\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 185, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/185\hfil Existence of traveling wavefronts] {Existence of traveling wavefronts for integrodifference equations with bilateral exponential kernel} \author[H. Peng, Z. Guo, Z. Wang \hfil EJDE-2016/185\hfilneg] {Huaqin Peng, Zhiming Guo, Zizi Wang} \address{Huaqin Peng \newline School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China} \email{huaqinpeng@126.com} \address{Zhiming Guo (corresponding author) \newline Key Laboratory of Mathematics and Interdisciplinary Science of Guangdong, Higher Education Institutes, Guangzhou University, Guangzhou 510006, China} \email{guozm@gzhu.edu.cn} \address{Zizi Wang \newline Key Laboratory of Mathematics and Interdisciplinary Science of Guangdong, Higher Education Institutes, Guangzhou University, Guangzhou 510006, China} \email{gdwangzizi@163.com} \thanks{Submitted October 15, 2015. Published July 12, 2016.} \subjclass[2010]{35C07, 39A12} \keywords{Integrodifference equation; traveling wavefronts; \hfill\break\indent upper and lower solutions} \begin{abstract} In this article, we study the existence of traveling wavefronts for integrodifference equation with a bilateral exponential kernel, namely, the Laplacian kernel. The existence of traveling wavefronts is proved by combining the monotone iteration technique with the upper and lower solution method. The minimal spreading speed $c^*$ is given, which can be figured out exactly when all parameters are given explicitly. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In 1937, a model for the spatial spread of an advantageous gene in a population living in a homogeneous one dimensional habitat was proposed by Fisher \cite{RAF}. In this model, the time evolution of the fraction $u(x,t)$ of the advantageous gene in the population at the point $x$ and at the time $t$ is governed by a partial differential equation of the form \begin{equation} \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + f(u), \label{1.1} \end{equation} where $f\in C^1[0,1]$ and $f(0)=f(1)=0$. In the same year, Kolmogorov, Petrovskii and Piskunov \cite{KPP} studied the same system, where $f\in C^2[0,1]$ and $f(0)=f(1)=0$. In previous few decades, there have been extensive investigations on traveling wave solutions and asymptotic behaviors in terms of spreading speeds for various evolution systems. Traveling waves were studied for nonlinear reaction-diffusion equations modeling physical and biological phenomena \cite{JDM1, JDM2, AV}, for lattice differential equation \cite{PXA, SJW, JZ, BGW} and for time-delayed reaction-diffusion equations \cite{KWS, HLX, HRX, JZ1}. Since the observation is often discontinuous, many discrete-time models are derived from different fields, such as difference equations \cite{VLG} and integrodifference equations \cite{HFW, HFW1}. As for the references mentioned above, much attention has been paid to the discrete-time model \begin{equation} u_{n+1}(x)= Q[u_n](x), \label{1.2} \end{equation} where $x\in \mathbb{H}\subseteq \mathbb{R}$, $\mathbb{H}$ is a habitat and $Q$ is a continuous mapping with respect to a proper topology. When we consider an organism with synchronous nonoverlapping generations, $u_n(x)$ can be viewed as the population density of the species at the point $x\in \mathbb{H}$ in population dynamics. System \eqref{1.2} implies that the evolution of the current individuals only depends on the individuals at the previous unit time or generation. When the life cycle of an organism consists of distinct growth and dispersal stages, and if these stages are synchronized within a population, the discrete-time models may be more accurate representations than continuous-time equation. Many plants, insect and migrating bird species in temperate climates fall into this category. Assume that there are two distinct stages that define the life cycle of these organisms, a sedentary stage and a dispersal stage. All growth occurs during the sedentary stage and all movement occurs during the dispersal stage. To formulate an integrodifference equation, when the population is continuously distributed, we denote the density of the population at time or generation $n$ at location $x$ as $u_{n}(x)$. The sedentary stage is described by some non-negative function $f(u)$, e.g. the Beverton-Holt Stock-recruitment curve \cite{RJHSJH} or the Ricker curve \cite{WER}, and the dispersal stage by a dispersal kernel, $k(x,y)$, where the product $k(x,y)d y$ is the probability that an individual who will move from the interval $(y,y+dy]$ to the point $x$ \cite{MMM}. The population density in the next generation is obtained by tallying arrivals at location $x$ from all possible locations $y$, or mathematically as the integral operator \begin{equation} u_{n+1}(x)= \int_{\Omega} k(x, y) f(u_n(y))dy, \label{1.3} \end{equation} where $\Omega$ is the habitat of the organism. If the environment is isotropic, one may hope that the kernels $k(x, y)$ is symmetric in $x$ and $y$, $k(x, y)=k(y, x)$. Dispersal tends to depend only on distance between source and destination, so the kernel may depend on absolute location or on relation distance. Let $k(x-y)$ be the spatial dispersal probability function of the species jumping from $y$ to $x$, then we obtain the following model \begin{equation} u_{n+1}(x)= \int_{\Omega} k(x-y) f(u_n(y))dy. \label{1.4} \end{equation} In the past three decades, the traveling wave solutions of \eqref{1.4} have been widely studied, we refer to Hsu and Zhao \cite{SBXQ}, Kot \cite{MK}, Liang and Zhao \cite{XXQ}, Neubert and Caswell \cite{MGH}, Weinberger \cite{HFW, HFW1}. In these papers, the monotonicity of the function $f$ plays a very important role. Recently, Lin and Li \cite{GWT1} and Lin et al \cite{GWS1} considered the existence of traveling wave solutions of a competitive system by a cross iteration scheme. In population dynamics, one typical integrodifference equation describing the age-structure and the birth function is (locally) monotone, then the traveling wave solutions and asymptotic spreading were studied by Lin and Li \cite{GWT} and Pan and Lin \cite{SG}. In 1992, Kot \cite{MK} studied the discrete-time traveling waves, when the integrodifference equation with the kernel being the bilateral exponential distribution $$ k(x,y)=\frac{1}{2}\alpha \exp(-\alpha |x-y|), $$ for a scalar equation with compensatory growth and two kinds of special recruitment cures. His research observed only simple traveling waves. However, in different biological systems there are kinds of recruitment cures, we cannot get the traveling waves following \cite{MK}. Motivated by the studies in \cite{XJS} and in \cite{GWS}, in this paper, we investigate the existence of traveling wavefronts to the following integrodifference equation \begin{equation} u_{n+1}(x)= \frac{\alpha}{2}\int_{\Omega} \exp(-\alpha |x-y|) f(u_n(y))dy. \label{1.5} \end{equation} For convenience, we only study the case that the environment $\Omega$ is $\mathbb{R}$. Thus equation \eqref{1.5} become \begin{equation} u_{n+1}(x)= \frac{\alpha}{2} \int_{\mathbb{R}} \exp(-\alpha |x-y|) f(u_n(y))dy. \label{1.6} \end{equation} The rest of this paper is organized as follows. In section 2, we obtain the existence of traveling wavefronts by using upper and lower solution method. In section 3, some numerical simulations are given to illustrate our main results. A brief conclusion will also be given in this section. \section{Existence of traveling wavefronts} In this section, we shall establish the existence of traveling wavefronts of \eqref{1.6} by combining the monotone iteration technique with the upper and lower solutions method. Let $$ C(\mathbb{R},\mathbb{R})=\{u|u: \mathbb{R}\to \mathbb{R} \text{ is uniformly continuous and bounded}\}. $$ Then $C(\mathbb{R}, \mathbb{R})$ is a Banach space equipped with supremum norm $|\cdot|$. If $a, b \in \mathbb{R}$ with $a< b$, then we denote $$ C_{[a, b]}=\{u\in C(\mathbb{R}, \mathbb{R}): a\leq u(x)\leq b \text{ for all } x\in \mathbb{R}\}. $$ Throughout the remainder of this paper, we assume that \begin{itemize} \item[(H1)] $f(0)=0$, $f(1)=1$, and $f(u)>u$ for any $u\in (0, 1)$. \item[(H2)] $f$ is a $C^2$ function and $0< f'(u)\leq f'(0)$ for $u \in [0, 1)$. \end{itemize} By (H2), there exists a constant $L > 0$ such that $| f''(u)| < f'(0)L$ for any $u \in [0, 1]$. \begin{definition} \label{def2.1} \rm A traveling wave solution of \eqref{1.6} is a special solution with the form $u_n(x)=\phi (x+cn)$, with $c>0$ is the wave speed that the wave profile $\phi\in C(\mathbb{R},\mathbb{R})$ spreads in $\mathbb{R}$. In particular, if $\phi(\xi)$ is monotone in $\xi\in \mathbb{R}$, then it is called a traveling wavefront. \end{definition} By Definition \ref{def2.1}, the traveling wavefront $\phi(\xi)$ of \eqref{1.6} must satisfy the integral equation \begin{equation} \begin{aligned} \phi(\xi+c) &= \frac{\alpha}{2} \int_{\mathbb{R}} \exp(-\alpha |x-y|) f(\phi(y+cn))dy\\ &= \frac{\alpha}{2} \int_{\mathbb{R}} \exp(-\alpha |x-y|) f(\phi(\xi-x+y))dy\\ &= \frac{\alpha}{2} \int_{\mathbb{R}} \exp(-\alpha |X|) f(\phi(\xi-X))dX, \end{aligned} \label{2.1} \end{equation} where $\xi=x+cn$, $X=x-y$. Because of the background of traveling wavefronts \cite{MK, MGH}, we also require that $\phi$ satisfies the asymptotic boundary value conditions, \begin{equation} \lim_{\xi\to -\infty} \phi(\xi)=0,\quad \lim_{\xi\to \infty} \phi(\xi)=1. \label{2.2} \end{equation} Thus, our intention is to prove the existence of a monotone solution of \eqref{2.1} with boundary value conditions \eqref{2.2}. For this purpose, we rewrite \eqref{2.1} as \begin{equation} \phi(\xi)=\frac{\alpha}{2} \int_{\mathbb{R}} \exp(-\alpha |x|) f(\phi(\xi-x-c))dx,\quad \xi\in \mathbb{R}. \label{2.3} \end{equation} The linearization of \eqref{2.3} in the neighborhood of $\phi=0$ is \begin{equation} \begin{aligned} \phi(\xi) &= \frac{\alpha}{2} \int_{\mathbb{R} }\exp(-\alpha |x|) f'(0)\phi(\xi-x-c)dx\\ &= \frac{f'(0) \alpha}{2} \int_{\mathbb{R}} \exp(-\alpha |x|) \phi(\xi-x-c)dx. \end{aligned}\label{2.4} \end{equation} One may attempt to find a solution of the form \begin{equation} \phi(\xi)= e^{\lambda \xi}, \label{2.5} \end{equation} where $\lambda$ is a positive number. Then $$ e^{\lambda \xi}=\frac{f'(0) \alpha}{2} \int_{\mathbb{R}} e^{-\alpha |x|} e^{\lambda(\xi-x-c)}dx. $$ Thus, $$ 1=\frac{f'(0) \alpha}{2} \int_{\mathbb{R}} e^{-\alpha |x|} e^{-\lambda(x+c)}dx. $$ We define \begin{equation} \Delta(\lambda, c)=\frac{f'(0) \alpha}{2} \int_{\mathbb{R} }e^{-\alpha |x|} e^{-\lambda(x+c)}dx \label{2.6} \end{equation} for any $\lambda\in (0, \alpha), c \in (0,\infty)$. Then $\Delta(\lambda, c)$ is well defined and the following result holds. \begin{lemma} \label{lem2.1} There exists a constant $c^\ast >0$ such that $\Delta(\lambda, c)=1$ has exactly two positive roots if $c>c^\ast$ while $\Delta(\lambda, c)=1$ has no real root if $cc^\ast$ holds and $\lambda_1(c)$ is the smaller root, $\lambda_2(c)$ is the other root, then for any $\eta \in (1, \frac{\lambda_2(c)}{\lambda_1(c)})$, $\Delta(\eta \lambda_1(c), c)<1$ holds. \end{lemma} \begin{proof} For any $\lambda\in (0, \alpha)$, \begin{align*} \Delta(\lambda, c) &= \frac{f'(0) \alpha}{2} \int_{-\infty}^{+\infty} e^{-\alpha |x|} e^{-\lambda(x+c)}dx\\ &= \frac{f'(0) \alpha}{2} \int_{-\infty}^{0} e^{\alpha x} e^{-\lambda(x+c)}dx + \frac{f'(0) \alpha}{2} \int_{0}^{+\infty} e^{-\alpha x} e^{-\lambda(x+c)}dx\\ &= \frac{f'(0) \alpha}{2e^{\lambda c}} \int_{-\infty}^{0} e^{(\alpha-\lambda) x}dx +\frac{f'(0) \alpha}{2e^{\lambda c}} \int_{0}^{+\infty} e^{-(\alpha+\lambda) x} dx\\ &= \frac{f'(0) \alpha}{2e^{\lambda c}}~\frac{1}{\alpha-\lambda} +\frac{f'(0) \alpha}{2e^{\lambda c}}~\frac{1}{\alpha+\lambda}\\ &= \frac{f'(0) \alpha^2}{e^{\lambda c}(\alpha-\lambda)(\alpha+\lambda)}. \end{align*} We see that $\Delta(\lambda, c)$ is continuous in $c>0$, $\lambda\in (0, \alpha)$. For fixed $c>0$, direct calculations show that \begin{gather*} \begin{aligned} \frac{\partial}{\partial \lambda}\Delta(\lambda, c) &=\frac{-cf'(0)\alpha^2e^{-\lambda c}(\alpha^2-\lambda^2) +2f'(0)\lambda \alpha^2e^{-\lambda c}} {(\alpha^2-\lambda^2)^2} \\ &=\frac{f'(0)\alpha^2[2\lambda-c(\alpha^2-\lambda^2)]} {e^{\lambda c}(\alpha^2-\lambda^2)^2}, \end{aligned} \\ \frac{\partial^2}{\partial \lambda^2}\Delta(\lambda, c) =\frac{f'(0)\alpha^2e^{-\lambda c}[(c(\alpha^2-\lambda^2)-2\lambda)^2 +4\lambda^2+2(\alpha^2-\lambda^2)]} {(\alpha^2-\lambda^2)^3}>0. \end{gather*} It is easy to see that $\Delta(\lambda, c)$ is convex in $\lambda\in (0, \alpha)$ for fixed $c>0$. Let $\frac{\partial}{\partial \lambda}\Delta(\lambda, c)=0$. Then $\lambda(c)=\frac{1}{c}(\sqrt{1+\alpha^2c^2}-1)$ attains the mimimun of $\Delta(\lambda, c)$ for fixed $c>0$. Also \begin{gather*} \Delta(\lambda(c), c)=\min_{\lambda\in (0, \alpha)}\Delta(\lambda, c) =\frac{f'(0)}{2}\exp[1-\sqrt{1+\alpha^2c^2}](\sqrt{1+\alpha^2c^2}+1). \\ \frac{d}{dc}\Delta(\lambda(c), c)=-\frac{c\alpha^2f'(0)}{2} \exp[1-\sqrt{1+\alpha^2c^2}]<0. \end{gather*} It means that $\Delta(\lambda(c), c)$ is strictly decreasing in $c$. Since $$ \lim_{c\to 0+}\Delta(\lambda(c), c)=f'(0)>1, \quad \lim_{c\to +\infty}\Delta(\lambda(c), c)= 0, $$ the continuity of $\Delta(\lambda(c), c)$ in $c$ implies that there exists unique $c^*$ such that $\Delta(\lambda(c^*), c^*)=1$. For any $c1$ for all $\lambda\in (0, \alpha)$, therefore, $\Delta(\lambda, c)=1$ has no real root. For $c>c^*$, $\Delta(\lambda(c), c)<1$. Since $$ \lim_{\lambda\to 0+}\Delta(\lambda, c)=f'(0)>1, \quad \lim_{\lambda\to \alpha-0}\Delta(\lambda, c)=+\infty, $$ and $\Delta(\lambda, c)$ is strictly deceasing in $\lambda\in(0, \lambda(c))$ and strictly increasing in $\lambda\in (\lambda(c), \alpha)$, then $\Delta(\lambda, c)=1$ has exactly two positive roots $\lambda_1(c)$ and $\lambda_2(c)$ with $\lambda_1(c)\in (0, \lambda(c))$, $\lambda_2(c)\in (\lambda(c), \alpha)$ and $\Delta(\lambda, c)<1$ for any $\lambda\in (\lambda_1(c), \lambda_2(c))$. This completes the proof. \end{proof} \begin{remark} \label{rmk2.1} \rm From the proof of Lemma \ref{lem2.1}, we know that $c^*$ can be formulated explicitly as follows. $$ c^*=\frac{\sqrt{(z^*)^2+2z^*}}{\alpha}, $$ where $z^*$ is the unique positive solution to the equation $$ \frac{1}{2}f'(0)(z+2)=e^z. $$ \end{remark} \begin{definition} \label{def2.2} \rm A continuous function $\phi(\xi)\in C_{[0,1]}$ is called an upper solution of \eqref{2.3}, if it satisfies $$ \phi(\xi)\geq\frac{\alpha}{2} \int_{\mathbb{R}} \exp(-\alpha |x|) f(\phi(\xi-x-c))dx,\quad \xi\in \mathbb{R}. $$ Similarly, a continuous function $\phi(\xi)\in C_{[0,1]}$ is called a lower solution of \eqref{2.3}, if it satisfies $$ \phi(\xi)\leq\frac{\alpha}{2} \int_{\mathbb{R} } \exp(-\alpha |x|) f(\phi(\xi-x-c))dx,\quad \xi\in \mathbb{R}. $$ \end{definition} For fixed $c>c^*$, let $q>1$, $\eta\in (1, \frac{\lambda_1(c)}{\lambda_2(c)})$ be given constants. We define continuous functions $\overline{\phi}(t)$ and $\underline{\phi}(t)$ as follows. \begin{gather*} \overline{\phi}(t)=\min\{1,e^{\lambda_1(c)t}+qe^{\eta \lambda_1(c)t}\}, \\ \underline{\phi}(t)=\max\{0,e^{\lambda_1(c)t}-qe^{\eta \lambda_1(c)t}\}. \end{gather*} It is easy to see that both $\overline{\phi}(t)$ and $\underline{\phi}(t)$ are continuous functions with $0<\overline{\phi}(t)\le 1$ and $0\le \underline{\phi}(t)<1$ for any $t\in (-\infty, +\infty)$. Clearly, there exists a constant $t^{\ast}<0$ such that $\overline{\phi}(t)$ is strictly increasing for $t< t^{\ast}$ and $\overline{\phi}(t)=1$ for $t\ge t^{\ast}$. Also there exists a constant $t_{*}<0$ such that $\underline{\phi}(t)>0$ for $t\frac{L\Delta(\eta\lambda_1(c),c)}{2(1-\Delta(\eta\lambda_1(c),c))}+1$, where $L$ satisfies $|f''(u)|\le f'(0)L$ for $u\in [0, 1]$. \end{proposition} \begin{proof} If $\underline{\phi}(t)=0$ for some $t$, then the result holds because $f(\underline{\phi}(t)) \geq 0$ for all $t \in \mathbb{R}$. If $\underline{\phi}(t)=e^{\lambda_1(c)t}-qe^{\eta \lambda_1(c)t}$ for some $t$, then, by Taylor expansion with Lagrangian remainder, we have by (H1) \begin{align*} &\frac{\alpha}{2} \int_{\mathbb{R}} e^{(-\alpha |x|)} f(\underline{\phi}(t-x-c))dx\\ &= \frac{\alpha}{2} \int_{\mathbb{R}} e^{(-\alpha |x|)} \left[f'(0)\underline{\phi}(t-x-c) +\frac{1}{2}f''(\theta\underline{\phi}(t-x-c))\underline{\phi}(t-x-c)^2\right]dx, \end{align*} with $0<\theta<1$. Then, \begin{align*} &\frac{\alpha}{2} \int_{\mathbb{R}} e^{(-\alpha |x|)} f(\underline{\phi}(t-x-c))dx\\ &\geq \frac{\alpha f'(0)}{2} \int_{\mathbb{R}} e^{(-\alpha |x|)}\underline{\phi}(t-x-c)dx -\frac{\alpha f'(0)L}{4} \int_{\mathbb{R}} e^{(-\alpha |x|)} \underline{\phi}(t-x-c)^2dx\\ &\geq \frac{\alpha f'(0)}{2} \int_{\mathbb{R}} e^{(-\alpha |x|)}[e^{\lambda_1(c)(t-x-c)} -qe^{\eta\lambda_1(c)(\xi-x-c)}]dx \\ &\quad -\frac{\alpha f'(0)L}{4} \int_{\mathbb{R}} e^{(-\alpha |x|)} \underline{\phi}(t-x-c)^{\eta}dx\\ &\ge e^{\lambda_1(c)t}\Delta(\lambda_1(c),c) -qe^{\eta\lambda_1(c)t}\Delta(\eta\lambda_1(c),c) -\frac{\alpha f'(0)L}{4} \int_{\mathbb{R}} e^{(-\alpha |x|)}e^{\eta\lambda_1(c)(t-x-c)}dx\\ &= e^{\lambda_1(c)t}-qe^{\eta\lambda_1(c)t}\Delta(\eta\lambda_1(c),c) -\frac{L}{2}e^{\eta\lambda_1(c)t}\Delta(\eta\lambda_1(c),c)\\ &\ge e^{\lambda_1(c)t}-qe^{\eta\lambda_1(c)t}=\underline{\phi}(t). \end{align*} This completes the proof. \end{proof} \begin{lemma} \label{lem2.2} Let $g(t)=e^{-\alpha|t|}$. Then $g$ is uniformly continuous on $\mathbb{R}$ \end{lemma} \begin{proof} Since $\lim_{|t|\to \infty}g(t)=0$, for any $\varepsilon>0$, there exists $K>0$, such that $g(t)<\frac{\varepsilon}{2}$ for $|t|>K$. The uniform continuity of $g$ on $[-K-1, K+1]$ means that there exists $\delta_1>0$, such that for any $t_1, t_2\in [-K-1, K+1]$, $|g(t_1)-g(t_2)|<\varepsilon$. Let $\delta=\min\{1, \delta_1\}$. Then for any $t_1, t_2\in \mathbb{R}, |t_1-t_2|<\delta$, we have $|g(t_1)-g(t_2)|<\varepsilon$. Then $g$ is uniformly continuous on $\mathbb{R}$. \end{proof} \begin{theorem} \label{thm2.1} Assume that $c>c^\ast$ holds. Then \eqref{2.3} with \eqref{2.2} has a monotone solution $\phi(t)$ such that $\lim_{t\to -\infty} \phi(t)e^{-\lambda_1(c)t}=1$. \end{theorem} \begin{proof} We now prove the result by standard iteration techniques \cite{OD, HRX}. According to Definition \ref{def2.2} and Proposition \ref{prop2.1}, we define continuous functions $\overline{\phi}_1(t)$ and $\underline{\phi}_1(t)$ as follows. \begin{gather*} \overline{\phi}_1(t)=\frac{\alpha}{2} \int_{\mathbb{R}} e^{(-\alpha |x|)} f(\overline{\phi}(t-x-c))dx,\quad t\in \mathbb{R},\\ \underline{\phi}_1(t)=\frac{\alpha}{2} \int_{\mathbb{R}} e^{(-\alpha |x|)} f(\underline{\phi}(t-x-c))dx,\quad t\in \mathbb{R}. \end{gather*} Then $\overline{\phi}_1(t)$, $\underline{\phi}_1(t)$ are well defined and $$ 1\ge \overline{\phi}(t)\geq \overline{\phi}_1(t) \geq \underline{\phi}_1(t)\geq \underline{\phi}(t)\ge 0,\quad t\in \mathbb{R}. $$ Let \begin{gather*} \overline{\phi}_{n+1}(t)=\frac{\alpha}{2} \int_{\mathbb{R}} e^{(-\alpha |x|)} f(\overline{\phi}_{n}(t-x-c))dx, \quad t\in \mathbb{R},\\ \underline{\phi}_{n+1}(t)=\frac{\alpha}{2} \int_{\mathbb{R}} e^{(-\alpha |x|)} f(\underline{\phi}_{n}(t-x-c))dx,\quad t\in \mathbb{R}, \end{gather*} for $n=1,2,\dots$. By mathematical induction and $(H2)$, we have $$ 1\ge\overline{\phi}_{n}(t)\geq \overline{\phi}_{n+1}(t)\geq \underline{\phi}_{n+1}(t)\geq \underline{\phi}_{n}(t)\ge 0,\quad t\in \mathbb{R}. $$ We rewrite $\overline{\phi}_{n}(t)$ and $\underline{\phi}_{n}(t)$ as follows. \begin{gather*} \overline{\phi}_{n}(t)=\frac{\alpha}{2} \int_{\mathbb{R}} e^{(-\alpha |t-y|)} f(\overline{\phi}_{n-1}(y-c))dy,\\ \underline{\phi}_{n}(t)=\frac{\alpha}{2} \int_{\mathbb{R}} e^{(-\alpha |t-y|)} f(\underline{\phi}_{n-1}(y-c))dy. \end{gather*} The monotonicity of $f$ and $\overline{\phi}$ means that $\overline{\phi}_{n}(t)$ is increasing in $t\in \mathbb{R}$ for each $n=1, 2, \dots$. For any given finite interval $[-M, M]$, we now prove the uniform convergence of sequence $\overline{\phi}_{n}$ on $[-M, M]$. For any $\varepsilon>0$, since $\frac{\alpha}{2}\int_{\mathbb{R}}e^{-\alpha|x|}dx=1$, there exists $K_1>0$, such that $$ \frac{\alpha}{2}\int_{|x|> K_1}e^{-\alpha|x|}dx<\frac{1}{4}\varepsilon. $$ By Lemma \ref{lem2.2}, there exists $\delta>0$, such that for any $t_1, t_2, |t_1-t_2|<\delta$, we have $|g(t_1)-g(t_2)|<\frac{\varepsilon}{2\alpha(M+K_1)}$. For any $t_1, t_2\in [-M, M], |t_1-t_2|<\delta$, we have \begin{align*} &|\overline{\phi}_{n}(t_1)-\overline{\phi}_{n}(t_2)|\\ &=|\frac{\alpha}{2} \int_{\mathbb{R}} e^{-\alpha |t_{1}-y|} f(\overline{\phi}_{n}(y-c))dy -\frac{\alpha}{2} \int_{\mathbb{R}} e^{-\alpha |t_{2}-y|} f(\overline{\phi}_{n}(y-c))dy|\\ &\leq\frac{\alpha}{2} \int_{\mathbb{R}} |(e^{-\alpha |t_{1}-y|} -e^{-\alpha |t_{2}-y|})|f(\overline{\phi}_{n}(y-c))dy\\ &\leq \frac{\alpha}{2} \int_{|y|>M+K_1} |(e^{-\alpha |t_{1}-y|} -e^{-\alpha |t_{2}-y|})|f(\overline{\phi}_{n}(y-c))dy\\ &\quad +\frac{\alpha}{2} \int_{-M-K_1}^{M+K_1} |(e^{-\alpha |t_{1}-y|} -e^{-\alpha |t_{2}-y|})|f(\overline{\phi}_{n}(y-c))dy\\ &\leq \alpha\int_{|x|>K_1} e^{-\alpha |x|}dx +\frac{\alpha}{2} \int_{-M-K_1}^{M+K_1} \frac{\varepsilon}{2\alpha(M+K_1)}dx\\ &=\varepsilon, \quad\text{for } n=1, 2, \dots. \end{align*} This implies that $\overline{\phi}_{n}(t)$ are equicontinuous for $n=1, 2, \dots$ and $t\in [-M, M]$. It is easy to know that $\overline{\phi}_{n}(t)$ converges to a nondecreasing continuous function uniformly on any compact subsets of $\mathbb{R}$. Let $\lim_{n\to \infty}\phi_n(t)=\phi(t)$. We claim that $\phi(t)$ satisfies $$ \phi(t)=\frac{\alpha}{2} \int_{\mathbb{R} }e^{(-\alpha |x|)} f(\phi(t-x-c))dx. $$ Actually, for a given $t$ and any $\varepsilon>0$, there exists $K_1>0$ such that $$ \frac{\alpha}{2} \int_{|x|>K_1}e^{(-\alpha |x|)}dx<\frac{\varepsilon}{4}. $$ By uniform convergence of $\{\overline{\phi}_n\}$ on $[t-c-K_1, t-c+K_1]$, there exists $N>0$ such that for any $n>N$ and $x\in [-K_1, K_1]$, $$ |f(\overline{\phi}_n(t-x-c))- f(\phi(t-x-c))|<\frac{\varepsilon}{2}. $$ Then \begin{align*} &\big|\frac{\alpha}{2} \int_{\mathbb{R} }e^{(-\alpha |x|)} f(\overline{\phi}_n(t-x-c))dx-\frac{\alpha}{2} \int_{\mathbb{R} }e^{(-\alpha |x|)} f(\phi(t-x-c))dx\big|\\ &\leq \frac{\alpha}{2} \int_{\mathbb{R} }e^{(-\alpha |x|)} |f(\overline{\phi}_n(t-x-c))-f(\phi(t-x-c))|dx\\ &\leq \alpha\int_{|x|>K_1}e^{(-\alpha |x|)}dx +\frac{\alpha}{2} \int_{-K_1}^{K_1}e^{(-\alpha |x|)} |f(\overline{\phi}_n(t-x-c))-f(\phi(t-x-c))|dx\\ &\leq \frac{\varepsilon}{2}+\frac{\varepsilon}{2} =\varepsilon \quad\text{for } n>N. \end{align*} Thus $\phi(t)$ is a monotone solution to \eqref{2.3}. To complete the proof of Theorem \ref{thm2.1}, we have to show that $\phi(t)$ satisfies \eqref{2.2}. The iteration scheme shows that $\overline{\phi}(t)\geq \phi(t)\geq \underline{\phi}(t), t\in \mathbb{R}$. By the monotonicity of $\phi(t)$, we obtain $$ \lim_{t\to -\infty} \phi(t)\in [0,\inf_{t\in \mathbb{R}}\overline{\phi}(t)], $$ By the properties of $\overline{\phi}(t)$, we obtain $\lim_{t\to -\infty} \phi(t)=0$. The monotonicity and the boundedness of $\phi$ imply the existence of $\lim_{t\to \infty}\phi(t)$. We claim that this limit is a constant solution of $\phi(t)=\frac{\alpha}{2} \int_{\mathbb{R} }e^{(-\alpha |x|)} f(\phi(t-x-c))dx$. Actually, let $\lim_{t\to\infty}\phi(t)=\phi^*$. Then by the continuity of $f$, for any $\varepsilon>0$, there exists $\delta>0$ such that $|f(u)-f(\phi^*)|<\frac{\varepsilon}{2}$ for $|u-\phi^*|<\delta$. Since $\lim_{t\to\infty}\phi(t)=\phi^*$, there exists $T>0$, such that $|\phi(t)-\phi^*|<\delta$ for $t>T$. Similar to the argument as above, there exists $K_1>0$, satisfies $$ \frac{\alpha}{2} \int_{x>K_1}e^{(-\alpha |x|)}dx<\frac{\varepsilon}{4}. $$ Then by the monotonicity of $\phi$ and $f$, for any $t>T+K_1+c$, \begin{align*} &\big|\frac{\alpha}{2} \int_{\mathbb{R} }e^{(-\alpha |x|)} f(\phi(t-x-c))dx-\frac{\alpha}{2} \int_{\mathbb{R} }e^{(-\alpha |x|)} f(\phi^*)dx\big|\\ &\leq \frac{\alpha}{2} \int_{K_1}^{\infty}e^{(-\alpha |x|)}| f(\phi(t-x-c))-f(\phi^*)|dx\\ &\quad +\frac{\alpha}{2} \int_{-\infty}^{K_1}e^{(-\alpha |x|)} |f(\phi(t-x-c))-f(\phi^*)|dx\\ &\leq \alpha\int_{K_1}^{\infty}e^{(-\alpha |x|)}dx +\frac{\alpha}{2} \int_{K_1}^{\infty}e^{(-\alpha |x|)}|f(\phi(t-K_1-c)) -f(\phi^*)|dx\leq\varepsilon. \end{align*} Thus we see that $$ \phi^*=\frac{\alpha}{2} \int_{\mathbb{R} }e^{(-\alpha |x|)} f(\phi^*)dx =f(\phi^*). $$ Clearly, $\phi^*=1$. From the definition of $\overline{\phi}(t)$ and $\underline{\phi}(t)$, we have \begin{gather*} \overline{\phi}(t)e^{-\lambda_1(c)t} = \begin{cases} 1+qe^{\lambda_1(c)t(\eta-1)} & t