\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 141, pp. 1--9.\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{9mm}} \begin{document} \title[\hfilneg EJDE-2011/141\hfil Bifurcation from infinity] {Bifurcation from infinity and multiple solutions for first-order periodic boundary-value problems} \author[Z. Wang, C. Gao \hfil EJDE-2011/141\hfilneg] {Zhenyan Wang, Chenghua Gao} \address{Zhenyan Wang \newline Department of Mathematics, Northwest Normal University, Lanzhou 730070, China} \email{wangzhenyan86714@163.com} \address{Chenghua Gao \newline Department of Mathematics, Northwest Normal University, Lanzhou 730070, China} \email{gaokuguo@163.com} \thanks{Submitted September 6, 2011. Published October 28, 2011.} \subjclass[2000]{34B18} \keywords{First-order periodic problems; Landsman-Lazer type condition; \hfill\break\indent Leray-Schauder degree; bifurcation; existence} \begin{abstract} In this article, we study the existence and multiplicity of solutions for the first-order periodic boundary-value problem \begin{gather*} u'(t)-a(t)u(t)=\lambda u(t)+g(u(t))-h(t), \quad t\in (0, T),\\ u(0)=u(T). \end{gather*} \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The first-order periodic differential equation $$u'(t)=a(t)u(t)-f(u(t-\tau(t)))$$ has been proposed as models for a variety of physiological processes and conditions including production of blood cells, respiration, and cardiac arrhythmias, see \cite{GBN,MG,WL}. Thus, the existence of periodic solutions of this periodic differential equation has been discussed by several authors; see for example \cite{AC,CZ,JW,MCC,M2,MS,PS1,PS2,WJX,W1,ZC} and the references therein. In these articles, the condition $\int_0^Ta(t)dt\neq0$ is used for showing the existence of solutions. A natural question is what would happen if $\int_0^Ta(t)dt=0$. It is easy to check that if $\int_0^Ta(t)dt=0$, then the equation $$-u'(t)+a(t)u(t)=0, \quad u(0)=u(T)$$ has nontrivial solutions. Thus, the operator $Lu=-u'(t)+a(t)u(t)$ is not invertible. In this article, using Leray-Schauder degree and bifurcation techniques and under the condition that $\int_0^Ta(t)dt=0$, we discuss the existence and multiplicity of solutions for the problem \begin{gather} u'(t)-a(t)u(t)=\lambda u(t)+g(u(t))-h(t), \quad t\in (0,T),\label{e1.1}\\ u(0)=u(T),\label{e1.2} \end{gather} where $g:\mathbb{R}\to\mathbb{R}$ is continuous, $h\in L^1(0, T)$, and the parameter $\lambda$ is close to $0$ which is the eigenvalue of $$-u'(t)+a(t)u(t)=\lambda u(t), \quad u(0)=u(T).$$ In this article, we use the following assumptions: \begin{itemize} \item[(H1)] $a(\cdot)\in C[0, T]$ and $\int^T_0a(t)dt=0$; \item[(H2)] $g:\mathbb{R}\to\mathbb{R}$ is continuous, and there exist $\alpha\in[0,1)$, $p,q\in(0,\infty)$, such that $$|g(u)|\leq p|u|^\alpha+q,\quad u\in\mathbb{R};$$ \item[(H3)] There exist constants $A, a, R, r$ such that $r<00>\lambda_-$ such that \begin{itemize} \item[(i)] \eqref{e1.1}, \eqref{e1.2} has at least one solution if $\lambda\in[0,\lambda_+]$; \item[(ii)] \eqref{e1.1}, \eqref{e1.2} has at least three solutions if $\lambda\in[\lambda_-,0)$. \end{itemize} \end{theorem} \begin{theorem} \label{thm1.2} Assume that {\rm (H1), (H2), (H3'),(H4')} hold. Then there exists $\lambda_+, \lambda_-$ with $\lambda_+>0>\lambda_-$ such that \begin{itemize} \item[(i)] \eqref{e1.1}, \eqref{e1.2} has at least one solution if $\lambda\in[\lambda_-,0]$; \item[(ii)] \eqref{e1.1}, \eqref{e1.2} has at least three solutions if $\lambda\in(0,\lambda_+]$. \end{itemize} \end{theorem} The rest of the paper is arranged as follows. In section 2, we discuss the Lyapunov-Schmidt procedure for \eqref{e1.1}, \eqref{e1.2}. In section 3, the existence of solutions of \eqref{e1.1}, \eqref{e1.2} is discussed under `Landesman-Lazer' type conditions. \section{Lyapunov-Schmidt procedure} Let $X,Y$ be the Banach spaces $C[0, T]$, $L^1[0,T]$ with the norm $\|x\|=\max\{|x(t)|:t\in [0,T]\}$, $\|u\|_1=\int_0^T|u(s)|ds$, respectively. Define linear operator $L: D(L)\subset X\to Y$ by $$Lu=-u'+a(t)u, u\in D(L),\label{e2.1}$$ where $D(L)=\{u\in W^{1,1}(0, T):u(0)=u(T)\}$. Let $N: X\to X$ be the nonlinear operator defined by $$(Nu)(t)=g(u(t)), \quad t\in[0, T],\; u\in D(L).\label{e2.2}$$ It is easy to see that $N$ is continuous. Note that \eqref{e1.1}, \eqref{e1.2} is equivalent to $$Lu+\lambda u+Nu=h, u\in D(L).\label{e2.3}$$ \begin{lemma} \label{lem2.1} Let $L$ be defined by \eqref{e2.1}. Then \begin{gather*} \ker L=\{x \in X: x(t)=c\psi(t):c\in\mathbb{R}\},\\ \operatorname{Im} L=\{y\in Y:\int_0^T\frac{y(s)}{\psi(s)}ds=0\}. \end{gather*} \end{lemma} \begin{proof} It is easy to see that $\ker L=\{c\psi(t):c\in\mathbb{R}\}$. The following will prove that $\operatorname{Im}L=\{y\in Y:\int_0^T\frac{y(s)}{\psi(s)}ds=0\}$. If $y\in\operatorname{Im}L$, then there exists $u\in D(L)$ such that $-u'(t)+a(t)u(t)=y(t)$. So $$u(t)=u(0)\psi(t)-\int_0^ty(s)e^{\int_s^ta(\tau)d\tau}ds.$$ Combining with $u(0)=u(T)$, we have $$\int_0^T\frac{y(s)}{\psi(s)}ds=0.$$ On the other hand, if $y\in Y$ satisfies $\int_0^T\frac{y(s)}{\psi(s)}ds=0$, then we set $$u(t):=-\int_0^ty(s)e^{\int_s^ta(\tau)d\tau}ds.$$ It is not difficult to prove that $x\in D(L)$ and $Lu=y$. \end{proof} Define operator $P: X\to \ker L$, $$(Pu)(t)=u(0)\psi(t), \quad u\in X. \label{e2.4}$$ Let $Q: Y\to Y$ be such that $$(Qy)(t)=\frac{1}{T}\psi(t)\int_0^T\frac{y(s)}{\psi(s)}ds.\label{e2.5}$$ Denote $X_1=\{u\in X:u(0)=0\}$. \begin{lemma} \label{lem2.2} Let operators $P$ and $Q$ be defined by \eqref{e2.4} and \eqref{e2.5}. Then $$X=X_1\oplus \ker L, \quad Y=\operatorname{Im}L\oplus\operatorname{Im}Q.$$ We define linear operator $K: \operatorname{Im}L\to D(L)\cap X_1$ $$(Ky)(t)=-\int_0^ty(s)e^{\int_s^ta(\tau)d\tau}ds, \quad y\in \operatorname{Im}L,\label{e2.6}$$ satisfying $K=L_p^{-1}$, where $L_p=L|_{D(L)\cap X_1}$. \end{lemma} \begin{proof} Let $y_1(t)=y(t)-(Qy)(t), y\in Y$, then it is easy to verify that $y_1 \in\operatorname{Im}L$. Thus $Y=\operatorname{Im}L + \operatorname{Im}Q$. Also $\operatorname{Im}L\cap\operatorname{Im}Q= \{0\}$. Hence $Y=\operatorname{Im}L\oplus\operatorname{Im}Q$. If $u\in D(L)\cap X_1$, then $$(KL_pu)(t)=K\big(-u'(t)+a(t)u(t)\big)=u(t).$$ If $y\in \operatorname{Im}L$, then $$(L_pKy)(t)=-\big(-\int_0^ty(s)e^{\int_s^ta(\tau)d\tau}ds\big)'-a(t)\int_0^ty(s)e^{\int_s^ta(\tau)d\tau}ds=y(t).$$ This indicates $K=L_p^{-1}$. \end{proof} Therefore, for every $u\in X$ , we have a unique decomposition $u(t) = \rho\psi(t) + v(t), t\in[0, T]$, where $\rho\in\mathbb{R}, v\in X_1$. Similarly, for every $h\in Y$, we have unique decomposition $h(t)=\tau\psi(t)+\bar{h}(t), t\in[0, T]$, where $\tau\in\mathbb{R}, \bar{h}\in \operatorname{Im}L$. The operator $Q, K$ be defined as \eqref{e2.5}, \eqref{e2.6}. Then $K(I-Q)N: X\to X$ is completely continuous, and \eqref{e2.3} is equivalent to the system \begin{gather} v(t)+\lambda Kv(t)+K(I-Q)N(\rho\psi(t)+v(t))=K\bar{h}(t),\label{e2.7}\\ \lambda\rho\psi(t)+QN(\rho\psi(t)+v(t))=\tau\psi(t).\label{e2.8} \end{gather} \begin{lemma}[\cite{IN}] \label{lem2.3} Assume that {\rm (H2), (H3)} hold. Then for each real number $s > 0$, there exists a decomposition $g(u)=q_s(u) + g_s(u)$ of $g$ by $q_s$ and $g_s$ satisfying the conditions: \begin{gather} uq_s(u)\geq 0, u\in \mathbb{R},\label{e2.9} \\ |q_s(u)|\leq p|u|+q+s, u\geq 1,\label{e2.10}, \end{gather} there exists $\sigma_s$ depending on $a, A$ and $g$ such that $$|g_s(u)| \leq\sigma_s, u\in \mathbb{R}.\label{e2.11}$$ \end{lemma} \begin{lemma} \label{lem2.4} Assume that {\rm (H1)--(H4)} hold, and $\lambda$ satisfies $$0\leq\lambda\leq\eta_1:=\frac{1}{2\|K\|_{\operatorname{Im}L\to X_1}}.\label{e2.12}$$ Then there exists constant $R_0>0$ such that any solution $u$ of \eqref{e1.1} \eqref{e1.2} satisfies $\|u\| 0$ and choose $B\in\mathbb{R}$ such that $$(b+1)|\frac{1}{u}|\leq\frac{1}{4}\bar{\delta}\label{e2.15}$$ for all $u\in\mathbb{R}$ with $|u|\geq B$. It follows from \eqref{e2.14} and \eqref{e2.15} that $$0\leq q_1(u)u^{-1}\leq p+\frac{1}{4}\bar{\delta}\label{e2.16}$$ for all $u\in\mathbb{R}$ with $|u|\geq B$. \textbf{Step 2.} Let us define $\gamma:\mathbb{R}\to\mathbb{R}$ by \gamma(u)= \begin{cases} u^{-1}q_1(u), &|u|\geq B;\\ B^{-1}q_1(B)(\frac{u}{B})+(1-\frac{u}{B})p, &0\leq u0that $$\frac{1}{\psi(s)}g(\rho_n\psi(s)+v_n(s))\geq \frac{1}{\psi(s)}\hat{K}(s), s\in[0,T].$$ Thus, applying Fatou's lemma to \eqref{e2.25}, we have \begin{align*} \tau&\geq \frac{1}{T}\liminf_{n\to\infty}\int_0^T \frac{g(\rho_n\psi(s)+v_n(s))}{\psi(s)}ds\\ &\geq \frac{1}{T}\int_0^T\liminf_{n\to\infty} \frac{g(\rho_n\psi(s)+v_n(s))}{\psi(s)}ds\\ &\geq \frac{1}{T}\int_0^T\frac{g_{+\infty}}{\psi(s)}ds. \end{align*} This contradicts with (H4). \end{proof} \begin{lemma} \label{lem2.4'} Assume that {\rm (H1), (H2), (H3'), (H4')} hold, and\lambda$satisfies $$0\leq\lambda\leq\eta_1:=\frac{1}{2\|K\|_{\operatorname{Im}L\to X_1}}.$$ Then there exists constant$R_0>0$such that any solution$u$of \eqref{e1.1} \eqref{e1.2} satisfy$\|u\|0$such that$\|u\|R_1$, $$\deg(L+\lambda I+N-h, B(R), 0)=\deg(L+\delta I, B(R), 0)=\pm 1.$$ \end{proof} \begin{lemma} \label{lem3.1'} Assume that {\rm (H1), (H2), (H3'),(H4')} hold. Then there exists$R_1: R_1\geq R_0$such that for$0\leq\lambda\leq\delta$, and$R\geq R_1$one has $$\deg(L+\lambda I+N-h, B(R), 0)=\deg(L+\delta I, B(R), 0)=\pm 1,$$ where$B(R)=\{u\in C[0, T]: \|u\|0$. Choosing sufficiently small$\mu>0$such that$\mu R<\tau_0$, then if$\lambda\in[-\mu, \mu]$, $$\deg(L+\lambda I+N-h, B(R), 0)=\deg(L+N-h, B(R), 0).$$ Combined with Lemma \ref{lem3.1}, the result can be proved. That is to see that if$\lambda\in[-\mu, \delta]$, \eqref{e2.3} has at least one solution in$\bar{B}(R)$. \end{proof} \begin{lemma} \label{lem3.2'} Assume that {\rm (H1), (H2), (H3')(H4')} hold. Then there exists$\mu\geq 0$such that for$-\mu\leq\lambda\leq 0$, one has $$\deg(L+\lambda I+N-h, B(R), 0)=\deg(L+\delta I, B(R), 0)=\pm 1,$$ where$R$be defined in Lemma \ref{lem3.1}. Then \eqref{e1.1}, \eqref{e1.2} has a solution in$B(R)$for$-\mu\leq\lambda\leq\delta$. \end{lemma} \begin{remark} \label{rmk1} \rm Since$g$is L-completely continuous and satisfies (H2) and since$\lambda=0$is a simple eigenvalue of L, it follows from bifurcation results of \cite{IN} that there exist two connected sets$\mathcal{C}_+, \mathcal{C}_-\subset \mathbb{R}\times X$of solutions of \eqref{e1.1}, \eqref{e1.2} such that for all sufficiently small$\epsilon>0$, $$\mathcal{C}_+\cap U_\epsilon\neq\emptyset,\quad \mathcal{C}_-\cap U_\epsilon\neq\emptyset,$$ where$U_\epsilon:=\{(\lambda, u)\in\mathbb{R}\times X, |\lambda|<\epsilon, \|u\|>1/\epsilon\}$. \end{remark} \begin{proof}[Proof of Theorem \ref{thm1.1}] Set$\lambda^+=\delta$, then it follows from Lemma \ref{lem3.1} and Lemma \ref{lem3.2} that \eqref{e1.1}, \eqref{e1.2} has at least one solution in$B(R)$for$\lambda\in[-\mu, \lambda_+]$. On the other hand, Remark \ref{rmk1} shows that there exists two connected sets$\mathcal{C}+$and$\mathcal{C}-$of solutions of \eqref{e1.1}, \eqref{e1.2} bifurcating from infinity at$\lambda=0$. Hence by Lemma \ref{lem2.4}, the connected sets$\mathcal{C}+$and$\mathcal{C}-$of Remark \ref{rmk1} must satisfy $$\mathcal{C}_+, \mathcal{C}_-\subset\{(\lambda, u): \|u\|\geq1/\epsilon, -\mu<\lambda<0\}.$$ and hence, if$1/\epsilon \geq R$; i.e.,$\epsilon\leq 1/k$. Choosing$\lambda_{-}=\max\{-\mu, -1/k\}$, we obtain two solutions$u_1, u_2: u_1\in\mathcal{C}_+, u_2\in\mathcal{C}_-$, and$\|u_i\|\geq R$($i=1, 2\$). \end{proof} Theorem \ref{thm1.2} can be proved by a similar method. \begin{thebibliography}{99} \bibitem{AC} R. P. Agarwal, Jinhai Chen; \emph{Periodic solutions for first order differential systems}. Appl. Math. Lett. 23 (2010), no. 3, 337-341. \bibitem{CZ} S. Cheng, G. 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