# Impulsive differential and impulsive integral inequalities with integral jump conditions

## Abstract

In this article, we establish some impulsive differential and impulsive integral inequalities for integral jump conditions. The new jump conditions for impulse effects are related to the integral conditions of the past state. Two examples are given to illustrate the advantage of our results.

2010 Mathematics Subject Classification: 34A37; 34A40.

## 1 Introduction

In , Lakshmikantham et al. developed a famous impulsive differential inequality given as Theorem A below.

Lakshmikantham et al. assume that 0 ≤ t0 < t1 < t2 <, limk→∞t k = ∞, R+ = [0, +∞) and I R. They define PC(R+, I) = {u: R+I; u(t) is continuous for tt k , and u(0+), $u\left({t}_{k}^{-}\right)$, and $u\left({t}_{k}^{+}\right)$ exist, and $u\left({t}_{k}^{-}\right)=u\left({t}_{k}\right),k=1,2,\dots \right\}$ and PC1(R+,I) = {u PC(R+, I): u'(t) is continuous everywhere for tt k , and u'(0+), ${u}^{\prime }\left({t}_{k}^{+}\right)$ and ${u}^{\prime }\left({t}_{k}^{-}\right)$ exist, and ${u}^{\prime }\left({t}_{k}^{-}\right)={u}^{\prime }\left({t}_{k}\right),k=1,2,\dots \right\}$.

Theorem A. Assume that

(H0) the sequence {t k } satisfies 0 ≤ t0 < t1 < t2 < , limk→∞t k = ∞;

(H1) m PC1[R+, R] and m(t) is left-continuous at t k , k = 1, 2,...;

(H2) for k = 1, 2,..., t ≥ t0,

${m}^{\prime }\left(t\right)\le p\left(t\right)m\left(t\right)+q\left(t\right),\phantom{\rule{1em}{0ex}}t\ne {t}_{k},$
(1.1)
$m\left({t}_{k}^{+}\right)\le {d}_{k}m\left({t}_{k}\right)+{b}_{k},$
(1.2)

where q, p C[R+, R], d k ≥ 0 and b k are constants.

Then,

$\begin{array}{cc}\hfill m\left(t\right)& \le m\left({t}_{0}\right)\prod _{{t}_{0}<{t}_{k}
(1.3)

Impulsive differential and impulsive integral inequalities play an important role in the study of the theory of impulsive differential equations (see ). In recent years, many authors have used impulsive (differential or integral) inequalities to investigate properties of solutions of various impulsive problems, such as existence, uniqueness, boundedness, stability, asymptotic behavior, and oscillation etc. (see, for example ). There are many good results on the impulsive differential and impulsive integral inequalities (see for example ). However, most of these articles deal with jump conditions at impulse point t k depending on the left-hand limit m(t k ) or a time-delay value, m(t k -τ), τ > 0. Our main goal is to extend the theory of impulsive differential and impulsive integral inequalities to include integral jump conditions.

In the present article, we will show that Theorem A can be generalized to obtain differential inequalities for integral jump conditions by replacing the inequality in (1.2) by the inequality in (1.4).

$m\left({t}_{k}^{+}\right)\le {d}_{k}m\left({t}_{k}\right)+{c}_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}m\left(s\right)ds+{b}_{k},\phantom{\rule{1em}{0ex}}k=1,2,\dots ,$
(1.4)

where 0 ≤ σ k τ k t k - tk-1. We note that if c k = 0 for all k = 1, 2,..., then condition (1.4) reduces to condition (1.2). If d k = 0, c k ≠ 0 and 0 ≤ σ k < τ k t k -tk-1, k = 1, 2,..., then condition (1.4) means that the bound of the jump condition at t k is a functional of past states on the interval (t k - τ k , t k - σ k ] before the impulse point t k . Moreover, we establish some new impulsive integral inequalities with integral jump conditions.

At the end of this article, we will show some applications of our results to prove a maximum principle and the boundedness of solutions for impulsive problems.

## 2 Main results

Denote l = max{k: tt k , k = 1, 2,...}. Now we are in the position to state and prove our results.

Theorem 2.1. Let (H 0 ) and (H1) hold. Suppose that p, q C[R+, R] and for k = 1, 2,..., tt0,

${m}^{\prime }\left(t\right)\le p\left(t\right)m\left(t\right)+q\left(t\right),\phantom{\rule{1em}{0ex}}t\ne {t}_{k},$
(2.1)
$m\left({t}_{k}^{+}\right)\le {d}_{k}m\left({t}_{k}\right)+{c}_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}m\left(s\right)ds+{b}_{k},$
(2.2)

where c k , d k ≥ 0, 0 ≤ σ k τ k t k - tk-1and b k are constants.

Then,

(2.3)

Proof. From (2.1) we have that

$\frac{d}{dt}\left[m\left(t\right){e}^{-{\int }_{{t}_{0}}^{t}p\left(\xi \right)d\xi }\right]\le q\left(t\right){e}^{-{\int }_{{t}_{0}}^{t}p\left(\xi \right)d\xi },$
(2.4)

for t [t0, t1]. Integrating (2.4) from t0 to t for t [t0, t1], we get

$m\left(t\right)\le m\left({t}_{0}\right){e}^{{\int }_{{t}_{0}}^{t}p\left(\xi \right)d\xi }+\underset{{t}_{0}}{\overset{t}{\int }}q\left(s\right){e}^{{\int }_{s}^{t}p\left(\xi \right)d\xi }ds.$
(2.5)

Hence (2.3) is valid on [t0, t1]. Assume that (2.3) holds for t [t0, t n ] for some integer n > 1. Then, for t [t n , tn+1], it follow from (2.1) and (2.5) that

$m\left(t\right)\le m\left({t}_{n}^{+}\right){e}^{{\int }_{{t}_{n}}^{t}p\left(\xi \right)d\xi }+\underset{{t}_{n}}{\overset{t}{\int }}q\left(s\right){e}^{{\int }_{s}^{t}p\left(\xi \right)d\xi }ds.$
(2.6)

Now using (2.2) and (2.6), we have

$m\left(t\right)\le \left({d}_{n}m\left({t}_{n}\right)+{c}_{n}\underset{{t}_{n}-{\tau }_{n}}{\overset{{t}_{n}-{\sigma }_{n}}{\int }}m\left(s\right)ds+{b}_{n}\right){e}^{{\int }_{{t}_{n}}^{t}p\left(\xi \right)d\xi }+\underset{{t}_{n}}{\overset{t}{\int }}q\left(s\right){e}^{{\int }_{s}^{t}p\left(\xi \right)d\xi }ds.$
(2.7)

By the principle of mathematical induction, (2.7) can be expressed as

$\begin{array}{l}m\left(t\right)\le \left\{{d}_{n}\left(\left\{m\left({t}_{0}\right)\prod _{{t}_{0}<{t}_{k}<{t}_{n}}\left({d}_{k}{e}^{{\int }_{{t}_{k-1}}^{{t}_{k}}p\left(\xi \right)d\xi }+{c}_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}{e}^{{\int }_{{t}_{k-1}}^{s}p\left(\xi \right)d\xi }ds\right)\\ +\sum _{{t}_{0}<{t}_{k}<{t}_{n}}\left[\prod _{{t}_{k}<{t}_{j}<{t}_{n}}\left({d}_{j}{e}^{{\int }_{{t}_{j-1}}^{{t}_{j}}p\left(\xi \right)d\xi }+{c}_{j}\underset{{t}_{j}-{\tau }_{j}}{\overset{{t}_{j}-{\sigma }_{j}}{\int }}{e}^{{\int }_{{t}_{j-1}}^{s}p\left(\xi \right)d\xi }ds\right)\\ ×\left({d}_{k}\underset{{t}_{k-1}}{\overset{{t}_{k}}{\int }}q\left(s\right){e}^{{\int }_{s}^{{t}_{k}}p\left(\xi \right)d\xi }ds\\ +{c}_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}\underset{{t}_{k-1}}{\overset{s}{\int }}q\left(r\right){e}^{{\int }_{r}^{s}p\left(\xi \right)d\xi }drds+{b}_{k}\right)\right]\right\}{e}^{{\int }_{{t}_{n-1}}^{{t}_{n}}p\left(\xi \right)d\xi }\\ +\underset{{t}_{n-1}}{\overset{{t}_{n}}{\int }}q\left(s\right){e}^{{\int }_{s}^{{t}_{n}}p\left(\xi \right)d\xi }ds\right)+{c}_{n}\underset{{t}_{n}-{\tau }_{n}}{\overset{{t}_{n}-{\sigma }_{n}}{\int }}\left\{\left\{m\left({t}_{0}\right)\prod _{{t}_{0}<{t}_{k}
(2.8)

Set

${E}_{k}={d}_{k}{e}^{{\int }_{{t}_{k-1}}^{{t}_{k}}p\left(\xi \right)d\xi }+{c}_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}{e}^{{\int }_{{t}_{k-1}}^{s}p\left(\xi \right)d\xi }ds$
(2.9)
${G}_{k}={d}_{k}\underset{{t}_{k-1}}{\overset{{t}_{k}}{\int }}q\left(s\right){e}^{{\int }_{s}^{{t}_{k}}p\left(\xi \right)d\xi }ds+{c}_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}\underset{{t}_{k-1}}{\overset{s}{\int }}q\left(r\right){e}^{{\int }_{r}^{s}p\left(\xi \right)d\xi }drds+{b}_{k}.$
(2.10)

Substituting (2.9), (2.10) into (2.8), we get that for t [t n , tn+1]

$\begin{array}{l}m\left(t\right)\le \left\{{d}_{n}\left(\left\{m\left({t}_{0}\right)\prod _{{t}_{0}<{t}_{k}<{t}_{n}}{E}_{k}+\sum _{{t}_{0}<{t}_{k}<{t}_{n}}\left[\prod _{{t}_{k}<{t}_{j}<{t}_{n}}{E}_{j}{G}_{k}\right]\right\}{e}^{{\int }_{{t}_{n-1}}^{{t}_{n}}p\left(\xi \right)d\xi }\\ +\underset{{t}_{n-1}}{\overset{{t}_{n}}{\int }}q\left(s\right){e}^{{\int }_{s}^{{t}_{n}}p\left(\xi \right)d\xi }ds\right)+{c}_{n}\underset{{t}_{n}-{\tau }_{n}}{\overset{{t}_{n}-{\sigma }_{n}}{\int }}\left\{\left\{m\left({t}_{0}\right)\prod _{{t}_{0}<{t}_{k}

Hence,

for t n ttn+1. Therefore, the estimate (2.3) holds for t0ttn+1. This completes the proof.

Remark 2.2. If c k = 0 for all k = 1, 2,..., then Theorem 2.1 reduces to Theorem A.

Corollary 2.3. Let (H0) and (H1) hold. Suppose that p, q C[R+, R] and for k = 1, 2,..., tt0,

${m}^{\prime }\left(t\right)\le p\left(t\right)m\left(t\right)+q\left(t\right),\phantom{\rule{1em}{0ex}}t\ne {t}_{k},$
(2.11)
$m\left({t}_{k}^{+}\right)\le {c}_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}m\left(s\right)ds+{b}_{k},$
(2.12)

where c k ≥ 0, 0 ≤ σ k τ k t k - tk-1and b k are constants.

Then,

(2.13)

The following corollary will be used in our examples. For convenience, we set

${A}_{k}=\frac{{c}_{k}}{p}\left({e}^{-p{\sigma }_{k}}-{e}^{-p{\tau }_{k}}\right),$
(2.14)
(2.15)

Corollary 2.4. Let (H0) and (H1) hold. Suppose that q C[R+, R], and for k = 1, 2,..., tt0,

${m}^{\prime }\left(t\right)\le pm\left(t\right)+q\left(t\right),\phantom{\rule{1em}{0ex}}t\ne {t}_{k},$
(2.16)
$m\left({t}_{k}^{+}\right)\le {c}_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}m\left(s\right)ds+{b}_{k},$
(2.17)

where p ≠ 0, c k ≥ 0, 0 ≤ σ k τ k t k - tk-1and b k are constants.

Then,

$m\left(t\right)\le m\left({t}_{0}\right)\left(\prod _{{t}_{0}<{t}_{k}
(2.18)

for tt0where A k , B k are defined by (2.14), (2.15), respectively.

Proof. By using Corollary 2.3 and reversing the order of double integration, we have the required result.

Corollary 2.5. Let (H0) and (H1) hold. Suppose that q C[R+, R], and for k = 1, 2,...,tt0,

${m}^{\prime }\left(t\right)\le q\left(t\right),\phantom{\rule{1em}{0ex}}t\ne {t}_{k},$
(2.19)
$\Delta m\left({t}_{k}\right)\le {c}_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}m\left(s\right)ds+{b}_{k},$
(2.20)

where$\Delta m\left({t}_{k}\right)=m\left({t}_{k}^{+}\right)-m\left({t}_{k}\right)$, c k ≥ 0, 0 ≤ σ k τ k t k - tk-1and b k are constants.

Then,

(2.21)

Proof. By setting p(t) ≡ 0 and d k = 1(k = 1, 2,...) in Theorem 2.1 and reversing the order of double integration, we have the required result.

Next, we give an application of Theorem 2.1 to the determination of a bound for the solutions of impulsive integral inequalities with integral jump conditions.

Theorem 2.6. Assume that (H0) and (H1) hold. Suppose that p C[R+, R+] and for k = 1, 2,...

$m\left(t\right)\le C+\underset{{t}_{0}}{\overset{t}{\int }}p\left(s\right)m\left(s\right)ds+\sum _{{t}_{0}<{t}_{k}
(2.22)

where α k , β k ≥ 0, 0 ≤ σ k τ k t k - tk-1and C are constants. Then

$m\left(t\right)\le C\prod _{{t}_{0}<{t}_{k}
(2.23)

Proof. Defining a function v(t) by the right side of (2.22), we have

$\begin{array}{c}{v}^{\prime }\left(t\right)=p\left(t\right)m\left(t\right),\phantom{\rule{1em}{0ex}}t\ne {t}_{k},\phantom{\rule{1em}{0ex}}v\left({t}_{0}\right)=C,\\ v\left({t}_{k}^{+}\right)=v\left({t}_{k}\right)+{\beta }_{k}m\left({t}_{k}\right)+{\alpha }_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}m\left(s\right)ds.\end{array}$

Since m(t) ≤ v(t), we get

$\begin{array}{c}{v}^{\prime }\left(t\right)\le p\left(t\right)v\left(t\right),\phantom{\rule{1em}{0ex}}t\ne {t}_{k},\phantom{\rule{1em}{0ex}}v\left({t}_{0}\right)=C,\\ v\left({t}_{k}^{+}\right)\le \left(1+{\beta }_{k}\right)v\left({t}_{k}\right)+{\alpha }_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}v\left(s\right)ds.\end{array}$

Applying Theorem 2.1, we obtain

$v\left(t\right)\le C\prod _{{t}_{0}<{t}_{k}

which results in (2.23).

Theorem 2.7 . Assume that (H0) and (H1) hold. Suppose that p C[R+, R+], h PC[R+, R] and for k = 1, 2,...

$m\left(t\right)\le h\left(t\right)+\underset{{t}_{0}}{\overset{t}{\int }}p\left(s\right)m\left(s\right)ds+\sum _{{t}_{0}<{t}_{k}
(2.24)

where α k , β k ≥ 0 and 0 ≤ σ k τ k t k - tk-1are constants.

Then,

(2.25)

Proof. Setting

$v\left(t\right)=\underset{{t}_{0}}{\overset{t}{\int }}p\left(s\right)m\left(s\right)ds+\sum _{{t}_{0}<{t}_{k}

and from the fact that m(t) ≤ h(t) + v(t), we obtain

$\begin{array}{c}{v}^{\prime }\left(t\right)\le p\left(t\right)v\left(t\right)+p\left(t\right)h\left(t\right),\phantom{\rule{1em}{0ex}}t\ne {t}_{k},\phantom{\rule{1em}{0ex}}v\left({t}_{0}\right)=0,\\ v\left({t}_{k}^{+}\right)\le \left(1+{\beta }_{k}\right)v\left({t}_{k}\right)+{\alpha }_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}v\left(s\right)ds+{\beta }_{k}h\left({t}_{k}\right)+{\alpha }_{k}\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}h\left(s\right)ds.\end{array}$

Using Theorem 2.1 together with m(t) ≤ h(t) + v(t), we then obtain the estimate (2.25).

Remark 2.8. If α k = 0 for all k = 1, 2,..., then Theorem 2.6 and Theorem 2.7 are reduced to the Theorems 1.5.1 and 1.5.2 in , respectively.

## 3 Some examples

In this section, two applications of impulsive differential and impulsive integral inequalities with integral jump conditions are given.

Corollary 3.1. Assume that u PC1[J, R] satisfies

$\left\{\begin{array}{c}\hfill {u}^{\prime }\left(t\right)-Mu\left(t\right)+a\left(t\right)\le 0,\phantom{\rule{1em}{0ex}}t\ne {t}_{k},\phantom{\rule{1em}{0ex}}t\in J=\left[0,T\right],\hfill \\ \hfill u\left({t}_{k}^{+}\right)\le {c}_{k}{\int }_{{t}_{k}-{\tau }_{k}}^{{t}_{k}-{\sigma }_{k}}u\left(s\right)ds,\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}k=1,\dots ,n,\hfill \\ \hfill u\left(0\right)=u\left(T\right)+\lambda ,\hfill \end{array}\right\$
(3.1)

where M > 0, a C[R+, R+], 0 < t1 < t2 < < t n < T. c k ≥ 0, 0 ≤ σ k τ k t k - tk-1, k = 1, 2,..., n.

(D1) ${\prod }_{k=1}^{n}\frac{{c}_{k}}{M}\left({e}^{-{\sigma }_{k}M}-{e}^{-{\tau }_{k}M}\right)<{e}^{-MT},$

(D2)

$\begin{array}{c}{e}^{\left({t}_{k}-{\tau }_{k}\right)M}\underset{{t}_{k-1}}{\overset{{t}_{k}-{\tau }_{k}}{\int }}a\left(r\right){e}^{-Mr}dr+\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}a\left(r\right)dr\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le {e}^{\left({t}_{k}-{\sigma }_{k}\right)M}\underset{{t}_{k-1}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}a\left(r\right){e}^{-Mr}dr,\phantom{\rule{1em}{0ex}}k=1,2,\dots ,n,\end{array}$

(D3) $\lambda \le {\int }_{{t}_{n}}^{T}a\left(s\right){e}^{M\left(T-s\right)}ds$.

Then u(t) ≤ 0 for t [0, T].

Proof. By Corollary 2.4 for t [0, T] we can write that

$u\left(t\right)\le u\left(0\right)\left(\prod _{{t}_{0}<{t}_{k}

where

${\overline{A}}_{k}=\frac{{c}_{k}}{M}\left({e}^{-{\sigma }_{k}M}-{e}^{-{\tau }_{k}M}\right)\ge 0,$

and

$\begin{array}{l}{\overline{B}}_{k}=\frac{{c}_{k}}{M}\left({e}^{\left({t}_{k}-{\tau }_{k}\right)M}\underset{{t}_{k-1}}{\overset{{t}_{k}-{\tau }_{k}}{\int }}a\left(r\right){e}^{-Mr}dr+\underset{{t}_{k}-{\tau }_{k}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}a\left(r\right)dr\\ -{e}^{\left({t}_{k}-{\sigma }_{k}\right)M}\underset{{t}_{k-1}}{\overset{{t}_{k}-{\sigma }_{k}}{\int }}a\left(r\right){e}^{-Mr}dr\right),k=1,2,\dots ,n.\end{array}$

Condition (D2) implies that ${\overline{B}}_{k}\le 0$ for k = 1, 2,..., n. Then, it is sufficient to show that u(0) ≤ 0. For t = T we have

$u\left(T\right)\le u\left(0\right)\left(\prod _{k=1}^{n}{\overline{A}}_{k}\right){e}^{MT}+\sum _{{t}_{0}<{t}_{k}

By the conditions (D1) and (D3), we see that

$\begin{array}{cc}\hfill u\left(0\right)\left[1-\left(\prod _{k=1}^{n}{\overline{A}}_{k}\right){e}^{MT}\right]& \le \lambda +\sum _{{t}_{0}<{t}_{k}

which implies that u(0) ≤ 0.

Corollary 3.2. Let v PC1[R+, R] such that

$\left\{\begin{array}{c}\hfill {v}^{\prime }\left(t\right)=f\left(t,v\left(t\right)\right),\phantom{\rule{1em}{0ex}}t\ne {t}_{k},\phantom{\rule{1em}{0ex}}t\in \left[{t}_{0},\infty \right),\hfill \\ \hfill \Delta v\left({t}_{k}\right)={I}_{k}\left({\int }_{{t}_{k}-{\tau }_{k}}^{{t}_{k}-{\sigma }_{k}}v\left(s\right)ds\right),\phantom{\rule{1em}{0ex}}k=1,2,\dots ,\hfill \\ \hfill v\left({t}_{0}\right)={v}_{0},\hfill \end{array}\right\$
(3.2)

where f C(R × R, R), I k C(R, R), 0 ≤ t0 < t1 < t2 < , limk→∞t k = ∞, 0 ≤ σ k τ k t k - tk-1, k = 1, 2,.... Assume that

(D4) there exists a constant N > 0, such that

$\left|f\left(t,v\left(t\right)\right)\right|\le N\left|v\left(t\right)\right|\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}t\ge {t}_{0},$

(D5) there exist constants L k ≥ 0 such that

$\left|{I}_{k}\left(x\right)\right|\le {L}_{k}\left|x\right|,\phantom{\rule{1em}{0ex}}x\in R,\phantom{\rule{1em}{0ex}}k=1,2,\dots .$

Then the following inequality is valid

$\left|v\left(t\right)\right|\le \left|{v}_{0}\right|\prod _{{t}_{0}<{t}_{k}
(3.3)

Proof. The solution v(t) of problem (3.2) satisfies the equation

$v\left(t\right)=v\left({t}_{0}\right)+\underset{{t}_{0}}{\overset{t}{\int }}f\left(s,v\left(s\right)\right)ds+\sum _{{t}_{0}<{t}_{k}

From the hypothesis (D4), (D5) it follows for tt0 that

$\begin{array}{cc}\hfill \left|v\left(t\right)\right|& \le \left|{v}_{0}\right|+\underset{{t}_{0}}{\overset{t}{\int }}\left|f\left(s,v\left(s\right)\right)\right|ds+\sum _{{t}_{0}<{t}_{k}

Hence Theorem 2.6 yields the estimate

$\left|v\left(t\right)\right|\le \left|{v}_{0}\right|\prod _{{t}_{0}<{t}_{k}

Therefore, the inequality (3.3) holds for tt0 and the proof is complete.

## References

1. Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.

2. Bainov DD, Simeonov PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific & Technical, Harlow; 1993.

3. Bainov DD, Simeonov PS: Impulsive Differential Equations: Asymptotic Properties of the Solutions. World Scientific, Singapore; 1995.

4. Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.

5. Bainov DD, Hristova SG: The method of quasilinearization for the periodic boundary value problem for systems of impulsive differential equations. Appl Math Comput 2001, 117: 73–85. 10.1016/S0096-3003(99)00156-3

6. Bonotto EM, Gimenes LP, Federson M: Oscillation for a second-order neutral equation with impulses. Appl Math Comput 2009, 215: 1–15. 10.1016/j.amc.2009.04.039

7. Cui BT, Han M, Yang H: Some sufficient conditions for oscillation of impulsive delay hyperbolic systems with Robin boundary conditions. J Comput Appl Math 2005, 180: 365–375. 10.1016/j.cam.2004.11.006

8. Ding W, Xing Y, Han M: Anti-periodic boundary value problems for first order impulsive functional differential equations. Appl Math Comput 2007, 186: 45–53. 10.1016/j.amc.2006.07.087

9. Franco D, Nieto JJ: First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions. Nonlinear Anal 2000, 42: 163–173. 10.1016/S0362-546X(98)00337-X

10. Franco D, Nieto JJ: Maximum principles for periodic impulsive first order problems. J Comput Appl Math 1998, 88: 149–159. 10.1016/S0377-0427(97)00212-4

11. Fu X, Zhang L: Forced oscillation for impulsive hyperbolic boundary value problems with delay. Appl Math Comput 2004, 158: 761–780. 10.1016/j.amc.2003.08.148

12. Gimenes LP, Federson M: Oscillation by impulses for a second-order delay differential equation. Comput Math Appl 2006, 52: 819–828. 10.1016/j.camwa.2006.06.001

13. He Z, Ge W: Oscillations of second-order nonlinear impulsive ordinary differential equations. J Comput Appl Math 2003, 158: 397–406. 10.1016/S0377-0427(03)00474-6

14. He Z, Ge W: Periodic boundary value problem for first order impulsive delay differential equations. Appl Math Comput 1999, 104: 51–63. 10.1016/S0096-3003(98)10059-0

15. He Z, He X: Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions. Comput Math Appl 2004, 48: 73–84. 10.1016/j.camwa.2004.01.005

16. He Z, He X: Periodic boundary value problems for first order impulsive integro-differential equations of mixed type. J Math Anal Appl 2004, 296: 8–20. 10.1016/j.jmaa.2003.12.047

17. He Z, Yu J: Periodic boundary value problem for first-order impulsive ordinary differential equations. J Math Anal Appl 2002, 272: 67–78. 10.1016/S0022-247X(02)00133-6

18. He Z, Yu J: Periodic boundary value problem for first-order impulsive functional differential equations. J Comput Appl Math 2002, 138: 205–217. 10.1016/S0377-0427(01)00381-8

19. Hristova SG, Kulev GK: Quasilinearization of a boundary value problem for impulsive differential equations. J Comput Appl Math 2001, 132: 399–407. 10.1016/S0377-0427(00)00442-8

20. Huang M: Oscillation criteria for second order nonlinear dynamic equations with impulses. Comput Math Appl 2010, 59: 31–41. 10.1016/j.camwa.2009.03.039

21. Jiao J, Chen L, Li L: Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations. J Math Anal Appl 2008, 337: 458–463. 10.1016/j.jmaa.2007.04.021

22. Li J: Periodic boundary value problems for second-order impulsive integro-differential equations. Appl Math Comput 2008, 198: 317–325. 10.1016/j.amc.2007.08.079

23. Li Q, Liang H, Zhang Z, Yu Y: Oscillation of second order self-conjugate differential equation with impulses. J Comput Appl Math 2006, 197: 78–88. 10.1016/j.cam.2005.10.035

24. Li J, Shen J: Periodic boundary value problems for delay differential equations with impulses. J Comput Appl Math 2006, 193: 563–573. 10.1016/j.cam.2005.05.037

25. Li J, Shen J: Periodic boundary value problems for impulsive integro-differential equations of mixed type. Appl Math Comput 2006, 183: 890–902. 10.1016/j.amc.2006.06.037

26. Li WN: On the forced oscillation of solutions for systems of impulsive parabolic differential equations with several delays. J Comput Appl Math 2005, 181: 46–57. 10.1016/j.cam.2004.11.016

27. Li WN, Han M: Oscillation of solutions for certain impulsive vector parabolic differential equations with delays. J Math Anal Appl 2007, 326: 363–371. 10.1016/j.jmaa.2006.03.005

28. Liu H, Li Q: Asymptotic behavior of second-order impulsive differential equations, Electron. J Diff Equ 2011, 33: 1–7.

29. Luo J: Oscillation of hyperbolic partial differential equations with impulses. Appl Math Comput 2002, 133: 309–318. 10.1016/S0096-3003(01)00217-X

30. Nieto JJ, Rodriguez-Lopez R: New comparison results for impulsive integro-differential equations and applications. J Math Anal Appl 2007, 328: 1343–1368. 10.1016/j.jmaa.2006.06.029

31. Pandian S, Purushothaman G: Asymptotic behavior of solutions of higher order nonlinear delay impulsive differential equations with damping. Int J Pure Appl Math 2011, 72: 401–414.

32. Peng M: Oscillation caused by impulses. J Math Anal Appl 2001, 255: 163–176. 10.1006/jmaa.2000.7218

33. Peng M: Oscillation criteria for second-order impulsive delay difference equations. Appl Math Comput 2003, 146: 227–235. 10.1016/S0096-3003(02)00539-8

34. Peng M: Oscillation theorems of second-order nonlinear neutral delay difference equations with impulses. Comput Math Appl 2002, 44: 741–748. 10.1016/S0898-1221(02)00187-6

35. Shen J: New maximum principles for first-order impulsive boundary value problems. Appl Math Lett 2003, 16: 105–112. 10.1016/S0893-9659(02)00151-9

36. Stamova IM: Lyapunov method for boundedness of solutions of nonlinear impulsive functional differential equations. Appl Math Comput 2006, 177: 714–719. 10.1016/j.amc.2005.09.107

37. Wang P, Wu Y: Oscillation criteria for impulsive parabolic differential equations of neutral type. Int J Pure Appl Math 2004, 14: 505–514.

38. Zhang C, Feng W, Yang F: Oscillations of higher order nonlinear functional differential equations with impulses. Appl Math Comput 2007, 190: 370–381. 10.1016/j.amc.2007.01.029

39. Zhang C, Feng W, Yang J, Huang M: Oscillations of second order impulses nonlinear FDE with forcing term. Appl Math Comput 2008, 198: 271–279. 10.1016/j.amc.2007.08.033

40. Ale SO, Oyelami BO, Sesay MS: Cone-valued impulsive differential and integrodifferential inequalities. Electron J Diff Equ 2005, 66: 1–14. 2005

41. Deng S, Prather C: Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay. J Inequal Pure Appl Math 2008., 9(2): Art. 34

42. Hristova SG: Nonlinear delay integral inequalities for piecewise continuous functions and applications. J Inequal Pure Appl Math 2004., 5(4): Art. 88

43. Li J: On some new impulsive integral inequalities. J Inequal Appl 2008., 312395(8):

44. Peng Y, Kang Y, Yuan M, Huang R, Yang L: Gronwall-type integral inequalities with impulses on time scales. Adv Diff Equ 2011., 2011(26):

45. Tatar NE: An impulsive nonlinear singular version of the Gronwall-Bihari inequality. J Inequal Appl 2006., 84561(12):

46. Wang H, Ding C: A new nonlinear impulsive delay differential inequality and its applications. J Inequal Appl 2011., 2011(11):

47. Wang WS, Li Z: A new class of impulsive integral inequalities and its application. IEEE 2011, 1897–1899. 2011 International Conference on Multimedia Technology

48. Yan J: Stability theorems of perturbed linear systems with impulse effect. Portuga Math 1996, 53: 43–51.

## Acknowledgements

The authors thank the referees for several useful remarks and interesting comments. This research was supported by the Centre of Excellence in Mathematics, Thailand.

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Correspondence to Phollakrit Thiramanus.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Thiramanus, P., Tariboon, J. Impulsive differential and impulsive integral inequalities with integral jump conditions. J Inequal Appl 2012, 25 (2012). https://doi.org/10.1186/1029-242X-2012-25 