Skip to main content

A new comparison principle and its application to nonlinear impulsive functional integro-differential equations

Abstract

In this article, we first create a new comparison principle for a nonlinear impulsive boundary problem involving different deviating arguments. Then we employ the new result and iterative method to study the existence of the max-minimal solution of a second-order impulsive functional integro-differential equation. The results achieved in this paper are more general and complement many previously known results.

Introduction

The comparison principle plays an important role since it is one of the basic tools to study ODE and PDE. Thus, how to create a new comparison principle is an interesting and important question. In this paper, we shall create a comparison principle with impulsive effect. By means of the comparison principle and monotone iterative method, the existence of the max-minimal solution of second-order impulsive functional integro-differential Eqs. (1.1) is investigated. Also, the iterative sequences of solutions of the system are given. The importance of this method does not need to be particularly pointed out [114]. The theorems achieved in this paper are more general and complement many previously known results.

Impulsive differential equations, arising in the mathematical modeling of complex systems and processes, have drawn more and more attention of the research community due to their numerous applications in various fields of science and engineering such as chemistry, physics, biology, medicine, mechanics, etc. (see [1524]). Boundary value problems (BVP) of differential equations have been investigated for many years. Now, nonlinear boundary conditions have drawn much attention, there exist many articles dealing with the problem for different kinds of boundary value conditions such as multi-point, integral boundary condition, and other conditions (see [2535]). On the other hand, deviated arguments also play an important role in nonlinear analysis. It should be noticed that such equations appear often in various fields of science and engineering such as mathematical physics, economics, mechanics, etc. (see [3638]). However, the relevant theory of this type of problem is still at its developing stage, and a great quantity of aspects remain to be explored. For a detailed description, see [3943].

Here, we use the new result we achieved in the article to investigate the existence theorems of max–min solutions for impulsive systems of the following:

$$ \textstyle\begin{cases} u''(t) = \Upsilon(t,u(t),u(\phi(t)), Xu(t), Yu(t),u(\psi(t,\mu (t)))),\quad t\in J',\\ \triangle u(t_{k})=I_{k}(u(t_{k})),\quad k=1,2,\ldots,m,\\ \triangle u'(t_{k})=I_{k}^{*}(u(t_{k}),u'(t_{k})),\quad k=1,2,\ldots,m,\\ \chi_{1}(u(0),u(T))=0,\qquad\chi_{2}(u'(0),u'(T))=0, \end{cases} $$
(1.1)

where \(t\in J=[0,T](T>0)\), \(\Upsilon\in C(J\times R^{5},R)\), \(I_{k}\in C(R,R)\), \(I_{k}^{*},\chi_{i}(i=1,2)\in C(R\times R,R)\), \(\phi\in C(J,J)\), \(\mu \in C(J,R)\), \(\psi\in C(J\times R,J)\), \(0=t_{0}< t_{1}<\cdots<t_{k}<\cdots<t_{m}<t_{m+1}=T\), \(J'=J\backslash \{t_{1},t_{2},\ldots,t_{m}\}\), and

$$Xu(t)= \int_{0}^{t}k(t,s)u(s)\,ds , \qquad Yu(t)= \int_{0}^{T}h(t,s)u(s)\,ds, $$

\(k(t,s)\in C(D,R^{+})\), \(h(t,s)\in C(J\times J,R^{+})\), \(D=\{(t,s)\in R^{2} |0\leq s\leq t,t\in J\}\), \(R^{+}=[0,+\infty)\). \(\triangle u(t_{k})=u(t_{k}^{+})-u(t_{k}^{-})\), where \(u(t_{k}^{+})\) and \(u(t_{k}^{-})\) denote the right and the left limits of \(u(t)\) at \(t= t_{k}(k=1,2,\ldots,m)\), respectively. \(\triangle u'(t_{k})\) has a similar meaning for \(u'(t)\). Let \(PC(J, R) = \{u: J \to R | u(t)\mbox{ is continuous at }t\neq t_{k}, \mbox{ left continuous at } t=t_{k}\mbox{ and }u(t_{k}^{+})\mbox{ exists}, k=1,2,\ldots,m \}\). In addition, \(PC^{1}(J, R)= \{u\in PC(J, R)| u(t)\mbox{ is continuously differentiable at }t\neq t_{k}, u'(t_{k}^{+})\mbox{ and } u'(t_{k}^{-})\mbox{ exist}, k=1,2,\ldots,m \}\). Obviously, \(PC(J, R)\) and \(PC^{1}(J, R)\) are Banach spaces with respective norms

$$\Vert u \Vert _{PC}=\sup_{t\in J} \bigl\vert u(t) \bigr\vert ,\qquad \Vert u \Vert _{PC^{1}}=\max\bigl\{ \Vert u \Vert _{PC}, \bigl\Vert u' \bigr\Vert _{PC}\bigr\} . $$

New comparison principle

Lemma 1

([16])

Let \(s\in[0,T)\), \(c_{k}\ge 0\), \(\phi_{k}\) (\(k=1,2,\ldots,m\)) be constants, and let \(a,b\in PC(J,R)\), \(w\in PC^{1}(J,R)\). If

$$\textstyle\begin{cases} w'(t) \le a(t)w(t)+b(t), \quad t\in[s,T), t\neq t_{k},\\ w(t_{k}^{+})\le c_{k}w(t_{k})+\phi_{k}, \quad t_{k}\in[s,T), \end{cases} $$

then for \(t\in[s,T]\)

$$\begin{aligned} w(t) \le& w\bigl(s^{+}\bigr) \biggl(\prod_{s< t_{k}< t} c_{k}\biggr)\exp\biggl( \int_{s}^{t}a(r)\,dr\biggr)+ \int _{s}^{t}\biggl(\prod _{r< t_{k}< t} c_{k}\biggr)\exp\biggl( \int_{r}^{t}a(\tau)\,d\tau\biggr)b(r)\,dr \\ &{}+\sum_{s< t_{k}< t}\biggl(\prod _{t_{k}< t_{i}< t} c_{k}\biggr)\exp\biggl( \int_{t_{k}}^{t}a(\tau)\,d\tau \biggr)\phi_{k}. \end{aligned}$$

Lemma 2

(New comparison principle)

Assume that \(u\in PC^{1}(J,R)\cap C^{2}(J',R)\) satisfies

$$ \textstyle\begin{cases} u''(t)\le-D_{1}(t)u(t)-D_{2}(t)u(\phi (t))-D_{3}(t)(Xu)(t)-D_{4}(t)(Yu)(t),\quad t\in J',\\ \triangle u(t_{k})\le-L_{k}u(t_{k}),\quad k=1,2,\ldots,m,\\ \triangle u'(t_{k})\le-L_{k}^{*}u'(t_{k}),\quad k=1,2,\ldots,m,\\ u(0)\le\lambda_{1}u(T),\qquad u'(0)\le\lambda_{2}u'(T), \end{cases} $$
(2.1)

where \(D_{i}\in C(J,R^{+})(i=1,2,3,4)\), \(0\le L_{k}<1\), \(0< \lambda_{1},\lambda_{2},L_{k}^{*}<1\), and

$$\begin{aligned} &\lambda_{1}\prod_{k=1}^{m}(1-L_{k})^{2} \prod_{k=1}^{m}\bigl(1-L_{k}^{*} \bigr) \Biggl(1-\lambda _{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr) \\ &\quad \ge \int_{0}^{T}q(t)\,dt \int_{0}^{T}\prod_{s< t_{k}< T}(1-L_{k}) \,ds, \end{aligned}$$
(2.2)

here \(q(t)=D_{1}(t)+D_{2}(t)+D_{3}(t)\int_{0}^{t}k(t,s)\,ds +D_{4}(t)\int_{0}^{T}h(t,s)\,ds\). Then \(u(t)\le0\), \(t\in J\).

Proof

Suppose the contrary. Then, for some \(t\in J\), \(u(t)>0\), thus there exist two cases:

Case a: For some \(\overline{t}\in J\), \(u(\overline{t})>0\), and \(u(t)\ge0\) for \(t\in J\).

Case b: For some \(t^{*},t_{*}\in J\) such that \(u(t^{*})>0\) and \(u(t_{*})<0\).

In Case a, it follows from (2.1) that \(u''(t)\le0\) for \(t\neq t_{k}\) and \(u'(t_{k}^{+})\le(1-L_{k}^{*})u'(t_{k})\). By means of Lemma 1, we have \(u'(t)\le u'(0)\prod_{0< t_{k}< t}(1-L_{k}^{*})\). From this, together with (2.1), we have \(u'(0)\le \lambda_{2}u'(T)\le\lambda_{2}u'(0)\prod_{k=1}^{m}(1-L_{k}^{*})\), which means \(u'(0)\le0\), thus \(u'(t)\le0\). Meanwhile, \(u(t_{k}^{+})\le(1-L_{k})u(t_{k})\le u(t_{k})\). So, \(u(t)\) is nonincreasing in J. Then \(u(0)\le \lambda_{1}u(T)\le\lambda_{1}u(0)\), which is a contradiction.

For Case b, put \(\inf_{t\in J}u(t)=-\gamma\), then \(\gamma>0\), and for some \(i\in\{1,2,\ldots,m\}\), there exists \(t_{*}\in(t_{i},t_{i+1}]\) such that \(u(t_{*})=-\gamma\) or \(u(t_{i}^{+})=-\gamma\). Only consider \(u(t_{*})=-\gamma\), the proof of the case \(u(t_{i}^{+})=-\gamma\) is similar.

By (2.1), we have

$$\textstyle\begin{cases} u''(t)\le\gamma[D_{1}(t)+D_{2}(t)+D_{3}(t)\int_{0}^{t}k(t,s)\,ds +D_{4}(t)\int_{0}^{T}h(t,s)\,ds]\equiv\gamma q(t),\\ u'(t_{k}^{+})\le(1-L_{k}^{*})u'(t_{k}). \end{cases} $$

By Lemma 1, we get

$$ u'(t)\le u'(0)\prod_{0< t_{k}< t} \bigl(1-L_{k}^{*}\bigr)+ \int_{0}^{t}\gamma\prod _{s< t_{k}< t}\bigl(1-L_{k}^{*}\bigr)q(s)\,ds. $$
(2.3)

In (2.3), let \(t=T\), then we have

$$\begin{aligned} u'(0) \le&\lambda_{2}u'(T)\\ \le& \lambda_{2}u'(0)\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)+\lambda_{2} \int_{0}^{T}\gamma\prod _{s< t_{k}< T}\bigl(1-L_{k}^{*}\bigr)q(s)\,ds \\ \le&\lambda_{2}u'(0)\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)+\lambda_{2} \int_{0}^{T}\gamma q(s)\,ds, \end{aligned}$$

which implies

$$ u'(0)\le\lambda_{2} \int_{0}^{T}\gamma q(s)\,ds\Biggl[1- \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr]^{-1}. $$
(2.4)

From (2.3) and (2.4), we have that

$$\begin{aligned} u'(t) \le&\lambda_{2} \int_{0}^{T}\gamma q(s)\,ds\Biggl[1- \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr]^{-1}\prod _{0< t_{k}< t}\bigl(1-L_{k}^{*}\bigr)\\ &{} + \int_{0}^{t}\gamma\prod _{s< t_{k}< t}\bigl(1-L_{k}^{*}\bigr)q(s)\,ds \\ \le& \int_{0}^{T}\gamma q(s)\,ds\Biggl\{ \lambda_{2}\prod_{0< t_{k}< t}\bigl(1-L_{k}^{*} \bigr)\Biggl[1-\lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr]^{-1}+1\Biggr\} \\ \le&\lambda_{2}\prod_{0< t_{k}< t} \bigl(1-L_{k}^{*}\bigr) \int_{0}^{T}\gamma q(s)\,ds\\ &{}\times\Biggl\{ \Biggl[1- \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr]^{-1}+\Biggl[ \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr]^{-1}\Biggr\} \\ \le& \int_{0}^{T}\gamma q(s)\,ds\Biggl[\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr) \Biggl(1- \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr)\Biggr]^{-1}, \end{aligned}$$

which and \(u(t_{k}^{+})\le(1-L_{k})u(t_{k})\) imply for \(t\in[t_{*},T]\)

$$ \begin{aligned}[b] u(t)\le&{} u(t_{*})\prod _{t_{*}< t_{k}< t}(1-L_{k})+ \int_{t_{*}}^{t}\prod_{s< t_{k}< t}(1-L_{k}) \int _{0}^{T}\gamma q(r)\,dr\\ &{}\times\Biggl[\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr) \Biggl(1-\lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr)\Biggr]^{-1}\,ds \\ ={}& u(t_{*})\prod_{t_{*}< t_{k}< t}(1-L_{k})+ \int_{0}^{T}\gamma q(r)\,dr\\ &{}\times\Biggl[\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr) \Biggl(1-\lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr)\Biggr]^{-1}\\ &{}\times \int _{t_{*}}^{t}\prod_{s< t_{k}< t}(1-L_{k}) \,ds. \end{aligned} $$
(2.5)

If \(t^{*}>t_{*}\), let \(t=t^{*}\) in (2.5), we have

$$\begin{aligned} 0 < & -\gamma\prod_{t_{*}< t_{k}< t^{*}}(1-L_{k})\\ &{}+\gamma \int_{0}^{T} q(r)\,dr\Biggl[\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr) \Biggl(1- \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr)\Biggr]^{-1}\\ &{}\times \int_{t_{*}}^{t^{*}}\prod_{s< t_{k}< t^{*}}(1-L_{k}) \,ds, \end{aligned}$$

so,

$$\begin{aligned} \frac{\prod_{k=1}^{m}(1-L_{k})}{\int_{0}^{T}\prod_{s< t_{k}< T}(1-L_{k})\,ds} \le&\frac{\prod_{t_{*}< t_{k}< t^{*}}(1-L_{k})}{\int_{t_{*}}^{t^{*}}\prod_{s< t_{k}< t^{*}}(1-L_{k})\,ds} \\ < & \int_{0}^{T}q(r)\,dr\Biggl[\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr) \Biggl(1- \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr)\Biggr]^{-1}, \end{aligned}$$

which contradicts (2.2). Hence, \(u(t)\le0\) on J.

If \(t^{*}< t_{*}\), without loss of generality, let \(t_{*}\in(t_{p-1},t_{p}]\) and \(t^{*}\in(t_{q},t_{q+1}]\), \(0\le q\le p-1\), \(p,q\in\{1,2,\ldots,m\}\). By Lemma 1, we have

$$ \begin{aligned}[b] u\bigl(t^{*}\bigr)\le{}& u(0)\prod _{0< t_{k}< t^{*}}(1-L_{k})+ \int_{0}^{T}\gamma q(r)\,dr\Biggl[\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr) \Biggl(1- \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr)\Biggr]^{-1} \\ &{}\times \int_{0}^{t^{*}}\prod_{s< t_{k}< t^{*}}(1-L_{k}) \,ds \\ ={}&u(0)\prod_{k=1}^{q}(1-L_{k})+ \int_{0}^{T}\gamma q(r)\,dr\Biggl[\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr) \\ &{}\times\Biggl(1-\lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr)\Biggr]^{-1} \int _{0}^{t^{*}}\prod_{s< t_{k}< t^{*}}(1-L_{k}) \,ds. \end{aligned} $$
(2.6)

On the other hand,

$$ \begin{aligned}[b] u(0)\le{}&\lambda_{1}u(T) \le \lambda_{1} u(t_{*})\prod_{t_{*}< t_{k}< T}(1-L_{k})+ \lambda_{1} \int_{0}^{T}\gamma q(r)\,dr \\ &{}\times\Biggl[\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr) \Biggl(1-\lambda_{2}\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr)\Biggr) \Biggr]^{-1} \int_{t_{*}}^{T}\prod_{s< t_{k}< T}(1-L_{k}) \,ds \\ ={}&{-}\lambda_{1}\gamma\prod_{k=p}^{m}(1-L_{k})+ \lambda_{1} \int_{0}^{T}\gamma q(r)\,dr \\ &{}\times\Biggl[\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr) \Biggl(1-\lambda_{2}\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr)\Biggr) \Biggr]^{-1} \int_{t_{*}}^{T}\prod_{s< t_{k}< T}(1-L_{k}) \,ds. \end{aligned} $$
(2.7)

From (2.6) and (2.7), we get that

$$\begin{aligned} &\lambda_{1}\prod_{k=p}^{m}(1-L_{k}) \prod_{j=1}^{q}(1-L_{j}) \\ &\quad < \int_{0}^{T}q(r)\,dr\Biggl[\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr) \Biggl(1- \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr)\Biggr]^{-1} \\ &\qquad {}\times\Biggl[\lambda_{1}\prod_{j=1}^{q}(1-L_{j}) \int_{t_{*}}^{T}\prod_{s< t_{k}< T}(1-L_{k}) \,ds+ \int_{0}^{t^{*}}\prod_{s< t_{k}< t^{*}}(1-L_{k}) \,ds\Biggr]. \end{aligned}$$

By \(\prod_{j=q+1}^{m}(1-L_{j})\) times the above inequality, then

$$\begin{aligned} &\lambda_{1}\prod_{k=p}^{m}(1-L_{k}) \prod_{j=1}^{m}(1-L_{j}) \\ &\quad < \int_{0}^{T}q(r)\,dr\Biggl[\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr) \Biggl(1- \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr)\Biggr]^{-1}\\ &\qquad {}\times\Biggl[ \lambda_{1}\prod_{j=1}^{m}(1-L_{j}) \int_{t_{*}}^{T}\prod_{s< t_{k}< T}(1-L_{k}) \,ds+\prod_{j=q+1}^{m}(1-L_{j}) \int _{0}^{t^{*}}\prod_{s< t_{k}< t^{*}}(1-L_{k}) \,ds\Biggr] \\ &\quad \le \int_{0}^{T}q(r)\,dr\Biggl[\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr) \Biggl(1- \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr)\Biggr]^{-1} \\ &\qquad {}\times\biggl[ \int_{t_{*}}^{T}\prod_{s< t_{k}< T}(1-L_{k}) \,ds+ \int_{0}^{t^{*}}\prod_{s< t_{k}< T}(1-L_{k}) \,ds\biggr] \\ &\quad \le \int_{0}^{T}q(r)\,dr\Biggl[\prod _{k=1}^{m}\bigl(1-L_{k}^{*}\bigr) \Biggl(1- \lambda_{2}\prod_{k=1}^{m} \bigl(1-L_{k}^{*}\bigr)\Biggr)\Biggr]^{-1} \\ &\qquad {}\times \int_{0}^{T}\prod_{s< t_{k}< T}(1-L_{k}) \,ds. \end{aligned}$$

So, \(\lambda_{1}\prod_{k=1}^{m}(1-L_{k})^{2}<\int_{0}^{T}q(r)\,dr[\prod_{k=1}^{m}(1-L_{k}^{*})(1-\lambda_{2}\prod_{k=1}^{m}(1-L_{k}^{*}))]^{-1}\int _{0}^{T}\prod_{s<t_{k}<T}(1-L_{k})\,ds\), which contradicts (2.2). Hence, \(u(t)\le0\) on J.

Consider the problem:

$$ \textstyle\begin{cases} u''(t)=\sigma(t)-D_{1}(t)u(t)-D_{2}(t)u(\phi (t))-D_{3}(t)(Xu)(t)-D_{4}(t)(Yu)(t),\quad t\in J',\\ \triangle u(t_{k})=\psi_{k}-L_{k}u(t_{k}),\quad k=1,2,\ldots,m,\\ \triangle u'(t_{k})=\nu_{k}-L_{k}^{*}u'(t_{k}),\quad k=1,2,\ldots,m,\\ u(0)= \lambda_{1}u(T)+m_{1},\qquad u'(0)=\lambda_{2}u'(T)+m_{2}, \end{cases} $$
(2.8)

where \(\sigma\in PC(J,R)\), \(\psi_{k},\nu_{k},m_{1},m_{2}\in R\). □

Lemma 3

\(u(t)\in PC^{1}(J,R)\cap C^{2}(J',R)\) is a solution of the impulsive differential system (2.8) iff \(u(t)\in PC^{1}(J,R)\) is a solution of the impulsive integral system

$$\begin{aligned} u(t)={}&G_{1}+tG_{2} \\ &{}+ \int_{0}^{t}(t-s)\bigl[\sigma(s)-D_{1}(s)u(s)-D_{2}(s)u \bigl(\phi (s)\bigr)-D_{3}(s) (Xu) (s) -D_{4}(s) (Yu) (s)\bigr]\,ds \\ &{}+\sum_{0< t_{k}< t} \bigl\{ \bigl[\psi _{k}-L_{k}u(t_{k}) \bigr]+(t-t_{k})\bigl[\nu_{k}-L_{k}^{*}u'(t_{k}) \bigr]\bigr\} , \end{aligned}$$
(2.9)

where

$$\begin{aligned} G_{1} =&\frac{\lambda_{1}}{1-\lambda_{1}}\biggl\{ \int_{0}^{T}(T-s)\bigl[\sigma (s)-D_{1}(s)u(s)-D_{2}(s)u \bigl(\phi(s)\bigr)-D_{3}(s) (Xu) (s) \\ &{}-D_{4}(s) (Yu) (s)\bigr]\,ds+\sum_{0< t_{k}< T} \bigl\{ \bigl[\psi _{k}-L_{k}u(t_{k}) \bigr]+(T-t_{k})\bigl[\nu_{k}-L_{k}^{*}u'(t_{k}) \bigr]\bigr\} \\ &{}+TG_{2}+m_{1}\biggr\} +m_{1}, \\ G_{2} =&\frac{\lambda_{2}}{1-\lambda_{2}}\biggl\{ \int_{0}^{T}\bigl[\sigma (s)-D_{1}(s)u(s)-D_{2}(s)u \bigl(\phi(s)\bigr)-D_{3}(s) (Xu) (s) \\ &{}-D_{4}(s) (Yu) (s)\bigr]\,ds+\sum_{0< t_{k}< T} \bigl[\nu _{k}-L_{k}^{*}u'(t_{k}) \bigr]+m_{2}\biggr\} +m_{2}. \end{aligned}$$

Lemma 3 is easy, so we omit its proof.

Lemma 4

For \(\sigma\in PC(J,R),~\psi_{k},\nu_{k},m_{1},m_{2}\in R,~0\le L_{k}<1,~0< \lambda_{1},\lambda_{2},L_{k}^{*}<1\) and functions \(M, K, N,L\in C(J,R^{+})\). If

$$ \textstyle\begin{cases} \frac{1}{1-\lambda_{1}}\{\int_{0}^{T}(T-s)p(s)\,ds+\sum_{k=1}^{m}[L_{k}+(T-t_{k})L_{k}^{*}]\}\\ \qquad {}+\frac{\lambda_{2}T}{(1-\lambda_{1})(1-\lambda_{2})}[\int_{0}^{T}p(s)\,ds+\sum_{k=1}^{m}L_{k}^{*}]< 1,\\ \frac{1}{1-\lambda_{2}}[\int_{0}^{T}p(s)\,ds+\sum_{k=1}^{m}L_{k}^{*}]< 1, \end{cases} $$
(2.10)

where \(p(t)=D_{1}(t)+D_{2}(t)+D_{3}(t)\int_{0}^{t}k(t,s)\,ds +D_{4}(t)\int_{0}^{T}h(t,s)\,ds\). (2.8) has a unique solution \(u(t)\in PC^{1}(J,R)\cap C^{2}(J',R)\).

A similar proof can be found in [22] (see Lemma 2.3), so we omit it.

Main results

Theorem 1

Assume that condition (2.10) holds. In addition, assume that

\((H_{1})\) There exist \(u_{0}(t)\le v_{0}(t)\in PC^{1}(J,R)\cap C^{2}(J',R)\) such that

$$\textstyle\begin{cases} u_{0}''(t) \le\Upsilon(t,u_{0}(t),u_{0}(\phi(t)), Xu_{0}(t), Yu_{0}(t),u_{0}(\psi (t,\mu(t)))),\\ \triangle u_{0}(t_{k})\le I_{k}(u_{0}(t_{k})),\quad k=1,2,\ldots,m,\\ \triangle u_{0}'(t_{k})\le I_{k}^{*}(u_{0}(t_{k}),u_{0}'(t_{k})),\quad k=1,2,\ldots,m,\\ \chi_{1}(u_{0}(0),u_{0}(T))\le0,\qquad \chi_{2}(u_{0}'(0),u_{0}'(T))\le0, \end{cases} $$

and

$$\textstyle\begin{cases} v_{0}''(t) \ge\Upsilon(t,v_{0}(t),v_{0}(\phi(t)), Xv_{0}(t), Yv_{0}(t),v_{0}(\psi (t,\mu(t)))),\\ \triangle v_{0}(t_{k})\ge I_{k}(v_{0}(t_{k})),\quad k=1,2,\ldots,m,\\ \triangle v_{0}'(t_{k})\ge I_{k}^{*}(v_{0}(t_{k}),v_{0}'(t_{k})),\quad k=1,2,\ldots,m,\\ \chi_{1}(v_{0}(0),v_{0}(T))\ge0,\qquad \chi_{2}(v_{0}'(0),v_{0}'(T))\ge0. \end{cases} $$

\((H_{2})\) Functions \(D_{i}\in C(J,R^{+})(i=1,2,3,4)\), which satisfy (2.2) such that

$$\begin{aligned} \Upsilon(t,u,v,w,z,\xi)-\Upsilon(t,\overline{u},\overline{v},\overline {w}, \overline{z},\overline{\xi}) \ge&-D_{1}(t) (u-\overline{u})-D_{2}(t) (v-\overline{v})\\ &{}-D_{3}(t) (w-\overline {w})-D_{4}(t) (z- \overline{z}), \end{aligned}$$

where \(u_{0}(t)\le\overline{u}\le u\le v_{0}(t)\), \(u_{0}(\phi(t))\le\overline {v}\le v\le v_{0}(\phi(t))\), \(Xu_{0}(t)\le\overline{w}\le w\le Xv_{0}(t)\), \(Yu_{0}(t)\le\overline{z}\le z\le Yv_{0}(t)\), \(u_{0}(\psi(t,\mu(t)))\le\overline{\xi}\le\xi\le v_{0}(\psi(t,\mu (t)))\), \(\forall t\in J\).

\((H_{3})\) There exist constants \(0\le L_{k}<1\), \(0< L_{k}^{*}<1\) (\(k=1,2,\ldots,m\)), and \(0< b_{1}< a_{1}\), \(0< b_{2}< a_{2}\) such that

$$\begin{aligned} &I_{k}(u)-I_{k}(\overline{u})\ge-L_{k}(u- \overline {u}),\\ & I_{k}^{*}\bigl(u,u' \bigr)-I_{k}^{*}\bigl(\overline{u},\overline{u}'\bigr)\ge -L_{k}^{*}\bigl(u'-\overline{u}'\bigr), \\ &\chi_{1}(u,v)-\chi_{1}(\overline{u},\overline{v})\le a_{1}(u-\overline {u})-b_{1}(v-\overline{v}),\\ &\chi_{2}\bigl(u',v'\bigr)-\chi_{2} \bigl(\overline{u}',\overline{v}'\bigr)\le a_{2}\bigl(u'-\overline{u}' \bigr)-b_{2}\bigl(v'-\overline{v}'\bigr), \end{aligned}$$

where \(u_{0}(t_{k})\le\overline{u}\le u\le v_{0}(t_{k})\) (\(k=1,2,\ldots,m\)), \(u_{0}(0)\le\overline{u}\le u\le v_{0}(0)\), \(u_{0}(T)\le\overline{v}\le v\le v_{0}(T)\).

Then the impulsive system (1.1) has the min-maximal solutions \(u^{*},v^{*}\) in \([u_{0},v_{0}]\), respectively. Moreover, there exist monotone iterative sequences \(\{u_{n}(t)\},\{v_{n}(t)\}\subset[u_{0},v_{0}]\) such that \(u_{n}\to u^{*},v_{n}\to v^{*}(n\to\infty)\) uniformly on \(t\in J\), where \(\{u_{n}(t)\},\{v_{n}(t)\}\) satisfy

$$\begin{aligned} &\textstyle\begin{cases} u_{n}''(t)=\Upsilon(t,u_{n-1}(t),u_{n-1}(\phi(t)), Xu_{n-1}(t), Yu_{n-1}(t),u_{n-1}(\psi(t,\mu(t))))\\ \hphantom{u_{n}''(t)=}{}-D_{1}(t)(u_{n}-u_{n-1})(t)-D_{2}(t)(u_{n}-u_{n-1})(\phi (t))-D_{3}(t)X(u_{n}-u_{n-1})(t)\\ \hphantom{u_{n}''(t)=}{}-D_{4}(t)Y(u_{n}-u_{n-1})(t),\quad t\in J',\\ \triangle u_{n}(t_{k})=I_{k}(u_{n-1}(t_{k}))-L_{k}(u_{n}-u_{n-1})(t_{k}),\quad k=1,2,\ldots,m,\\ \triangle u_{n}'(t_{k})=I_{k}^{*}(u_{n-1}(t_{k}),u_{n-1}'(t_{k}))-L_{k}^{*}(u_{n}'-u_{n-1}')(t_{k}),\quad k=1,2,\ldots ,m,\\ u_{n}(0)=u_{n-1}(0)+\lambda_{1}[u_{n}(T)-u_{n-1}(T)]-\frac{1}{a_{1}}\chi _{1}(u_{n-1}(0),u_{n-1}(T)),\quad n=1,2,\ldots,\\ u_{n}'(0)=u_{n-1}'(0)+\lambda_{2}[u_{n}'(T)-u_{n-1}'(T)]-\frac{1}{a_{2}}\chi _{2}(u_{n-1}'(0),u_{n-1}'(T)),\quad n=1,2,\ldots, \end{cases}\displaystyle \end{aligned}$$
(3.1)
$$\begin{aligned} & \textstyle\begin{cases} v_{n}''(t)=\Upsilon(t,v_{n-1}(t),v_{n-1}(\phi(t)), Xv_{n-1}(t), Yv_{n-1}(t),v_{n-1}(\psi(t,\mu(t))))\\ \hphantom{v_{n}''(t)=}{} -D_{1}(t)(v_{n}-v_{n-1})(t)-D_{2}(t)(v_{n}-v_{n-1})(\phi (t))-D_{3}(t)X(v_{n}-v_{n-1})(t)\\ \hphantom{v_{n}''(t)=}{}-D_{4}(t)Y(v_{n}-v_{n-1})(t),\quad t\in J',\\ \triangle v_{n}(t_{k})=I_{k}(v_{n-1}(t_{k}))-L_{k}(v_{n}-v_{n-1})(t_{k}),\quad k=1,2,\ldots,m,\\ \triangle v_{n}'(t_{k})=I_{k}^{*}(v_{n-1}(t_{k}),v_{n-1}'(t_{k}))-L_{k}^{*}(v_{n}'-v_{n-1}')(t_{k}),\quad k=1,2,\ldots ,m,\\ v_{n}(0)=v_{n-1}(0)+\lambda_{1}[v_{n}(T)-v_{n-1}(T)]-\frac{1}{a_{1}}\chi _{1}(v_{n-1}(0),v_{n-1}(T)),\quad n=1,2,\ldots,\\ v_{n}'(0)=v_{n-1}'(0)+\lambda_{2}[v_{n}'(T)-v_{n-1}'(T)]-\frac{1}{a_{2}}\chi _{2}(v_{n-1}'(0),v_{n-1}'(T)),\quad n=1,2,\ldots, \end{cases}\displaystyle \end{aligned}$$
(3.2)

and

$$ u_{0}\le u_{1} \le\cdots\le u_{n}\le\cdots\le u^{*} \le v^{*}\le \cdots\le v_{n}\le\cdots\le v_{1}\le v_{0}, $$
(3.3)

here \(\lambda_{i}=b_{i}/a_{i}(i=1,2)\).

Proof

For any \(u_{n-1},v_{n-1}\in PC^{1}(J,R)\cap C^{2}(J',R)\), it follows from Lemma 4 that (3.1) and (3.2) have unique solutions \(u_{n}\) and \(v_{n}\) in \(PC^{1}(J,R)\cap C^{2}(J',R)\), respectively.

Now, we verify that

$$ u_{n-1}\le u_{n}\le v_{n}\le v_{n-1}, \quad n=1,2,\ldots. $$
(3.4)

Let \(p(t)=u_{0}(t)-u_{1}(t)\), \(q(t)=v_{1}(t)-v_{0}(t)\), \(w(t)=u_{1}(t)-v_{1}(t)\), by (3.1), (3.2) and \((H_{1})-(H_{4})\), we have that

$$\begin{aligned} & \textstyle\begin{cases} p''(t)\le-D_{1}(t)p(t)-D_{2}(t)p(\phi (t))-D_{3}(t)(Xp)(t)-D_{4}(t)(Yp)(t),\quad t\in J',\\ \triangle p(t_{k})\le-L_{k}p(t_{k}),\quad k=1,2,\ldots,m,\\ \triangle p'(t_{k})\le-L_{k}^{*}p'(t_{k}),\quad k=1,2,\ldots,m,\\ p(0)\le\lambda_{1}p(T),\qquad p'(0)\le\lambda_{2}p'(T), \end{cases}\displaystyle \\ & \textstyle\begin{cases} q''(t)\le-D_{1}(t)q(t)-D_{2}(t)q(\phi (t))-D_{3}(t)(Xq)(t)-D_{4}(t)(Yq)(t),\quad t\in J',\\ \triangle q(t_{k})\le-L_{k}q(t_{k}),\quad k=1,2,\ldots,m,\\ \triangle q'(t_{k})\le-L_{k}^{*}q'(t_{k}),\quad k=1,2,\ldots,m,\\ q(0)\le\lambda_{1}q(T),\qquad q'(0)\le\lambda_{2}q'(T), \end{cases}\displaystyle \\ & \textstyle\begin{cases} w''(t)=\Upsilon(t,u_{0}(t),u_{0}(\phi(t)), Xu_{0}(t), Yu_{0}(t),u_{0}(\psi(t,\mu (t))))-D_{1}(t)(u_{1}-u_{0})(t)\\ \hphantom{w''(t)=}{}-D_{2}(t)(u_{1}-u_{0})(\phi (t))-D_{3}(t)X(u_{1}-u_{0})(t)-D_{4}(t)Y(u_{1}-u_{0})(t)\\ \hphantom{w''(t)=}{}-\Upsilon(t,v_{0}(t),v_{0}(\phi(t)), Xv_{0}(t), Yv_{0}(t),v_{0}(\psi (t,\mu(t))))+D_{1}(t)(v_{1}-v_{0})(t)\\ \hphantom{w''(t)=}{}+D_{2}(t)(v_{1}-v_{0})(\phi (t))+D_{3}(t)X(v_{1}-v_{0})(t)+D_{4}(t)Y(v_{1}-v_{0})(t)\\ \hphantom{w''(t)}\le-D_{1}(t)w(t)-D_{2}(t)w(\phi (t))-D_{3}(t)(Xw)(t)-D_{4}(t)(Yw)(t),\quad t\in J',\\ \triangle w(t_{k})=I_{k}(u_{0}(t_{k}))-I_{k}(v_{0}(t_{k}))-L_{k}(u_{1}-u_{0})(t_{k})+L_{k}(v_{1}-v_{0})(t_{k})\\ \hphantom{w(t_{k})}\le-L_{k}w(t_{k}),\quad k=1,2,\ldots,m,\\ \triangle w'(t_{k})=I_{k}^{*}(u_{0}(t_{k}),u_{0}'(t_{k}))-I_{k}^{*}(v_{0}(t_{k}),v_{0}'(t_{k}))-L_{k}^{*}(u_{1}'-u_{0}')(t_{k})+L_{k}^{*}(v_{1}'-v_{0}')(t_{k})\\ \hphantom{w'(t_{k})}\le-L_{k}^{*}w'(t_{k}),\quad k=1,2,\ldots,m,\\ w(0)=u_{0}(0)-v_{0}(0)+\lambda_{1}[u_{1}(T)-u_{0}(T)]-\lambda_{1}[v_{1}(T)-v_{0}(T)]\\ \hphantom{w(0)=}{}-\frac{1}{a_{1}}\chi_{1}(u_{0}(0),u_{0}(T))+\frac{1}{a_{1}}\chi _{1}(v_{0}(0),v_{0}(T))\\ \hphantom{w(0)}\le u_{0}(0)-v_{0}(0)+\lambda_{1}w(T)-\lambda_{1}[u_{0}(T)-v_{0}(T)]\\ \hphantom{w(0)\le}{}+[v_{0}(0)-u_{0}(0)]-\lambda_{1}[v_{0}(T)-u_{0}(T)]\\ \hphantom{w(0)}\le\lambda_{1}w(T),\\ w'(0)\le\lambda_{2}w'(T). \end{cases}\displaystyle \end{aligned}$$

Thus, by means of Lemma 2, we have \(p(t)\le0\), \(q(t)\le0\), \(w(t)\le 0\), \(\forall t\in J\), \(i.e\)., \(u_{0}\le u_{1}\le v_{1}\le v_{0}\).

Assume that \(u_{k-1}\le u_{k}\le v_{k}\le v_{k-1}\) for some \(k\ge1\). Thus, employing the same technique once again, by Lemma 2, one can get \(u_{k}\le u_{k+1}\le v_{k+1}\le v_{k}\). Thus, one can easily show that

$$ u_{0}\le u_{1} \le\cdots\le u_{n}\le\cdots\le v_{n}\le\cdots \le v_{1}\le v_{0},\quad n=1,2, \ldots. $$
(3.5)

Employing the standard arguments, we have

$$\begin{aligned} &\lim_{n\to\infty}u_{n}(t)=u^{*}(t),\\ &\lim_{n\to\infty}v_{n}(t)=v^{*}(t) \end{aligned}$$

uniformly on \(t\in J\), and the limit functions \(u^{*},v^{*}\) satisfy (1.1). Moreover, \(u^{*},v^{*}\in[u_{0},v_{0}]\).

Next, we prove that \(u^{*},v^{*}\) are the min-maximal solutions of impulsive differential system (1.1) in \([u_{0},v_{0}]\). If \(w\in[u_{0},v_{0}]\) is any solution of (1.1). Let \(u_{n-1}(t)\le w(t)\le v_{n-1}(t),\forall t\in J\), for some positive integer n. Put \(p=u_{n}-w\). Then

$$\textstyle\begin{cases} p''(t)=\Upsilon(t,u_{n-1}(t),u_{n-1}(\phi(t)), Xu_{n-1}(t), Yu_{n-1}(t),u_{n-1}(\psi(t,\mu(t))))\\ \hphantom{p''(t)=}{}-D_{1}(t)(u_{n}-u_{n-1})(t)-D_{2}(t)(u_{n}-u_{n-1})(\phi (t))\\ \hphantom{p''(t)=}{}-D_{3}(t)X(u_{n}-u_{n-1})(t)-D_{4}(t)Y(u_{n}-u_{n-1})(t)\\ \hphantom{p''(t)=}{}-\Upsilon(t,w(t),w(\phi(t)), Xw(t), Yw(t),w(\psi(t,\mu(t))))\\ \hphantom{p''(t)}\le D_{1}(t)(w-u_{n-1})(t)+D_{2}(t)(w-u_{n-1})(\phi(t))+D_{3}(t)X(w-u_{n-1})(t)\\ \hphantom{p''(t)=}{}+D_{4}(t)Y(w-u_{n-1})(t)-D_{1}(t)(u_{n}-u_{n-1})(t)-D_{2}(t)(u_{n}-u_{n-1})(\phi (t))\\ \hphantom{p''(t)=}{}-D_{3}(t)X(u_{n}-u_{n-1})(t)-D_{4}(t)Y(u_{n}-u_{n-1})(t)\\ \hphantom{p''(t)}= -D_{1}(t)p(t)-D_{2}(t)p(\phi (t))-D_{3}(t)(Xp)(t)-D_{4}(t)(Yp)(t),\quad t\in J',\\ \triangle p(t_{k})=I_{k}(u_{n-1}(t_{k}))-I_{k}(w(t_{k}))-L_{k}(u_{n}-u_{n-1})(t_{k})\\ \hphantom{\triangle p(t_{k})}\le-L_{k}p(t_{k}),\quad k=1,2,\ldots,m,\\ \triangle p'(t_{k})=I_{k}^{*}(u_{n-1}(t_{k}),u_{n-1}'(t_{k}))-I_{k}^{*}(w(t_{k}),w'(t_{k}))-L_{k}^{*}(u_{n}'-u_{n-1}')(t_{k})\\ \hphantom{\triangle p'(t_{k})}\le-L_{k}^{*}p'(t_{k}),\quad k=1,2,\ldots,m,\\ p(0)=u_{n-1}(0)+\lambda_{1}[u_{n}(T)-u_{n-1}(T)]-\frac{1}{a_{1}}\chi _{1}(u_{n-1}(0),u_{n-1}(T))\\ \hphantom{p(0)=} {}+\frac{1}{a_{1}}\chi_{1}(w(0),w(T))-w(0)\\ \hphantom{p(0)}\le u_{n-1}(0)+\lambda _{1}[u_{n}(T)-u_{n-1}(T)]+[w(0)-u_{n-1}(0)]\\ \hphantom{p(0)=}{}-\lambda _{1}[w(T)-u_{n-1}(T)]-w(0)\\ \hphantom{p(0)}= \lambda_{1}p(T),\\ p'(0)=u_{n-1}'(0)+\lambda_{2}[u_{n}'(T)-u_{n-1}'(T)]-\frac{1}{a_{2}}\chi _{2}(u_{n-1}'(0),u_{n-1}'(T))\\ \hphantom{p(0)=}{}+\frac{1}{a_{2}}\chi_{2}(w(0),w(T))-w(0)\\ \hphantom{p(0)}\le\lambda_{2}p'(T). \end{cases} $$

By Lemma 2, we have \(u_{n}(t)\le w(t),\forall t\in J\). By the same way as above, we can show \(w(t)\le v_{n}(t),\forall t\in J\). That is, \(u_{n}(t)\le w(t)\le v_{n}(t)\), \(\forall t\in J\).

Now, if \(n\to\infty\), then

$$u_{0}(t)\le u^{*}(t)\le w(t)\le v^{*}(t)\le v_{0}(t),\quad \forall t\in J. $$

That is, \(u^{*},v^{*}\) are the min-maximal solutions of (1.1) in \([u_{0},v_{0}]\). □

Example

Consider

$$ \textstyle\begin{cases} u''(t)= \frac{1}{100}t^{3}[t-u(t)]+\frac{1}{300}t^{3}[t-u(t^{2})]^{3} +\frac{1}{500}t[t^{3}-\int_{0}^{t}tsu(s)\,ds]^{5}\\ \hphantom{u''(t)= }{}+\frac{1}{700}t^{2}[t^{2}-\int_{0}^{1}t^{2}su(s)\,ds]^{7}-\frac {1}{100}t^{4}e^{-u(\frac{1}{2}t^{3}+\frac{1}{2}t^{2}e^{-t})},\quad t\in [0,1],t\neq\frac{1}{2},\\ \triangle u(\frac{1}{2})=-\frac{1}{4}u(\frac{1}{2}),\\ \triangle u'(\frac{1}{2})=bu(\frac{1}{2})-\frac{3}{8}u'(\frac{1}{2}),\\ \chi_{1}(u(0),u(1))=u^{2}(0)-\frac{1}{2}u(1)=0,\\ \chi_{2}(u'(0),u'(1))=u'(0)-\frac{1}{3}u'(1)=0, \end{cases} $$
(4.1)

where \(0\le b\le\frac{1}{26}\), \(m=1\), \(t_{1}=\frac{1}{2}\), \(\phi(t)=t^{2}\), \(\psi(t,\mu (t))=\frac{1}{2}t^{3}+\frac{1}{2}t^{2}e^{-t}\), \(\forall t\in J\).

Take

$$ u_{0}(t)=0,\quad \forall t\in J, \qquad v_{0}(t)= \textstyle\begin{cases} 1+t+\frac{1}{2}t^{2},\quad t\in[0,\frac{1}{2}],\\ 1+t^{2},\quad t\in(\frac{1}{2},1]. \end{cases} $$
(4.2)

Then

$$ u_{0}(t)\le v_{0}(t),\qquad u_{0}'(t)=0, \qquad v_{0}'(t)= \textstyle\begin{cases} 1+t,\quad t\in[0,\frac{1}{2}],\\ 2t,\quad t\in(\frac{1}{2},1], \end{cases} $$
(4.3)

and

$$\begin{aligned} &\textstyle\begin{cases} u_{0}''(t)=0\le\frac{1}{100}t^{4}+\frac{1}{300}t^{6}+\frac {1}{500}t^{16}+\frac{1}{700}t^{16}-\frac{1}{100}t^{4} \\ \hphantom{u_{0}''(t)}=\Upsilon(t,u_{0}(t),u_{0}(\phi(t)), Xu_{0}(t), Yu_{0}(t),u_{0}(\psi (t,\mu(t)))),\\ \triangle u_{0}(\frac{1}{2})=0= I_{1}(u_{0}(\frac{1}{2})),\\ \triangle u_{0}'(\frac{1}{2})=0= I_{1}^{*}(u_{0}(\frac{1}{2}),u_{0}'(\frac {1}{2})),\\ \chi_{1}(u_{0}(0),u_{0}(1))=u_{0}^{2}(0)-\frac{1}{2}u_{0}(1)=0,\\ \chi_{2}(u_{0}'(0),u_{0}'(1))=u_{0}'(0)-\frac{1}{3}u_{0}'(1)=0, \end{cases}\displaystyle \\ & \textstyle\begin{cases} v_{0}''(t)\ge1>\frac{1}{100}+\frac{1}{300}+\frac{1}{500}+\frac {1}{700}\\ \hphantom{v_{0}''(t)}\ge\Upsilon(t,v_{0}(t),v_{0}(\phi(t)), Xv_{0}(t), Yv_{0}(t),v_{0}(\psi (t,\mu(t)))),\\ \triangle v_{0}(\frac{1}{2})=-\frac{3}{8}\ge-\frac{13}{32}=I_{1}(v_{0}(\frac {1}{2})),\\ \triangle v_{0}'(\frac{1}{2})=-\frac{1}{2}\ge\frac{13b}{8}-\frac {9}{16}=I_{1}^{*}(v_{0}(\frac{1}{2}),v_{0}'(\frac{1}{2})),\\ \chi_{1}(v_{0}(0),v_{0}(1))=v_{0}^{2}(0)-\frac{1}{2}v_{0}(1)=0,\\ \chi_{2}(v_{0}'(0),v_{0}'(1))=v_{0}'(0)-\frac{1}{3}v_{0}'(1)=\frac{1}{3}>0. \end{cases}\displaystyle \end{aligned}$$

Consequently, \(u_{0}\), \(v_{0}\) satisfy \((H_{1})\). Let

$$\begin{aligned} \Upsilon(t,u,v,w,z,\xi) =&\frac{1}{100}t^{3}(t-u)+ \frac{1}{300}t^{3}(t-v)^{3} +\frac{1}{500}t \bigl(t^{3}-w\bigr)^{5}\\ &{}+\frac{1}{700}t^{2} \bigl(t^{2}-z\bigr)^{7}-\frac {1}{100}t^{4}e^{-\xi}, \end{aligned}$$

we have

$$\begin{aligned} &\Upsilon(t,u,v,w,z,\xi)-\Upsilon(t,\overline{u},\overline{v},\overline {w}, \overline{z},\overline{\xi}) \\ &\quad =\frac{1}{100}t^{3}\bigl[(t-u)-(t-\overline{u})\bigr]+ \frac {1}{300}t^{3}\bigl[(t-v)^{3}-(t- \overline{v})^{3}\bigr]+\frac {1}{500}t\bigl[\bigl(t^{3}-w \bigr)^{5}-\bigl(t^{3}-\overline{w}\bigr)^{5}\bigr] \\ &\qquad{} +\frac{1}{700}t^{2}\bigl[\bigl(t^{2}-z \bigr)^{7}-\bigl(t^{2}-\overline{z}\bigr)^{7}\bigr]- \frac {1}{100}t^{4}\bigl(e^{-\xi}-e^{-\overline{\xi}}\bigr) \\ &\quad \ge-\frac{1}{100}t^{3}(u-\overline{u})- \frac{1}{25}t^{3}(v-\overline {v})-\frac{1}{100}t^{13}(w- \overline{w})-\frac{1}{100}t^{14}(z-\overline{z}), \end{aligned}$$

where \(u_{0}(t)\le\overline{u}\le u\le v_{0}(t)\), \(u_{0}(\phi(t))\le\overline {v}\le v\le v_{0}(\phi(t))\), \(Xu_{0}(t)\le\overline{w}\le w\le Xv_{0}(t)\), \(Yu_{0}(t)\le\overline{z}\le z\le Yv_{0}(t)\), \(u_{0}(\psi(t,\mu(t)))\le\overline{\xi}\le\xi\le v_{0}(\psi(t,\mu (t)))\), \(\forall t\in J\). For \(L_{1}=\frac{1}{4}\), \(L_{1}^{*}=\frac{3}{8}\), \(a_{1}=2\), \(b_{1}=\frac {1}{2}\), \(a_{2}=1\), \(b_{2}=\frac{1}{3}\), obviously, \((H_{3})\) and \((H_{4})\) hold. On the other hand, put \(D_{1}(t)=\frac{1}{100}t^{3}\), \(D_{2}(t)=\frac{1}{25}t^{3}\), \(D_{3}(t)=\frac {1}{100}t^{13}\), \(D_{4}(t)=\frac{1}{100}t^{14}\), \(\lambda_{1}=\frac{1}{4}\), \(\lambda_{2}=\frac{1}{3}\), \(L_{1}=\frac {1}{4}\), \(L_{1}^{*}=\frac{3}{8}\), it is easy to see that conditions (2.2) and (2.10) hold. So, \((H_{2})\) also holds.

Thus, Theorem 1 is satisfied. Therefore, our conclusions come from Theorem 1 that (4.1) has the min-maximal solution \(u^{*},v^{*}\in[u_{0},v_{0}]\).

References

  1. Ladde, G.S., Lakshmikantham, V., Vatsala, A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, London (1985)

    MATH  Google Scholar 

  2. Cui, Y., Zou, Y.: Existence of solutions for second-order integral boundary value problems. Nonlinear Anal., Model. Control 21(6), 828–838 (2016)

    MathSciNet  Article  Google Scholar 

  3. Bai, Z., Zhang, S., Sun, S., Yin, C.: Monotone iterative method for a class of fractional differential equations. Electron. J. Differ. Equ. 2016, 06 (2016)

    Article  Google Scholar 

  4. Cui, Y.: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48–54 (2016)

    MathSciNet  Article  Google Scholar 

  5. Wang, G.: Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval. Appl. Math. Lett. 47, 1–7 (2015)

    MathSciNet  Article  Google Scholar 

  6. Zhang, X., Liu, L., Wu, Y.: The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl. Math. Lett. 37, 26–33 (2014)

    MathSciNet  Article  Google Scholar 

  7. Zhang, X., Liu, L., Wu, Y.: Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equations. J. Nonlinear Sci. Appl. 10, 3364–3380 (2017)

    MathSciNet  Article  Google Scholar 

  8. Wu, J., Zhang, X., Liu, L., Wu, Y.: Twin iterative solutions for a fractional differential turbulent flow model. Bound. Value Probl. 2016, 98 (2016)

    MathSciNet  Article  Google Scholar 

  9. Wang, G.: Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments. J. Comput. Appl. Math. 236, 2425–2430 (2012)

    MathSciNet  Article  Google Scholar 

  10. Pei, K., Wang, G., Sun, Y.: Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain. Appl. Math. Comput. 312, 158–168 (2017)

    MathSciNet  Google Scholar 

  11. Wang, G.: Twin iterative positive solutions of fractional q-difference Schrodinger equations. Appl. Math. Lett. 76, 103–109 (2018)

    MathSciNet  Article  Google Scholar 

  12. Wang, G., Pei, K., Baleanu, D.: Explicit iteration to Hadamard fractional integro-differential equations on infinite domain. Adv. Differ. Equ. 2016, 299 (2016)

    MathSciNet  Article  Google Scholar 

  13. Zhang, L., Ahmad, B., Wang, G.: Successive iterations for positive extremal solutions of nonlinear fractional differential equations on a half-line. Bull. Aust. Math. Soc. 91, 116–128 (2015)

    MathSciNet  Article  Google Scholar 

  14. Zhang, L., Ahmad, B., Wang, G.: Existence and approximation of positive solutions for nonlinear fractional integro-differential boundary value problems on an unbounded domain. Appl. Comput. Math. 15, 149–158 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions, vol. 2. Hindawi Publishing Corporation, New York (2006)

    Book  Google Scholar 

  16. Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical, Harlow (1993)

    MATH  Google Scholar 

  17. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    Book  Google Scholar 

  18. Guo, D.J., Lakshmikantham, V., Liu, X.Z.: Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic Publishers, Dordrecht (1996)

    Book  Google Scholar 

  19. Yan, J., Zhao, A., Nieto, J.J.: Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems. Math. Comput. Model. 40, 509–518 (2004)

    MathSciNet  Article  Google Scholar 

  20. Ahmad, B., Alsaed, A.: Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type. Nonlinear Anal. Hybrid Syst. 3, 501–509 (2009)

    MathSciNet  Article  Google Scholar 

  21. Bai, Z., Dong, X., Yin, C.: Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions. Bound. Value Probl. 2016, 63 (2016)

    MathSciNet  Article  Google Scholar 

  22. Nieto, J.J., O’Regan, D.: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 10, 680–690 (2009)

    MathSciNet  Article  Google Scholar 

  23. Zhang, H., Chen, L.S., Nieto, J.J.: A delayed epidemic model with stage structure and pulses for management strategy. Nonlinear Anal., Real World Appl. 9, 1714–1726 (2008)

    MathSciNet  Article  Google Scholar 

  24. Liu, Z., Liang, J.: A class of boundary value problems for first-order impulsive integro-differential equations with deviating arguments. J. Comput. Appl. Math. 237, 477–486 (2013)

    MathSciNet  Article  Google Scholar 

  25. Franco, D., Nieto, J.J., O’Regan, D.: Existence of solutions for first order ordinary differential equations with nonlinear boundary conditions. Appl. Math. Comput. 153, 793–802 (2004)

    MathSciNet  MATH  Google Scholar 

  26. Sun, F., Liu, L., Zhang, X., Wu, Y.: Spectral analysis for a singular differential system with integral boundary conditions. Mediterr. J. Math. 13, 4763–4782 (2016)

    MathSciNet  Article  Google Scholar 

  27. Hao, X., Liu, L., Wu, Y.: Iterative solution to singular nth-order nonlocal boundary value problems. Bound. Value Probl. 2015, 125 (2015)

    Article  Google Scholar 

  28. Liang, J., Wang, L., Wang, X.: A class of BVPs for second-order impulsive integro-differential equations of mixed type in Banach space. J. Comput. Anal. Appl. 21, 331–344 (2016)

    MathSciNet  MATH  Google Scholar 

  29. Zhang, X., Wu, Y., Lou, C.: Nonlocal fractional order differential equations with changing-sign singular perturbation. Appl. Math. Model. 39, 6543–6552 (2015)

    MathSciNet  Article  Google Scholar 

  30. Jiang, J., Liu, L., Wu, Y.: Positive solutions for second order impulsive differential equations with Stieltjes integral boundary conditions. Adv. Differ. Equ. 2012, 124 (2012)

    MathSciNet  Article  Google Scholar 

  31. Wang, W., Shen, J.H., Luo, Z.G.: Multi-point boundary value problems for second-order functional differential equations. Comput. Math. Appl. 56, 2065–2072 (2008)

    MathSciNet  Article  Google Scholar 

  32. Yang, X.X., Shen, J.H.: Periodic boundary value problems for second-order impulsive integro-differential equations. J. Comput. Appl. Math. 209, 176–186 (2007)

    MathSciNet  Article  Google Scholar 

  33. Li, J.L.: Periodic boundary value problems for second-order impulsive integro-differential equations. Appl. Math. Comput. 198, 317–325 (2008)

    MathSciNet  MATH  Google Scholar 

  34. Liu, Y.J.: Positive solutions of periodic boundary value problems for nonlinear first-order impulsive differential equations. Nonlinear Anal. 70, 2106–2122 (2009)

    MathSciNet  Article  Google Scholar 

  35. Luo, Z.G., Jing, Z.J.: Periodic boundary value problem for first-order impulsive functional differential equations. Comput. Math. Appl. 55, 2094–2107 (2008)

    MathSciNet  Article  Google Scholar 

  36. Agarwal, R.P., O’Regan, D., Wong, P.J.Y.: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht (1999)

    Book  Google Scholar 

  37. Burton, T.A.: Differential inequalities for integral and delay differential equations. In: Liu, X., Siegel, D. (eds.) Comparison Methods and Stability Theory. Lecture Notes in Pure and Appl. Math. Dekker, New York (1994)

    Google Scholar 

  38. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Book  Google Scholar 

  39. Wang, G., Zhang, L., Song, G.: Integral boundary value problems for first order integro-differential equations with deviating arguments. J. Comput. Appl. Math. 225, 602–611 (2009)

    MathSciNet  Article  Google Scholar 

  40. Wang, G., Zhang, L., Song, G.: Extremal solutions for the first order impulsive functional differential equations with upper and lower solutions in reversed order. J. Comput. Appl. Math. 235, 325–333 (2010)

    MathSciNet  Article  Google Scholar 

  41. Wang, G., Zhang, L., Song, G.: Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions. Nonlinear Anal. 74, 974–982 (2011)

    MathSciNet  Article  Google Scholar 

  42. Wang, G.: Boundary value problems for systems of nonlinear integro-differential equations with deviating arguments. J. Comput. Appl. Math. 234, 1356–1363 (2010)

    MathSciNet  Article  Google Scholar 

  43. Wang, G., Pei, K., Agarwal, R.P., Zhang, L., Ahmad, B.: Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line. J. Comput. Appl. Math. 343, 230–239 (2018)

    MathSciNet  Article  Google Scholar 

Download references

Funding

The authors are supported financially by the NNSF of China (No. 11501342).

Author information

Authors and Affiliations

Authors

Contributions

All authors contributed equally. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Guotao Wang.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hou, Y., Zhang, L. & Wang, G. A new comparison principle and its application to nonlinear impulsive functional integro-differential equations. Adv Differ Equ 2018, 380 (2018). https://doi.org/10.1186/s13662-018-1849-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-018-1849-7

Keywords

  • Comparison principle
  • Impulsive functional integro-differential equations
  • Monotone iterative technique
  • Upper and lower solutions