\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 166, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/166\hfil Impulsive boundary-value problems] {Impulsive boundary-value problems for first-order integro-differential equations} \author[X. Wang, C. Bai \hfil EJDE-2010/166\hfilneg] {Xiaojing Wang, Chuanzhi Bai} \address{Xiaojing Wang \newline Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsi 223300, China} \email{wangxj2010106@sohu.com} \address{Chuanzhi Bai \newline Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsi 223300, China} \email{czbai8@sohu.com} \thanks{Submitted November 1, 2010. Published November 17, 2010.} \thanks{Supported by grant 10771212 from the National Natural Science Foundation of China} \subjclass[2000]{34A37, 34B15} \keywords{Impulsive integro-differential equation; \hfill\break\indent coupled lower-upper quasi-solutions; monotone iterative technique} \begin{abstract} This article concerns boundary-value problems of first-order nonlinear impulsive integro-differential equations: \begin{gather*} y'(t) + a(t)y(t) = f(t, y(t), (Ty)(t), (Sy)(t)), \quad t \in J_0, \\ \Delta y(t_k) = I_k(y(t_k)), \quad k = 1, 2, \dots , p, \\ y(0) + \lambda \int_0^c y(s) ds = - y(c), \quad \lambda \le 0, \end{gather*} where $J_0 = [0, c] \setminus \{t_1, t_2, \dots , t_p\}$, $f \in C(J \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}, \mathbb{R})$, $I_k \in C(\mathbb{R}, \mathbb{R})$, $a \in C(\mathbb{R}, \mathbb{R})$ and $a(t) \le 0$ for $t \in [0, c]$. Sufficient conditions for the existence of coupled extreme quasi-solutions are established by using the method of lower and upper solutions and monotone iterative technique. Wang and Zhang \cite{w} studied the existence of extremal solutions for a particular case of this problem, but their solution is incorrect. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In recent years, many authors have paid attention to the research of differential equations with impulsive boundary conditions, because of their potential applications; see for example \cite{b2, d1, h1, l2, l3, l5, n}. First-order and second-order impulsive differential equations with anti-periodic boundary conditions have also drawn much attention; see \cite{a1, a2, b1, c, d2, f,l4, l6, y}. Recently, Wang and Zhang \cite{w} studied the existence of extremal solutions of the following nonlinear anti-periodic boundary value problem of first-order integro-differential equation with impulse at fixed points $$\begin{gathered} y'(t) = f(t, y(t), (Ty)(t), (Sy)(t)), \quad t \in J_0, \\ \Delta y(t_k) = I_k(y(t_k)), \quad k = 1, 2, \dots , p, \\ y(0) = - y(T), \end{gathered} \label{e1.1}$$ where $J = [0, T]$, $J_0 = J \setminus \{t_1, t_2, \dots , t_p\}$, $0 < t_1 < t_2 < \dots < t_p < T$, $f \in C(J \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}, \mathbb{R})$, $I_k \in C(\mathbb{R}, \mathbb{R})$, $\Delta y(t_k) = y(t_k^+) - y(t_k^-)$ denotes the jump of $y(t)$ at $t = t_k$; $y(t_k^+)$ and $y(t_k^-)$ represent the right and left limits of $y(t)$ at $t = t_k$, respectively. $$(Ty)(t) = \int_0^t k(t, s)y(s) ds, \quad (Sy)(t) = \int_0^T h(t, s) y(s) ds,$$ $k \in C(D, \mathbb{R}^+)$, $D = \{(t, s) \in J \times J : t \ge s\}$, $h \in C(J \times J, \mathbb{R}^+)$. Unfortunately, their extremal solutions $y_*(t), y^*(t)$ are wrong. In fact, by \cite[Theorem 3.1]{w} we obtain \begin{gather*} y_{*}'(t) = f(t, y_*(t), (Ty_*)(t), (Sy_*)(t)), \quad t \in J_0, \\ \Delta y_*(t_k) = I_k(y_*(t_k)), \quad k = 1, 2, \dots , p, \\ y_*(0) = - y^*(T), \end{gather*} and \begin{gather*} y^{*\prime}(t) = f(t, y^*(t), (Ty^*)(t), (Sy^*)(t)), \quad t \in J_0, \\ \Delta y^*(t_k) = I_k(y^*(t_k)), \quad k = 1, 2, \dots , p, \\ y^*(0) = - y_*(T), \end{gather*} which implies that $y_*(t), y^*(t)$ are not solutions of \eqref{e1.1}. So the conclusions of \cite{w} are reconsidered here, for a more general equation. In this paper, we investigate the following integral boundary value problem for first-order integro-differential equation with impulses at fixed points $$\begin{gathered} y'(t) + a(t)y(t) = f(t, y(t), (Ty)(t), (Sy)(t)), \quad t \in J_0, \\ \Delta y(t_k) = I_k(y(t_k)), \quad k = 1, 2, \dots , p, \\ y(0) + \lambda \int_0^c y(s) ds = - y(c), \quad \lambda \le 0, \end{gathered} \label{e1.2}$$ where $J = [0, c]$, $J_0 = J \setminus \{t_1, t_2, \dots , t_p\}$, $0 < t_1 < t_2 < \dots < t_p < c$, $f \in C(J \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}, \mathbb{R})$, $I_k \in C(\mathbb{R}, \mathbb{R})$, $a \in C(\mathbb{R}, \mathbb{R})$ and $a(t) \le 0$ for $t \in J$. $$(Ty)(t) = \int_0^t k(t, s)y(s) ds, \quad (Sy)(t) = \int_0^c h(t, s) y(s) ds,$$ $k \in C(D, \mathbb{R}^+)$, $D = \{(t, s) \in J \times J : t \ge s\}$, $h \in C(J \times J, \mathbb{R}^+)$. \begin{remark} \label{rmk1.1}\rm If $a(t) \equiv 0$ and $\lambda \equiv 0$, then \eqref{e1.2} reduces to \eqref{e1.1}. \end{remark} We will give the concept of coupled quasi-solutions of BVP \eqref{e1.2} in next section. It is well known that the monotone iterative technique offers an approach for obtaining approximate solutions of nonlinear differential equations, for details, see \cite{h2, l1} and the references therein. The aim of this paper is to investigate the existence of coupled quasi-solutions of \eqref{e1.2} by using the method of upper and lower solutions combined with a monotone iterative technique. Our result correct and generalize the main result of \cite{w}. \section{Preliminaries} In this section, we present some definitions needed for introducing the concept of quasi-solutions for \eqref{e1.2}. Let \begin{align*} PC(J) = \{&y : J \to \mathbb{R} : y \text{ is continuous at } t \in J_0; \\ & y(0^+), y(T^-), y(t_k^+), y(t_k^-) \text{ exist and } y(t_k^-) = y(t_k),\; k = 1,\dots , p\}, \end{align*} \begin{align*} PC^1(J) = \{&y \in PC(J) : y \text{ is continuously differentiable for } t \in J_0; \\ & y'(0^+), y'(T^-), y'(t_k^+), y'(t_k^-) \text{ exist},\; k = 1, \dots, p\}, \end{align*} The sets $PC(J)$ and $PC^1(J)$ are Banach spaces with the norms $$\|y\|_{PC(J)} = \sup \{|y(t)| : t \in J\}, \quad \|y\|_{PC^1(J)} = \|y\|_{PC(J)} + \|y'\|_{PC(J)}.$$ \begin{definition} \label{def2.1} \rm Functions $\alpha_0, \beta_0 \in PC^1(J)$ are said to be coupled lower-upper quasi-solutions to the problem \eqref{e1.2} if $$\begin{gathered} \alpha_0'(t) + a(t) \alpha_0(t) \le f(t, \alpha_0(t), (T\alpha_0)(t), (S\alpha_0)(t)), \quad t \in J_0, \\ \Delta \alpha_0(t_k) \le I_k(\alpha_0(t_k)), \quad k = 1, 2, \dots , p, \\ \alpha_0(0) + \lambda \int_0^c \alpha_0(s) ds \le - \beta_0(c), \quad \lambda \le 0, \\ \beta_0'(t) + a(t) \beta_0(t) \ge f(t, \beta_0(t), (T\beta_0)(t), (S\beta_0)(t)), \quad t \in J_0, \\ \Delta \beta_0(t_k) \ge I_k(\beta_0(t_k)), \quad k = 1, 2, \dots , p, \\ \beta_0(0) + \lambda \int_0^c \beta_0(s) ds \ge - \alpha_0(c), \quad \lambda \le 0. \end{gathered} \label{e2.1}$$ \end{definition} Note that if $\alpha_0(c) = \beta_0(c)$, then the above definition reduces to the notion of lower and upper solutions of \eqref{e1.2}. \begin{definition} \label{def2.2}\rm Functions $v, w \in PC^1(J)$ are said to be coupled quasi-solutions to \eqref{e1.2} if $$\begin{gathered} v'(t) + a(t) v(t) = f(t, v(t), (Tv)(t), (Sv)(t)), \quad t \in J_0, \\ \Delta v(t_k) = I_k(v(t_k)), \quad k = 1, 2, \dots , p, \\ v(0) + \lambda \int_0^c v(s) ds = - w(c), \quad \lambda \le 0, \\ w'(t) + a(t) w(t) = f(t, w(t), (Tw)(t), (Sw)(t)), \quad t \in J_0, \\ \Delta w(t_k) = I_k(w(t_k)), \quad k = 1, 2, \dots , p, \\ w(0) + \lambda \int_0^c w(s) ds = - v(c), \quad \lambda \le 0. \end{gathered} \label{e2.2}$$ \end{definition} Let $\alpha_0, \beta_0 \in PC^1(J)$ and $\alpha_0(t) \le \beta_0(t)$ for $t \in J_0$. In what follows we define the segment $$[\alpha_0, \beta_0] = \{u \in PC^1(J) : \alpha_0(t) \le u(t) \le \beta_0(t), \ t \in J\}.$$ \begin{definition} \rm Let $u, v$ be coupled quasi-solutions of \eqref{e1.2} such as $u(t) \le v(t)$ for $t \in J_0$. Assume that $\alpha_0, \beta_0 \in PC^1(J)$ and $\alpha_0(t) \le \beta_0(t)$ for $t \in J_0$. Coupled quasi-solutions $u, v$ of \eqref{e1.2} are called coupled minimal-maximal quasi-solutions in segment $[\alpha_0, \beta_0]$ if $\alpha_0(t) \le u(t)$, $v(t) \le \beta_0(t)$ for $t \in J_0$ and for any $U, V$ coupled quasi-solutions of \eqref{e1.2}, such as $\alpha_0(t) \le U(t)$, $V(t) \le \beta_0(t)$ for $t \in J_0$ we have $u(t) \le U(t)$ and $V(t) \le v(t)$, $t \in J_0$. \end{definition} For convenience, we assume the following conditions are satisfied \begin{itemize} \item[(H1)] Functions $\alpha_0(t), \beta_0(t)$ are coupled lower-upper quasi-solutions of \eqref{e1.2} such that $\alpha_0(t) \le \beta_0(t)$ for $t \in J_0$. \item[(H2)] There exist $M > 0, N, N_1 \ge 0$ such that $$f(t, x_1, y_1, z_1) - f(t, x_2, y_2, z_2) \ge - M(x_1 - x_2) - N(y_1 - y_2) - N_1(z_1 - z_2),$$ for $\alpha_0 \le x_2 \le x_1 \le \beta_0$, $T\alpha_0 \le y_2 \le y_1 \le T\beta_0$, $S\alpha_0 \le z_2 \le z_1 \le S\beta_0$, $t \in J$. \item[(H3)] There exist $0 \le L_k < 1$, $k = 1, 2, \dots , p$, satisfy $$I_k(x) - I_k(y) \ge - L_k (x - y),$$ for $\alpha_0 \le y \le x \le \beta_0$, $t \in J$. \end{itemize} Now we consider the problem $$\begin{gathered} y'(t) + My(t) + N(Ty)(t) + N_1(Sy)(t) = \sigma(t), \quad t \in J_0, \\ \Delta y(t_k) = - L_k y(t_k) + b_k, \quad k = 1, 2, \dots , p, \\ y(0) = b, \end{gathered} \label{e2.3}$$ where $M > 0$, $N, N_1 \ge 0$, $L_k < 1$, $k = 1, 2, \dots , p$. \begin{lemma}\label{lem2.4} If $y \in PC^1(J)$, $M > 0$, $N, N_1 \ge 0$, $L_k < 1$, $k = 1, 2, \dots , p$, and $$\bar{k} + \bar{h} + \sum _{i=1}^p L_i < 1, \label{e2.4}$$ where \begin{gather*} \bar{k} = \begin{cases} k_0 c M^{-1}(1-e^{-Mc}), & \text{if }M > 1,\\ k_0 c M^{-1}(1-M e^{-Mc}), & \text{if }0 < M \le 1,\\ \frac{1}{2}k_0 c^2, & \text{if }M = 0. \end{cases} \\ \bar{h} = \begin{cases} h_0 c M^{-1}(1-e^{-Mc}), & \text{if }M > 0,\\ h_0 c^2, & \text{if }M = 0, \end{cases} \end{gather*} where $k_0 = \max_{0 \le s \le t \le c} k(t, s)$ and $h_0 = \max_{0 \le t, s \le c} h(t, s)$. Then \eqref{e2.3} has a unique solution. \end{lemma} \begin{proof} If $y \in PC^1(J)$ is a solution of \eqref{e2.3}, then, by integrating, we obtain \begin{aligned} y(t) &= b e^{-Mt} + \int_0^t e^{-M(t-s)} [\sigma(s) - N(Ty)(s) - N_1(Sy)(s)]ds \\ &\quad + \sum _{0 < t_i < t} e^{-M(t-t_i)} (- L_i y(t_i) + b_i). \label{e2.5} \end{aligned} Conversely, if $y(t) \in PC(J)$ is solution of the above-mentioned integral equation \eqref{e2.5}, then it is easy to check that $y'(t) = - M y(t) - N(Ty)(t) - N_1(Sy)(t) + \sigma(t)$, $t \neq t_k$, $\Delta y(t_k) = - L_k y(t_k) +b_k$, $k = 1, 2, \dots , p$, and $y(0) = b$. So \eqref{e2.3} is equivalent to the integral equation \eqref{e2.5}. Now, we define operator $B : PC(J) \to PC(J)$ as \begin{aligned} (By)(t) &= b e^{-Mt} + \int_0^t e^{-M(t-s)}[\sigma(s) - N(Ty)(s) - N_1(Sy)(s)] ds \\ &\quad + \sum _{0 < t_i < t} e^{-M(t-t_i)} (- L_i y(t_i) + b_i). \label{e2.6} \end{aligned} For each $u, v \in PC(J)$, we have \begin{aligned} |(Bu)(t) - (Bv)(t)| \le & N \Big|\int_0^t e^{-M(t-s)}(Tu - Tv)(s) ds\Big| \\ & + N_1 \Big|\int_0^t e^{-M(t-s)}(Su - Sv)(s) ds\Big| \\ & + \sum _{0 < t_i < t} L_i |e^{-M(t-t_i)} (u(t_i) - v(t_i))|. \label{e2.7} \end{aligned} We easily check that \begin{aligned} &\big|\int_0^t e^{-M(t-s)}(Tu-Tv)(s) ds\big| \\ &\le \begin{cases} k_0 t M^{-1}(1-e^{-Mt}) \|u-v\|_{PC}, & \text{if }M > 1,\\ k_0 t M^{-1}(1-M e^{-Mt}) \|u-v\|_{PC}, & \text{if }0 < M \le 1,\\ k_0 \frac{1}{2} t^2 \|u-v\|_{PC}, & \text{if }M = 0, \end{cases} \end{aligned} \label{e2.8} and $$\big|\int_0^t e^{-M(t-s)}(Su-Sv)(s) ds\big| \le \begin{cases} h_0 c M^{-1}(1-e^{-Mt}) \|u-v\|_{PC}, & \text{if }M > 0,\\ h_0 c t \|u-v\|_{PC}, & \text{if }M = 0. \end{cases} \label{e2.9}$$ Substituting \eqref{e2.8} and \eqref{e2.9} into \eqref{e2.7}, we obtain $$\|Bu - Bv\|_{PC} \le (\bar{k} + \bar{h} + \sum _{i=1}^p L_i) \|u - v\|_{PC}.$$ This indicates that $B$ is a contraction mapping (by \eqref{e2.4}). Then there is one unique $y \in PC(J)$ such that $By = y$, that is, \eqref{e2.3} has a unique solution. \end{proof} \begin{lemma}[\cite{w}] \label{lem2.5} Assume that $y \in PC^1(J)$ satisfies $$\begin{gathered} y'(t) + My(t) + N(Ty)(t) + N_1(Sy)(t) \le 0, \quad t \in J_0, \\ \Delta y(t_k) \le - L_k y(t_k), \quad k = 1, 2, \dots , p, \\ y(0) \le 0, \end{gathered} \label{e2.10}$$ where $M > 0$, $N, N_1 \ge 0$, $L_k < 1$, $k = 1, 2, \dots , p$, and $$\int_0^c q(s) ds \le \prod _{j=1}^p (1 - \bar{L}_j) \label{e2.11}$$ with $\bar{L}_k = \max \{L_k, 0\}$, $k = 1, 2, \dots , p$, $$q(t) = N \int_0^t k(t, s) e^{M(t-s)} \prod _{s < t_k < c} (1-L_k) ds + N_1 \int_0^c h(t, s) e^{M(t-s)} \prod _{s < t_k < c} (1-L_k) ds,$$ then $y \le 0$. \end{lemma} \section{Main result} \begin{theorem}\label{thm3.1} If {\rm (H1),(H2),(H3)} are satisfied, and, in addition, if there exist $M > 0$, $N, N_1 \ge 0$, $L_k < 1$, $k = 1, 2, \dots , p$, such that \eqref{e2.4} and \eqref{e2.11} hold, then \eqref{e1.2} has, in segment $[\alpha_0, \beta_0]$ the coupled minimal-maximal quasi-solutions. \end{theorem} \begin{proof} For convenience, let $(K\phi)(t) = N (T\phi)(t) + N_1 (S\phi)(t)$. We now construct two sequences $\{\alpha_n(t)\}$ and $\{\beta_n(t)\}$ that satisfy the following problems \begin{gathered} \begin{aligned} &\alpha_i'(t) + a(t) \alpha_{i-1}(t) + M \alpha_i(t) + (K \alpha_i)(t) \\ &= f(t, \alpha_{i-1}(t), (T\alpha_{i-1})(t), (S\alpha_{i-1})(t)) + M \alpha_{i-1}(t) + (K \alpha_{i-1})(t), \quad t \in J_0, \end{aligned} \\ \Delta \alpha_i(t_k) = I_k(\alpha_{i-1}(t_k)) - L_k(\alpha_i(t_k) - \alpha_{i-1}(t_k)), \quad k = 1, 2, \dots , p, \\ \alpha_i(0) + \lambda \int_0^c \alpha_{i-1}(s) ds = - \beta_{i-1}(c), \end{gathered} \label{e3.1} and \begin{gathered} \begin{aligned} &\beta_i'(t) + a(t) \beta_{i-1}(t) + M \beta_i(t) + (K \beta_i)(t) \\ &= f(t, \beta_{i-1}(t), (T\beta_{i-1})(t), (S\beta_{i-1})(t)) + M \beta_{i-1}(t) + (K \beta_{i-1})(t), \quad t \in J_0, \end{aligned} \\ \Delta \beta_i(t_k) = I_k(\beta_{i-1}(t_k)) - L_k(\beta_i(t_k) - \beta_{i-1}(t_k)), \quad k = 1, 2, \dots , p, \\ \beta_i(0) + \lambda \int_0^c \beta_{i-1}(s) ds = - \alpha_{i-1}(c). \end{gathered} \label{e3.2} For each $\phi, \psi \in [\alpha_0, \beta_0]$, we consider the equation \begin{gathered} \begin{aligned} &y'(t) + M y(t) + (K y)(t) \\ &= f(t, \phi(t), (T\phi)(t), (S\phi)(t)) - a(t) \phi(t) + M \phi(t) + (K \phi)(t), \quad t \in J_0, \end{aligned} \\ \Delta y(t_k) = I_k(\phi(t_k)) - L_k(y(t_k) - \phi(t_k)), \quad k = 1, 2, \dots , p, \\ y(0) + \lambda \int_0^c \phi(s) ds = - \psi(c). \end{gathered} \label{e3.3} By condition \eqref{e2.4} and Lemma \ref{lem2.4}, we know that \eqref{e3.3} has a unique solution $y(t) \in PC^1(J)$. Define the operator $A : PC^1(J) \times PC^1(J) \to PC^1(J)$ as $A(\phi, \psi) = y$. Let $\alpha_n(t) = A (\alpha_{n-1}, \beta_{n-1})(t)$ and $\beta_n(t) = A (\beta_{n-1}, \alpha_{n-1})(t)$, $n = 1, 2, \dots ,$ we will prove that $\{\alpha_n\}$, $\{\beta_n\}$ have the following properties. \begin{itemize} \item[(i)] $\alpha_{i-1} \le \alpha_i$, $\beta_i \le \beta_{i-1}$; \item[(ii)] $\alpha_i \le \beta_i$, $i = 1, 2, ...$. \end{itemize} Firstly, we prove that $\alpha_0 \le \alpha_1$. Set $p(t) = \alpha_0(t) - \alpha_1(t)$, it follows that $$\begin{gathered} p'(t) + M p(t) + N (T p)(t) + N_1 (S p)(t) = p'(t) + M p(t) + (Kp)(t) \le 0, \\ \Delta p(t_k) \le - L_k p(t_k), \quad k = 1, 2, \dots , p, \\ p(0) \le 0. \end{gathered} \label{e3.4}$$ Then by condition \eqref{e2.11} and Lemma \ref{lem2.5}, we get $p(t) \le 0$, which implies that $\alpha_0(t)\le \alpha_1(t)$, for all $t \in J_0$. In a similar way, it can be proved that $\beta_1(t) \le \beta_0(t)$, for all $t \in J_0$. Now we prove that $\alpha_1(t) \le \beta_1(t)$, for all $t \in J_0$. In fact, setting $p(t) = \alpha_1(t) - \beta_1(t)$ and using assumption, we obtain \begin{align*} & p'(t) + M p(t) + N (T p)(t) + N_1 (S p)(t)\\ &= \alpha_1'(t) - \beta_1'(t) + M(\alpha_1(t) - \beta_1(t)) + N (T\alpha_1(t) - T \beta_1(t)) + N_1 (S\alpha_1(t) - S\beta_1(t))\\ &= f(t, \alpha_0(t), (T\alpha_0)(t), (S\alpha_0)(t)) - a(t) \alpha_0(t) + M \alpha_0(t) + N (T\alpha_0)(t) + N_1 (S\alpha_0)(t)\\ & \quad - f(t, \beta_0(t), (T\beta_0)(t), (S\beta_0)(t)) + a(t) \beta_0(t) - M \beta_0(t) - N (T\beta_0)(t) - N_1 (S\beta_0)(t)\\ &\leq a(t)(\beta_0(t) - \alpha_0(t)) \le 0, \quad t \in J_0, \end{align*} and \begin{gather*} \Delta p(t_k) = - L_kp(t_k) + I_k(\alpha_0(t_k)) - I_k(\beta_0(t_k)) + L_k \alpha_0(t_k) - L_k \beta_0(t_k) \le - L_k p(t_k), \\ p(0) = \alpha_1(0) - \beta_1(0) = \lambda \int_0^c (\beta_0(s) - \alpha_0(s))ds + \alpha_0(c) - \beta_0(c) \le 0. \end{gather*} Again by Lemma \ref{lem2.5}, we obtain $p(t) \le 0$, that is, $\alpha_1(t) \le \beta_1(t)$ for all $t \in J_0$. Thus we have $\alpha_0(t) \le \alpha_1(t) \le \beta_1(t) \le \beta_0(t)$ for all $t \in J_0$. Continuing this process, by induction, one can obtain monotone sequence $\{\alpha_n(t)\}$ and $\{\beta_n(t)\}$ such that $$\alpha_0(t) \le \alpha_1(t) \le \dots \le \alpha_n(t) \le \dots \le \beta_n(t) \le \dots \beta_1(t) \le \beta_0(t), \quad t \in J_0,$$ where each $\alpha_i(t), \beta_i(t) \in PC^1(J)$ satisfies \eqref{e3.1} and \eqref{e3.2}. As the sequences $\{\alpha_n\}$, $\{\beta_n\}$ are uniformly bounded and equi-continuous, by employing the standard arguments Ascoli-Arzela criterion \cite{l2}, we conclude that the sequences $\{\alpha_n\}$ and $\{\beta_n\}$ converge uniformly on $J_0$ with $$\lim _{n \to \infty} \alpha_n(t) = y_*(t), \quad \lim _{n \to \infty} \beta_n(t) = y^*(t).$$ Obviously, $y_*(t), y^*(t)$ are coupled lower-upper quasi-solutions of \eqref{e1.2}. Now we have to prove that $(y_*, y^*)$ are coupled minimal-maximal quasi-solutions of problem \eqref{e1.2} in segment $[\alpha_0, \beta_0]$. Let $x, z$ be coupled quasi-solutions of \eqref{e1.2} such that $$\alpha_n(t) \le x(t), \quad z(t) \le \beta_n(t), \quad t \in J_0$$ for some $n \in {\bf N }$. Put $q(t) = \alpha_{n+1}(t) - x(t)$, for $t \in J_0$. Form definition of $\alpha_{n+1}$ and properties of quasi-solution $x(t)$, we obtain \begin{align*} & q'(t) + M q(t) + N (T q)(t) + N_1 (S q)(t)\\ &= f\big(t, \alpha_n(t), (T\alpha_n)(t), (S\alpha_n)(t)\big) - a(t) \alpha_n(t) + M \alpha_n(t) + N (T\alpha_n)(t) \\ &\quad + N_1 (S\alpha_n)(t) - f\big(t, x(t), (Tx)(t), (Sx)(t)\big) + a(t) x(t) - M x(t) \\ &\quad - N (Tx)(t) - N_1(Sx)(t)\\ & \leq a(t)(x(t) - \alpha_n(t)) \le 0, \quad t \in J_0, \end{align*} and \begin{gather*} \Delta q(t_k) = - L_k q(t_k) + I_k(\alpha_n(t_k)) - I_k(x(t_k)) + L_k \alpha_n(t_k) - L_k x(t_k) \le - L_k q(t_k), \\ q(0) = \alpha_{n+1}(0) - x(0) = \lambda \int_0^c (x(s) - \alpha_n(s))ds + z(c) - \beta_n(c) \le 0. \end{gather*} By Lemma \ref{lem2.5}, we have $q(t) \le 0$ for all $t \in J_0$, that is $\alpha_{n+1}(t) \le x(t)$. Similarly, we can prove that $z(t) \le \beta_{n+1}(t)$ for all $t \in J_0$. By induction, we obtain $$\alpha_m(t) \le x(t), \quad z(t) \le \beta_m(t), \quad t \in J_0, \quad \text{for } m \in {\bf N}.$$ If $m \to \infty$, it yields $$y_*(t) \le x(t), \quad z(t) \le y^*(t), \quad t \in J_0.$$ It shows that $(y_*, y^*)$ are coupled minimal-maximal quasi-solutions of problem \eqref{e1.2} in segment $[\alpha_0, \beta_0]$. \end{proof} \begin{example} \rm Consider the problem $$\begin{gathered} y'(t) - \frac{t}{4}(1-e^{-t}) y(t) = - y(t) - \frac{1}{8} \int_0^t t e^{-(t-s)}y(s) ds - \frac{5}{6} \int_0^1 y(s)ds,\\ t \in [0, t_1) \cup (t_1, 1], \\ \Delta y(t_1) = - \frac{1}{9} y(t_1), \quad t_1 = \frac{1}{3} \\ y(0) - \frac{1}{6} \int_0^1 y(s) ds = - y(1). \end{gathered} \label{e3.5}$$ where $a(t) = - \frac{t}{4}(1-e^{-t}) \le 0$, $I_1(x) = -\frac{1}{9} x$, $L_1 = \frac{1}{9}$ and $\lambda = - \frac{1}{6} < 0$. Let $f(t, x, y, z) = - M x - N y - N_1 z$, $M = 1, N = \frac{3}{8}$, $N_1 = \frac{5}{6}$, $J = [0, 1]$, $c = 1$, $k(t, s) = \frac{t}{3} e^{-(t-s)}$, $h(t, s) = 1$, then for $t \in J$, $x_i, y_i, z_i \in \mathbb{R}$, $i = 1, 2$, $x_1 \ge x_2$, $y_1 \ge y_2$, $z_1 \ge z_2$, $$f(t, x_1, y_1, z_1) - f(t, x_2, y_2, z_2) = -(x_1 - x_2) - \frac{3}{8}(y_1 - y_2) - \frac{5}{6}(z_1 - z_2).$$ Thus the condition (H2) holds. It is easy to see that $k_0 = \frac{1}{3}$, $h_0 = 1$, $\bar{k} = \frac{1}{3} \bar{h} = \frac{1}{3} (1-e^{-1})$ and $$\bar{h} + \bar{k} + L_1 = 0.9359 < 1.$$ Hence the condition \eqref{e2.4} holds. Moreover, we have \begin{align*} \int_0^1 q(s)ds & \le \int_0^1\Big(\frac{3}{8} \int_0^t \frac{t}{3}e^{-(t-s)} e^{(t-s)}(1-L_1)ds + \frac{5}{6} \int_0^1 e^{(t-s)}(1-L_1)ds\Big)dt\\ & = \int_0^1\Big(\frac{t^2}{18} + \frac{20}{27}(1-e^{-1})e^t\Big)dt\\ &= \frac{1}{54} + \frac{20}{27}(e+e^{-1}-2) = 0.8231 < 0.8889 = 1 - L_1, \end{align*} which implies that the condition \eqref{e2.11} holds. Let $$\alpha_0(t) = - \frac{5}{4}, \quad \beta_0(t) = 2 - t, \quad t \in [0, 1].$$ Then $\alpha_0(t)$ and $\beta_0(t)$ are coupled lower-upper quasi-solutions of problem \eqref{e4.1}. In fact, \begin{gather*} \begin{aligned} \alpha_0'(t) + a(t) \alpha_0(t) & = \frac{5}{16} t(1-e^{-t}) \le 2 + \frac{5}{32}t(1-e^{-t}) \\ & < \frac{5}{4} + \frac{5}{32} \int_0^t t e^{-(t-s)} ds + \frac{25}{24} \int_0^1 ds \\ & = f(t, \alpha_0(t), (T\alpha_0)(t), (S\alpha_0)(t)), \end{aligned}\\ \Delta \alpha_0(1/3) = 0 < \frac{5}{36} = - L_1 \alpha_0(1/3) \\ \alpha_0(0) - \frac{1}{6} \int_0^1 \alpha_0(s)ds = - \frac{25}{24} < -1 = - \beta_0(1), \end{gather*} and \begin{gather*} \begin{aligned} \beta_0'(t) + a(t) \beta_0(t) & = - 1 - \frac{1}{4}t(1-e^{-t})(2-t)\\ & \ge - 1 - \frac{1}{4}(1-e^{-1}) \\ & > - \frac{27}{12} + \frac{3}{8} e^{-1} \\ & \ge t - 2 - \frac{1}{8}t(3-t) + \frac{3}{8}te^{-t} - \frac{15}{12}\\ & = t - 2 - \frac{1}{8} \int_0^t t e^{-(t-s)}(2-s) ds - \frac{5}{6} \int_0^1 (2-s) ds\\ & = f(t, \beta_0(t), (T\beta_0)(t), (S\beta_0)(t)), \end{aligned} \\ \Delta \beta_0(1/3) = 0 > - \frac{5}{27} = - L_1 \beta_0(1/3) \\ \beta_0(0) - \frac{1}{6} \int_0^1 \beta_0(s)ds = \frac{7}{4} > \frac{5}{4} = - \alpha_0(1). \end{gather*} Obviously, $\alpha_0(t) \le \beta_0(t)$. 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