\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 102, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/102\hfil Uniqueness of traveling wave solutions] {Uniqueness of traveling wave solutions for non-monotone cellular neural networks with distributed delays} \author[H.-L. Zhou, Z. Yu \hfil EJDE-2017/102\hfilneg] {Hui-Ling Zhou, Zhixian Yu} \address{Hui-Ling Zhou \newline College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China} \email{hlzhou@163.com} \address{Zhixian Yu (corresponding author) \newline College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China} \email{zxyu0902@163.com} \thanks{Submitted March 1, 2017. Published April 11, 2017.} \subjclass[2010]{35C07, 92D25, 35B35} \keywords{ Cellular neural network; uniqueness; non-monotone; \hfill\break\indent asymptotic behavior; distributed delays} \begin{abstract} In this article, we study the uniqueness of traveling wave solutions for non-monotone cellular neural networks with distributed delays. First we establish a priori asymptotic behavior of the traveling wave solutions at infinity. Then, based on Ikehara's theorem, we prove the uniqueness of the solution $\psi(n-ct)$ with $c\leq c_*$, where $c_*<0$ is the critical wave speed. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article, we study the uniqueness of traveling wave solution for the non-monotone cellular neural networks with distributed delays \begin{equation} \label{1.1} \begin{aligned} x'_n(t)&=-x_n(t)+\sum_{i=1}^m\int_0^\tau a_iJ_i(y)f(x_{n-i}(t-y))dy\\ &\quad +\alpha\int_0^\tau J_{m+1}(y)f(x_n(t-y))dy +\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}(y)f(x_{n+j}(t-y))dy, \end{aligned} \end{equation} where the constants $n\in Z$, $m,l\in N$, $\tau\geq 0$, and the varible $t\in \mathbb{R}$. We use the following assumptions: \begin{itemize} \item[(H0)] (i) $\alpha>0$, $a_1>0$, $a_i\geq 0$ ($i=2,\ldots,m$), $\beta_1>0$ and $\beta_j\geq0$ $(j=2,\ldots,l)$. $a=\sum_{i=1}^ma_i$ and $\beta=\sum_{j=1}^l\beta_j$. \\ (ii) $J_i:[0,\tau]\to (0,+\infty)$ is the piecewise continuous function satisfying $\int_0^{\tau}J_i(y)dy=1$, where $0<\tau<\infty$. \item[(H1)] $f\in C([0,b],[0,\frac{b}{a+\alpha+\beta}])$, $f(0)=0$, $\alpha f'(0)\geq 1$ and there exists $K>0$ with $K\leq b$ such that \[ (a+\alpha+\beta)f(K)=K,\quad |f(u)-f(v)|\leq f'(0)|u-v|\quad \text{for }u,v\in[0,b]. \] \item[(H2)] $(a+\alpha+\beta)f(u)>u$ for $u\in(0,K)$ and $(a+\alpha+\beta)f(u)0$, $\delta>0$ and $M>0$ such that \begin{equation*} f(u)\geq f'(0)u-Mu^{1+\sigma} \quad\text{for }u\in[0,\delta]. \end{equation*} \end{itemize} A traveling wave solution \eqref{1.1} with speed $c$ is a nonnegative bounded solution of the form $u_n(t) = \psi(n -ct)$ satisfying $\psi(-\infty)=0$ and $\liminf_{\xi\to +\infty}\psi(\xi)>0$. Substituting $u_n(t) = \psi(n -ct)$ in \eqref{1.1}, we have the wave profile equation \begin{equation} \label{1.2} \begin{aligned} -c\psi'(\xi) &=-\psi(\xi)+\sum_{i=1}^m\int_0^\tau a_iJ_i(y)f(\psi(\xi-i+cy))dy \\ &\quad +\alpha\int_0^\tau J_{m+1}(y)f(\psi(\xi+cy))dy \\ &\quad +\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}(y)f(\psi(\xi+j+cy))dy. \end{aligned} \end{equation} When the output function $f$ is monotone, the existence of traveling wave solutions for many versions of CNNs \eqref{1.1} with delays or without delays has been widely investigated. see for example \cite{Hsu-Li-Yang,Hsu-Lin-Shen,Hsu-Yang1,Hsu-Yang2,Hsu-Yang,Li-Huang, Li-Li,Liu-Weng-Xu,Weng-Wu,Wu-Hsu-12,Wu-Hsu-14,Yu-Mei,Yu-Yuan-T,Yu-Yuan-Hsu-Jiang}. The existence of entire solutions for \eqref{1.1} has been investigated by Wu and Hsu \cite{Wu-Hsu-12,Wu-Hsu-14}. Letting $J_i=\delta(y-\tau_i)$, $i=1,\dots,m+l+1$, \eqref{1.1} reduces to the multiple discrete delays equation \begin{equation} \label{0.2} \begin{aligned} w_n'(t)&=-w_n(t)+\sum_{i=1}^ma_if(w_{n-i}(t-\tau_i)) +\alpha f(w_n(t-\tau_{m+1})) \\ &\quad +\sum_{j=1}^l\beta_jf(w_{n+j}(t-\tau_{m+1+j})). \end{aligned} \end{equation} Yu and Mei \cite{Yu-Mei} investigated uniqueness and stability of traveling wave solutions for \eqref{0.2} with the monotone output function. In \cite{Yu-Mei} the authors used the technique in \cite{Carr-Chmaj} to study uniqueness of travelling wave soluitons for \eqref{0.2} with discrete delays. We will extend this method to \eqref{1.1} with distributed delays. For the non-monotone output function $f$, Yu et al.\ \cite{Yu-Yuan-Hsu-Peng} only established the existence of non-critical traveling wave solutions. Yu and Zhao \cite{Yu-Zhao} further established the existence of the spreading speed, its coincidence with the minimal wave speed and the existence of critical waves for the non-monotone DCNNs \eqref{1.1}. We summarize the existence of traveling wave solutions of \eqref{1.1} with the non-monotone output function in \cite{Yu-Yuan-Hsu-Peng,Yu-Zhao} as follows. \begin{proposition}\label{prop1.1} Assume that {\rm (H0)-(H3)} hold. Then there exists $c^*<0$ (which is given in Lemma \ref{lem2.1}) such that for any $c\leq c^*$, \eqref{1.1} admits a non-negative traveling wave solution $\psi(n-ct)$ with the wave speed $c^*<0$ and satisfying \begin{equation}\label{1.4} \psi(-\infty)=0\quad \text{and}\quad 0<\liminf_{\xi\to +\infty}\psi(\xi)\leq\limsup_{\xi\to +\infty}\psi(\xi)\leq b. \end{equation} \end{proposition} The uniqueness of monotone travelling wave solutions for various evolution systems has been established; see for example \cite{Aguerrea,Berestyski-Nirenberg,Chen,Chen-Fu-Guo,Ma-Wu, Ma-Zou} and the references therein. The proof of uniqueness strongly relies on the monotonicity of travelling waves. It seems very difficult to extend the techniques in those literatures to the non-monotone evolution systems because the wave profile may lose the monotonicity and the study of the corresponding uniqueness is very limited, see, e.g., \cite{Diekmann-Kaper,Fang-Zhao,Fang-Wei-Zhao1}. Recently, the authors in \cite{Yu-12,Yu-Mei-C13} extend the technique in \cite{Carr-Chmaj} to non-monotone lattice equations with discrete delays. In this article, we extend the technique in \cite{Carr-Chmaj} to non-monotone CNNs with distributed delays. The rest of this article is organized as follows. Section 2 is devoted to studying the asymptotic behavior of the traveling wave solutions. In Section 3, we prove the uniqueness of the solution. \section{Asymptotic behavior of traveling wave solutions} In this section, we consider the asymptotic behavior at negative infinity of any traveling wave solutions of \eqref{1.1}. The characteristic equation of \eqref{1.2} at 0 is \begin{equation} \label{2.1} \begin{aligned} \Delta(c,\lambda) &=-c\lambda+1-f'(0)\Big[\sum_{i=1}^m a_i\int_0^\tau J_i(y)e^{\lambda(-i+cy)}dy\\ &\quad +\alpha\int_0^\tau J_{m+1}(y)e^{\lambda cy}dy +\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}(y)e^{\lambda(j+cy)}dy\Big]. \end{aligned} \end{equation} \begin{lemma}[{\cite[Lemma 2.1]{Yu-Yuan-Hsu-Jiang}}] \label{lem2.1} Assume that {\rm(H0)} and $\alpha f'(0)\geq1$ hold. Then there exist a unique pair of $c_*<0$ and $\lambda_*>0$ such that \begin{itemize} \item[(i)] $\Delta(c_*,\lambda_*)=0$, $\frac{\partial\Delta(c,\lambda)}{\partial\lambda}|_{c=c_*,\lambda=\lambda_*}=0$; \item[(ii)] For any $c>c_*$ and $\lambda\in[0,+\infty)$, $\triangle(c,\lambda)<0$; \item[(iii)] For any $c0$. Moreover, if $c0$ for any $\lambda\in (\lambda_1,\lambda_2)$; if $c=c_*$, then $\lambda_1=\lambda_2=\lambda_*$. \end{itemize} \end{lemma} Now we give a different version of Ikehara's Theorem, which can be found in \cite{Carr-Chmaj}. \begin{proposition}\label{prop2.1} Let $F(\lambda):=\int_0^{+\infty}u(x)e^{-\lambda x}dx$, where $u(x)$ is a positive decreasing function. Assume $F(\lambda)$ can be written as \[ F(\lambda)=\frac{h(\lambda)}{(\lambda+\mu)^{k+1}}, \] where $k>-1$ and $h(\lambda)$ is analytic in the strip $-\mu\leq \Re\lambda<0$. Then \begin{align*} \lim_{x\to +\infty }\frac{u(x)}{x^k e^{-\mu x}}=\frac{h(-\mu)}{\Gamma(\mu+1)}. \end{align*} \end{proposition} \begin{remark}\label{remark2.1} \rm Changing the variable $t=-x$, and modifying the proof for Ikehara's Theorem given in \cite{Ellison}, we can show the following version of Proposition \ref{prop2.1}. Let $F(\lambda):=\int_{-\infty}^0 u(t)e^{-\lambda t}dt$, where $u(t)$ is a positive increasing function. Assume $F(\lambda)$ can be written as \begin{align*} F(\lambda)=\frac{h(\lambda)}{(\mu-\lambda)^{k+1}}, \end{align*} where $k>-1$ and $h(\lambda)$ is analytic in the strip $\mu-\epsilon< \Re\lambda\leq\mu$ for some $0<\epsilon<\mu$. Then \begin{align*} \lim_{x\to -\infty }\frac{u(x)}{|x|^k e^{\mu x}}=\frac{h(\mu)}{\Gamma(\mu+1)}. \end{align*} \end{remark} To apply Ikehara's Theorem, we need to assure that traveling wave solutions are positive. \begin{lemma}\label{lem2.2} Assume that {\rm (H0)--(H3)} hold and let $\psi(n-ct)$ be a non-negative traveling wave of \eqref{1.1} with $c\leq c_*$ satisfying \eqref{1.4}. Then $\psi(\xi)>0$ for $\xi\in\mathbb{R}$. \end{lemma} \begin{proof} Assume that there exists $\xi_0$ such that $\psi(\xi_0)=0$. Without loss of generality, we may assume $\xi_0$ is the left-most point. According to $\psi(\xi)\geq0$ for $\xi\in\mathbb{R}$, we can easily see that $\psi(\xi)$ attains the minimum at $\xi_0$ and $\psi'(\xi_0)=0$. According to (H0) and (H1), it follows from \eqref{1.2} that $$ \int_0^\tau J_{m+1}(y)f(\psi(\xi_0+cy))dy=0, $$ which implies that $f(\psi(\xi_0+cy))=0$ for any $y\in[0,\tau]$. Thus, choosing some sufficiently small number $y_0>0$, we can obtain $\psi(\xi_0+cy_0)=0$ according to the continuity of $\psi(\xi)$ and $c<0$. This contradicts to the choice of $\xi_0$, and completes the proof. \end{proof} \begin{lemma}\label{lem2.3} Assume that {\rm (H1)--(H3)} hold and let $\psi(n-ct)$ be any non-negative traveling wave of \eqref{1.1} with $c\leq c_*$ and satisfy \eqref{1.4}. Then there exists a positive number $\rho>0$ such that $\psi(\xi)=O(e^{\rho\xi})$ as $\xi\to -\infty$. \end{lemma} \begin{proof} Since $f'(0)(a+\alpha+\beta)>1$, there exists $\epsilon_0 >0$ such that \begin{align*} A:=(1-\epsilon_0)f'(0)(a+\alpha+\beta)-1>0. \end{align*} For such $\epsilon_0 >0$, there exist $\delta_1>0$ such that $f(u)\geq(1-\epsilon_0)f'(0)u$ for any $u\in[0,\delta_1]$. Since $\psi(-\infty)=0$, there exists $M>0$ and $\forall\xi\leq -M$ such that $\psi(\xi)<\delta_1 $. Integrating \eqref{1.2} from $\eta$ to $\xi $ with $\xi\leq-l-M$, it follows that \begin{equation} \label{2.2} \begin{aligned} &-c[\psi(\xi)-\psi(\eta)] \\ &=-\int_\eta^\xi \psi(x)dx+\sum_{i=1}^m a_i\int_\eta^\xi \int_0^\tau J_i(y)f(\psi(x-i+cy))\,dy\,dx \\ &\quad+\alpha\int_\eta^\xi \int_0^\tau J_{m+1}(y)f(\psi(x+cy))\,dy\,dx \\ &\quad +\sum_{j=1}^l \beta_j\int_\eta^\xi \int_0^\tau J_{m+1+j}(y)f(\psi(x+j+cy))\,dy\,dx \\ &\geq -\int_\eta^\xi \psi(x)dx+f'(0)(1-\epsilon_0)\Big[\sum_{i=1}^m a_i\int_\eta^\xi \int_0^\tau J_i(y)\psi(x-i+cy)\,dy\,dx \\ &\quad +\alpha\int_\eta^\xi \int_0^\tau J_{m+1}(y)\psi(x+cy)\,dy\,dx \\ &\quad +\sum_{j=1}^l \beta_j\int_\eta^\xi \int_0^\tau J_{m+1+j}(y)\psi(x+j+cy)\,dy\,dx\Big] \\ &=A\int_\eta^\xi \psi(x)dx+f'(0)(1-\epsilon_0)\Big[\alpha\int_\eta^\xi \int_0^\tau J_{m+1}(y)(\psi(x+cy)-\psi(x))\,dy\,dx \\ &\quad +\sum_{i=1}^m a_i\int_\eta^\xi \int_0^\tau J_i(y)(\psi(x-i+cy)-\psi(x))\,dy\,dx \\ &\quad +\sum_{j=1}^l \beta_j\int_\eta^\xi \int_0^\tau J_{m+1+j}(y)(\psi(x+j+cy)-\psi(x))\,dy\,dx\Big]. \end{aligned} \end{equation} Since $\psi(x)$ is differentiable, we have \begin{align*} \int_\eta^\xi(\psi(x-i+cy)-\psi(x))dx &=\int_\eta^\xi\int_0^{-i+cy}\psi'(x+s)dsdx\\ &=\int_0^{-i+cy}(\psi(\xi+s)-\psi(\eta+s))ds. \end{align*} Similarly, \begin{gather*} \int_\eta^\xi(\psi(x+cy)-\psi(x))dx=\int_0^{cy}(\psi(\xi+s)-\psi(\eta+s))ds,\\ \int_\eta^\xi(\psi(x+j+cy)-\psi(x))dx=\int_0^{j+cy}(\psi(\xi+s)-\psi(\eta+s))ds. \end{gather*} Letting $\eta\to -\infty$ in \eqref{2.2}, we obtain \begin{equation} \label{2.3} \begin{aligned} &A\int_{-\infty}^\xi\psi(x)dx \\ &\leq-c\psi(\xi)-f'(0)(1-\epsilon_0)\Big[\sum_{i=1}^m a_i\int_0^\tau\int_0^{-i+cy}J_i(y)\psi(\xi+s)\,ds\,dy\\ &\quad +\alpha\int_0^\tau\int_0^{cy}J_{m+1}(y)\psi(\xi+s)\,ds\,dy\\ &\quad +\sum_{j=1}^l \beta_j\int_0^\tau\int_0^{j+cy}J_{m+1+j}(y)\psi(\xi+s)\,ds\,dy\Big]. \end{aligned} \end{equation} From \eqref{2.3}, we know that $\int_{-\infty}^\xi\psi(x)dx<{+\infty}$. Letting $\Phi(\xi)=\int_{-\infty}^\xi\psi(x)dx $ and integrating \eqref{2.3} from ${-\infty}$ to $\xi$, we have \begin{equation} \label{2.4} \begin{aligned} &A\int_{-\infty}^\xi\Phi(x)dx \\ &\leq-c\Phi(\xi)-f'(0)(1-\epsilon_0)\Big[\sum_{i=1}^ma_i \int_0^\tau\int_0^{-i+cy}J_{i}(y)\Phi(\xi+s)\,ds\,dy\\ &\quad +\alpha\int_{0}^\tau\int_{0}^{cy}J_{m+1}(y)\Phi(\xi+s)\,ds\,dy \\ &\quad +\sum_{j=1}^l\beta_j\int_0^\tau\int_0^{j+cy}J_{m+1+j}(y)\Phi(\xi+s)\,ds\,dy \Big]\\ &\leq\varrho\Phi(\xi+\kappa) \end{aligned} \end{equation} for some $\kappa>0$ and $\varrho>0$ according to the monotonicity of $\Phi(\xi)$, Letting $\varpi>0$ such that $\varrho0$. Hence, \begin{align*} h(\xi-\varpi)=\Phi(\xi-\varpi)e^{-\rho(\xi-\varpi)} \leq\frac{\varrho}{A\varpi}e^{\rho(\kappa+\varpi)}h(\xi+\kappa)=h(\xi+\kappa), \end{align*} which implies $h$ is bounded. Therefore, $\Phi(\xi)=O(e^{\rho\xi})$ when $\xi\to {-\infty}$. Integrating \eqref{1.2} from $-\infty$ to $\xi$, it follows from (H2) that \begin{equation} \label{2.6} \begin{aligned} -c\psi(\xi) &=\sum_{i=1}^ma_i\int_{0}^\tau\int_{-\infty}^\xi J_i(y)f(\psi(x-i+cy))\,dx\,dy\\ &\quad +\alpha\int_0^\tau\int_{-\infty}^\xi J_{m+1}(y)f(\psi(x+cy))\,dx\,dy\\ &\quad +\sum_{j=1}^l\beta_j\int_0^\tau\int_{-\infty}^\xi J_{m+1+j}(y)f(\psi(x+j+cy)) \,dx\,dy-\Phi(\xi)\\ &\leq f'(0)\sum_{i=1}^ma_i\int_{0}^\tau\int_{-\infty}^\xi J_i(y)\psi(x-i+cy)\,dx\,dy\\ &\quad +\alpha f'(0)\int_0^\tau\int_{-\infty}^\xi J_{m+1}(y)\psi(x+cy)\,dx\,dy\\ &\quad +f'(0)\sum_{j=1}^l\beta_j\int_0^\tau\int_{-\infty}^\xi J_{m+1+j}(y) \psi(x+j+cy)\,dx\,dy-\Phi(\xi)\\ &=-\Phi(\xi)+f'(0)\sum_{i=1}^ma_i\int_{0}^\tau J_i(y)\Phi(\xi-i+cy)dy\\ &\quad +\alpha f'(0)\int_0^\tau J_{m+1}(y)\Phi(\xi+cy)dy\\ &\quad +f'(0)\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}(y)\Phi(\xi+j+cy)dy. \end{aligned} \end{equation} Thus, we have $\psi(\xi)=O(e^{\rho\xi})$ when $\xi\to -\infty$. With the help of Ikehara's theorem, we obtain the asymptotic behavior of traveling wave solutions at $-\infty$. \end{proof} \begin{proposition}\label{prop2.2} Assume that {\rm (H1)--(H3)} hold and let $\psi(n-ct)$ be any non-negative traveling wave of \eqref{1.1} with the wave speed $c\leq c_*$ and satisfy \eqref{1.4}. Then \begin{equation}\label{2.7} \lim_{\xi\to -\infty}\frac{\psi(\xi)}{e^{\lambda_1 \xi}}\, \text{ exists for } c< c_*,\quad \lim_{\xi\to -\infty}\frac{\psi(\xi)}{|\xi|e^{\lambda_* \xi}}\, \text{ exists for } c= c_*. \end{equation} \end{proposition} \begin{proof} According to Lemma \ref{lem2.3}, we define the two-sided Laplace transform of $\psi$ for $0<\Re\lambda<\rho$, \begin{align*} L(\lambda)\equiv\int_{-\infty}^{+\infty}\psi(x)e^{-\lambda x}dx. \end{align*} We claim that $L(\lambda)$ is analytic for $0<\Re\lambda<\lambda_1$ and has a singularity at $\lambda=\lambda_1$. Note that \begin{align*} &-c\psi'(\xi)+\psi(\xi)-f'(0)\sum_{i=1}^ma_i\int_0^\tau J_i(y)\psi(\xi-i+cy)dy\\ &-\alpha f'(0)\int_0^\tau J_{m+1}(y)\psi(\xi+cy)dy\\ &-f'(0)\sum_{j=1}^l\beta_j\int_0^\tau J_i{m+1+j}\psi(\xi+j+cy)dy\\ &=\sum_{i=1}^ma_i\int_0^\tau J_i(y)[f(\psi(\xi-i+cy))-f'(0)\psi(\xi-i+cy)]dy\\ &\quad +\alpha \int_0^\tau J_{m+1}(y)[f(\psi(\xi+cy))-f'(0)\psi(\xi-i+cy)]dy\\ &\quad +\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}[f(\psi(\xi+j+cy))-f'(0)\psi(\xi-i+cy)]dy\\ &=:Q(\psi)(\xi). \end{align*} Multiplying the two sides of the above equality by $e^{-\lambda\xi}$ and integrating $\xi$ on $\mathbb{R}$, we obtain \begin{align}\label{2.8} \Delta(c,\lambda)L(\lambda)=\int_{-\infty}^{+\infty}e^{-\lambda x}Q(\psi)(x)dx. \end{align} We know that the left-hand side of \eqref{2.8} is analytic for $0<\Re\lambda<\rho$. According to (H3), for any $\overline{u}>0$, there exists $\overline{d}>0$ such that $f(u)\geq f'(0)u-\overline{d}u^{\sigma+1}$, for all $u\in[0,\overline{u}]$, where $\overline{d}:=\max\{d,\gamma^{-(\sigma+1)} \max_{u\in[\gamma,\overline{u}]}\{f'(0)u-f(u)\}\}$. Thus, \begin{equation} \label{2.800} \begin{aligned} &-\overline{d}\Big[\sum_{i=1}^ma_i\int_0^\tau J_i(y)\psi^{\sigma+1}(\xi-i+cy)dy+\alpha\int_0^\tau J_{m+1}(y)\psi^{\sigma+1}(\xi+cy)dy \\ &+\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}\psi^{\sigma+1}(\xi+j+cy)dy\Big] \\ &\leq Q(\psi)(\xi)\leq0. \end{aligned} \end{equation} Choose $\upsilon>0$ such that $\frac{\upsilon}{\sigma}<\rho$. Then for any $\Re\lambda\in(0,\rho+\upsilon)$, we have % \label{2.9} \begin{align*} &\big|\int_{-\infty}^{+\infty}e^{-\lambda x}Q(\psi)(x)dx\big|\\ &\leq \overline{d}\int_{-\infty}^{+\infty}e^{-\lambda \xi} \Big[\sum_{i=1}^ma_i\int_0^\tau J_i(y)\psi^{\sigma+1}(\xi-i+cy)dy \\ &\quad +\alpha\int_0^\tau J_{m+1}(y)\psi^{\sigma+1}(\xi+cy)dy\\ &\quad +\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}\psi^{\sigma+1}(\xi+j+cy)dy\Big]d\xi\\ &=\overline{d}[\sum_{i=1}^ma_i\int_0^\tau J_i(y)e^{\lambda(-i+cy)}dy+\alpha\int_0^\tau J_{m+1}(y)e^{\lambda cy}dy\\ &\quad +\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}e^{\lambda(j+cy)}dy]\int_{-\infty}^{+\infty}e^{-\lambda x} \psi^{\sigma+1}(x)dx\\ &\leq \overline{d}\Big[\sum_{i=1}^ma_i\int_0^\tau J_i(y)e^{\lambda(-i+cy)}dy+\alpha\int_0^\tau J_{m+1}(y)e^{\lambda cy}dy\\ &\quad +\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}e^{\lambda(j+cy)}dy\Big]L(\lambda-\upsilon) \Big(\sup_{\xi\in\mathbb{R}}\psi(\xi)e^{\frac{-\upsilon\xi}{\sigma}} \Big)^\sigma<+\infty. \end{align*} We use properties of Laplace transform \cite[p. 58]{Widder}. Since $\psi>0$ according to Lemma \ref{lem2.2}, there exists a real number $D$ such that $L(\lambda)$ is analytic for $0<\Re\lambda0$. Indeed, notice that $H(\lambda_1)=G(\lambda_1)$. On the other hand, $\int_{-\infty}^{+\infty}e^{-\lambda_1 x}Q(\psi)(x)dx<0$ according to \eqref{2.800} and $\lim_{\lambda\to \lambda_1^-}\Delta(c,\lambda)/(\lambda_1-\lambda)^{k+1}<0$ according to Lemma \ref{lem2.1}. Since $\psi(\xi)$ may be non-monotone, Ikehara's Theorem could be directly used. Thus, we need to make a function transformation, i.e., $\widehat{\psi}(\xi)=\psi(\xi)e^{p\xi}$, where $p=\frac{-1}{c}>0$. It follows from \eqref{1.2} that \begin{align*} \widehat{\psi}'(\xi) &= \frac{-1}{c}\Big[\sum_{i=1}^m\int_0^\tau a_iJ_i(y)f(\psi(\xi-i+cy))dy +\alpha\int_0^\tau J_{m+1}(y)f(\psi(\xi+cy))dy\\ &\quad +\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}(y)f(\psi(\xi+j+cy))dy\Big]e^{p\xi}>0. \end{align*} Therefore, $\widehat{\psi}(\xi)$ is increasing and $\widehat{\psi}(\xi)>0$. Now we apply the Ikehara's Theorem to $\widehat{\psi}(\xi)$. Let \[ \widehat{F}(\lambda):=\int_{-\infty}^0\widehat{\psi}(x)e^{-\lambda x}dx =F(\lambda-p). \] Then \[ \widehat{F}(\lambda)=\frac{\widehat{H}(\lambda)}{((\lambda_1+p)-\lambda)^{k+1}}, \] where $\widehat{H}(\lambda)=H(\lambda-p)$ is analytic for $p<\Re\lambda\leq\lambda_1+p$ and $\widehat{H}(\lambda_1+p)=H(\lambda_1)>0$. According to Remark \ref{remark2.1}, the limits exists and \[ \lim_{\xi\to -\infty}\frac{\widehat{\psi}(\xi)}{|\xi|^ke^{(\lambda_1+p)\xi}} =\lim_{\xi\to -\infty}\frac{\psi(\xi)}{|\xi|^ke^{\lambda_1\xi}}, \] i.e., \[ \lim_{\xi\to -\infty}\frac{\psi(\xi)}{e^{\lambda_1 \xi}}\, \text{ exists for } c< c_*,\quad \lim_{\xi\to -\infty}\frac{\psi(\xi)}{|\xi|e^{\lambda_* \xi}} \text{ exists for } c= c_*. \] This completes the proof. \end{proof} \section{Uniqueness of traveling wave solutions} In this section, we show the following unique result of traveling wave solutions of \eqref{1.1}. \begin{theorem} \label{thm3.1} Assume that {\rm (H1)--(H3)} hold. Let $\psi(n-ct)$ be a traveling wave of \eqref{1.1} with the wave speed $c\leq c_*$, which is given in Proposition \ref{prop1.1}. If $\phi(n-ct)$ is any non-negative traveling wave of \eqref{1.1} with the same wave speed $c$ satisfying \eqref{1.4}, then $\phi$ is a translation of $\psi$; more precisely, there exists $\bar{\xi}\in\mathbb{R}$ such that $\phi(n-ct)=\psi(n-ct+\bar{\xi})$. \end{theorem} \begin{proof} From Proposition \ref{prop2.2}, there exist two positive numbers $\vartheta_1$ and $\vartheta_2$ such that \[ \lim_{\xi\to -\infty}\frac{\phi(\xi)}{|\xi|^k e^{\lambda_1 \xi}}=\vartheta_1, \quad \lim_{\xi\to -\infty}\frac{\psi(\xi)}{|\xi|^k e^{\lambda_1 \xi}}=\vartheta_2 \] where $k=0$ for $c0$, define \begin{equation} \label{3.1} \omega(\xi):=\frac{\phi(\xi)-\psi(\xi+\overline{\xi})}{e^{\lambda_1\xi}} \text{ for }c0 ,\quad \omega'(\xi_0)=0. \] We claim that for all $i,j\in\mathbb{Z}$, we have \[ \omega(\xi_0-i+cy)=\omega(\xi_0+cy)=\omega(\xi_0+j+cy)=\omega(\xi_0) \] for $y\in[0,\tau]$. Suppose on the contrary that one of three inequalities $\omega(\xi_0-i+cy)<\omega(\xi_0)$, $\omega(\xi_0+cy)<\omega(\xi_0)$ and $\omega(\xi_0+j+cy)<\omega(\xi_0)$ for some $i_0$, $j_0$ must hold. According to \eqref{1.2}, \eqref{3.1} and (H2), we obtain %\label{3.2} \begin{align*} 0&=c\omega'(\xi_0)\\ &=-c\lambda_1\omega(\xi_0)+\omega(\xi_0) \\ &\quad -e^{-\lambda_1\xi_0}\sum_{i=1}^ma_i\int_0^\tau J_i(y)[f(\phi(\xi_0-i+cy))-f(\psi(\xi_0+\overline{\xi}-i+cy))]dy\\ &\quad -e^{-\lambda_1\xi_0}\alpha\int_0^\tau J_{m+1}(y)[f(\phi(\xi_0+cy))-f(\psi(\xi_0+\overline{\xi}+cy))]dy\\ &\quad -e^{-\lambda_1\xi_0}\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}(y)[f(\phi(\xi_0+j+cy))-f(\psi(\xi_0+\overline{\xi}+j+cy))]dy\\ &>-c\lambda_1\omega(\xi_0)+\omega(\xi_0)-f'(0)\omega(\xi_0) \Big[\sum_{i=1}^ma_i\int_0^\tau J_i(y)e^{\lambda_1(-i+cy)}dy\\ &\quad +\alpha\int_0^\tau J_{m+1}(y)e^{\lambda_1cy}dy+\sum_{j=1}^l\beta_j \int_0^\tau J_{m+1+j}(y)e^{\lambda_1(cy+j)}dy\Big]\\ &=-\omega(\xi_0)\triangle(c,\lambda_1)=0, \end{align*} which is a contradiction. Thus, $\omega(\xi_0+cy_0)=\omega(\xi_0)$ also holds for $y_0\in (0,\tau)$. Again by bootstrapping, $\omega(\xi_0+kcy_0)=\omega(\xi_0)$ for all $k\in \mathbb{Z}$ and $\omega(+\infty)=0$. Therefore, we have $\phi(\xi)\equiv\psi(\xi+\overline{\xi})$ for $\xi\in \mathbb{R}$, which contradicts to $\omega(\xi)\not\equiv0$. Next, we consider $c=c_*$. Assume $\sup_{\xi\in \mathbb{R}}\{\omega(\xi)\}\geq |\inf_{\xi\in \mathbb{R}}\{\omega(\xi)\}|$. If $\omega_\epsilon(\xi)\neq0$, there exists $\xi_0^\epsilon$ such that \begin{align*} \omega_\epsilon(\xi_0^\epsilon)=\max_{\xi\in \mathbb{R}}\{\omega_\epsilon(\xi)\}=\sup_{\xi\in \mathbb{R}}\{\omega_\epsilon(\xi)\}>0 ,\quad \omega_\epsilon'(\xi_0^\epsilon)=0. \end{align*} Now we divide this part into three cases: \smallskip \noindent\textbf{Case 1:} Suppose that $\xi_0^\epsilon\to +\infty$ as $\epsilon\to 0$. It follows from \eqref{3.1} and \eqref{1.2} that \begin{align*} &c_*[\phi'(\xi_0^\epsilon)-\psi'(\xi_0^\epsilon+\overline{\xi})] \\ &= \phi(\xi_0^\epsilon)-\sum_{i=1}^ma_i\int_0^\tau J_i(y)[f(\phi(\xi_0^\epsilon-i+c_*y)) -f(\psi(\xi_0^\epsilon+\overline{\xi}-i+c_*y))]dy\\ &\quad -\alpha\int_0^\tau J_{m+1}(y)[f(\phi(\xi_0^\epsilon+c_*y)) -f(\psi(\xi_0^\epsilon+\overline{\xi}+c_*y))]dy -\psi(\xi_0^\epsilon+\overline{\xi})\\ &\quad -\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}(y) f(\phi(\xi_0^\epsilon+j+c_*y)) -f(\psi(\xi_0^\epsilon+\overline{\xi}+j+c_*y))]dy. \end{align*} We claim that for all $i,j\in\mathbb{Z}$, \begin{align*} \omega_\epsilon(\xi_0^\epsilon-i+c_*y)=\omega_\epsilon(\xi_0^\epsilon+c_*y)=\omega_\epsilon(\xi_0^\epsilon+j+c_*y)=\omega_\epsilon(\xi_0^\epsilon) \end{align*} for $y\in[0,\tau]$. Suppose for the contrary that one of three inequalities $\omega_\epsilon(\xi_0^\epsilon-i+c_*y)<\omega_\epsilon(\xi_0^\epsilon)$, $\omega_\epsilon(\xi_0^\epsilon+c_*y)<\omega_\epsilon(\xi_0^\epsilon)$ and $\omega_\epsilon(\xi_0^\epsilon+j+c_*y)<\omega_\epsilon(\xi_0^\epsilon)$ for some $i_0$ and $j_0$ must hold. Choose $\epsilon>0$ sufficiently small such that $\xi_0^\epsilon-m+c_*\tau>0$. Thus, % \label{3.3} \begin{align*} &-c_*\omega_\epsilon(\xi_0^\epsilon)\epsilon -c_*\lambda_*\omega_\epsilon(\xi_0^\epsilon)(\epsilon\xi_0^\epsilon+1)\\ &\leq-\omega_\epsilon(\xi_0^\epsilon)(\epsilon|\xi_0^\epsilon|+1) \\ &\quad +f'(0)\sum_{i=1}^ma_i\int_0^\tau J_i(y)e^{\lambda_*(-i+c_*y)}[\epsilon|\xi_0^\epsilon-i+c_*y|+1] \omega_\epsilon(\xi_0^\epsilon-i+c_*y)dy\\ &\quad +\alpha f'(0)\int_0^\tau J_{m+1}(y)e^{\lambda_*c_*y}[\epsilon|\xi_0^\epsilon+c_*y|+1] \omega_\epsilon(\xi_0^\epsilon+c_*y)dy\\ &\quad +f'(0)\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}(y)e^{\lambda_*(j+c_*y)}[\epsilon|\xi_0^\epsilon+j+c_*y|+1] \omega_\epsilon(\xi_0^\epsilon+j+c_*y)dy\\ &<-\omega_\epsilon(\xi_0^\epsilon)(\epsilon|\xi_0^\epsilon|+1) +f'(0)\sum_{i=1}^ma_i\int_0^\tau J_i(y)e^{\lambda_*(-i+c_*y)}[\epsilon|\xi_0^\epsilon-i+c_*y|+1]\omega_\epsilon(\xi_0^\epsilon)dy\\ &\quad +\alpha f'(0)\int_0^\tau J_{m+1}(y)e^{\lambda_*c_*y}[\epsilon|\xi_0^\epsilon+c_*y|+1]\omega_\epsilon(\xi_0^\epsilon)dy\\ &\quad +f'(0)\sum_{j=1}^l\beta_j\int_0^\tau J_{m+1+j}(y)e^{\lambda_*(j+c_*y)}[\epsilon|\xi_0^\epsilon+j+c_*y|+1] \omega_\epsilon(\xi_0^\epsilon)dy, \end{align*} it follows that \begin{equation} \label{3.4} \begin{aligned} &-c_*\omega_\epsilon(\xi_0^\epsilon)\epsilon+\omega_\epsilon(\xi_0^\epsilon)(\epsilon\xi_0^\epsilon+1)\Delta(c_*;\lambda_*)\\ &