\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 159, 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 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/159\hfil Existence of solutions] {Existence of solutions for nonlinear second-order two-point boundary-value problems} \author[R.-J. Du \hfil EJDE-2009/159\hfilneg] {Rui-Juan Du} \address{Rui-Juan Du \newline Department of Computer Science, Gansu Political Science and Law Institute\\ Lanzhou, Gansu, 730070, China} \email{drjlucky@163.com} \thanks{Submitted May 25, 2009. Published December 15, 2009.} \thanks{Supported by the Gansu Political Science and Law Institute Research Projects} \subjclass[2000]{34B15} \keywords{Two-point boundary value problem; existence; Leray-Schauder theory} \begin{abstract} We consider the existence of solutions for the nonlinear second-order two-point ordinary differential equations \begin{gather*} u''(t)+\lambda u(t)+g(u(t))=h(t),\quad t\in[0,1] \\ u(0)=u(1)=0, \quad\text{or} \quad u'(0)=u'(1)=0 \end{gather*} where $g:\mathbb{R}\to \mathbb{R}$ is continuous, and $h\in L^1(0,1)$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} We consider the existence of solutions for the seconder-order two-point ordinary differential equation $$\ u''(t)+\lambda u(t)+g(u(t))=h(t),\quad t\in[0,1]\label{e1.1}$$ satisfying either $$u(0)=u(1)=0,\label{e1.2}$$ or $$u'(0)=u'(1)=0\label{e1.2'}$$ where $g:\mathbb{R}\to \mathbb{R}$ is continuous, $h\in L^1(0,1)$. The parameter $\lambda\in \mathbb{R}$ is allowed change near $m^2\pi^2(m=1,2,\dots)$, the $m$-th eigenvalue of the linear eigenvalue problem $$\begin{gathered} u''(t)+\lambda u(t)=0,\quad t\in[0,1],\\ u(0)=u(1)=0, \end{gathered} \label{e1.1lamb}$$ and $$\begin{gathered} u''(t)+\lambda u(t)=0,\quad t\in[0,1],\\ u'(0)=u'(1)=0. \end{gathered} \label{e1.1lamb'}$$ The linear problem associated with $\eqref{e1.1lamb}, \eqref{e1.1lamb'}$ are $$\begin{gathered} u''(t)+\lambda u(t)=h(t),\quad t\in[0,1],\\ u(0)=u(1)=0, \end{gathered} \label{e1.3}$$ and $$\begin{gathered} u''(t)+\lambda u(t)=h(t),\quad t\in[0,1],\\ u'(0)=u'(1)=0,\quad \end{gathered} \label{e1.3'}$$ and the corresponding existence results are known from the linear theory. Namely, if $\lambda \neq m^2\pi^2(m=1,2,\dots)$, then $\eqref{e1.1lamb}, \eqref{e1.1lamb'}$ have a unique solution for each given $h$; While for $\lambda= m^2\pi^2(m=1,2,\dots)$ a solution exists if, and only if, $h$ satisfies the orthogonality conditions $$\int_0^1h(t)\phi_i(t)dt=0 \quad(i=1,2),$$ where $\phi_1(t)=\sin m\pi t$, $\phi_2(t)=\cos m\pi t$ are the eigenfunctions associated with the eigenvalue $m^2\pi^2$. In this case, there are infinity many solutions $u(t)=u_0(t)+a \sin \pi t$, $v(t)=v_0(t)+b \cos \pi t$, $a,b \in \mathbb{R}$ with $u_0, v_0$ are the any particular solution of \eqref{e1.1lamb}, \eqref{e1.1lamb'}. A similar situation arises when introducing a sufficiently nonlinearity $g$. Assuming for the moment $g$ uniformly bounded, it is easy to see that $\lambda\neq m^2\pi^2$, \eqref{e1.1}-\eqref{e1.2}, \eqref{e1.1}-\eqref{e1.2'} again have a solution for each given $h$. If $\lambda= m^2\pi^2$, there are more difficulties to hold the existence of solutions of \eqref{e1.1}-\eqref{e1.2}, \eqref{e1.1}-\eqref{e1.2'}. In \cite{i1}, only when $m=1$, \eqref{e1.1}-\eqref{e1.2} is solvable if $h$ satisfies so called the Landesman-Lazer condition $$\limsup_{ t\to -\infty}g(t)<\int_0^1h(t)\phi_1(t)dt < \liminf_{ t\to +\infty}g(t)\,.$$ In \cite{s1}, the authors assumed the nonlinearity $f(t, u)=g(u)-h(t)$ did not satisfy Landesman-Lazer conditions, were also proved that the boundary value problem \eqref{e1.1}-\eqref{e1.2} has at least one solution , but $m$ is only allowed equal to 1. It is not difficulty to see that when $m=2,3,\dots$, the case became more complex, there are only a few scholars to study it. In addition, in most of the papers about second-order two-point are using the same method as \cite{i1,s1}. There aren't much more method to solve those problems. Inspired by the above results, in this paper, we try to establish the existence results of boundary value problems \eqref{e1.1}-\eqref{e1.2}, \eqref{e1.1}-\eqref{e1.2'}, $\lambda$ is allowed to change near $m^2\pi ^2(m=1,2,\dots)$, the nonlinearity $g$ has weaker conditions than \cite{s1}, and the methods are different from the methods in \cite{s1}. \section{Preliminaries} In this paper, we use the following assumptions in $g$ and $h$: \begin{itemize} \item[(H1)] $g:\mathbb{R}\to\mathbb{R}$ is continuous, there is $\alpha\in[0, 1)$, $c, d\in (0, +\infty)$ such that $$|g(u)|\leq c|u|^{\alpha}+d,\quad u\in \mathbb{R};\label{e2.1}$$ \item[(H2)] There exists $r>0$ such that $$ug(u)>0,\quad |u|>r;\label{e2.2}$$ \item[(H2')] There exists $r>0$ such that $$ug(u)<0,\quad |u|>r;\label{e2.2'}$$ \item[(H3)] $h:[0, 1]\to \mathbb{R}$, $h\in L^1(0, 1)$ satisfying $$\int_0^1h(t)\phi_1(t)dt=0.\label{e2.3}$$ \item[(H3')] $h:[0, 1]\to \mathbb{R}$, $h\in L^1(0, 1)$ satisfying $$\int_0^1h(t)\phi_2(t)dt=0. \label{e2.3'}$$ \end{itemize} \textbf{Remark.} % 2.1} For convenience, we rewrite $\lambda=\lambda_m+\bar{\lambda}$, where $\lambda_m$ is the $m$-th eigenvalue of the linear eigenvalue problem \eqref{e1.1}-\eqref{e1.2}, or of \eqref{e1.1}-\eqref{e1.2'}. Then the ordinary differential equation \eqref{e1.1} is equivalent to $$u''(t)+\lambda_m u(t)+\bar{\lambda}u(t)+g(u(t))=h(t).\label{e2.4}$$ Let $X, Y$ be the linear Banach space $C^1[0,1]$, $L^1(0,1)$, whose norms are denoted by $$\|u\|=\max\{\|u\|_0,\|u'\|_0\},\quad \|u\|_1=\int_0^1|u(s)|ds,$$ where $\|u\|_0$ denotes the max norm $\|u\|_0=\max\{u(t), t\in[0,1]\}$. Let $L_i: \mathop{\rm dom}L_i\subset X\to Y$ ($i=1,2$) be linear operators defined for $u\in \mathop{\rm dom}L_i$ as $$L_iu:=u''+\lambda_m,\label{e2.5}$$ where $\mathop{\rm dom}L_1=\{u\in W^{2,1}(0,1): u(0)=u(1)=0\}$, $\mathop{\rm dom}L_2=\{u\in W^{2,1}(0,1): u'(0)=u'(1)=0\}$. \begin{lemma} \label{lem2.1} Let $L_i(i=1,2)$ be the linear operator as defined in \eqref{e2.5}. Then \begin{gather*} \ker L_i=\{u\in X: u(t)=\rho\phi_i(t), \rho\in \mathbb{R}\}, \\ \mathop{\rm Im}L_i=\{u\in Y: \int_0^1 u(t)\phi_i(t)dt=0\}. \end{gather*} \end{lemma} Defined the operator $P_i: X\to \ker L_i\cap X$, $Q_i: X\to \mathop{\rm Im}Q_i\cap Y$, \begin{gather} (P_iu)(t)=\phi_i(t)\int_0^1u(s)\phi_i(s)ds,\label{e2.6} \\ (Q_iu)(t)=u(t)-(\int_0^1u(s)\phi_i(s)ds)\phi_i(t).\label{e2.7} \end{gather} It is easy to check that $\mathop{\rm Im}P_i=\ker L_i$, $Y/\mathop{\rm Im}Q_i=\mathop{\rm Im}L_i$ ($i=1,2$), and to show the following Lemma. \begin{lemma} \label{lem2.2} Let $X_{P_i}=\ker L_i$, $X_{I-{P_i}}=\ker P_i$, $Y_{Q_i}=\mathop{\rm Im}L_i$, $Y_{I-{P_i}}=\mathop{\rm Im}{Q_i}$. Then $$X=X_{P_i}\oplus X_{I-{P_i}}, \quad Y=Y_{I-{Q_i}}\oplus Y_{Q_i}.$$ \end{lemma} It is easy to check that the restriction of $L_i$ to $X_{I-{P_i}}$ is a bijection from $X_{I-{P_i}}$ onto $\mathop{\rm Im}L_i$ ($i=1,2$). We define $K_i: \mathop{\rm Im}L_i\to X_{I-{P_i}}$ by $$K_i=L_i|^{-1}_{X_{I-{P_i}}}.\label{e2.8}$$ Define the nonlinear operator $G: X\to Y$ by $$(Gu)(t)=g(u(t))\quad t\in[0,1].$$ It is easy to check that $G: X\to Y$ is completely continuous. Obviously \eqref{e1.1}-\eqref{e1.2}, \eqref{e1.1}-\eqref{e1.2'} are equivalent to $$L_iu+\bar{\lambda }u+Gu=h,\quad u\in D(L_i).\label{e2.9}$$ Since $\ker L_i= \mathop{\rm span} \{\phi_i(t)\}$ ($i=1,2$), we see that each $x\in X$ can be uniquely decomposed as $$x(t)=\rho \phi_i(t)+v(t)\quad t\in[0, 1],$$ for some $\rho\in \mathbb{R}$ and $v\in X_{I-{P_i}}$. For $y\in Y$, we also have the decomposition $$y(t)=\tau \phi_i(t)+w(t),\quad t\in[0, 1],$$ with $\tau \in \mathbb{R}$, $w\in Y_{Q_i}$ ($i=1,2$). \begin{lemma} \label{lem2.3} The boundary-value problems \eqref{e2.9} are equivalent to the system $$\begin{gathered} L_iv(t)+\bar{\lambda} v(t)+Q_iG(\rho \phi_i(t)+v(t))=h(t),\\ \bar{\lambda }\int_0^1( \phi_i(t))^2dt +\int_0^1\phi_i(t)G(\rho \phi_i(t)+v(t))dt=0. \end{gathered} \label{e2.10}$$ \end{lemma} \section{Main results} \begin{theorem} \label{thm3.1} Assume {\rm (H1), (H2), (H3)}. Then there exists $\lambda_+>0$ such that \eqref{e1.1}-\eqref{e1.2} has at least one solutions in $C^1[0,1]$ if $\lambda\in[0, \lambda_+]$. \end{theorem} \begin{theorem} \label{thm3.2} Assume {\rm (H1), (H2'), (H3)}. Then there exists $\lambda_-<0$ such that \eqref{e1.1}-\eqref{e1.2} has at least one solutions in $C^1[0,1]$ if $\lambda\in[\lambda_-,0]$. \end{theorem} \begin{theorem} \label{thm3.1'} Assume {\rm (H1), (H2), (H3')}. Then there exists $\lambda_+>0$ such that \eqref{e1.1}-\eqref{e1.2'} has at least one solutions in $C^1[0,1]$ if $\lambda\in[0, \lambda_+]$. \end{theorem} \begin{theorem} \label{thm3.2'} Assume {\rm (H1), (H2'), (H3')}. Then there exists $\lambda_-<0$ such that \eqref{e1.1}-\eqref{e1.2'} has at least one solutions in $C^1[0,1]$ if $\lambda\in[\lambda_-,0]$. \end{theorem} In this article, we prove only Theorem \ref{thm3.1}; the other theorems can be proved by using the similarly method. \begin{lemma} \label{lem3.1} Assume {\rm (H1), (H2), (H3)}. Then there exists $M>0$, such that any solution $u\in D(L_1)$ of \eqref{e2.9} satisfies $\|u\|< M$, as long as $$0\leq\bar{\lambda} \leq \delta :=\frac{1}{2\|K_1J_1\|_{Y_{Q_1}\to X_{I-{P_1}}}} \label{e2.11}$$ where $J_1: X\to Y$ is defined by $(J_1u)(t)=u(t), t\in[0, 1]$. \end{lemma} \begin{proof} We divide the proof into two steps. \textbf{Step I.} Obviously $(L_1+\bar{\lambda} J_1)|_{X_{I-{P_1}}}:X_{I-{P_1}}\to Y_{Q_1}$ is invertible for $\bar{\lambda}\leq \delta$. Moreover, by \eqref{e2.11}, \begin{align*} \|(L_1+\bar{\lambda} J_1)|^{-1}_{X_{I-{P_1}}}\|_{Y_{Q_1}\to X_{I -{P_1}}} &=\|L_1|^{-1}_{X_{I-{P_1}}}(I+\bar{\lambda} K_1J_1)^{-1}\|_{Y_{Q_1}\to X_{I-{P_1}}}\\ &=\|K_1\|_{Y_{Q_1}\to X_{I -{P_1}}}\|(I+\bar{\lambda} K_1J_1)^{-1}\|_{X_{I-P}\to X_{I-{P_1}}}\\ &\leq 2\|K_1\|_{Y_{Q_1}\to X_{I -{P_1}}}. \end{align*} Let $u(t)=\rho \phi_1(t)+v(t)=\rho \sin m\pi t+v(t)$ is a solution of \eqref{e2.9} for some $\rho\neq 0$. Then \begin{align*} \|v\| &=\|(L_1+\bar{\lambda} J_1)|^{-1}_{X_{I-{P_1}}}Q_1(h-g(\rho \sin m\pi t+v(t)))\|\\ &\leq \|(L_1+\bar{\lambda} J_1)|^{-1}_{X_{I-{P_1}}}\|_{Y_{Q_1}\to X_{I -{P_1}}}\|Q_1\|_{Y\to Y_{Q_1}}\\ &\quad\times \big[\|h\|_1+c(|\rho|\|\sin m\pi t\|_1+\|v\|_1)^\alpha+d\big ]\\ &\leq 2\|K_1\|_{Y_{Q_1}\to X_{I -{P_1}}}\|Q_1\|_{Y\to Y_{Q_1}}[\|h\|_1+c(|\rho|\|\sin m\pi t\|+\|v\|)^\alpha+d]\\ &\leq 2\|K_1\|_{Y_{Q_1}\to X_{I -{P_1}}}\|Q_1\|_{Y\to Y_{Q_1}}[\|h\|_1+c(|\rho| m\pi +\|v\|)^\alpha+d]\\ &= 2\|K\|_{Y_{Q_1}\to X_{I -{P_1}}}\|Q_1\|_{Y\to Y_{Q_1}}[\|h\|_1+c m\pi|\rho|^\alpha (1 +\frac{\|v\|}{m\pi|\rho|})^\alpha+d]\\ &\leq 2\|K_1\|_{Y_{Q_1}\to X_{I -P}}\|Q_1\|_{Y\to Y_{Q_1}}[\|h\|_1+c m\pi|\rho|^\alpha (1 +\frac{\alpha\|v\|}{m\pi|\rho|})+d]\\ &= 2\|K_1\|_{Y_{Q_1}\to X_{I -{P_1}}}\|Q_1\|_{Y\to Y_{Q_1}}\\ &\quad\times\big[\|h\|_1+c m\pi|\rho|^\alpha (1 +\frac{\alpha}{(m\pi|\rho|)^{1-\alpha}}\cdot \frac{\|v\|}{(m\pi|\rho|)^{\alpha}} )+d\big] \end{align*} Hence, $\frac{\|v\|}{(m\pi|\rho|)^{\alpha}}\leq \frac{c_0}{(m\pi|\rho|)^{\alpha}}+c_1+\frac{\alpha c_1}{(m\pi|\rho|)^{1-\alpha}}\cdot \frac{\|v\|}{(m\pi|\rho|)^{\alpha}}.$ where \begin{gather*} c_0=2\|K_1\|_{Y_{Q_1}\to X_{I-{P_1}}}\|Q_1\|_{Y\to Y_{Q_1}}(\|h\|_1+d), \\ c_1=2c\|K_1\|_{Y_{Q_1}\to X_{I-{P_1}}}\|Q_1\|_{Y\to Y_{Q_1}}. \end{gather*} If $$|\rho|\geq \frac{(2\alpha c_1)^{-\frac{1}{1-\alpha}}}{m\pi}:=\tilde{c},$$ then $$\frac{\|v\|}{(m\pi|\rho|)^{\alpha}}\leq \frac{2c_0}{(m\pi \tilde{c})^{\alpha}}+2c_1:=\bar{c}.\label{e3.1}$$ \textbf{Step II.} If we assume that the conclusion of the lemma is false, we obtain a sequence $\{\bar{\lambda}_n\}$ with $0\leq \bar{\lambda}_n \leq \delta, \bar{\lambda}_n\to 0$ and a sequence of corresponding solutions $\{u_n\}: u_n=\rho_n\phi_1(t) dt +(t), \rho_n\in \mathbb{R}, v_n\in X_{I-{P_1}}, n\in N$, such that $$\|u_n\|\to +\infty.$$ From \eqref{e3.1} $$\|v\|\leq \bar{c}(m\pi)^{\alpha}(|\rho|)^{\alpha}:=\hat{c}|\rho|^{\alpha}. \label{e3.2}$$ we conclude that $|\rho_n|\to +\infty$. We may assume that $\rho_n\to +\infty$, the other case can be treated in the same way. Then for all $n\in N$, we get that $\rho_n\geq \tilde{c}$. Now, from \eqref{e2.9} we obtain $$\bar{\lambda}_n \rho_n\int_0^1( \sin m\pi t)^2dt+\int_0^1\sin m\pi tg(\rho_n \sin m\pi t+v_n(t))dt=0.\label{e3.3}$$ Since $\bar{\lambda}_n\geq 0$, $\int_0^1\bar{\lambda}_n\rho_n( \sin m\pi t)^2dt\geq0$, for all $n\in N$, so we have $$\int_0^1\sin m\pi tg(\rho_n \sin m\pi t+v_n(t))dt\leq0.\label{e3.4}$$ Let $I^+:=\{t: t\in[0, 1], \sin \pi t>0\}$, $I^-:=\{t: t\in[0, 1], \sin \pi t<0\}$. It is easy to see that $I^+\cap I^-\neq 0,$ and $$\min\{|\sin m\pi t|t\in I^+\cap I^-\}>0.\label{e3.5}$$ Combining \eqref{e3.5} and \eqref{e3.2},we conclude \begin{gather} \lim_{\rho_n\to +\infty}\min\{\rho_n\sin m\pi t+v_n(t)|t\in I^+\}=+\infty.\label{e3.6} \\ \lim_{\rho_n\to +\infty}\min\{\rho_n\sin m\pi t+v_n(t)|t\in I^-\}=-\infty.\label{e3.7} \end{gather} Applying \eqref{e3.3},\eqref{e3.6} and \eqref{e3.7} and (H2), we conclude \begin{align*} \int_0^1\sin m\pi tg(\rho_n \sin m\pi t+v_n(t))dt &=\int_{t\in I^+}\sin m\pi tg(\rho_n \sin m\pi t+v_n(t))dt\\ &\quad +\int_{t\in I^-}\sin m\pi tg(\rho_n \sin m\pi t+v_n(t))dt>0 \end{align*} hold for some $n$ large enough. This contradicts \eqref{e3.4}. \end{proof} Similarly, we obtain the following result. \begin{lemma} \label{lem3.2} Assume {\rm (H1), (H2'), (H3)}. Then there exists $M'>0$, such that any solution $u\in D(L_1)$ of \eqref{e2.9} satisfies $$\|u\|< M',$$ as long as $-\delta\leq \lambda \leq 0$, where $\delta$ and $J_1$ as lemma \ref{lem3.1} \end{lemma} \begin{proof}[Proof of Theorem \ref{thm3.1}] Consider the linear operator $L: X\to Y$, defined for $u\in \mathop{\rm dom}L$ by $$Lu=L_1u+\bar{\lambda}u=\lambda_m u+\bar{\lambda}u,$$ and the family maps $T_{\mu}: X\to Y$ ($0\leq\mu\leq 1$), $$(T_{\mu}u)(t)=\mu(h(t)-g(u(t))),\quad t\in[0,1].$$ where $\mathop{\rm dom}L:= \{u\in W^{2,1}(0,1): u(0)=u(1)=0\}$. Observe that $L$ is invertible with, let $K: Y\to X$, then $K=L^{-1}$, and $$u(t)=K(G(u(t))-h(t)),\quad t\in[0,1].\label{e3.8}$$ If $$R=\{u\in X: \|u\|\leq M+1\},$$ we can define a compact homotopy $H_{\mu}: R\to \mathop{\rm dom}L$, $$H_{\mu}=L^{-1}\circ(T_{\mu}u)\circ J_1.$$ We can see that the fixed points of $H_{\mu}$ are exactly the solution of \eqref{e1.1}-\eqref{e1.2}, and the choice of $R$ enables us to say that the homotopy $H_{\mu}$ is fixed-point free on the boundary of $R$. since $H_0=0$, by the Leray-Schauder theory \cite{c1}, we obtain that $H_1$ has a fixed point and so there is a solution to \eqref{e1.1}-\eqref{e1.2}. \end{proof} \begin{thebibliography}{0} \bibitem{a1} S. Ahmad; \emph{A resonane problem in which the nonlinearly may grow linearly}, Proc. Am. math. Soc. 1984, 92: 381-383. \bibitem{a2} S. Ahmad; \emph{Nonselfadjoint resonance problems with unbounded perturbations}. Nonlinear Analysis 1986, 10:147-156. \bibitem{c1} A. Constantin; \emph{On a Two-Point Boundary Value Problem}, J. Math. Anal. Appl. 193(1995)318-328. \bibitem{i1} R. Iannacci, M. N. Nkashama; \emph{Unbounded perturbations of forced second order ordinary differential equations at resonance}. J. Differential Equations 69 (1987), no. 3, 289--309. \bibitem{k1} P. Kelevedjiev; \emph{Existence of solutions for two-point boundary value problems}, Nonlinear Analysis, 1994, 22: 217-224. \bibitem{l1} B. Lin; \emph{Solvability of multi-point boundary value problems at resonance(IV)}, Appl. Math. Comput. 2003, 143: 275-299. \bibitem{m1} J. Mawhin, K. Schmitt; \emph{Landesman-Lazer type problems at an eigenvalue off odd multiplicity}, Result in Maths. 14(1988)138-146. \bibitem{p1} H. Peitgen, K. Schmitt; \emph{Global analysis of two-parameter linear elliptic eigenvalue problems}, Trans. Amer. Math. Soc. 283(1984)57-95. \bibitem{s1} J. Santanilla; \emph{Solvability of a Nonlinear Boundary Value Problem Without Landesman-Lazer Condition}. Nonlinear Analysis 1989, 13(6):683-693. \end{thebibliography} \end{document}