\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 35, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2011/35\hfil Impulsive differential equations] {Mixed two-point boundary-value problems for impulsive differential equations} \author[Z. Han, S. Wang \hfil EJDE-2011/35\hfilneg] {Zhiqing Han, Suqin Wang} % in alphabetical order \address{Zhiqing Han \newline School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, 116024, China} \email{hanzhiq@dlut.edu.cn} \address{Suqin Wang \newline School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning, 116024, China} \email{wangsuqinwsq@126.com} \thanks{Submitted November 22, 2010. Published March 3, 2011.} \subjclass[2000]{34B15, 34B25} \keywords{Impulsive differential equations; mixed boundary value problem; \hfill\break\indent variational methods; resonant and non-resonant} \begin{abstract} In this article, we prove the existence of solutions to mixed two-point boundary-value problem for impulsive differential equations by variational methods, in both resonant and the non resonant cases. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article, we study the existence of solutions to the impulsive problem $$\label{e1.1} \begin{gathered} -u''+\lambda u=f(t,u), \quad\text{a.e. }t\in (0,T),\\ u(0)=0,\quad u'(T)=0,\\ \Delta u'(t_j)=I_j(u(t_j)), \quad j=1,2,\dots p, \end{gathered}$$ where $I_j: \mathbb{R}\to\mathbb{R}$, $j=1,2,\dots, p$, are continuous functions and $f(t,u)$ satisfies the condition \begin{itemize} \item[(A)] $f(t,u)$ is measurable in $t$ for each $u\in\mathbb{R}$, continuous in $u$ for a.e. $t\in[0,T]$, and there exist functions $g\in C(\mathbb{R}^{+},\mathbb{R}^{+})$ and $h\in L^{1} (0,T;\mathbb{R}^{+})$ such that $|f(t,u)|\leq g(|u|)h(t)$, for all $u\in \mathbb{R}$ and a.e. $t\in[0,T]$. \end{itemize} The theory of impulsive differential equations describes processes which experience a sudden change of their states at certain time. It can be successfully used for mathematical simulation in some problems from theoretical physics, chemistry, medicine, population dynamics, optimal control and in some other processes and phenomena in science and technology, see \cite{VL,SP} for the general aspects of the theory. Some classical tools such as upper and lower solutions, monotone iterative technique, fixed point theory, degree theory and so on have been widely used to such equations, we refer to \cite{AR,CTY,CS,CN,CH,GD,VL,NJA,QL} for some references. In recent years, the variational methods \cite{MW,RA} have been applied to such equations and are proved to be very effective, we refer to \cite{NJ,TYG,TYW,HZ,ZR} for some results. Let us recall some results related to the impulsive problem \eqref{e1.1} obtained by the variational methods. Nieto and O'Regan (\cite{NJ}) first noticed that the equation in \eqref{e1.1} coupled with the Dirichlet boundary value condition and the impulsive conditions has a variational structure and obtained some existence results for the problem. The results are extended to more general nonlinearities in \cite{ZR}. Recently, Tian and Ge \cite{TYG,TYW} investigated the equation with the more general (Sturm-Liouville) boundary value conditions. That is they considered the equation with the Sturm-Liouville boundary value conditions $\alpha u'(0)-\beta u(0)=0$, $\gamma u'(T)+\sigma u(T)=0$ but with a restriction $\alpha,\gamma>0$, $\beta,\sigma \geq0$. The restriction excludes the boundary value conditions of the problem \eqref{e1.1} investigated in this paper. In this paper, we investigate the problem \eqref{e1.1} still by the variational methods. The framework involved is different from those for the other kinds of boundary value conditions. We obtain some existence results both by the Ambrosetti-Rabinowitz type condition in the non-resonant case and the generalized Ahmad-Lazer-Paul type condition in the resonant case. \section{Preliminaries} Denote $H=\{u(t)|u(t)$ is absolutely continuous on $[0,T]$, $u(0)=0$, $u'(t)\in L^2(0,T;\mathbb{R})\}$. It is easy to see that $H_0^{1}(0,T)\subset H\subset H^{1}(0,T)$ and $H$ is a closed subset of $H^{1}(0,T)$. So $H$ is a Hilbert space with the usual inner product in $H^{1}(0,T)$. \begin{proposition} \label{prop2.1} If $u\in H$, then $\|u\|_{c}\leq \sqrt{T}\|u'\|_{L^2}$, where $$\|u\|_{c}=\max_{t\in[0,T]}|u(t)|.$$ \end{proposition} \begin{proof} For $u\in H$, we have $$|u(t)|=|\int_0^{t}u'(s)ds|\leq\sqrt{t} \Big(\int_0^{t}|u'(s)|^2ds\Big)^{1/2} \leq\sqrt{T}\|u'\|_{L^2}.$$ \end{proof} For the linear problem $$\label{LP} \begin{gathered} -u''(t)=\lambda u(t), \quad t\in (0,T),\\ u(0)=0,\quad u'(T)=0, \end{gathered}$$ there is a sequence of eigenvalues $\lambda_k=(2k+1)^2\pi^2/(4T^2)$ and the corresponding $L^2(0,T)$-normalized eigenfunctions $\phi_k(t)=\sqrt{(2/T)} \sin((2k+1)\pi t/(2T)$, $k=0,1,2\dots$. If the equation in the problem \eqref{LP} is coupled with the boundary value condition $u'(0)=0$, $u(T)=0$, the eigenvalues are as before and the corresponding $L^2(0,T)$-normalized eigenfunctions are $\psi_k(t)=\sqrt{(2/T)}\cos((2k+1)\pi t/(2T))$, $k=0,1,2 \dots$. According to the Sturm-Liouville theory, $\{\phi_k\}$ and $\{\psi_k\}$ are complete bases in the Hilbert space $L^2(0,T)$. \noindent\textbf{Claim} If $u\in H$, then $\|u'\|^2_{L^2}\geq\lambda_0\|u\|^2_{L^2}$ (Poincar\'e inequality). In fact, for any $u\in H$, set $u=\sum_{k=0}^{\infty}c_k\phi_k$, where $c_k=(u,\phi_k)=\frac{2}{T}\int_0^Tu\phi_kdt.$ By the Parseval identity, $\int_0^Tu^2dt=\sum_{k=0}^{\infty}c_k^2$. Let $u'=\sum_{k=0}^{\infty}a_k\psi_k$, where $a_k=(u',\psi_k)=\frac{2}{T}\int_0^Tu'\psi_kdt$. By an easy calculation, $$a_k=\frac{2}{T}\int_0^Tu'\psi_kdt =-\frac{2}{T}\int_0^Tu\psi'_kdt=\frac{(2k+1)\pi}{2T} \cdot\frac{2}{T}\int_0^Tu\phi_kdt =\frac{(2k+1)\pi}{2T}c_k.$$ Then we have $$\int_0^T|u'|^2dt=\sum_{k=0}^{\infty}a_k^2 =\sum_{k=0}^{\infty}\frac{(2k+1)^2\pi^2}{4T^2}c_k^2 \geq\frac{\pi^2}{4T^2}\sum_{k=0}^{\infty}c_k^2 =\lambda_0\int_0^T|u|^2dt.$$ Similarly, we can prove that $\|u'\|^2_{L^2}\geq\lambda_1\|u\|^2_{L^2}$, for all $u\in H\ominus \operatorname{span}\{\phi_0\}$, which will be used later. According to the Poincar\'e inequality, we can define an inner product in $H$, $(u,v)=\int_0^Tu'(t)v'(t)dt$. The induced norm $\|u\|=\sqrt{\int_0^Tu'^2dt}$ is equivalent to the old one. Throughout the paper, we will use this norm. \begin{proposition} \label{prop2.2} Assume that $\mu(t)\in L^1(0,T)$ satisfies $$0\leq\mu(t)\leq\lambda_0,\label{e2.1}$$ for a.e. $t\in [0,T]$ and $\mu(t)<\lambda_0$ holds on a subset of $[0,T]$ with positive measure. Then there exists $\delta>0$ such that for all $u\in H$, one has $$\int_0^T[u'^2(t)-\mu(t)u^2(t)]dt\geq\delta\int_0^Tu'^2(t)dt.$$ \end{proposition} \begin{proof} We use some known arguments \cite{RMN} to prove it. By \eqref{e2.1} and the Poincar\'e inequality, we see that for all $u\in H$, $$\int_0^T[u'^2(t)-\mu(t)u^2(t)]dt\geq\int_0^T[u'^2(t) -\lambda_0u^2(t)]dt\geq0.\label{e2.2}$$ Now we assume that the proposition were false. Then we can find a sequence $\{u_n\}$ in $H$, $u_n\neq0$ such that $$\int_0^T[u_n'^2(t)-\mu(t)u_n^2(t)]dt <\frac{1}{n}\int_0^Tu_n'^2(t)dt.$$ Let $v_n=u_n/\|u_n\|$. We obtain $$\int_0^T[v_n'^2(t)-\mu(t)v_n^2(t)]dt<\frac{1}{n}.$$ We can assume that $v_n\rightharpoonup v$ in $H$ and $v_n\to v$ in $C[0,T]$. We obtain $$1\leq\int_0^T\mu(t)v^2(t)dt.\label{e2.3}$$ From \eqref{e2.3}, we have $$v\neq0.\label{e2.4}$$ Since $v_n\rightharpoonup v$ in $H$, we have $$\|v\|^2\leq\liminf_{n\to \infty}\|v_n\|^2=1.$$ Hence, we have $$\int_0^T[v'^2(t)-\mu(t)v^2(t)]dt\leq0.$$ Therefore, $$\int_0^T[v'^2(t)-\mu(t)v^2(t)]dt=0.\label{e2.5}$$ From \eqref{e2.2}, we have $$\int_0^T[v'^2(t)-\lambda_0v^2(t)]dt=0,$$ which shows that $v(t)$ is an eigenfunction corresponding to $\lambda_0$. So we set $v(t)=C\phi_0(t)$, where $C$ is a constant. Substituting it to \eqref{e2.5} and noticing that $\mu(t)<\lambda_0$ holds on a subset of $[0,T]$ with positive measure, we obtain $C=0$, which contradicts to \eqref{e2.4}. This completes the proof. \end{proof} \section{Non-resonant case} In this section, we study the existence of solutions for the non-resonant impulsive problem $$\label{e3.1} \begin{gathered} -u''+\lambda u=f(t,u), \quad\text{a.e. }t\in (0,T),\\ u(0)=0,\quad u'(T)=0, \\ \Delta u'(t_j)=I_j(u(t_j)), \quad j=1,2,\dots p \end{gathered}$$ with $\lambda>-\lambda_0$, where $0=t_00$; \item[(I2)] $a(t)\leq\lambda_0[\frac{1}{2}-\frac{T}{2}(\sum_{j=1}^pb_j)] +\frac{\lambda}{2} =a^{*}$ with $a(t)< a^{*}$ holds on a subset of $[0,T]$ with positive measure. \end{itemize} Then problem \eqref{e3.1} has at least one solution. \end{theorem} \begin{proof} For all $u\in H$, \begin{align*} \sum_{j=1}^p\int_0^{u(t_j)}I_j(s)ds &\leq \sum_{j=1}^p\int_0^{|u(t_j)|}|I_j(s)|ds \leq\sum_{j=1}^p\int_0^{|u(t_j)|}(a_j+b_j|s|)ds\\ &\leq (\sum_{j=1}^pa_j)\|u\|_{c}+\frac{1}{2}(\sum_{j=1}^pb_j)\|u\|_{c}^2\\ &\leq (\sqrt{T}\sum_{j=1}^pa_j)\|u\|+\frac{T}{2}(\sum_{j=1}^pb_j)\|u\|^2 \end{align*} and $\int_0^TF(t,u)dt\leq\int_0^T(a(t)|u|^2+b(t)|u|+c(t))dt\leq\int_0^Ta(t)|u|^2dt+C\|u\|+C$ where and in the following $C$ denotes a universal constant. Then \begin{align*} J(u) &= \frac{1}{2}\int_0^Tu'^2dt+\frac{\lambda}{2}\int_0^Tu^2dt +\sum_{j=1}^p\int_0^{u(t_j)}I_j(s)ds-\int_0^TF(t,u)dt\\ &\geq \frac{1}{2}\int_0^Tu'^2dt+\frac{\lambda}{2}\int_0^Tu^2dt -\sum_{j=1}^p\int_0^{|u(t_j)|}|I_j(s)|ds-\int_0^Ta(t)|u|^2dt\\ &\quad -C\|u\|-C\\ &\geq \frac{1}{2}\int_0^Tu'^2dt+\int_0^T (\frac{\lambda}{2}-a(t))|u|^2dt-(\sqrt{T}\sum_{j=1}^pa_j)\|u\| -\frac{T}{2}(\sum_{j=1}^pb_j)\|u\|^2\\ &\quad -C\|u\|-C\\&= [\frac{1}{2}-\frac{T}{2}(\sum_{j=1}^pb_j)] \int_0^Tu'^2dt -\int_0^T(a(t)-\frac{\lambda}{2})|u|^2dt-C\|u\|-C. \end{align*} By Proposition \ref{prop2.2} and the condition (I2), there exists $\delta>0$ such that $$\Big[\frac{1}{2}-\frac{T}{2}(\sum_{j=1}^pb_j)\Big] \int_0^Tu'^2dt -\int_0^T(a(t)-\frac{\lambda}{2})|u|^2dt \geq\delta\int_0^Tu'^2dt,\quad \forall u\in H.$$ Thus $$J(u)\geq\delta\|u\|^2-C\|u\|-C,\quad \forall u\in H.$$ So $\lim_{\|u\|\to\infty,u\in H}J(u)=+\infty.$ Hence every minimizing sequence is bounded. It is easily verified by the condition (A) and a compact imbedding result that $J(u)$ is weakly lower semi-continuous. Hence by a standard result, $J(u)$ has a minimizing point $u$, which is a classical solution of the problem \eqref{e3.1}. \end{proof} Since $\lambda>-\lambda_0$, we can choose the equivalent norm $\|u\|^2_1=\int_0^T(u'^2+\lambda u^2)dt$ in $H$. Hence there exist positive constants $m_1$ and $m_2$ such that $m_1\|u\|\leq\|u\|_1\leq m_2\|u\|$. \begin{theorem} \label{thm3.5} Suppose that $f(t,u)$ satisfies the condition {\rm (A)}. Moreover, we assume that {\rm (I1)} and the following conditions hold: \begin{itemize} \item[(F2)] $\lim_{u\to 0}\frac{f(t,u)}{u}2$ and $R>0$ such that for a.e. $t\in[0,T]$ and $|u|\geq R$, $0<\mu F(t,u)\leq uf(t,u)$ (Ambrosetti-Rabinowitz type condition); \item[(I3)] $\lim_{u\to0}\frac{I_j(u)}{u}\to0$ for all $1\leq j\leq p$; \item[(I*)] $[(\frac{1}{2}-\frac{1}{\mu})-\frac{1}{m_1^2}(\frac{1}{2} +\frac{1}{\mu})T(\sum_{j=1}^pb_j)]>0$. \end{itemize} Then problem \eqref{e3.1} has at least one nontrivial solution. \end{theorem} \begin{proof} Obviously, $J(0)=0$. By (F3) and condition (A), there exist nonnegative functions $d_1(t)$, $d_2(t)\in L^1(0,T)$ such that $d_1(t)>0$ a.e. on $[0,T]$ and $F(t,u)\geq d_1(t)|u|^{\mu}-d_2(t)$ for a.e. $t\in[0,T]$ and all $u \in \mathbb{R}$. Choosing $u\in H\backslash\{0\}$, then for $q>0$, we have \begin{align*} J(qu) &\leq \frac{q^2}{2}\|u\|^2_1 +\sum_{j=1}^p\int_0^{|qu(t_j)|}|I_j(s)|ds-\int_0^TF(t,qu)dt\\ &\leq \frac{q^2}{2}\|u\|^2_1+C q\|u\|+C q^2\|u\|^2 -\int_0^T(d_1(t)|qu|^{\mu}-d_2(t))dt\\ &\leq \frac{q^2}{2}\|u\|^2_1+C q\|u\|_1+C q^2\|u\|_1^2 -|q|^{\mu} \int_0^T d_1(t)|u|^{\mu}dt+C\\ &\to -\infty \end{align*} as $q\to\infty$. Setting $e=qu$, then for $q$ large, we obtain $\|e\|_1>R$ and $J(e)\leq0$. By (F2) and (I3), for some proper $\varepsilon>0$, there exists $00$ such that for $\|u\|_1=\rho$, we have $J(u)>\sigma$. Let $\{u_k\}$ be a PS sequence in $H$; that is, $$|J({u}_k)|\leq C,\; \forall k\in \mathbb{N},\quad |(J'(u_k),h)|\leq o(1)\|h\|_1\quad \text{for all }k\in \mathbb{N}\text{ and }h\in H.$$ We only need to prove that $\{u_k\}$ is bounded. For $k$ large, \begin{align*} C+\frac{1}{\mu}\|u_k\|_1 &\geq J({u}_k)-\frac{1}{\mu}J'(u_k)u_k=(\frac{1}{2} -\frac{1}{\mu})\int_0^T(|u'_k|^2+\lambda u_k^2)dt\\ &\quad +\sum_{j=1}^p\int_0^{u_k(t_j)}I_j(s)ds -\frac{1}{\mu}\sum_{j=1}^pI_j(u_k(t_j))u_k(t_j) -\int_0^TF(t,u_k)dt\\ &\quad +\frac{1}{\mu}\int_0^Tf(t,u_k)u_kdt, \end{align*} where \begin{align*} -\int_0^TF(t,u_k)dt+\frac{1}{\mu}\int_0^Tf(t,u_k)u_k\,dt &=-\frac{1}{\mu}\int_0^T(\mu F(t,u_k)dt-f(t,u_k)u_k)dt \\ &= -\frac{1}{\mu}\int_{|u_k|\geq R}(\mu F(t,u_k)dt-f(t,u_k)u_k)dt\\ &\quad -\frac{1}{\mu}\int_{|u_k|\leq R}(\mu F(t,u_k)dt-f(t,u_k)u_k)dt. \end{align*} The above first term is nonnegative by (F3), the second is bounded by the condition (A). Moreover, \begin{align*} & |\sum_{j=1}^p\int_0^{u_k(t_j)}I_j(s)ds -\frac{1}{\mu}\sum_{j=1}^pI_j(u_k(t_j))u_k(t_j)|\\ &\leq |\sum_{j=1}^p\int_0^{u_k(t_j)}I_j(s)ds| +\frac{1}{\mu}|\sum_{j=1}^pI_j(u_k(t_j))u_k(t_j)| \\ &\leq (\sum_{j=1}^pa_j)\|u_k\|_{c}+\frac{1}{2} (\sum_{j=1}^pb_j)\|u_k\|_{c}^2+ \frac{1}{\mu}(\sum_{j=1}^pa_j)\|u_k\|_{c}+\frac{1}{\mu} (\sum_{j=1}^pb_j)\|u_k\|_{c}^2\\ &\leq (1+\frac{1}{\mu})\sqrt{T}(\sum_{j=1}^pa_j)\|u_k\| +(\frac{1}{2}+\frac{1}{\mu})T(\sum_{j=1}^pb_j)\|u_k\|^2\\ &\leq \frac{1}{m_1}(1+\frac{1}{\mu})\sqrt{T} (\sum_{j=1}^pa_j)\|u_k\|_1+\frac{1}{m_1^2}(\frac{1}{2} +\frac{1}{\mu})T(\sum_{j=1}^pb_j)\|u_k\|^2_1. \end{align*} Hence \begin{align*} C+\frac{1}{\mu}\|u_k\|_1 &\geq (\frac{1}{2}-\frac{1}{\mu})\|u_k\|_1^2 -\frac{1}{m_1}(1+\frac{1}{\mu})\sqrt{T}(\sum_{j=1}^pa_j)\|u_k\|_1 \\ &\quad -\frac{1}{m_1^2}(\frac{1}{2}+\frac{1}{\mu})T(\sum_{j=1}^pb_j)\|u_k\|_1^2\\ &= [(\frac{1}{2}-\frac{1}{\mu}) -\frac{1}{m_1^2}(\frac{1}{2}+\frac{1}{\mu}) T(\sum_{j=1}^pb_j)]\|u_k\|_1^2\\ &\quad -\frac{1}{m_1}(1+\frac{1}{\mu})\sqrt{T} (\sum_{j=1}^pa_j)\|u_k\|_1. \end{align*} Therefore, by the condition (I*), $\{u_k\}$ is bounded in $H$. By the Mountain Pass Lemma \cite{MW,RA}, $J(u)$ possesses a critical point $u\in H$ such that $J(u)\geq \sigma>0$; hence $u$ is a nontrivial weak solution of \eqref{e3.1}. \end{proof} \section{Resonance case} In this section, we study the existence of solutions for the resonant impulsive problem $$\label{e4.1} \begin{gathered} u''+\lambda_0u=f(t,u), \quad\text{a.e. }t\in (0,T),\\ u(0)=0,\quad u'(T)=0, \\ \Delta u'(t_j)=I_j(u(t_j)), \quad j=1,2,\dots,p \end{gathered}$$ where $f(t,u)$ satisfies condition (A), $I_j:\mathbb{R}\to \mathbb{R}$, $j=1,2,\dots, p$, are continuous functions. The corresponding functional $J:H\to \mathbb{R}$ is defined by $$J(u)=\frac{1}{2}\int_0^Tu'^2dt-\frac{\lambda_0}{2} \int_0^Tu^2dt+\sum_{j=1}^p\int_0^{u(t_j)}I_j(s)ds +\int_0^TF(t,u)dt.$$ We will consider the case where the nonlinearity satisfies some kind of sublinear and generalized Ahmad, Lazer, Paul type coercive conditions; e.g. see \cite{ZH} and \cite{HZ} and the references therein for some applications to the periodic boundary value problems. Decompose $H$ as $H=\bar{H}\oplus\widetilde{H}$, where $\bar{H}=\operatorname{span}\{\phi_0\}$ and $\widetilde{H}=\overline{\operatorname{span}\{\phi_1,\phi_2,\dots\}}$. For all $u\in H$, we write as $u=\bar{u}\oplus\widetilde{u}$, $\bar{u}\in\bar{H}$, $\widetilde{u}\in\widetilde{H}$. We recall the inequality, $$\int_0^T\widetilde{u}'^2dt\geq\lambda_1\int_0^T\widetilde{u}^2dt$$ for all $\widetilde{u}\in\widetilde{H}$. \begin{theorem}\label{thm4.1} Suppose that $f(t,u)$ satisfies condition (A). Moreover, the following conditions hold. \begin{itemize} \item[(F4)] There exists $\alpha$ with $\frac{1}{2}\leq\alpha<1$ such that $|f(t,u)|\leq g(t)|u|^{\alpha}+h(t)$, where $g(t),h(t)\in L^{1}(0,T)$; \item[(F5)] $\lim_{|\bar{u}|\to\infty}\int_0^TF(t,\bar{u})dt /|\bar{u}|^{2\alpha}=+\infty$, $\bar{u}\in\bar{H}$; \item[(I4)] $|I_j(u)|\leq p_j+q_j|u|^{\gamma_j}$ where $p_j,q_j\geq 0$, $j=1,2,\dots,p$, and $0\leq\gamma_j\leq2\alpha-1$. \end{itemize} Then problem \eqref{e4.1} has at least one solution. \end{theorem} \begin{proof} For all $u\in H$, $u=\bar{u}\oplus\widetilde{u}$, $\bar{u}\in\bar{H}$, $\widetilde{u}\in\widetilde{H}$, we have $$\label{e4.2} \begin{split} |\sum_{j=1}^p\int_0^{u(t_j)}I_j(s)ds| &\leq \sum_{j=1}^p\int_0^{|u(t_j)|}|I_j(s)|ds \leq\sum_{j=1}^p\int_0^{|u(t_j)|}(p_j+q_j|s|^{\gamma_j})ds\\ &\leq (\sum_{j=1}^pp_j)\|u\|_{c}+\sum_{j=1}^pq_j \frac{1}{\gamma_j+1}\|u\|^{\gamma_j+1}\\ &\leq C|\bar{u}|+C\|\widetilde{u}\|+C|\bar{u}|^{\gamma+1} +C\|\widetilde{u}\|^{\gamma+1}+C, \end{split}$$ where $\gamma=\max\{\gamma_j, j=1,2,\dots,p\}$. By (F4), we have $$\begin{split} \int_0^T(F(t,u)-F(t,\bar{u}))dt &\leq \int_0^T[\int_0^{1}f(t,\bar{u}+s\widetilde{u}) \widetilde{u}ds]dt\\ &\leq \int_0^T[\int_0^{1}(g(t)|\bar{u}+s\widetilde{u}|^{\alpha} +h(t))\widetilde{u}ds]dt\\ &\leq C|\bar{u}|^{\alpha}\|\widetilde{u}\|_{c}+C\|\widetilde{u} \|_{c}^{\alpha+1}+C\|\widetilde{u}\|_{c}\\ &\leq C|\bar{u}|^{\alpha}\|\widetilde{u}\|+C\|\widetilde{u}\|^{\alpha+1} +C\|\widetilde{u}\|\\ &\leq \varepsilon\|\widetilde{u}\|^2+C(\varepsilon) |\bar{u}|^{2\alpha}+C\|\widetilde{u}\|^{\alpha+1} +C\|\widetilde{u}\|. \end{split}\label{e4.3}$$ Then \begin{align*} J(u)&= \frac{1}{2}\int_0^Tu'^2dt-\frac{\lambda_0}{2}\int_0^Tu^2dt +\sum_{j=1}^p\int_0^{u(t_j)}I_j(s)ds\\ &\quad +\int_0^T(F(t,u)-F(t,\bar{u}))dt+\int_0^TF(t,\bar{u})dt\\ &\geq \frac{1}{2}\int_0^T|\widetilde{u}'|^2dt-\frac{\lambda_0}{2}\cdot\frac{1}{\lambda_1}\int_0^T|\widetilde{u}'|^2dt-C|\bar{u}|-C\|\widetilde{u}\| -C|\bar{u}|^{\gamma+1}-C\|\widetilde{u}\|^{\gamma+1}\\ &\quad -\varepsilon\|\widetilde{u}\|^2 -C(\varepsilon)|\bar{u}|^{2\alpha}-C\|\widetilde{u}\|^{\alpha+1} -C\|\widetilde{u}\|+\int_0^TF(t,\bar{u})dt-C\\ &\geq \big[\frac{1}{2}(1-\frac{\lambda_0}{\lambda_1}) -\varepsilon\big]\|\widetilde{u}\|^2-C\|\widetilde{u}\|^{\alpha+1} -C\|\widetilde{u}\|^{\gamma+1}-C\|\widetilde{u}\| -C(\varepsilon)|\bar{u}|^{2\alpha}-C\\ &\quad +\int_0^TF(t,\bar{u})dt\\ &\geq \big[\frac{1}{2}(1-\frac{\lambda_0}{\lambda_1}) -\varepsilon\big]\|\widetilde{u}\|^2-C\|\widetilde{u}\|^{\alpha+1} -C\|\widetilde{u}\|^{\gamma+1}-C\|\widetilde{u}\| +|\bar{u}|^{2\alpha}[-C(\varepsilon)\\ &\quad +\frac{1}{|\bar{u}|^{2\alpha}}\int_0^TF(t,\bar{u})dt]-C, \end{align*} where we have used $0\leq\gamma\leq 2\alpha-1$. Fixing some $0<\epsilon<\frac{1}{2}(1-\frac{\lambda_0}{\lambda_1})$, by (F5), we have $J(u)\to+\infty$ as $\|u\|\to\infty$. Noticing also the weak lower semi-continuity of $J(u)$, we complete the proof. \end{proof} \begin{theorem} \label{thm4.2} Suppose that {\rm (F4), (I4)} and the following condition are satisfied: \begin{itemize} \item[(F6)] $\lim_{|\bar{u}|\to\infty}\int_0^TF(t,\bar{u})dt/|\bar{u}|^{2\alpha} =-\infty$, $\bar{u}\in\bar{H}$. \end{itemize} Then problem \eqref{e4.1} has at least one solution. \end{theorem} \begin{proof} We apply the saddle point theorem \cite{MW,RA} to prove the theorem. Step 1. For $u=\bar{u}\in \bar{H}$, \begin{align*} J(\bar{u}) &= \frac{1}{2}\int_0^T|\bar{u}'|^2dt -\frac{\lambda_0}{2}\int_0^T|\bar{u}|^2dt -\sum_{j=1}^p\int_0^{\bar{u}(t_j)}I_j(s)ds +\int_0^TF(t,\bar{u})dt\\ &\leq C|\bar{u}|+C|\bar{u}|^{\gamma+1}+\int_0^TF(t,\bar{u})dt+C\\ &\leq C|\bar{u}|^{2\alpha}+\int_0^TF(t,\bar{u})dt+C\\ &= |\bar{u}|^{2\alpha}(C+\frac{1}{|\bar{u}|^{2\alpha}} \int_0^TF(t,\bar{u})dt)+C. \end{align*} Hence, we have $J(\bar{u})\to-\infty$ as $|\bar{u}|\to\infty$. Step 2. For $u=\widetilde{u}\in \widetilde{H}$, we have \begin{align*} \int_0^T(F(t,\widetilde{u})-F(t,0))dt &= \int_0^T[\int_0^{1}f(t,s\widetilde{u})\widetilde{u}ds]dt\\ &\leq \int_0^T[\int_0^{1}(g(t)|s\widetilde{u}|^{\alpha} +h(t))\widetilde{u}ds]dt\\ &\leq C\|\widetilde{u}\|^{\alpha+1}+C\|\widetilde{u}\|. \end{align*} By some arguments in the proof of Theorem \ref{thm4.1}, we obtain \begin{align*} J(\widetilde{u}) &\geq \frac{1}{2}(1-\frac{\lambda_0}{\lambda_1}) \int_0^T|\widetilde{u}'|^2dt +\sum_{j=1}^p\int_0^{\widetilde{u}(t_j)}I_j(s)ds +\int_0^TF(t,\widetilde{u})dt\\ &\geq \frac{1}{2}(1-\frac{\lambda_0}{\lambda_1}) \int_0^T|\widetilde{u}'|^2dt-C\|\widetilde{u}\| -C\|\widetilde{u}\|^{\gamma+1}+\int_0^T(F(t,\widetilde{u}) -F(t,0))dt \\ &\quad +\int_0^TF(t,0)dt\\ &\geq \frac{1}{2}(1-\frac{\lambda_0}{\lambda_1})\|\widetilde{u}\|^2 -C\|\widetilde{u}\|-C\|\widetilde{u}\|^{\gamma+1} -C\|\widetilde{u}\|^{\alpha+1} -C\|\widetilde{u}\|-C. \end{align*} Since $\frac{1}{2}\leq\alpha<1$ and $0\leq\gamma\leq2\alpha-1$, we obtain that $J(u)$ is bounded below on $\widetilde{H}$. So there exists $R>0$ such that $$\sup_{u\in S_{R}}J(u)<\inf_{u\in \widetilde{H}}J(u)$$ where $S_{R}=\{u|~\|u\|=R,~~u\in \bar{H}\}$. Step 3. We show that $J$ satisfies the PS condition. Let $\{u_k\}$ be a PS sequence in $H$, then there exists constant $C$ such that $$|J({u}_k)|\leq C,\; \forall k\in \mathbb{N},\quad |(J'(u_k),h)|\leq o(1)\|h\|\text{ for all }k\in \mathbb{N}, \; h\in H.$$ Since \begin{align*} &|\sum_{j=1}^pI_j(u_k(t_j))\widetilde{u}_k(t_j)|\\ &\leq \sum_{j=1}^pp_j|\widetilde{u}_k(t_j)|+\sum_{j=1}^pq_j|\bar{u}_k(t_j)|^{\gamma_j}|\widetilde{u}_k(t_j)| +\sum_{j=1}^pq_j|\widetilde{u}_k(t_j)|^{\gamma_j+1}\\ &\leq C\|\widetilde{u}_k\|+C|\bar{u}_k|^{\gamma}\|\widetilde{u}_k\| +C\|\widetilde{u}_k\|^{\gamma+1} \end{align*} and \begin{align*}| \int_0^Tf(t,u_k)\widetilde{u}_kdt| &\leq \int_0^T(g(t)|u_k|^{\alpha}+h(t))|\widetilde{u}_k|dt\\ &\leq C|\bar{u}_k|^{\alpha}\|\widetilde{u}_k\| +C\|\widetilde{u}_k\|^{\alpha+1}+C\|\widetilde{u}_k\|, \end{align*} we have, for $k$ large, \begin{align*} \|\widetilde{u}_k\| &\geq |(J'(u_k),\widetilde{u}_k)|=|\int_0^Tu'_k\widetilde{u}'_kdt-\lambda_0\int_0^Tu_k\widetilde{u}_kdt -\sum_{j=1}^pI_j(u_k(t_j))\widetilde{u}_k(t_j)\\ &\quad +\int_0^Tf(t,u_k)\widetilde{u}_kdt|\\ &\geq (1-\frac{\lambda_0}{\lambda_1})\|\widetilde{u}_k\|^2 -C\|\widetilde{u}_k\|-C|\bar{u}_k|^{\gamma}\|\widetilde{u}_k\| -C\|\widetilde{u}_k\|^{\gamma+1}\\ &\quad -C|\bar{u}_k|^{\alpha}\|\widetilde{u}_k\| -C\|\widetilde{u}_k\|^{\alpha+1}-C\|\widetilde{u}_k\|. \end{align*} Therefore, we obtain $$\|\widetilde{u}_k\|\leq C|\bar{u}_k|^{\gamma} +C|\bar{u}_k|^{\alpha}+C\leq C|\bar{u}_k|^{\alpha}+C.\label{e4.4}$$ Hence, noticing the arguments in \eqref{e4.2} and \eqref{e4.3}, we have \begin{align*} J(u_k) &= \frac{1}{2}\int_0^T|\tilde{u}_k'|^2dt -\frac{\lambda_0}{2}\int_0^T\tilde{u}_k^2dt +\sum_{j=1}^p\int_0^{u_k(t_j)}I_j(s)ds \\ &\quad +\int_0^T(F(t,u_k)-F(t,\bar{u}_k))dt +\int_0^TF(t,\bar{u}_k)dt\\ &\leq \frac{1}{2}\|\widetilde{u}_k\|^2 +C|\bar{u}_k|+C\|\widetilde{u}_k\| +C|\bar{u}_k|^{\gamma+1}+C\|\widetilde{u}_k\|^{\gamma+1}\\ &\quad +\varepsilon\|\widetilde{u}_k\|^2 +C(\varepsilon)|\bar{u}_k|^{2\alpha} +C\|\widetilde{u}_k\|^{\alpha+1}+C\|\widetilde{u}_k\| +\int_0^TF(t,\bar{u}_k)dt\\ &\leq (\frac{1}{2}+\varepsilon)\|\widetilde{u}_k\|^2 +C\|\widetilde{u}_k\|^{\gamma+1}+C\|\widetilde{u}_k\|^{\alpha+1} +C\|\widetilde{u}_k\|+C|\bar{u}_k| +C|\bar{u}_k|^{\gamma+1}\\ &\quad +C(\varepsilon)|\bar{u}_k|^{2\alpha} +\int_0^TF(t,\bar{u}_k)dt+C. \end{align*} From \eqref{e4.4}, we obtain \begin{gather*} \|\widetilde{u}_k\|^2\leq (C|\bar{u}_k|^{\alpha}+C)^2 \leq C|\bar{u}_k|^{2\alpha}+C,\\ \|\widetilde{u}_k\|^{\gamma+1}\leq (C|\bar{u}_k|^{\alpha} +C)^{\gamma+1}\leq C|\bar{u}_k|^{2\alpha}+C,\\ \|\widetilde{u}_k\|^{\alpha+1}\leq (C|\bar{u}_k|^{\alpha}+C)^{\alpha+1} \leq C|\bar{u}_k|^{2\alpha}+C. \end{gather*} So \begin{align*} J(u_k)&\leq C(\varepsilon)|\bar{u}_k|^{2\alpha} +\int_0^TF(t,\bar{u}_k)dt+C\\ &\leq |\bar{u}_k|^{2\alpha}[C(\varepsilon) +\frac{1}{|\bar{u}_k|^{2\alpha}}\int_0^TF(t,\bar{u}_k)dt]+C. \end{align*} Hence, if $\{|\bar{u}_k|\}$ has a unbounded subsequence, we will get a contradiction by ($F_6$). Therefore, $\{|\bar{u}_k|\}$ is bounded and moreover $\{u_k\}$ is bounded in $H$. By a standard argument, $\{u_k\}$ has a convergent subsequence. We obtain that $J(u)$ satisfies the PS condition. Then existence of a critical point for $J$ then follows from the saddle point theorem. The proof is complete. \end{proof} Examining carefully the proofs of the Theorems \ref{thm4.1} and \ref{thm4.2}, we can also get the following two theorems. \begin{theorem} \label{thm4.3} If the conditions of Theorem \ref{thm4.1} are replaced by the following conditions: \begin{itemize} \item[(F4')] There exists $\alpha$ with $0\leq\alpha<1$ such that $|f(t,u)|\leq g(t)|u|^{\alpha}+h(t)$, where $g(t),h(t)\in L^{1}(0,T)$; \item[(I4')] $|I_j(u)|\leq p_j+q_j|u|^{\gamma_j}$, where $p_j,q_j\geq 0$, $j=1,2,\dots,p$, $0\leq\gamma_j<1$; \item[(F5')] $\lim_{|\bar{u}|\to\infty,\bar{u}\in\bar{H} }(\int_0^TF(t,\bar{u})dt)/(|\bar{u}|^{\beta})=+\infty$, where $\beta=\max\{\gamma+1,2\alpha\}$, $\gamma=\max\{\gamma_j$, $j=1,2,\dots,p\}$. \end{itemize} Then problem \eqref{e4.1} has at least one solution. \end{theorem} \begin{theorem} \label{thm4.4} Suppose that the conditions {\rm (F4'), (I4')} and the following condition are satisfied: \begin{itemize} \item[(F5'')] $\lim_{|\bar{u}|\to\infty}(\int_0^TF(t,\bar{u})dt)/(|\bar{u}|^{\beta}) =-\infty$, $\bar{u}\in\bar{H}$ where $\beta=\max\{\gamma+1,2\alpha\}$. \end{itemize} Then problem \eqref{e4.1} has at least one solution. \end{theorem} \begin{thebibliography}{00} \bibitem{AR}Ravi P. Agarwal, Donal O'Regan; \emph{Multiple nonnegative solutions for second order impulsive differential equations}. Appl. Math. Comput. 114 (2000) 51-59. \bibitem{CTY} Jinhai Chen, Christopher C. Tisdell, Rong Yuan; \emph{On the solvability of periodic boundary value problems with impulse}. J. Math. Anal. Appl. 331 (2) (2007) 902-912. \bibitem{CS}Lijing Chen, Jitao Sun; \emph{Nonlinear boundary value problem for first order impulsive functional differential equations}. J. Math. Anal. Appl. 318 (2006) 726-741. \bibitem{CN} Jifeng Chu, Juan J. Nieto; \emph{Impulsive periodic solutions of first-order singular differential equations}. Bull. London Math. Soc. 40 (1) (2008) 143-150. \bibitem{CH} C. De Coster, P. Habets; \emph{Upper and lower solutions in the theory of ODE boundary value problems}. classical and recent results, in: F. Zanolin (Ed.), Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations CISM-ICMS, vol. 371, Springer, New York, 1996, pp. 1-78. \bibitem{ZH} Zhiqing Han; \emph{$2\pi$-periodic solutions to n-dimensional systems of Duffing's type}. J. Qingdao Univ. 7 (1994) 19-26. \bibitem{GD} Zhiming He, Jianshe Yu; \emph{Periodic boundary value problem for first-order impulsive functional differential equations}. J. Comput. Appl. Math. 138 (2002) 205-217. \bibitem{RMN} R. Iannacci, M. N. Nkashama; \emph{Unbounded Perturbations of Forced Second Order Ordinary Differential Equations at Resonance}. J.Differential Equations 69 (1987) 289-309. \bibitem{VL} V. Lakshmikantham, D. D. Bainov, P. S. Simeonov; \emph{Theory of impulsive differential equations}. Series in Modern Applied Mathematics, Volume 6 (World Scientific, Teaneck, NJ, 1989). \bibitem{MW} J. Mawhin, M. Willem; \emph{Critical Point Theory and Hamiltonian Systems}. Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1989. \bibitem{NJA} Juan J. Nieto; \emph{Basic theory for nonresonance impulsive periodic problems of first order}. J. Math. Anal. Appl. 205 (1997) 423-433. \bibitem{NJ} Juan J. Nieto, Donal O'Regan; \emph{Variational approach to impulsive differential equations}. Nonlinear Anal. RWA 10 (2009) 680-690. \bibitem{QL} Dingbian Qian, Xinyu Li; \emph{Periodic solutions for ordinary differential equations with sublinear impulsive effects}. J. Math. Anal. Appl. 303 (2005) 288-303. \bibitem{RA} P. Rabinowitz; \emph{Minimax Methods in Critical Point Theory with Applications to Differential Equations}. AMS, Providence, RI, 1986. \bibitem{SP} A. M. Samoilenko, N. A. Perestyuk; \emph{Impulsive Differential Equations}. World Science, Singapore, 1995. \bibitem{TYG} Yu Tian, Weigao Ge; \emph{Applications of variational methods to boundary value problem for impulsive differential equations}. Proc. Edinburgh Math. Soc. 51 (2008) 509-527. \bibitem{TYW} Yu Tian, Weigao Ge; \emph{Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations}. Nonlinear Anal. 72 (2010) 277-287. \bibitem{HZ} Hao Zhang, Zhixiang Li; \emph{Variational approach to impulsive differential equations with periodic boundary conditions}. Nonlinear Anal. 11 (2010) 67-78. \bibitem{ZR} Ziheng Zhang, Rong Yuan; \emph{An application of variational methods to Dirichlet boundary value problem with impulses}. Nonlinear Anal. 11 (2010) 155-162. \end{thebibliography} \end{document}