.19}) and we get $$\label{3.20} \lim_{n,i\, \to \infty} \Bigl[ \| v_n-v_i\|^2-\int_0^{\pi}\left[\,[\lambda_+ (v_n^+-v_i^+) -\lambda_- (v_n^--v_i^-)](v_n-v_i)\,\right] \,dx\Bigr]=0\,.$$ Since $v_n\to v_0$ strongly in $L^2(0,\pi)$ the integral in (\ref{3.20}) converges to 0 and then $v_n$ is a Cauchy sequence in $H$ and $v_n\to v_0$ strongly in $H$ and $\|v_0\|=1$. It follows from (\ref{3.17}) and the usual regularity argument for ordinary differential equations (see Fu\v{c}\'{\i}k \cite{lit2}) that $v_0$ is the solution of the equation $$v_0''+\lambda_+ v_0^+ -\lambda_- v_0^-=0\,.$$ From the assumption $[\lambda_+,\lambda_-]\in \Sigma_m$ it follows that $v_0=a_0\varphi_{m}, a_0>0$. We set $u_n=a_n\varphi_m+\widehat{u}_n$, where $a_n\ge 0, \widehat{u}_n\in H^-\oplus S$. We remark that $u=u^+-u^-$ and using (\ref{2.3}) in the first integral in (\ref{3.16}) we obtain \label{3.21} \begin{aligned} I_1 &= \int_0^{\pi} \left[ (a_n \varphi_{m}+\widehat{u}_n)'w_k' - (\lambda_+ u_n^+ - \lambda_-u_n^-) w_k \right] \,dx \\ &= \int_0^{\pi} \left[ a_n \varphi_{m}'w_k' + (\widehat{u}_n)'w_k' - ((\lambda_+ -\lambda_-)u_n^+ + \lambda_-u_n)\,w_k \right] \,dx\\ &= \int_0^{\pi} \left[ a_n(\lambda_+ \varphi_{m}^+-\lambda_- \varphi_{m}^-)w_k + (\widehat{u}_n)'w_k' - ((\lambda_+ -\lambda_-)u_n^+ + \lambda_-u_n)\,w_k \right] \,dx\\ &= \int_0^{\pi} \big\{ a_n[(\lambda_+-\lambda_-) \varphi_{m}^+ +\lambda_- \varphi_{m}]w_k + (\widehat{u}_n)'w_k' \\ &\quad - [(\lambda_+ -\lambda_-)(a_n\varphi_m+\widehat{u}_n)^+ + \lambda_-(a_n\varphi_m+\widehat{u}_n)]\,w_k \big\} \,dx \\ &= \int_0^{\pi} \left[ (\lambda_+-\lambda_-) (a_n\varphi_{m}^+ - (a_n\varphi_m+\widehat{u}_n)^+)w_k + (\widehat{u}_n)'w_k' - \lambda_- \widehat{u}_n w_k \right] \,dx\,. \end{aligned} Similarly we obtain $$\label{3.22} I_1= \int_0^{\pi} \left[ (\lambda_+-\lambda_-) (a_n\varphi_{m}^- - (a_n\varphi_m+\widehat{u}_n)^-)w_k + (\widehat{u}_n)'w_k' - \lambda_+ \widehat{u}_n w_k \right] \,dx\,.$$ Adding (\ref{3.21}) and (\ref{3.22}) and we have $$\label{3.23} 2 I_1 = \int_0^{\pi} [ (\lambda_+ - \lambda_-) (|a_n\varphi_{m}| - |a_n\varphi_m+\widehat{u}_n|) w_k + 2 (\widehat{u}_n)'w_k' - (\lambda_+ + \lambda_-)\widehat{u}_n w_k ]\,dx\,.$$ We set \, $\widehat{u}_n=\overline{u}_n+\widetilde{u}_n$ \, where \, $\overline{u}_n\in H^- ,\, \widetilde{u}_n\in S$ \, and we put in (\ref{3.23}) \, $w_k=(\overline{u}_n-\widetilde{u}_n)/\|\widehat{u}_n\|$ then we have \label{3.24} \begin{aligned} 2 I_1&= \frac{1}{\|\widehat{u}_n\|}\int_0^{\pi} \big[ (\lambda_+-\lambda_-)(|a_n\varphi_{m}| - |a_n\varphi_m+\overline{u}_n+\widetilde{u}_n|) (\overline{u}_n-\widetilde{u}_n) \\ &\quad + 2 \left.(\overline{u}_n')^2 -2(\widetilde{u}_n')^2 - (\lambda_+ +\lambda_-)(\overline{u}_n^2-\widetilde{u}_n^2) \right] \,dx\,. \end{aligned} Hence \label{3.25} \begin{aligned} 2 I_1 &\le \frac{1}{\|\widehat{u}_n\|} \Bigl( \int_0^{\pi} [(\lambda_+-\lambda_-)\,|\overline{u}_n+\widetilde{u}_n| \, |\overline{u}_n-\widetilde{u}_n| ]\,dx \\ &\quad + 2 \|\overline{u}_n\|^2 -2\|\widetilde{u}_n\|^2 - (\lambda_+ +\lambda_-)(\|\overline{u}_n\|_2^2-\|\widetilde{u}_n\|_2^2) \Bigr) \\ &= \frac{1}{\|\widehat{u}_n\|} \Bigl( \int_0^{\pi} [(\lambda_+-\lambda_-)\,|\overline{u}_n^2-\widetilde{u}_n^2| ]\,dx \\ & \quad+ 2 \|\overline{u}_n\|^2 -(\lambda_+ +\lambda_-) \|\overline{u}_n\|_2^2 -2\|\widetilde{u}_n\|^2 + (\lambda_+ +\lambda_-) \|\widetilde{u}_n\|_2^2 \Bigr) \,. \end{aligned} The inequality $|a^2-b^2|\le \max\{a^2, b^2\}$, (\ref{3.25}) and (\ref{1.3}) yield $$\label{3.26} I_1 \le \max \bigl\{\|\overline{u}_n\|^2 - \lambda_- \|\overline{u}_n\|_2^2, -\|\widetilde{u}_n\|^2 + \lambda_+ \|\widetilde{u}_n\|_2^2\bigr\}\, \frac{1}{\|\widehat{u}_n\|} \,.$$ We note that the following relations hold \, $\|\overline{u}_n\|^2 \le (m-1)^2\|\overline{u}_n\|_2^2,$ \, $\|\widetilde{u}_n\|^2 \ge$ $(m+1)^2\|\widetilde{u}_n\|_2^2$. Hence from assumption (\ref{1.3}) and (\ref{3.26}) it follows that there is $\varepsilon>0$ such that $$\label{3.27} I_1 \le -\varepsilon \max \bigl\{\|\overline{u}_n\|^2 , \|\widetilde{u}_n\|^2 \bigr\}\frac{1}{\|\widehat{u}_n\|} \,.$$ From (\ref{3.16}), (\ref{3.27}) it follows $$\label{3.28} \lim_{n\to \infty} -\varepsilon \frac{\max \bigl\{\|\overline{u}_n\|^2 , \|\widetilde{u}_n\|^2 \bigr\}}{\|\widehat{u}_n\|} -\int_0^{\pi} \big[ (g(x,u_n)- f)\frac{\overline{u}_n-\widetilde{u}_n}{\|\widehat{u}_n\|} \big] \,dx \ge 0.$$ Now we suppose that $\|\widehat{u}_n\|\to\infty$. We note that $\|\widehat{u}_n\|^2=\|\overline{u}_n\|^2+\|\widetilde{u}_n\|^2$, we divide (\ref{3.28}) by $\|\widehat{u}_n\|$ and using (\ref{1.5}) we have $$\label{3.29} -\frac{\varepsilon}{2}\ge\lim_{n\to \infty} -\varepsilon \frac{\max \bigl\{\|\overline{u}_n\|^2 , \|\widetilde{u}_n\|^2 \bigr\} }{\|\widehat{u}_n\|^2}-\int_0^{\pi} \frac{g(x,u_n)- f}{\|\widehat{u}_n\|}\, \frac{\overline{u}_n-\widetilde{u}_n}{\|\widehat{u}_n\|} \,dx \ge 0$$ a contradiction to $\varepsilon>0$. This implies that the sequence $(\widehat{u}_n)$ is bounded. We use (\ref{2.2}) from Lemma \ref{l1} with $w=\widehat{u}_n$ and we obtain $$\int_0^{\pi }\bigl[(\widehat{u}_n')^2-\lambda_+ \widehat{u}_n^2 \bigr] \,dx \le J(u_n)\le \int_0^{\pi }\bigl[(\widehat{u}_n')^2-\lambda_- \widehat{u}_n^2 \bigr] \,dx\,.$$ Hence $$\label{3.30} \lim_{n\to\infty} \frac{J(u_n)}{\|u_n\|}=\lim_{n\to\infty}\frac{ \int_0^{\pi }\left[ (u_n')^2-\lambda_+ u_n^2-\lambda_- u_n^2 \right] \,dx}{\|u_n\|}=0\,.$$ We divide (\ref{3.14}) by $\|u_n\|$ and (\ref{3.30}) yield $$\label{3.31} \lim_{n\to \infty} \int_0^{\pi } \bigl[ \frac{-G(x,u_n)+f u_n}{\|u_n\|} \bigr] \,dx= 0$$ and using Fatou's lemma in (\ref{3.31}) we obtain a contradiction to (\ref{1.4}). This implies that the sequence $(u_n)$ is bounded. Then there exists $u_0\in H$ such that $u_n\rightharpoonup u_0$ in $H$, $u_n\to u_0$ in $L^2(0,\pi)$ (up to subsequence). It follows from the equality (\ref{3.16}) that $$\label{3.32} \lim_{n,i,k\, \to \infty} \int_0^{\pi}\left[(u_n-u_i)'w_k'- [\lambda_+ (u_n^+-u_i^+) -\lambda_- (u_n^--u_i^-)]w_k\right] \,dx =0\,.$$ We put $w_k=u_n-u_i$ in (\ref{3.32}) and the strong convergence $u_n \to u_0$ in $L^2(0,\pi)$ and (\ref{3.32}) imply the strong convergence $u_n \to u_0$ in $H$. This shows that the functional $I$ satisfies Palais-Smale condition and the proof of Theorem\,\ref{thm2} for $m$ even is complete. \smallskip Now we suppose that $m$ is odd. We have $[\lambda_+,\lambda_-]\in\Sigma_{m2}$ and the nontrivial solution $\varphi_{m2}$ of (\ref{1.2}) corresponding to $[\lambda_+,\lambda_-]$. Then there is $k>0$ such that $[\lambda_+-k,\lambda_--k]\in\Sigma_{m1}$ and solution $\varphi_{m1}$ corresponding to $[\lambda_+-k,\lambda_--k]=[\lambda_+',\lambda_-']$ (see Remark \ref{rem2})\,. We define the sets $Q$ and $S$ like for $m$ even and the proof of the steps (i), (ii) of theorem \ref{thm2} is the same. In the step (iii) we change inequality (\ref{3.10}) if $v_0=a_0\varphi_{m1}$ as it follows \label{3.10p} \begin{aligned} &\int_0^{\pi} \left[ (v_0')^2- \lambda_+(v_0^+)^2-\lambda_-(v_0^-)^2 \right] \,dx \\ &=\int_0^{\pi} \left[ (v_0')^2- (\lambda_+-k)(v_0^+)^2-(\lambda_--k)(v_0^-)^2 \right] \,dx- k \int_0^{\pi} v_0^2 \,dx \\ &\le -k \int_0^{\pi} v_0^2 \,dx + \int_0^{\pi} \left[ (w_0')^2- \lambda_-(w_0)^2 \right] \,dx\,. \end{aligned} Then by (\ref{3.9}), (\ref{3.10p}) and (\ref{3.11}) we obtain $k \int_0^{\pi} v_0^2 \,dx=0$, a contradiction to $\|v_0\|=1$. The proof of the step (iv) is similar to the prove for $m$ even. The proof of the theorem \ref{thm2} is complete. \end{proof} \begin{thebibliography}{00} \bibitem{lit1} A. K. Ben-Naoum, C. Fabry, \& D. Smets; {\em Resonance with respect to the Fu\v{c}\'{\i}k spectrum}, Electron J. Diff. Eqns., Vol. {\bf 2000}(2000), No. 37, pp. 1-21. \bibitem{lit2} S. 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