1 Introduction

Impulsive differential equations, that is, differential equations involving impulse effect, appear as a natural description of observed evolution phenomena of several real world problems. Many processes studied in applied sciences are represented by impulsive differential equations. However, the situation is quite different in many physical phenomena that have a sudden change in their states such as mechanical systems with impact, biological systems such as heart beats, blood flows, population dynamics theoretical physics, pharmacokinetics, mathematical economy, chemical technology, electric technology, metallurgy, ecology, industrial robotics, biotechnology processes, and so on (see [1–3] and [4] for details).

In spite of the importance of impulsive differential equations, the development of the theory of impulsive differential equations has been quite slow due to special features possessed by impulsive differential equations in general, such as pulse phenomena, confluence, and loss of autonomy. Among these results, differential inequalities and integral inequalities with impulsive effects play increasingly important roles in the study of quantitative properties of solutions of impulsive differential systems. However, most of these results involving impulsive effects are point-discontinuous, i.e., impulsive effects are added at a sequence of discontinuous points (see [5–12] for details). For example, in 2004, Borysenko [13] considered the following integral inequality with impulsive effect:

$$u(t)\leq a(t)+ \int_{t_{0}}^{t} f(s)u(s)\,ds+\sum _{t_{0}< t_{i}< t}{\alpha _{i}}u^{r}(t_{i}-0), $$

in 2007, Iovane [14] studied the following integral inequalities:

$$\begin{aligned}& u(t)\leq a(t)+ \int_{t_{0}}^{t} f(s)u\bigl(\lambda(s)\bigr)\,ds+\sum _{t_{0}< t_{i}< t}{\alpha_{i}}u^{r}(t_{i}-0), \\& u(t)\leq a(t)+q(t) \biggl[ \int_{t_{0}}^{t}f(s)u\bigl(\alpha(s)\bigr)\,ds+ \int _{t_{0}}^{t}f(s) \int_{t_{0}}^{s} g(t)u\bigl(\tau(t)\bigr)\,dt\,ds\\& \hphantom{u(t)\leq}{}+\sum _{t_{0}< t_{i}< t}{\alpha _{i}}u^{r}(t_{i}-0) \biggr], \end{aligned}$$

in 2011, Wu-Sheng Wang [5] gave the upper bound for the nonlinear inequality

$$v^{p}(t)\leq A_{0}(t)+\frac{p}{p-q} \int_{t_{0}}^{t}f(s)v^{q}\bigl(\tau(s) \bigr)\,ds+\sum_{t_{0}< t_{i}< t}{\alpha_{i}}v^{q}(t_{i}-0). $$

As we know, most of the phenomena occurring in the natural world do not suddenly change, so the impulsive differential equations with integral jump conditions are more accurate than impulsive differential equations with stationary discontinuous points in characterizing the nature. In 2012, based on a well-known result given by Lakshmikantham et al. [1], Thiramanns and Tarboon [15] studied the following impulsive linear differential inequalities:

$$ \textstyle\begin{cases} m'(t)\leq p(t)m(t)+q(t),\quad t\neq t_{k}, \\ m\bigl(t_{k}^{+}\bigr)\leq d_{k}m(t_{k})+c_{k} \int_{t_{k}-\tau _{k}}^{t_{k}-\sigma_{k}}m(s)\,ds+b_{k}, \end{cases} $$
(1.1)

and gave the upper-bound estimation of the unknown function \(m(t)\).

Theorem 1.1

Suppose that (\(\mathrm{H}_{0}\)) and (\(\mathrm{H}_{1}\)) hold. If \(p, q\in C[\mathbb{R}_{+}, \mathbb{R}]\) and for \(k=1, 2, \ldots, t\geq t_{0}\), the impulsive linear differential inequality (1.1) holds, where \(c_{k}\); \(d_{k}\geq0\), \(0\leq\sigma_{k}\leq\tau_{k}\leq t_{k}-t_{k-1}\), \(b_{k}\) are constants. Then

$$\begin{aligned} m(t) \leq& \biggl\{ m(t_{0})\prod _{t_{0}< t_{k}< t} \biggl(d_{k}e^{\int _{t_{k-1}}^{t_{k}}p(\tau)\,d\tau} +c_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}}e^{\int _{t_{k-1}}^{s}p(\tau)\,d\tau}\,ds \biggr) \\ &{}+\sum_{t_{0}< t_{k}< t} \biggl[\prod _{t_{k}< t_{j}< t} \biggl(d_{j}e^{\int _{t_{j-1}}^{t_{j}}p(\tau)\,d\tau} +c_{j} \int_{t_{j}-\tau_{j}}^{t_{j}-\sigma_{j}}e^{\int _{t_{j-1}}^{s}p(\tau)\,d\tau}\,ds \biggr) \\ &{}\times \biggl(d_{k} \int_{t_{k-1}}^{t_{k}}q(s)e^{\int _{s}^{t_{k}}p(\tau)\,d\tau}\,ds \\ &{}+c_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma _{k}} \int^{s}_{t_{k-1}}q(\nu)e^{\int_{\nu}^{s}p(\tau)\,d\tau}\,d\nu \,ds+b_{k} \biggr) \biggr] \biggr\} e^{\int_{t_{i}}^{t}p(\tau)\,d\tau} \\ &{}+ \int^{t}_{t_{i}}q(s)e^{\int_{s}^{t}p(\tau)\,d\tau} \,ds,\quad t\geq t_{0}. \end{aligned}$$
(1.2)

This result can be used to investigate the qualitative properties of certain linear impulsive differential equations.

A natural question arises, that is, how about the upper bound if the inequality is of nonlinearity? In this paper, under different jump conditions, we will study the upper-bound estimation of the nonlinear inequality

$$ m'(t)\leq p(t)m(t)+q(t)m^{\alpha}(t). $$

2 Main results

In this paper, let \(0\leq t_{0}< t_{1}< t_{2}<\cdots\) be a sequence. For \(I\subset\mathbb{R}\), we denote by \(\mathrm{PC}(I, \mathbb{R})\) the functions \(u(t)\) defined on I, which is continuous for \(t\neq t_{k}\), \(u(0+)\), \(u(t_{k}+)\), \(u(t_{k}-)\) exist and \(u(t)\) is left continuous at \(t_{k}\), \(k=1, 2, \ldots\) , \(\mathrm{PC}^{1}(I, \mathbb{R}_{+})\) is the collection of functions \(u(t)\) such that \(u, u'\in\mathrm{PC}(I, \mathbb {R}_{+})\). Throughout this paper, we assume the following hypotheses:

(\(\mathrm{H}_{0}\)):

the sequence \(\{t_{k}\}\) satisfies \(0\leq t_{0}\leq t_{1}\leq t_{2}\leq\cdots\) , \(\lim_{k\to\infty}t_{k}=+\infty\).

(\(\mathrm{H}_{1}\)):

\(m\in\mathrm{PC}^{1}(I, \mathbb{R}_{+})\), and \(m(t)\) is left continuous at \(t_{k}\), \(k=1, 2, \ldots\) .

Lemma 2.1

(see [11])

Suppose that \(a, b\in\mathbb{R}\), \(p>0\). Then

$$ \bigl(|a|+|b|\bigr)^{p} \leq C_{p}\bigl(|a|^{p}+|b|^{p} \bigr), $$

where \(C_{p}=1\) for \(0< p\leq1\), and \(C_{p}=2^{p-1}\) for \(p>1\).

Theorem 2.1

Suppose that (\(\mathrm{H}_{0}\)) and (\(\mathrm{H}_{1}\)) hold. If for \(k=1, 2, \ldots, t\geq t_{0}\),

$$\begin{aligned}& m'(t)\leq p(t)m(t)+q(t)m^{\alpha}(t),\quad t\neq t_{k}, \end{aligned}$$
(2.1)
$$\begin{aligned}& m^{1-\alpha}\bigl(t_{k}^{+}\bigr)\leq d_{k}m^{1-\alpha}(t_{k})+c_{k} \int _{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}}m^{1-\alpha}(s)\,ds+b_{k}, \end{aligned}$$
(2.2)

here \(0<\alpha<1\), \(p, q\in C[\mathbb{R}_{+}, \mathbb{R}]\), and for \(k=1, 2, \ldots, t\geq t_{0}\), \(c_{k}\); \(d_{k}\geq0\), \(0\leq\sigma_{k}\leq\tau_{k}\leq t_{k}-t_{k-1}\), \(b_{k}\) are constants. We have the estimation

$$\begin{aligned} m(t) \leq& \biggl\{ \biggl[m^{1-\alpha}(t_{0}) \prod_{t_{0}< t_{k}< t}E_{k}+\sum _{t_{0}< t_{k}< t}G_{k}\prod_{t_{k}< t_{j}< t}E_{j} \biggr]e^{\int _{t_{i}}^{t}(1-\alpha)p(\tau)\,d\tau} \\ &{}+(1-\alpha) \int^{t}_{t_{i}}q(s)e^{\int_{s}^{t}(1-\alpha )p(\tau)\,d\tau} \,ds \biggr\} ^{\frac{1}{1-\alpha}}, \quad t\geq t_{0}, \end{aligned}$$
(2.3)

where

$$\begin{aligned}& E_{k}=d_{k}e^{\int_{t_{k-1}}^{t_{k}}(1-\alpha)p(\tau)\,d\tau }+c_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}}e^{\int _{t_{k-1}}^{s}(1-\alpha)p(\tau)\,d\tau}\,ds , \end{aligned}$$
(2.4)
$$\begin{aligned}& G_{k}=d_{k} \int_{t_{k-1}}^{t_{k}}(1-\alpha)q(s)e^{\int _{s}^{t_{k}}(1-\alpha)p(\tau)\,d\tau}\,ds \\& \hphantom{G_{k}=}{}+(1-\alpha)c_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}} \int ^{s}_{t_{k-1}}q(\nu)e^{\int_{\nu}^{s}(1-\alpha)p(\tau)\,d\tau}\,d\nu \,ds+b_{k}. \end{aligned}$$
(2.5)

Proof

For \(t\in[t_{0}, t_{1}]\), we have

$$ \frac{d}{dt} \bigl[e^{\int_{t_{0}}^{t}-(1-\alpha)p(\tau)\,d\tau }m^{1-\alpha}(t) \bigr] \leq(1-\alpha)q(t)e^{-\int _{t_{0}}^{t}(1-\alpha)p(\tau)\,d\tau}, $$
(2.6)

integrating (2.6) implies

$$ m^{1-\alpha}(t)\leq m^{1-\alpha}(t_{0})e^{\int_{t_{0}}^{t}(1-\alpha )p(\tau)\,d\tau}+(1- \alpha) \int_{t_{0}}^{t}q(s)e^{\int _{s}^{t}(1-\alpha)p(\tau)\,d\tau}\,ds, $$
(2.7)

which shows that (2.3) holds for \(t\in[t_{0}, t_{1}]\).

Now we suppose that (2.3) holds for \(t\in[t_{0}, t_{n}]\), then we need only prove that (2.3) holds for \(t\in(t_{n}, t_{n+1}]\) by mathematical induction. Since

$$\begin{aligned}& m^{1-\alpha}(t_{n})\leq \biggl[m^{1-\alpha}(t_{0}) \prod_{t_{0}< t_{k}< t_{n}}E_{k}+\sum _{t_{0}< t_{k}< t_{n}}G_{k}\prod_{t_{k}< t_{j}< t_{n}}E_{j} \biggr]e^{\int _{t_{n-1}}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \\& \hphantom{ m^{1-\alpha}(t_{n})\leq}{} +(1-\alpha) \int^{t_{n}}_{t_{n-1}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha )p(\tau)\,d\tau} \,ds \\& \hphantom{m^{1-\alpha}(t_{n})}= \Biggl[m^{1-\alpha}(t_{0})\prod_{i=1}^{n-1}E_{i}+ \sum_{i=1}^{n-1}G_{i}\prod _{j=i+1}^{n-1}E_{j} \Biggr]e^{\int _{t_{n-1}}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \\& \hphantom{ m^{1-\alpha}(t_{n})\leq}{} +(1-\alpha) \int^{t_{n}}_{t_{n-1}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha )p(\tau)\,d\tau} \,ds, \\& m^{1-\alpha}\bigl(t_{n}^{+}\bigr)\leq d_{n}m^{1-\alpha }(t_{n})+c_{n} \int_{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}}m^{1-\alpha }(s)\,ds+b_{n} \\& \hphantom{m^{1-\alpha}\bigl(t_{n}^{+}\bigr)}\leq d_{n} \Biggl\{ \Biggl[m^{1-\alpha}(t_{0})\prod _{i=1}^{n-1}E_{i}+\sum _{i=1}^{n-1}G_{i}\prod _{j=i+1}^{n-1}E_{j} \Biggr]e^{\int_{t_{n-1}}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \\& \hphantom{ m^{1-\alpha}\bigl(t_{n}^{+}\bigr)\leq}{} +(1-\alpha) \int^{t_{n}}_{t_{n-1}}q(s)e^{\int _{s}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \,ds \Biggr\} \\& \hphantom{ m^{1-\alpha}\bigl(t_{n}^{+}\bigr)\leq}{}+c_{n} \int_{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}} \Biggl[m^{1-\alpha }(t_{0}) \prod_{i=1}^{n-1}E_{i}+\sum _{i=1}^{n-1}G_{i}\prod _{j=i+1}^{n-1}E_{j} \Biggr]e^{\int_{t_{n-1}}^{s}(1-\alpha)p(\tau )\,d\tau}\,ds \\& \hphantom{ m^{1-\alpha}\bigl(t_{n}^{+}\bigr)\leq}{}+c_{n} \int_{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}}(1-\alpha) \int _{t_{n-1}}^{s}q(\nu)e^{\int_{\nu}^{s}(1-\alpha)p(\tau)\,d\tau}\,d\nu \,ds+b_{n} \\& \hphantom{m^{1-\alpha}\bigl(t_{n}^{+}\bigr)} = \Biggl[m^{1-\alpha}(t_{0})\prod _{i=1}^{n-1}E_{i}+\sum _{i=1}^{n-1}G_{i}\prod _{j=i+1}^{n-1}E_{j} \Biggr] \\& \hphantom{ m^{1-\alpha}\bigl(t_{n}^{+}\bigr)\leq}{}\times \biggl[d_{n}e^{\int_{t_{n-1}}^{t_{n}}(1-\alpha)p(\tau)\,d\tau}+c_{n} \int_{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}}e^{\int _{t_{n-1}}^{s}(1-\alpha)p(\tau)\,d\tau}\,ds \biggr] \\& \hphantom{ m^{1-\alpha}\bigl(t_{n}^{+}\bigr)\leq}{} +d_{n}(1-\alpha) \int^{t_{n}}_{t_{n-1}}q(s)e^{\int_{s}^{t_{n}}p(\tau )\,d\tau}\,ds \\& \hphantom{ m^{1-\alpha}\bigl(t_{n}^{+}\bigr)\leq}{}+(1-\alpha)c_{n} \int_{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}} \int _{t_{n}-1}^{s}q(\nu)e^{\int_{\nu}^{s}(1-\alpha)p(\tau)\,d\tau}\,d\nu \,ds+b_{n} \\& \hphantom{m^{1-\alpha}\bigl(t_{n}^{+}\bigr)}= \Biggl[m^{1-\alpha}(t_{0})\prod_{i=1}^{n-1}E_{i}+ \sum_{i=1}^{n-1}G_{i}\prod _{j=i+1}^{n-1}E_{j} \Biggr]E_{n}+G_{n} \\& \hphantom{m^{1-\alpha}\bigl(t_{n}^{+}\bigr)}=m^{1-\alpha}(t_{0})\prod_{i=1}^{n}E_{i}+ \Biggl(\sum_{i=1}^{n-1}G_{i} \prod_{j=i+1}^{n-1}E_{j} \Biggr)E_{n}+G_{n} \\& \hphantom{m^{1-\alpha}\bigl(t_{n}^{+}\bigr)}=m^{1-\alpha}(t_{0})\prod_{i=1}^{n}E_{i}+ \sum_{i=1}^{n}G_{i}\prod _{j=i+1}^{n}E_{j}, \end{aligned}$$
(2.8)

substituting (2.8) into (2.7), with \(t_{0}\) being replaced by \(t_{n}^{+}\), we obtain, for \(t\in(t_{n}, t_{n+1}]\),

$$\begin{aligned} m^{1-\alpha}(t) \leq& m^{1-\alpha}\bigl(t_{n}^{+} \bigr)e^{\int _{t_{n}}^{t}(1-\alpha)p(\tau)\,d\tau}+(1-\alpha) \int _{t_{n}}^{t}q(s)e^{\int_{s}^{t}(1-\alpha)p(\tau)\,d\tau}\,ds \\ \leq& \Biggl[m^{1-\alpha}(t_{0})\prod _{i=1}^{n}E_{i}+\sum _{i=1}^{n}G_{i}\prod _{j=i+1}^{n}E_{j} \Biggr]e^{\int _{t_{n}}^{t}(1-\alpha)p(\tau)\,d\tau} \\ &{}+(1-\alpha) \int^{t}_{t_{n}}q(s)e^{\int_{s}^{t}(1-\alpha)p(\tau )\,d\tau} \,ds \\ =& \biggl[m^{1-\alpha}(t_{0})\prod_{t_{0}< t_{k}< t_{n+1}}E_{k}+ \sum_{t_{0}< t_{k}< t_{n+1}}G_{k}\prod _{t_{k}< t_{j}< t_{n+1}}E_{j} \biggr]e^{\int_{t_{n}}^{t}(1-\alpha)p(\tau)\,d\tau} \\ &{}+(1-\alpha) \int^{t}_{t_{n}}q(s)e^{\int_{s}^{t}(1-\alpha)p(\tau )\,d\tau} \,ds. \end{aligned}$$

This completes the proof of Theorem 2.1. □

If \(d_{k}\equiv0\) in Theorem 2.1, we obtain the following corollary.

Corollary 2.1

Suppose that (\(\mathrm{H}_{0}\)) and (\(\mathrm{H}_{1}\)) hold, \(p, q\in C[\mathbb{R}_{+}, \mathbb{R}]\) and for \(k=1, 2, \ldots, t\geq t_{0}\),

$$\begin{aligned}& m'(t)\leq p(t)m(t)+q(t)m^{\alpha}(t),\quad 0< \alpha< 1, \\& m^{1-\alpha}\bigl(t_{k}^{+}\bigr)\leq c_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma _{k}}m^{1-\alpha}(s)\,ds+b_{k}, \end{aligned}$$

where \(c_{k}\), \(b_{k}\), \(\sigma_{k}\), \(\tau_{k}\) are defined as in Theorem 2.1, then we have

$$\begin{aligned} m(t) \leq& \biggl\{ \biggl[m^{1-\alpha}(t_{0})\prod _{t_{0}< t_{k}< t}c_{k} \int _{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}}e^{\int_{t_{k-1}}^{s}(1-\alpha )p(\tau)\,d\tau}\,ds \\ &{}+\sum_{t_{0}< t_{k}< t}(1-\alpha)c_{k} \int_{t_{k}-\tau _{k}}^{t_{k}-\sigma_{k}} \int^{s}_{t_{k-1}}q(\nu)e^{\int_{\nu }^{s}(1-\alpha)p(\tau)\,d\tau}\,d\nu \,ds \\ &{}\times\prod_{t_{k}< t_{j}< t}c_{j} \int_{t_{j}-\tau _{j}}^{t_{j}-\sigma_{j}}e^{\int_{t_{j-1}}^{s}(1-\alpha)p(\tau)\,d\tau }\,ds \biggr]e^{\int_{t_{i}}^{t}(1-\alpha)p(\tau)\,d\tau} \\ &{}+(1-\alpha) \int^{t}_{t_{i}-1}q(s)e^{\int_{s}^{t}(1-\alpha )p(\tau)\,d\tau} \,ds \biggr\} ^{\frac{1}{1-\alpha}}. \end{aligned}$$

If \(d_{k}\equiv1\), we obtain the following theorem.

Theorem 2.2

Suppose that (\(\mathrm{H}_{0}\)) and (\(\mathrm{H}_{1}\)) hold. If, for \(k=1, 2, \ldots, t\geq t_{0}\),

$$ \textstyle\begin{cases} m'(t)\leq p(t)m(t)+q(t)m^{\alpha}(t),\quad t\neq t_{k}, \\ \Delta m^{1-\alpha}(t_{k})\leq c_{k} \int_{t_{k}-\tau _{k}}^{t_{k}-\sigma_{k}}m^{1-\alpha}(s)\,ds+b_{k}, \end{cases} $$
(2.9)

where \(0<\alpha<1\), \(p, q\in C[\mathbb{R}_{+}, \mathbb{R}]\), and for \(k=1, 2, \ldots, t\geq t_{0}\), \(c_{k} \geq0\), \(0\leq\sigma_{k}\leq\tau_{k}\leq t_{k}-t_{k-1}\), \(b_{k}\) are constants, \(\Delta m^{1-\alpha}(t_{k})=m^{1-\alpha}(t_{k}^{+})-m^{1-\alpha}(t_{k})\). We have the estimation

$$\begin{aligned} m(t) \leq& \biggl\{ \biggl[m^{1-\alpha}(t_{0}) \prod_{t_{0}< t_{k}< t}E_{k}+\sum _{t_{0}< t_{k}< t}H_{k}\prod_{t_{k}< t_{j}< t}E_{j} \biggr]e^{\int _{t_{i}}^{t}(1-\alpha)p(\tau)\,d\tau} \\ &{}+(1-\alpha) \int^{t}_{t_{0}}q(s)e^{\int_{s}^{t}(1-\alpha )p(\tau)\,d\tau} \,ds \biggr\} ^{\frac{1}{1-\alpha}},\quad t\geq t_{0}, \end{aligned}$$
(2.10)

where \(E_{k}\) is defined as (2.4) (with \(d_{k}\equiv1\)),

$$ H_{k}=(1-\alpha)c_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}} \int ^{s}_{t_{0}}q(\nu)e^{\int_{\nu}^{s}(1-\alpha)p(\tau)\,d\tau}\,d\nu \,ds+b_{k}. $$

Proof

As the proof of Theorem 2.1, we prove (2.7) holds for \(t\in[t_{0}, t_{1}]\), which means that (2.10) holds for \(t\in[t_{0}, t_{1}]\). Now suppose that (2.10) holds for \(t\in[t_{0}, t_{n}]\), then

$$\begin{aligned} m^{1-\alpha}(t_{n}) \leq& \biggl[m^{1-\alpha}(t_{0}) \prod_{t_{0}< t_{k}< t}E_{k}+\sum _{t_{0}< t_{k}< t}H_{k}\prod_{t_{k}< t_{j}< t}E_{j} \biggr]e^{\int _{t_{n-1}}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \\ &{}+(1-\alpha) \int^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha )p(\tau)\,d\tau} \,ds \\ =& \Biggl[m^{1-\alpha}(t_{0})\prod_{i=1}^{n-1}E_{i}+ \sum_{i=1}^{n-1}H_{i}\prod _{j=i+1}^{n-1}E_{j} \Biggr]e^{\int _{t_{n-1}}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \\ &{}+(1-\alpha) \int^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha )p(\tau)\,d\tau} \,ds. \end{aligned}$$

So

$$\begin{aligned} m^{1-\alpha}\bigl(t_{n}^{+}\bigr) \leq& m^{1-\alpha}(t_{n})+c_{n} \int _{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}}m^{1-\alpha}(s)\,ds+b_{n} \\ \leq& \Biggl[m^{1-\alpha}(t_{0})\prod _{i=1}^{n-1}E_{i}+\sum _{i=1}^{n-1}H_{i}\prod _{j=i+1}^{n-1}E_{j} \Biggr]e^{\int _{t_{n-1}}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \\ &{}+(1-\alpha) \int^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha )p(\tau)\,d\tau} \,ds \\ &{}+c_{n} \int_{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}} \Biggl[m^{1-\alpha }(t_{0}) \prod_{i=1}^{n-1}E_{i}+\sum _{i=1}^{n-1}H_{i}\prod _{j=i+1}^{n-1}E_{j} \Biggr]e^{\int_{t_{n-1}}^{s}(1-\alpha)p(\tau )\,d\tau}\,ds \\ &{}+c_{n} \int_{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}}(1-\alpha) \int _{t_{0}}^{s}q(\nu)e^{\int_{\nu}^{s}(1-\alpha)p(\tau)\,d\tau}\,d\nu \,ds+b_{n} \\ =& \Biggl[m^{1-\alpha}(t_{0})\prod_{i=1}^{n-1}E_{i}+ \sum_{i=1}^{n-1}H_{i}\prod _{j=i+1}^{n-1}E_{j} \Biggr] \biggl[e^{\int_{t_{n-1}}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \\ &{}+c_{n} \int_{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}}e^{\int _{t_{n-1}}^{s}(1-\alpha)p(\tau)\,d\tau}\,ds \biggr] \\ &{}+(1-\alpha) \int^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha )p(\tau)\,d\tau}\,ds+H_{n} \\ =&m^{1-\alpha}(t_{0})\prod_{i=1}^{n}E_{i}+ \Biggl(\sum_{i=1}^{n-1}H_{i} \prod_{j=i+1}^{n-1}E_{j} \Biggr)E_{n} \\ &{}+H_{n}+(1-\alpha) \int^{t_{n}}_{t_{0}}q(s)e^{\int _{s}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \,ds \\ =&m^{1-\alpha}(t_{0})\prod_{i=1}^{n}E_{i}+ \sum_{i=1}^{n}H_{i}\prod _{j=i+1}^{n}E_{j} \\ &{}+(1-\alpha) \int^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha )p(\tau)\,d\tau} \,ds. \end{aligned}$$

Using (2.7) (with \(t_{0}\) being replaced by \(t_{n}^{+}\)), we obtain, for \(t\in(t_{n}, t_{n+1}]\),

$$\begin{aligned} m^{1-\alpha}(t) \leq& \Biggl[m^{1-\alpha}(t_{0}) \prod_{i=1}^{n}E_{i}+\sum _{i=1}^{n}H_{i}\prod _{j=i+1}^{n}E_{j}+(1-\alpha) \int ^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \,ds \Biggr] \\ &{}\times e^{\int_{t_{n}}^{t}(1-\alpha)p(\tau)\,d\tau}+(1-\alpha) \int ^{t}_{t_{n}}q(s)e^{\int_{s}^{t}(1-\alpha)p(\tau)\,d\tau} \,ds \\ =& \Biggl[m^{1-\alpha}(t_{0})\prod_{i=1}^{n}E_{i}+ \sum_{i=1}^{n}H_{i}\prod _{j=i+1}^{n}E_{j} \Biggr]e^{\int _{t_{n}}^{t}(1-\alpha)p(\tau)\,d\tau} \\ &{}+(1-\alpha) \int^{t}_{t_{0}}q(s)e^{\int_{s}^{t}(1-\alpha)p(\tau )\,d\tau} \,ds \\ =& \biggl[m^{1-\alpha}(t_{0})\prod_{t_{0}< t_{k}< t_{n+1}}E_{k}+ \sum_{t_{0}< t_{k}< t_{n+1}}H_{k}\prod _{t_{k}< t_{j}< t_{n+1}}E_{j} \biggr]e^{\int_{t_{n}}^{t}(1-\alpha)p(\tau)\,d\tau} \\ &{}+(1-\alpha) \int^{t}_{t_{0}}q(s)e^{\int_{s}^{t}(1-\alpha)p(\tau )\,d\tau} \,ds. \end{aligned}$$

This completes the proof. □

Remark 2.1

If \(\alpha=0\), then Theorem 2.1 reduces to Theorem 1.1, and Theorem 2.2 improves Theorem 1.1.

If \(p(t)\equiv0\) in Theorem 2.2, we obtain the following useful corollary.

Corollary 2.2

If (\(\mathrm{H}_{0}\)) and (\(\mathrm{H}_{1}\)) hold and for \(k=1, 2, \ldots, t\geq t_{0}\),

$$ \textstyle\begin{cases}m'(t)\leq q(t)m^{\alpha}(t), \\ \Delta m^{1-\alpha}(t)\leq c_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma _{k}}m^{1-\alpha}(s)\,ds+b_{k}, \end{cases} $$
(2.11)

then

$$\begin{aligned} m(t) \leq& \biggl\{ m^{1-\alpha}(t_{0})\prod _{t_{0}< t_{k}< t}\bigl(1+c_{k}(\tau _{k}- \sigma_{k})\bigr) \\ &{}+\sum_{t_{0}< t_{k}< t} \biggl[(1-\alpha)c_{k} \biggl( \int_{t_{k}-\tau _{k}}^{t_{k}-\sigma_{k}} \int^{s}_{t_{k-1}}q(\nu)\,d\nu \,ds+b_{k} \biggr)\prod_{t_{k}< t_{j}< t}\bigl(1+c_{j}( \tau_{j}-\sigma_{j})\bigr) \biggr] \\ &{}+(1-\alpha) \int^{t}_{t_{0}}q(s)\,ds \biggr\} ^{\frac {1}{1-\alpha}}. \end{aligned}$$

Next, we will give another kind of nonlinear impulsive differential inequalities.

Theorem 2.3

Suppose that (\(\mathrm{H}_{0}\)) holds, and \(m\in\mathrm{PC}^{1}[\mathbb {R}_{+}, \mathbb{R}_{+}]\), \(m(t)\) is left continuous at \(t_{k}\), \(k=1, 2, \ldots, p(t)\), \(q(t)\in C[\mathbb{R}_{+}, \mathbb{R}_{+}]\). Assume

$$ \textstyle\begin{cases} m'(t)\leq p(t)m(t)+q(t)m^{\alpha}(t),\quad t\neq t_{k}, \\ \Delta m(t_{k})\leq c_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma _{k}}m(s)\,ds+b_{k}, \end{cases} $$
(2.12)

where \(\Delta m(t_{k})=m(t_{k}^{+})-m(t_{k})\), \(0<\alpha<1\), \(c_{k} \geq0\), \(0\leq\sigma_{k}\leq\tau_{k}\leq t_{k}-t_{k-1}\), \(b_{k}\) are constants. We have the estimation

$$\begin{aligned} m^{1-\alpha}(t) \leq& \biggl(m(t_{0})\prod _{t_{0}< t_{k}< t}F_{k}+\sum _{t_{0}< t_{k}< t}R_{k}\prod_{t_{k}< t_{j}< t}F_{j} \biggr)^{1-\alpha}e^{\int_{t_{i}}^{t}(1-\alpha )p(\tau)\,d\tau} \\ &{}+2^{(k-1)\alpha}(1-\alpha) \int^{t}_{t_{0}}q(s)e^{\int _{s}^{t}(1-\alpha)p(\tau)\,d\tau} \,ds,\quad t\geq t_{0}, \end{aligned}$$
(2.13)

where

$$\begin{aligned}& F_{k}=2^{\frac{\alpha}{1-\alpha}} \biggl[e^{\int_{t_{k-1}}^{t_{k}}p(\tau )\,d\tau}+c_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}} e^{\int_{t_{k-1}}^{s}p(\tau)\,d\tau}\,ds \biggr], \\& R_{k}=c_{k}2^{\frac{(k-1)\alpha}{1-\alpha}}(1-\alpha)^{\frac {1}{1-\alpha}} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}} \biggl( \int_{t_{0}}^{v}q(s)e^{\int_{s}^{v}(1-\alpha)p(\tau)\,d\tau}\,ds \biggr)^{\frac{1}{1-\alpha}}\,dv+b_{k}. \end{aligned}$$

Proof

Obviously, (2.13) holds for \(t\in[t_{0}, t_{1}]\) as (2.7). Now we suppose (2.13) holds for \(t\in[t_{0}, t_{n}]\), then by mathematical induction, we see that

$$\begin{aligned} m^{1-\alpha}(t_{n}) \leq& \biggl(m(t_{0}) \prod_{t_{0}< t_{k}< t_{n}}F_{k}+\sum _{t_{0}< t_{k}< t_{n}}R_{k}\prod_{t_{k}< t_{j}< t_{n}}F_{j} \biggr)^{1-\alpha}e^{\int_{t_{n-1}}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \\ &{}+2^{(n-1)\alpha}(1-\alpha) \int^{t_{n}}_{t_{0}}q(s)e^{\int _{s}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \,ds \\ =& \Biggl(m(t_{0})\prod_{i=1}^{n-1}F_{i}+ \sum_{i=1}^{n-1}R_{i}\prod _{j=i+1}^{n-1}F_{j} \Biggr)^{1-\alpha}e^{\int _{t_{n-1}}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \\ &{}+2^{(n-1)\alpha}(1-\alpha) \int^{t_{n}}_{t_{0}}q(s)e^{\int _{s}^{t_{n}}(1-\alpha)p(\tau)\,d\tau} \,ds. \end{aligned}$$

Since \(\frac{1}{1-\alpha}>1\), by Lemma 2.1,

$$\begin{aligned}& m(t_{n})\leq 2^{\frac{\alpha}{1-\alpha}} \Biggl(m(t_{0}) \prod_{i=1}^{n-1}F_{i}+\sum _{i=1}^{n-1}R_{i}\prod _{j=i+1}^{n-1}F_{j} \Biggr)e^{\int_{t_{n-1}}^{t_{n}}p(\tau)\,d\tau} \\& \hphantom{ m(t_{n})\leq}{}+2^{\frac{\alpha}{1-\alpha}}2^{\frac{(n-1)\alpha}{1-\alpha }}(1-\alpha)^{\frac{1}{1-\alpha}} \biggl( \int ^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha)p(\tau)\,d\tau}\,ds \biggr)^{\frac{1}{1-\alpha}}, \\& m\bigl(t_{n}^{+}\bigr)\leq m(t_{n})+c_{n} \int_{t_{n}-\tau _{n}}^{t_{n}-\sigma_{n}}m(s)\,ds+b_{n} \\& \hphantom{ m\bigl(t_{n}^{+}\bigr)}\leq2^{\frac{\alpha}{1-\alpha}} \Biggl(m(t_{0})\prod _{i=1}^{n-1}F_{i}+\sum _{i=1}^{n-1}R_{i}\prod _{j=i+1}^{n-1}F_{j} \Biggr)e^{\int_{t_{n-1}}^{t_{n}}p(\tau)\,d\tau} \\& \hphantom{m\bigl(t_{n}^{+}\bigr)\leq}{}+2^{\frac{n\alpha}{1-\alpha}}(1-\alpha)^{\frac{1}{1-\alpha }} \biggl( \int^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha)p(\tau )\,d\tau}\,ds \biggr)^{\frac{1}{1-\alpha}} \\& \hphantom{m\bigl(t_{n}^{+}\bigr)\leq}{}+c_{n} \int_{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}}2^{\frac{\alpha }{1-\alpha}} \Biggl(m(t_{0}) \prod_{i=1}^{n-1}F_{i}+\sum _{i=1}^{n-1}R_{i}\prod _{j=i+1}^{n-1}F_{j} \Biggr)e^{\int_{t_{n-1}}^{s}p(\tau)\,d\tau}\,ds \\& \hphantom{m\bigl(t_{n}^{+}\bigr)\leq}{}+c_{n}2^{\frac{n\alpha}{1-\alpha}}(1-\alpha)^{\frac{1}{1-\alpha }} \int_{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}} \biggl( \int^{\nu}_{t_{0}}q(s)e^{\int_{s}^{\nu}(1-\alpha)p(\tau )\,d\tau}\,ds \biggr)^{\frac{1}{1-\alpha}}\,d\nu+b_{n} \\& \hphantom{ m\bigl(t_{n}^{+}\bigr)}= \Biggl(m(t_{0})\prod_{i=1}^{n-1}F_{i}+ \sum_{i=1}^{n-1}R_{i}\prod _{j=i+1}^{n-1}F_{j} \Biggr)\\& \hphantom{m\bigl(t_{n}^{+}\bigr)\leq}{}\times \biggl[2^{\frac{\alpha}{1-\alpha}} \biggl(e^{\int _{t_{n-1}}^{t_{n}}p(\tau)\,d\tau} +c_{n} \int_{t_{n}-\tau_{n}}^{t_{n}-\sigma_{n}}e^{\int ^{s}_{t_{n-1}}p(\tau)\,d\tau}\,ds \biggr) \biggr] \\& \hphantom{m(t_{n}^{+})\leq}{}+R_{n}+2^{\frac{n\alpha}{1-\alpha}}(1-\alpha)^{\frac{1}{1-\alpha }} \biggl( \int^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha)p(\tau )\,d\tau}\,ds \biggr)^{\frac{1}{1-\alpha}} \\& \hphantom{ m(t_{n}^{+})}= \Biggl(m(t_{0})\prod_{i=1}^{n-1}F_{i}+ \sum_{i=1}^{n-1}R_{i}\prod _{j=i+1}^{n-1}F_{j} \Biggr)F_{n}+R_{n} \\& \hphantom{m(t_{n}^{+})\leq}{}+2^{\frac{n\alpha}{1-\alpha}}(1-\alpha)^{\frac{1}{1-\alpha }} \biggl( \int^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha)p(\tau )\,d\tau}\,ds \biggr)^{\frac{1}{1-\alpha}} \\& \hphantom{ m(t_{n}^{+})}=m(t_{0})\prod_{i=1}^{n}F_{i}+ \sum_{i=1}^{n}R_{i}\prod _{j=i+1}^{n}F_{j} \\& \hphantom{m(t_{n}^{+})\leq}{}+2^{\frac{n\alpha}{1-\alpha}}(1-\alpha)^{\frac{1}{1-\alpha }} \biggl( \int^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha)p(\tau )\,d\tau}\,ds \biggr)^{\frac{1}{1-\alpha}}. \end{aligned}$$

So for \(t\in(t_{n}, t_{n+1}]\), since \(0<1-\alpha<1\), by Lemma 2.1 and (2.7) (with \(t_{0}\) being replaced by \(t_{n}^{+}\)), we obtain

$$\begin{aligned} m^{1-\alpha}(t) \leq&m^{1-\alpha}\bigl(t_{n}^{+} \bigr)e^{\int^{t}_{t_{n}}(1-\alpha)p(\tau )\,d\tau}\\ &{}+(1-\alpha) \int^{t}_{t_{n}}q(s)e^{\int_{s}^{t}(1-\alpha )p(\tau)\,d\tau} \,ds \\ =& \Biggl[m(t_{0})\prod_{i=1}^{n}F_{i}+ \sum_{i=1}^{n}R_{i}\prod _{j=i+1}^{n}F_{j} \\ &{}+2^{\frac{n\alpha}{1-\alpha}}(1-\alpha)^{\frac{1}{1-\alpha }} \biggl( \int^{t_{n}}_{t_{0}}q(s)e^{\int_{s}^{t_{n}}(1-\alpha)p(\tau )\,d\tau}\,ds \biggr)^{\frac{1}{1-\alpha}} \Biggr]^{1-\alpha} \\ &{}\times e^{\int^{t}_{t_{n}}(1-\alpha)p(\tau)\,d\tau}+(1-\alpha) \int ^{t}_{t_{n}}q(s)e^{\int_{s}^{t}(1-\alpha)p(\tau)\,d\tau} \,ds \\ \leq& \Biggl[ \Biggl(m(t_{0})\prod _{i=1}^{n}F_{i}+\sum _{i=1}^{n}R_{i}\prod _{j=i+1}^{n}F_{j} \Biggr)^{1-\alpha} \\ &{}+2^{n\alpha}(1-\alpha) \int^{t_{n}}_{t_{0}}q(s)e^{\int _{s}^{t_{n}}(1-\alpha)p(\tau)\,d\tau}\,ds \Biggr] \\ &{}\times e^{\int^{t}_{t_{n}}(1-\alpha)p(\tau)\,d\tau}+(1-\alpha) \int ^{t}_{t_{n}}q(s)e^{\int_{s}^{t}(1-\alpha)p(\tau)\,d\tau} \,ds \\ \leq& \Biggl(m(t_{0})\prod_{i=1}^{n}F_{i}+ \sum_{i=1}^{n}R_{i}\prod _{j=i+1}^{n}F_{j} \Biggr)^{1-\alpha}e^{\int^{t}_{t_{n}}(1-\alpha )p(\tau)\,d\tau} \\ &{}+2^{n\alpha}(1-\alpha) \int^{t}_{t_{0}}q(s)e^{\int_{s}^{t}(1-\alpha )p(\tau)\,d\tau}\,ds \\ =& \biggl(m(t_{0})\prod_{t_{0}< t_{k}< t_{n+1}}F_{k} +\sum_{t_{0}< t_{k}< t_{n+1}}R_{k}\prod _{t_{k}< t_{j}< t_{n+1}}F_{j} \biggr)^{1-\alpha}e^{\int ^{t}_{t_{n}}(1-\alpha)p(\tau)\,d\tau} \\ &{}+2^{n\alpha}(1-\alpha) \int^{t}_{t_{0}}q(s)e^{\int_{s}^{t}(1-\alpha )p(\tau)\,d\tau}\,ds, \end{aligned}$$

which shows that (2.13) holds for \(k=n+1\). This completes the proof. □

Now we give an upper-bound estimation of a nonlinear integral inequality with integral jump conditions.

Theorem 2.4

Suppose that (\(\mathrm{H}_{0}\)) holds, and suppose \(m, p, q\in C[\mathbb {R}_{+}, \mathbb{R}_{+}]\). For \(t\geq t_{0}\), if

$$ m(t)\leq c+ \int_{t_{0}}^{t} p(s)m(s)\,ds+ \int_{t_{0}}^{t} q(s)m^{\alpha }(s)\,ds+\sum _{t_{0}< t_{k}< t}\alpha_{k} \int_{t_{k}-\tau _{k}}^{t_{k}-\sigma_{k}}m(s)\,ds, $$
(2.14)

where \(\alpha_{k} \geq0\), \(0\leq\sigma_{k}\leq\tau_{k}\leq t_{k}-t_{k-1}\), \(c\geq0\), \(0<\alpha<1\) are constants. Then we have the estimation

$$\begin{aligned} m(t) \leq& \biggl\{ \biggl(c\prod _{t_{0}< t_{k}< t}F_{k}+\sum_{t_{0}< t_{k}< t}R_{k} \prod_{t_{k}< t_{j}< t}F_{j} \biggr)^{1-\alpha }e^{\int_{t_{i}}^{t}(1-\alpha)p(\tau)\,d\tau} \\ &{}+2^{(k-1)\alpha}(1-\alpha) \int^{t}_{t_{0}}q(s)e^{\int _{s}^{t}(1-\alpha)p(\tau)\,d\tau} \,ds \biggr\} ^{1/(1-\alpha)}, \end{aligned}$$
(2.15)

where \(F_{k}\) and \(R_{k}\) are defined as that in Theorem 2.3, with \(c_{k}\) being replaced by \(\alpha_{k}\).

Proof

Defined the right-hand side of (2.14) as a new function \(v(t)\), we have \(m(t)\leq v(t)\) and \(v(t_{0})=c\). Since

$$\begin{aligned}& v'(t)=p(t)m(t)+q(t)m^{\alpha}(t),\quad t\neq t_{k}, \\& v(t_{k}+)=v(t_{k})+\alpha_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma _{k}}m(s)\,ds, \end{aligned}$$

we obtain further

$$\begin{aligned}& v'(t)\leq p(t)v(t)+q(t)v^{\alpha}(t),\quad t\neq t_{k}, \\& v(t_{k}+)=v(t_{k})+\alpha_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma _{k}}v(s)\,ds. \end{aligned}$$

Then using Theorem 2.3 implies the estimation of \(v(t)\), the estimation of the unknown function \(m(t)\) is obtained since \(m(t)\leq v(t)\), and this completes the proof. □

3 Application to impulsive differential equations

As an application of Theorem 2.4, we give an upper-bound estimation of certain nonlinear impulsive differential equation as follows:

$$ \textstyle\begin{cases} v'(t)=f(t, v), \quad t\neq t_{k}, \\ \Delta v(t_{k})=I_{k} \bigl( \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma _{k}}v(s)\,ds \bigr),\quad t\in[t_{0}, \infty), \\ v(t_{0})=v_{0}, \end{cases} $$
(3.1)

where \(f\in C(\mathbb{R}\times\mathbb{R}, \mathbb{R})\), \(I_{k}\in C(\mathbb{R}, \mathbb{R})\), \(0< t_{0}< t_{1}<\cdots\) , \(\lim_{t\to\infty}t_{k}=+\infty\), \(0\leq\sigma _{k}\leq\tau_{k}\leq t_{k}-t_{k-1}\), \(k=1, 2,\ldots\) . If there exists \(L>0\) such that

$$ \bigl|f(t, v)\bigr|\leq L|v|^{\alpha},\quad 0< \alpha< 1; $$
(3.2)

and there exist \(\iota_{k}\geq0\), such that

$$ \bigl|I_{k}(v)\bigr|\leq\iota_{k}|v|,\quad k=1, 2, \ldots, $$
(3.3)

then for any solution \(v(t)\) of (3.1), we have

$$\begin{aligned} \bigl|v(t)\bigr| \leq& \biggl\{ \biggl(|v_{0}|\prod _{t_{0}< t_{k}< t}2^{\alpha/(1-\alpha)}\bigl(1+\iota_{k}( \tau_{k}-\sigma_{k})\bigr) \\ &{}+\sum_{t_{0}< t_{k}< t}2^{\frac{(k-1)\alpha}{1-\alpha}}\bigl(L(1-\alpha ) \bigr)^{\frac{1}{1-\alpha}}\iota_{k}(t_{k}-t_{0})^{\frac{2-\alpha}{1-\alpha}} \\ &{}\times \prod_{t_{k}< t_{j}< t}2^{\frac{\alpha}{1-\alpha}}\bigl(1+\iota _{j}(\tau_{j}-\sigma_{j})\bigr) \biggr)^{1-\alpha} \\ &{}+2^{(k-1)\alpha}(1-\alpha)L(t-t_{0}) \biggr\} ^{1/(1-\alpha)}. \end{aligned}$$
(3.4)

Proof

Suppose \(v=v(t)\) is a solution of (3.1), we integrate (3.1) to obtain

$$\begin{aligned} v(t) =&v(t_{0})+ \int_{t_{0}}^{t}f\bigl(s, v(s)\bigr)\,ds+\sum _{t_{0}< t_{k}< t}I_{k} \biggl( \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}}v(s)\,ds \biggr) \\ =&v_{0}+ \int_{t_{0}}^{t}f\bigl(s, v(s)\bigr)\,ds+\sum _{t_{0}< t_{k}< t}I_{k} \biggl( \int _{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}}v(s)\,ds \biggr). \end{aligned}$$
(3.5)

By (3.2) and (3.3), we obtain

$$\begin{aligned} \bigl|v(t)\bigr| \leq& |v_{0}|+ \int_{t_{0}}^{t}\bigl|f\bigl(s, v(s)\bigr)\bigr|\,ds+\sum _{t_{0}< t_{k}< t}\biggl\vert I_{k} \biggl( \int_{t_{k}-\tau _{k}}^{t_{k}-\sigma_{k}}v(s)\,ds \biggr)\biggr\vert \\ \leq& |v_{0}|+L \int_{t_{0}}^{t}\bigl|v(s)\bigr|^{\alpha}\,ds+\sum _{t_{0}< t_{k}< t}\iota_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma _{k}}\bigl|v(s)\bigr|\,ds. \end{aligned}$$
(3.6)

Then by Theorem 2.4, we compute that

$$\begin{aligned}& F_{k}=2^{\alpha/(1-\alpha)}\bigl(1+\iota_{k}( \tau_{k}-\sigma_{k})\bigr); \\& R_{k}=\iota_{k}2^{\frac{(k-1)\alpha}{1-\alpha}}(1-\alpha)^{\frac {1}{1-\alpha}} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}}\biggl( \int _{t_{0}}^{v}L\,ds\biggr)^{1/(1-\alpha)}\,dv \\& \hphantom{R_{k}}=\frac{1-\alpha}{2-\alpha}2^{\frac{(k-1)\alpha}{1-\alpha }}\bigl(L(1-\alpha)\bigr)^{\frac{1}{1-\alpha}} \iota_{k} \bigl[(t_{k}-\sigma_{k}-t_{0})^{\frac{2-\alpha}{1-\alpha}} -(t_{k}-\tau_{k}-t_{0})^{\frac{2-\alpha}{1-\alpha}} \bigr] \\& \hphantom{R_{k}}\leq 2^{\frac{(k-1)\alpha}{1-\alpha}}\bigl(L(1-\alpha)\bigr)^{\frac {1}{1-\alpha}}\iota_{k}(t_{k}-t_{0})^{\frac{2-\alpha}{1-\alpha}}. \end{aligned}$$

Substituting \(F_{k}\) and \(R_{k}\) in (2.15), we obtain (3.4). This completes the proof. □