1 Introduction

Impulsive differential and integral inequalities play a fundamental role in the global existence, uniqueness, oscillation, stability, and other properties of the solutions of various nonlinear impulsive differential and integral equations; see [1–27] and the references given therein.

Let \(0\leq t_{0}< t_{1}< t_{2}<\cdots\) , \(\lim_{k\rightarrow\infty}t_{k}=\infty\), \({\mathbb{R}}_{+}=[0,+\infty )\), and \(I\subset{\mathbb{R}}\). We introduce the following function spaces: \(PC({\mathbb{R}}_{+}, I)=\{u : {\mathbb{R}}_{+}\rightarrow I ; u(t) \mbox{ is continuous for } t\neq t_{k}\mbox{, and }u(0^{+}), u(t_{k}^{-})\mbox{ and }u(t_{k}^{+})\mbox{ exist, and }u(t_{k}^{-})=u(t_{k}), k=1, 2, \ldots\}\) and \(PC^{1}({\mathbb{R}}_{+}, I)=\{ u\in PC({\mathbb{R}}_{+}, I) : u'(t)\mbox{ }\mbox{is continuous everywhere for } t\neq t_{k}\mbox{, and }u'(0^{+}), u'(t_{k}^{+})\mbox{ and }u'(t_{k}^{-})\mbox{ exist, and }u'(t_{k}^{-})=u'(t_{k}), k=1, 2, \ldots\}\).

In [1], Lakshmikantham et al. developed a famous impulsive differential inequality given in the next theorem.

Theorem 1.1

Assume that:

(H0):

the sequence \(\{t_{k}\}\) satisfies \(0\leq t_{0}< t_{1}< t_{2}<\cdots\) , \(\lim_{k\rightarrow\infty}t_{k}=\infty\);

(H1):

\(m\in PC^{1}[{\mathbb{R}}_{+},{\mathbb{R}}]\) and \(m(t)\) is left-continuous at \(t_{k}\), \(k=1, 2, \ldots\) ;

(H2):

for \(k=1, 2, \ldots\) , \(t\geq t_{0}\),

$$\begin{aligned}& m'(t) \leq p(t)m(t)+q(t),\quad t\neq t_{k}, \end{aligned}$$
(1.1)
$$\begin{aligned}& m\bigl(t_{k}^{+}\bigr) \leq d_{k}m(t_{k})+b_{k}, \end{aligned}$$
(1.2)

where \(q, p \in C[{\mathbb{R}}_{+},{\mathbb{R}}]\), \(d_{k}\geq0\) and \(b_{k}\), \(k=1,2,\ldots\) , are constants.

Then

$$\begin{aligned} m(t) \leq& m(t_{0})\prod_{t_{0}< t_{k}< t}d_{k}e^{\int_{t_{0}}^{t}p(s)\,ds}+ \sum_{t_{0}< t_{k}< t} \biggl(\prod_{t_{k}< t_{j}< t}d_{j}e^{\int_{t_{k}}^{t}p(s)\,ds} \biggr)b_{k} \\ &{} +\int_{t_{0}}^{t}\prod _{s< t_{k}< t}d_{k}e^{\int_{s}^{t}p(\sigma)\,d\sigma}q(s)\,ds,\quad t\geq t_{0}. \end{aligned}$$
(1.3)

There are many results on the impulsive differential and integral inequalities (see for example [28–36]). However, most of these papers deal with jump conditions at impulse point \(t_{k}\) depending on the left hand limit \(m(t_{k})\) or a time-delay value, \(m(t_{k}-\tau)\), \(\tau>0\).

Recently, in [37], Theorem 1.1 was generalized to obtain differential inequalities for integral jump conditions by replacing the inequality in (1.2) by the following inequality:

$$ m\bigl(t_{k}^{+}\bigr)\leq d_{k}m(t_{k})+c_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma_{k}}m(s)\,ds+b_{k},\quad k=1, 2, \ldots, $$
(1.4)

where \(0\leq\sigma_{k}\leq\tau_{k}\leq t_{k}-t_{k-1}\).

In the present paper we generalize further Theorem 1.1 by replacing the inequality in (1.2) by the inequality

$$ m\bigl(t_{k}^{+}\bigr) \leq \frac{c_{k}}{\Gamma(\beta_{k})} \int _{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}m(s)\,ds+d_{k}m(t_{k})+b_{k}, $$
(1.5)

where \(c_{k},d_{k}\geq0\), \(\beta_{k}>0\) and \(b_{k}\), \(k=1,2,\ldots\) , are constants. Some new impulsive differential and integral inequalities are obtained. Two applications to impulsive differential and integral inequalities with Riemann-Liouville fractional integral jump conditions are given. In the first one we study the maximum principle of an impulsive differential inequality and in the second one we show the boundedness of solution of impulsive differential equation with Riemann-Liouville fractional integral jump conditions.

Nonlocality and memory effects can be represented by the concepts of fractional calculus which contains definitions of fractional derivatives and fractional integrals in the form of weighted integrals. It is learnt through experimentation that the integral operators of fractional order take care of some of the hereditary properties of many phenomena and processes. Impulsive equations and inequalities with nonlocal fractional jump conditions provide a tool to describe systems which have a sudden change of the state values via memorizing previous events. For details of nonlocal theory and memory effects, we refer to [38].

2 Impulsive inequalities with nonlocal jumps

In this section, we state and prove some new impulsive differential and integral inequalities with nonlocal jumps. Throughout of this paper we denote \(t_{l}=\max\{t_{k} : t\geq t_{k}, k= 1, 2, \ldots\}\).

Theorem 2.1

Let (H0) and (H1) hold. Suppose that \(p, q\in C[{\mathbb{R}}_{+}, {\mathbb{R}}]\) and, for \(k=1, 2, \ldots\) , \(t\geq t_{0}\),

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

where \(c_{k},d_{k}\geq0\), \(\beta_{k}>0\) and \(b_{k}\), \(k=1,2,\ldots\) are constants.

Then, for \(t\geq t_{0}\),

$$\begin{aligned} m(t) \leq& \biggl\{ m(t_{0})\prod_{t_{0}< t_{k}< t} \biggl(\frac{c_{k}}{\Gamma (\beta_{k})} \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}e^{\int_{t_{k-1}}^{s}p(\xi )\,d\xi}\,ds+d_{k}e^{\int_{t_{k-1}}^{t_{k}}p(\xi)\,d\xi} \biggr) \\ &{}+\sum_{t_{0}< t_{k}< t} \biggl[\prod _{t_{k}< t_{j}< t} \biggl(\frac{c_{j}}{\Gamma (\beta_{j})}\int_{t_{j-1}}^{t_{j}}(t_{j}-s)^{\beta_{j}-1}e^{\int _{t_{j-1}}^{s}p(\xi)\,d\xi}\,ds +d_{j}e^{\int_{t_{j-1}}^{t_{j}}p(\xi)\,d\xi} \biggr) \\ &{}\times \biggl(\frac{c_{k}}{\Gamma(\beta_{k})}\int_{t_{k-1}}^{t_{k}} \int_{t_{k-1}}^{s}(t_{k}-s)^{\beta_{k}-1}q(v)e^{\int_{v}^{s}p(\xi)\,d\xi}\,dv\,ds \\ &{}+ d_{k}\int_{t_{k-1}}^{t_{k}}q(s)e^{\int_{s}^{t_{k}}p(\xi)\,d\xi}\,ds+ b_{k} \biggr) \biggr] \biggr\} e^{\int_{t_{l}}^{t}p(\xi)\,d\xi}+\int _{t_{l}}^{t}q(s)e^{\int_{s}^{t}p(\xi)\,d\xi}\,ds. \end{aligned}$$
(2.3)

Proof

For \(t\in[t_{0}, t_{1}]\), inequality (2.1) can be written as

$$ \frac{d}{dt} \bigl[m(t)e^{-\int_{t_{0}}^{t}p(\xi)\,d\xi} \bigr]\leq q(t)e^{-\int_{t_{0}}^{t}p(\xi)\,d\xi}. $$
(2.4)

Integrating (2.4) from \(t_{0}\) to t for \(t\in[t_{0}, t_{1}]\), we have

$$ m(t)\leq m(t_{0})e^{\int_{t_{0}}^{t}p(\xi)\,d\xi}+\int _{t_{0}}^{t}q(s)e^{\int_{s}^{t}p(\xi )\,d\xi}\,ds. $$
(2.5)

Hence (2.3) is valid on \([t_{0},t_{1}]\). Assume that (2.3) holds for \(t\in[t_{0},t_{n}]\) for some integer \(n>1\). Then, for \(t\in[t_{n}, t_{n+1}]\), it follows from (2.1) and (2.5) that

$$ m(t)\leq m\bigl(t_{n}^{+}\bigr)e^{\int_{t_{n}}^{t}p(\xi)\,d\xi}+\int _{t_{n}}^{t}q(s)e^{\int _{s}^{t}p(\xi)\,d\xi}\,ds. $$
(2.6)

Applying (2.2) with (2.6), one has

$$\begin{aligned} m(t) \leq& \biggl(\frac{c_{n}}{\Gamma(\beta_{n})}\int _{t_{n-1}}^{t_{n}}(t_{n}-s)^{\beta_{n}-1}m(s)\,ds+ d_{n}m(t_{n})+b_{n} \biggr) e^{\int_{t_{n}}^{t}p(\xi)\,d\xi} \\ &{}+\int_{t_{n}}^{t}q(s)e^{\int_{s}^{t}p(\xi)\,d\xi}\,ds. \end{aligned}$$
(2.7)

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

$$\begin{aligned} m(t) \leq& \biggl\{ \frac{c_{n}}{\Gamma(\beta_{n})}\int_{t_{n-1}}^{t_{n}}(t_{n}-s)^{\beta_{n}-1} \\ &{}\times \biggl\{ \biggl\{ m(t_{0})\prod _{t_{0}< t_{k}< s} \biggl( \frac{c_{k}}{\Gamma(\beta_{k})}\int_{t_{k-1}}^{t_{k}}(t_{k}-v)^{\beta _{k}-1}e^{\int_{t_{k-1}}^{v}p(\xi)\,d\xi}\,dv+d_{k}e^{\int _{t_{k-1}}^{t_{k}}p(\xi)\,d\xi} \biggr) \\ &{}+ \sum_{t_{0}< t_{k}< s} \biggl[\prod _{t_{k}< t_{j}< s} \biggl(\frac{c_{j}}{\Gamma(\beta_{j})}\int_{t_{j-1}}^{t_{j}}(t_{j}-v)^{\beta_{j}-1}e^{\int_{t_{j-1}}^{v}p(\xi)\,d\xi}\,dv +d_{j}e^{\int_{t_{j-1}}^{t_{j}}p(\xi)\,d\xi} \biggr) \\ &{}\times \biggl(\frac{c_{k}}{\Gamma(\beta_{k})}\int_{t_{k-1}}^{t_{k}} \int_{t_{k-1}}^{v}(t_{k}-v)^{\beta_{k}-1}q(r)e^{\int_{r}^{v}p(\xi)\,d\xi }\,dr\,dv \\ &{}+d_{k} \int_{t_{k-1}}^{t_{k}}q(v)e^{\int_{v}^{t_{k}}p(\xi)\,d\xi }\,dv+b_{k} \biggr) \biggr] \biggr\} e^{\int_{t_{n-1}}^{s}p(\xi)\,d\xi}+ \int_{t_{n-1}}^{s}q(v)e^{\int_{v}^{s}p(\xi)\,d\xi}\,dv \biggr\} \,ds \\ &{}+ d_{n} \biggl( \biggl\{ m(t_{0})\prod _{t_{0}< t_{k}< t_{n}} \biggl(\frac {c_{k}}{\Gamma(\beta_{k})}\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta _{k}-1}e^{\int_{t_{k-1}}^{s}p(\xi)\,d\xi}\,ds +d_{k}e^{\int_{t_{k-1}}^{t_{k}}p(\xi)\,d\xi} \biggr) \\ &{}+ \sum_{t_{0}< t_{k}< t_{n}} \biggl[\prod _{t_{k}< t_{j}< t_{n}} \biggl(\frac {c_{j}}{\Gamma(\beta_{j})}\int_{t_{j-1}}^{t_{j}}(t_{j}-s)^{\beta _{j}-1}e^{\int_{t_{j-1}}^{s}p(\xi)\,d\xi}\,ds + d_{j}e^{\int_{t_{j-1}}^{t_{j}}p(\xi)\,d\xi} \biggr) \\ &{}\times \biggl(\frac{c_{k}}{\Gamma(\beta_{k})}\int_{t_{k-1}}^{t_{k}} \int_{t_{k-1}}^{s}(t_{k}-s)^{\beta_{k}-1}q(v)e^{\int _{v}^{s}p(\xi)\,d\xi}\,dv\,ds \\ &{}+ d_{k}\int_{t_{k-1}}^{t_{k}}q(s)e^{\int_{s}^{t_{k}}p(\xi)\,d\xi }\,ds+b_{k} \biggr) \biggr] \biggr\} e^{\int_{t_{n-1}}^{t_{n}}p(\xi)\,d\xi} \\ &{}+ \int_{t_{n-1}}^{t_{n}}q(s)e^{\int_{s}^{t_{n}}p(\xi)\,d\xi}\,ds \biggr)+b_{n} \biggr\} e^{\int_{t_{n}}^{t}p(\xi)\,d\xi}+\int_{t_{n}}^{t}q(s)e^{\int_{s}^{t}p(\xi)\,d\xi }\,ds. \end{aligned}$$
(2.8)

Set

$$\begin{aligned}& E_{k} = \frac{c_{k}}{\Gamma(\beta_{k})}\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta _{k}-1}e^{\int_{t_{k-1}}^{s}p(\xi)\,d\xi}\,ds+d_{k}e^{\int _{t_{k-1}}^{t_{k}}p(\xi)\,d\xi}, \end{aligned}$$
(2.9)
$$\begin{aligned}& G_{k} = \frac{c_{k}}{\Gamma(\beta_{k})}\int_{t_{k-1}}^{t_{k}} \int_{t_{k-1}}^{s}(t_{k}-s)^{\beta_{k}-1}q(v)e^{\int_{v}^{s}p(\xi)\,d\xi }\,dv\,ds \\& \hphantom{E_{k} =}{}+d_{k}\int_{t_{k-1}}^{t_{k}}q(s)e^{\int_{s}^{t_{k}}p(\xi)\,d\xi }\,ds+b_{k}. \end{aligned}$$
(2.10)

Substituting (2.9), (2.10) into (2.8), we get for \(t\in[t_{n}, t_{n+1}]\)

$$\begin{aligned} m(t) \leq& \biggl\{ \frac{c_{n}}{\Gamma(\beta_{n})}\int_{t_{n-1}}^{t_{n}}(t_{n}-s)^{\beta_{n}-1} \biggl\{ \biggl\{ m(t_{0})\prod_{t_{0}< t_{k}< s}E_{k} + \sum_{t_{0}< t_{k}< s} \biggl[\prod _{t_{k}< t_{j}< s}E_{j} G_{k} \biggr] \biggr\} e^{\int_{t_{n-1}}^{s}p(\xi)\,d\xi}\\ &{}+\int_{t_{n-1}}^{s}q(v)e^{\int_{v}^{s}p(\xi)\,d\xi}\,dv \biggr\} \,ds \\ &{}+ d_{n} \biggl( \biggl\{ m(t_{0})\prod _{t_{0}< t_{k}< t_{n}}E_{k} +\sum_{t_{0}< t_{k}< t_{n}} \biggl[\prod_{t_{k}< t_{j}< t_{n}}E_{j}G_{k} \biggr] \biggr\} e^{\int_{t_{n-1}}^{t_{n}}p(\xi)\,d\xi} \\ &{}+ \int_{t_{n-1}}^{t_{n}}q(s)e^{\int_{s}^{t_{n}}p(\xi)\,d\xi}\,ds \biggr)+b_{n} \biggr\} e^{\int_{t_{n}}^{t}p(\xi)\,d\xi}+\int_{t_{n}}^{t}q(s)e^{\int_{s}^{t}p(\xi)\,d\xi }\,ds \\ =& \biggl\{ \biggl(m(t_{0})\prod_{t_{0}< t_{k}< t_{n}}E_{k} +\sum_{t_{0}< t_{k}< t_{n}} \biggl[\prod _{t_{k}< t_{j}< t_{n}}E_{j}G_{k} \biggr] \biggr) \frac{c_{n}}{\Gamma(\beta_{n})}\int_{t_{n-1}}^{t_{n}}(t_{n}-s)^{\beta_{n}-1} e^{\int_{t_{n-1}}^{s}p(\xi)\,d\xi }\,ds \\ &{}+ \frac{c_{n}}{\Gamma(\beta_{n})}\int_{t_{n-1}}^{t_{n}}\int _{t_{n-1}}^{s}(t_{n}-s)^{\beta_{n}-1}q(v)e^{\int_{v}^{s}p(\xi)\,d\xi}\,dv\,ds \\ &{}+ \biggl(m(t_{0})\prod_{t_{0}< t_{k}< t_{n}}E_{k} +\sum_{t_{0}< t_{k}< t_{n}} \biggl[\prod _{t_{k}< t_{j}< t_{n}}E_{j}G_{k} \biggr] \biggr)d_{n}e^{\int_{t_{n-1}}^{t_{n}}p(\xi)\,d\xi} \\ &{}+ d_{n}\int_{t_{n-1}}^{t_{n}}q(s)e^{\int_{s}^{t_{n}}p(\xi)\,d\xi }\,ds+b_{n} \biggr\} e^{\int_{t_{n}}^{t}p(\xi)\,d\xi}+\int_{t_{n}}^{t}q(s)e^{\int_{s}^{t}p(\xi)\,d\xi }\,ds \\ =& \biggl\{ \biggl(m(t_{0})\prod_{t_{0}< t_{k}< t_{n}}E_{k} +\sum_{t_{0}< t_{k}< t_{n}} \biggl[\prod _{t_{k}< t_{j}< t_{n}}E_{j}G_{k} \biggr] \biggr)E_{n}+G_{n} \biggr\} e^{\int_{t_{n}}^{t}p(\xi)\,d\xi}\,ds \\ &{}+ \int_{t_{n}}^{t}q(s)e^{\int_{s}^{t}p(\xi)\,d\xi} \\ =& \biggl\{ m(t_{0})\prod_{t_{0}< t_{k}< t}E_{k} +\sum_{t_{0}< t_{k}< t} \biggl[\prod _{t_{k}< t_{j}< t}E_{j}G_{k} \biggr] \biggr\} e^{\int_{t_{n}}^{t}p(\xi)\,d\xi}\,ds+\int_{t_{n}}^{t}q(s)e^{\int _{s}^{t}p(\xi)\,d\xi}. \end{aligned}$$

Hence,

$$\begin{aligned} m(t) \leq& \biggl\{ m(t_{0})\prod_{t_{0}< t_{k}< t} \biggl(\frac{c_{k}}{\Gamma (\beta_{k})} \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}e^{\int_{t_{k-1}}^{s}p(\xi )\,d\xi}\,ds+d_{k}e^{\int_{t_{k-1}}^{t_{k}}p(\xi)\,d\xi} \biggr) \\ &{}+\sum_{t_{0}< t_{k}< t} \biggl[\prod _{t_{k}< t_{j}< t} \biggl(\frac{c_{j}}{\Gamma (\beta_{j})}\int_{t_{j-1}}^{t_{j}}(t_{j}-s)^{\beta_{j}-1}e^{\int _{t_{j-1}}^{s}p(\xi)\,d\xi}\,ds +d_{j}e^{\int_{t_{j-1}}^{t_{j}}p(\xi)\,d\xi} \biggr) \\ &{}\times \biggl(\frac{c_{k}}{\Gamma(\beta_{k})}\int_{t_{k-1}}^{t_{k}} \int_{t_{k-1}}^{s}(t_{k}-s)^{\beta_{k}-1}q(v)e^{\int_{v}^{s}p(\xi)\,d\xi}\,dv\,ds \\ &{}+ d_{k}\int_{t_{k-1}}^{t_{k}}q(s)e^{\int_{s}^{t_{k}}p(\xi)\,d\xi }\,ds+b_{k} \biggr) \biggr] \biggr\} e^{\int_{t_{n}}^{t}p(\xi)\,d\xi}+\int_{t_{n}}^{t}q(s)e^{\int_{s}^{t}p(\xi)\,d\xi}\,ds, \end{aligned}$$

for \(t_{n}\leq t\leq t_{n+1}\). Therefore, the estimate (2.3) holds for \(t_{0}\leq t\leq t_{n+1}\). This completes the proof. □

Theorem 2.2

Assume that the hypotheses of Theorem 2.1 are fullfilled. Then, for \(t\geq t_{0}\), we have:

  1. (i)

    \(0<\beta_{k}\leq\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} m(t) \leq& \biggl\{ m(t_{0})\prod_{t_{0}< t_{k}< t} \biggl[ \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta _{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int _{t_{k-1}}^{t_{k}}e^{\nu_{k} (\int_{t_{k-1}}^{s}p(\xi )\,d\xi-s )}\,ds \biggr)^{\frac{1}{\nu_{k}}} \\ &{}+d_{k}e^{\int_{t_{k-1}}^{t_{k}}p(\xi)\,d\xi} \biggr] \\ &{}+\sum_{t_{0}< t_{k}< t} \biggl[\prod _{t_{k}< t_{j}< t} \biggl(\frac{c_{j}}{\Gamma (\beta_{j})} \biggl(\frac{e^{\mu_{j}t_{j}}\Gamma(\beta_{j}^{2})}{\mu _{j}^{\beta_{j}^{2}}} \biggr)^{\frac{1}{\mu_{j}}} \biggl(\int_{t_{j-1}}^{t_{j}}e^{\nu_{j} (\int_{t_{j-1}}^{s}p(\xi )\,d\xi-s )}\,ds \biggr)^{\frac{1}{\nu_{j}}} \\ &{}+d_{j}e^{\int_{t_{j-1}}^{t_{j}}p(\xi)\,d\xi} \biggr) \\ &{}\times \biggl(\frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{\mu _{k}t_{k}}\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int_{t_{k-1}}^{t_{k}}e^{-\nu_{k}s} \biggl\{ \int_{t_{k-1}}^{s}q(v)e^{\int_{v}^{s}p(\xi)\,d\xi}\,dv \biggr\} ^{\nu_{k}}\,ds \biggr)^{\frac{1}{\nu_{k}}} \\ &{}+ d_{k}\int_{t_{k-1}}^{t_{k}}q(s)e^{\int_{s}^{t_{k}}p(\xi)\,d\xi }\,ds+b_{k} \biggr) \biggr] \biggr\} e^{\int_{t_{l}}^{t}p(\xi)\,d\xi}+ \int_{t_{l}}^{t}q(s)e^{\int_{s}^{t}p(\xi)\,d\xi}\,ds, \end{aligned}$$
    (2.11)

    where \(\mu_{k}=\beta_{k}+1\) and \(\nu_{k}=1+\frac{1}{\beta _{k}}\),

  2. (ii)

    \(\beta_{k}>\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} m(t) \leq& \biggl\{ m(t_{0})\prod_{t_{0}< t_{k}< t} \biggl[ \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{2t_{k}}\Gamma(2\beta_{k}-1)}{2^{2\beta_{k}-1}} \biggr)^{\frac{1}{2}} \biggl(\int _{t_{k-1}}^{t_{k}}e^{2 (\int_{t_{k-1}}^{s}p(\xi)\,d\xi -s )}\,ds \biggr)^{\frac{1}{2}} \\ &{}+d_{k}e^{\int_{t_{k-1}}^{t_{k}}p(\xi)\,d\xi} \biggr] \\ &{}+\sum_{t_{0}< t_{k}< t} \biggl[\prod _{t_{k}< t_{j}< t} \biggl(\frac{c_{j}}{\Gamma (\beta_{j})} \biggl(\frac{e^{2t_{j}}\Gamma(2\beta_{j}-1)}{2^{2\beta _{j}-1}} \biggr)^{\frac{1}{2}} \biggl(\int_{t_{j-1}}^{t_{j}}e^{2 (\int_{t_{j-1}}^{s}p(\xi)\,d\xi -s )}\,ds \biggr)^{\frac{1}{2}} \\ &{} +d_{j}e^{\int_{t_{j-1}}^{t_{j}}p(\xi)\,d\xi} \biggr) \\ &{}\times \biggl(\frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac {e^{2t_{k}}\Gamma(2\beta_{k}-1)}{2^{2\beta_{k}-1}} \biggr)^{\frac{1}{2}} \biggl(\int_{t_{k-1}}^{t_{k}}e^{-2s} \biggl\{ \int_{t_{k-1}}^{s}q(v)e^{\int _{v}^{s}p(\xi)\,d\xi}\,dv \biggr\} ^{2}\,ds \biggr)^{\frac{1}{2}} \\ &{}+ d_{k}\int_{t_{k-1}}^{t_{k}}q(s)e^{\int_{s}^{t_{k}}p(\xi)\,d\xi }\,ds+b_{k} \biggr) \biggr] \biggr\} e^{\int_{t_{l}}^{t}p(\xi)\,d\xi} + \int_{t_{l}}^{t}q(s)e^{\int_{s}^{t}p(\xi)\,d\xi}\,ds. \end{aligned}$$
    (2.12)

Proof

To prove (i) we apply the Hölder inequality. We have for \(k\in\mathbb{N}\)

$$\begin{aligned}& \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}e^{\int_{t_{k-1}}^{s}p(\xi )\,d\xi}\,ds \\& \quad\leq \biggl(\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\mu_{k}(\beta_{k}-1)}e^{\mu _{k}s}\,ds \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int_{t_{k-1}}^{t_{k}}e^{-\nu_{k}s}e^{\nu_{k}\int_{t_{k-1}}^{s}p(\xi )\,d\xi}\,ds \biggr)^{\frac{1}{\nu_{k}}} \\& \quad< \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma(1-\mu_{k}(1-\beta_{k}))}{\mu _{k}^{1-\mu_{k}(1-\beta_{k})}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int _{t_{k-1}}^{t_{k}}e^{\nu_{k} (\int_{t_{k-1}}^{s}p(\xi )\,d\xi-s )}\,ds \biggr)^{\frac{1}{\nu_{k}}} \\& \quad= \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta _{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int _{t_{k-1}}^{t_{k}}e^{\nu_{k} (\int_{t_{k-1}}^{s}p(\xi )\,d\xi-s )}\,ds \biggr)^{\frac{1}{\nu_{k}}}, \end{aligned}$$

using

$$\begin{aligned} \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\mu_{k}(\beta_{k}-1)}e^{\mu_{k}s}\,ds =&e^{\mu _{k}t_{k}} \int_{0}^{t_{k}-t_{k-1}}\tau^{\mu_{k}(\beta_{k}-1)}e^{-\mu_{k}\tau }\,d\tau \\ =&\frac{e^{\mu_{k}t_{k}}}{\mu_{k}^{1-\mu_{k}(1-\beta_{k})}}\int_{0}^{\mu _{k}(t_{k}-t_{k-1})} \sigma^{\mu_{k}(\beta_{k}-1)}e^{-\sigma}\,d\sigma \\ < &\frac{e^{\mu_{k}t_{k}}}{\mu_{k}^{1-\mu_{k}(1-\beta_{k})}}\Gamma\bigl(1-\mu _{k}(1-\beta_{k}) \bigr) \\ =&\frac{e^{\mu_{k}t_{k}}}{\mu_{k}^{\beta_{k}^{2}}}\Gamma\bigl(\beta_{k}^{2}\bigr), \end{aligned}$$

and

$$\begin{aligned}& \int_{t_{k-1}}^{t_{k}}\int_{t_{k-1}}^{s}(t_{k}-s)^{\beta_{k}-1}q(v)e^{\int _{v}^{s}p(\xi)\,d\xi}\,dv\,ds \\& \quad\leq \biggl(\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\mu_{k}(\beta_{k}-1)}e^{\mu _{k}s}\,ds \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int_{t_{k-1}}^{t_{k}}e^{-\nu_{k}s} \biggl\{ \int_{t_{k-1}}^{s}q(v)e^{\int_{v}^{s}p(\xi)\,d\xi}\,dv \biggr\} ^{\nu_{k}}\,ds \biggr)^{\frac{1}{\nu_{k}}} \\& \quad < \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta _{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int _{t_{k-1}}^{t_{k}}e^{-\nu_{k}s} \biggl\{ \int _{t_{k-1}}^{s}q(v)e^{\int_{v}^{s}p(\xi)\,d\xi}\,dv \biggr\} ^{\nu_{k}}\,ds \biggr)^{\frac{1}{\nu_{k}}}. \end{aligned}$$

Substituting the above inequalities in (2.3), we obtain the desired inequality in (2.11).

To prove (ii), applying the Cauchy-Schwarz inequality, we get for \(k\in\mathbb{N}\)

$$\begin{aligned}& \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}e^{\int_{t_{k-1}}^{s}p(\xi )\,d\xi}\,ds \\& \quad\leq \biggl(\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{2(\beta_{k}-1)}e^{2s}\,ds \biggr)^{\frac{1}{2}} \biggl(\int_{t_{k-1}}^{t_{k}}e^{2 (\int_{t_{k-1}}^{s}p(\xi)\,d\xi -s )}\,ds \biggr)^{\frac{1}{2}} \\& \quad< \biggl(\frac{e^{2t_{k}}\Gamma(2\beta_{k}-1)}{2^{2\beta_{k}-1}} \biggr)^{\frac{1}{2}} \biggl(\int _{t_{k-1}}^{t_{k}}e^{2 (\int_{t_{k-1}}^{s}p(\xi)\,d\xi -s )}\,ds \biggr)^{\frac{1}{2}}, \end{aligned}$$

and

$$\begin{aligned}& \int_{t_{k-1}}^{t_{k}}\int_{t_{k-1}}^{s}(t_{k}-s)^{\beta_{k}-1}q(v)e^{\int _{v}^{s}p(\xi)\,d\xi}\,dv\,ds \\& \quad\leq \biggl(\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{2(\beta _{k}-1)}e^{2s}\,ds \biggr)^{\frac{1}{2}} \biggl(\int_{t_{k-1}}^{t_{k}}e^{-2s} \biggl\{ \int_{t_{k-1}}^{s}q(v)e^{\int _{v}^{s}p(\xi)\,d\xi}\,dv \biggr\} ^{2}\,ds \biggr)^{\frac{1}{2}} \\& \quad< \biggl(\frac{e^{2t_{k}}\Gamma(2\beta_{k}-1)}{2^{2\beta_{k}-1}} \biggr)^{\frac{1}{2}} \biggl(\int _{t_{k-1}}^{t_{k}}e^{-2s} \biggl\{ \int _{t_{k-1}}^{s}q(v)e^{\int _{v}^{s}p(\xi)\,d\xi}\,dv \biggr\} ^{2}\,ds \biggr)^{\frac{1}{2}}. \end{aligned}$$

Substituting these two inequalities in (2.3), we get the required inequality in (2.12). The proof is completed. □

Corollary 2.3

Let (H0) and (H1) hold. Suppose that \(q\in C[\mathbb{R}_{+},\mathbb{R}]\) and, for \(k=1,2,\ldots\) , \(t\geq t_{0}\),

$$ \textstyle\begin{cases} m'(t)\leq\lambda m(t)+q(t), \quad t\neq t_{k},\\ m(t_{k}^{+})\leq\frac{c_{k}}{\Gamma(\beta_{k})} \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}m(s)\,ds+b_{k}, \end{cases} $$
(2.13)

where λ, \(c_{k}\geq0\), \(\beta_{k}>0\) and \(b_{k}\), \(k=1,2,\ldots\) are constants. Then, for \(t\geq t_{0}\), we have the following two cases.

Case I: \(\lambda\neq1\),

  1. (i)

    \(0<\beta_{k}\leq\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} m(t) \leq& m(t_{0}) \biggl(\prod _{t_{0}< t_{k}< t}A_{k} \biggr)e^{\lambda (t-t_{0})}+\sum _{t_{0}< t_{k}< t} \biggl(\prod_{t_{k}< t_{j}< t}A_{j}B_{k}e^{\lambda (t-t_{k})} \biggr) \\ &{}+ \int_{t_{l}}^{t}q(s)e^{\lambda(t-s)}\,ds, \end{aligned}$$
    (2.14)

    where

    $$\begin{aligned}& A_{k} = \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{\Gamma(\beta _{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\frac{1-e^{\nu_{k}(t_{k}-t_{k-1})(1-\lambda)}}{\nu_{k}(\lambda -1)} \biggr)^{\frac{1}{\nu_{k}}}, \\& B_{k} = \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma (\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int_{t_{k-1}}^{t_{k}} e^{\nu_{k}(\lambda-1)s} \biggl\{ \int_{t_{k-1}}^{s}q(v)e^{-\lambda v}\,dv \biggr\} ^{\nu_{k}}\,ds \biggr)^{\frac{1}{\nu_{k}}}+b_{k}, \end{aligned}$$
  2. (ii)

    \(\beta_{k}>\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} m(t) \leq& m(t_{0}) \biggl(\prod _{t_{0}< t_{k}< t}C_{k} \biggr)e^{\lambda (t-t_{0})}+\sum _{t_{0}< t_{k}< t} \biggl(\prod_{t_{k}< t_{j}< t}C_{j}D_{k}e^{\lambda (t-t_{k})} \biggr) \\ &{}+ \int_{t_{l}}^{t}q(s)e^{\lambda(t-s)}\,ds, \end{aligned}$$
    (2.15)

    where

    $$\begin{aligned}& C_{k} = \frac{c_{k}}{2^{\beta_{k}}\Gamma(\beta_{k})} \biggl(\frac{\Gamma (2\beta_{k}-1)}{\lambda-1} \bigl[1-e^{2(t_{k}-t_{k-1})(1-\lambda)} \bigr] \biggr)^{\frac{1}{2}}, \\& D_{k} = \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{2t_{k}}\Gamma (2\beta_{k}-1)}{2^{2\beta_{k}-1}} \biggr)^{\frac{1}{2}} \biggl(\int_{t_{k-1}}^{t_{k}} e^{2(\lambda-1)s} \biggl\{ \int_{t_{k-1}}^{s}q(v)e^{-\lambda v}\,dv \biggr\} ^{2}\,ds \biggr)^{\frac{1}{2}}+b_{k}. \end{aligned}$$

Case II: \(\lambda= 1\),

  1. (i)

    \(0<\beta_{k}\leq\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} m(t) \leq& m(t_{0}) \biggl(\prod _{t_{0}< t_{k}< t}P_{k} \biggr)e^{(t-t_{0})}+\sum _{t_{0}< t_{k}< t} \biggl(\prod_{t_{k}< t_{j}< t}P_{j}V_{k}e^{(t-t_{k})} \biggr) \\ &{}+ \int_{t_{l}}^{t}q(s)e^{(t-s)}\,ds, \end{aligned}$$
    (2.16)

    where

    $$\begin{aligned}& P_{k} = \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{\Gamma(\beta _{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} (t_{k}-t_{k-1} )^{\frac{1}{\nu_{k}}}, \\& V_{k} = \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma (\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int_{t_{k-1}}^{t_{k}} \biggl\{ \int_{t_{k-1}}^{s}q(v)e^{-v}\,dv \biggr\} ^{\nu_{k}}\,ds \biggr)^{\frac{1}{\nu_{k}}}+b_{k}, \end{aligned}$$
  2. (ii)

    \(\beta_{k}>\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} m(t) \leq& m(t_{0}) \biggl(\prod _{t_{0}< t_{k}< t}S_{k} \biggr)e^{(t-t_{0})}+\sum _{t_{0}< t_{k}< t} \biggl(\prod_{t_{k}< t_{j}< t}S_{j}U_{k}e^{(t-t_{k})} \biggr) \\ &{}+ \int_{t_{l}}^{t}q(s)e^{(t-s)}\,ds, \end{aligned}$$
    (2.17)

    where

    $$\begin{aligned}& S_{k} = \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{\Gamma(2\beta _{k}-1)(t_{k}-t_{k-1})}{2^{2\beta_{k}-1}} \biggr)^{\frac{1}{2}}, \\& U_{k} = \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{2t_{k}}\Gamma (2\beta_{k}-1)}{2^{2\beta_{k}-1}} \biggr)^{\frac{1}{2}} \biggl(\int_{t_{k-1}}^{t_{k}} \biggl\{ \int_{t_{k-1}}^{s}q(v)e^{-v}\,dv \biggr\} ^{2}\,ds \biggr)^{\frac{1}{2}}+b_{k}. \end{aligned}$$

Corollary 2.4

Let (H0) and (H1) hold. Suppose that \(q\in C[\mathbb{R}_{+},\mathbb{R}]\) and, for \(k=1,2,\ldots\) , \(t\geq t_{0}\),

$$ \textstyle\begin{cases} m'(t)\leq q(t),\quad t\neq t_{k},\\ \Delta m(t_{k})\leq\frac{c_{k}}{\Gamma(\beta_{k})} \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}m(s)\,ds+b_{k}, \end{cases} $$
(2.18)

where \(c_{k}\geq0\), \(\beta_{k}>0\) and \(b_{k}\), \(k=1,2,\ldots\) are constants, \(\Delta m(t_{k})=m(t_{k}^{+})-m(t_{k})\). Then, for \(t\geq t_{0}\), the following assertions hold:

  1. (i)

    \(0<\beta_{k}\leq\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$ m(t) \leq m(t_{0}) \biggl(\prod _{t_{0}< t_{k}< t}F_{k} \biggr)+\sum _{t_{0}< t_{k}< t} \biggl(\prod_{t_{k}< t_{j}< t}F_{j}H_{k} \biggr)+\int_{t_{l}}^{t}q(s)\,ds, $$
    (2.19)

    where

    $$\begin{aligned}& F_{k} = 1+\frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{\Gamma(\beta _{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\frac{e^{\nu_{k}(t_{k}-t_{k-1})}-1}{\nu_{k}} \biggr)^{\frac{1}{\nu _{k}}}, \\& H_{k} = \int_{t_{k-1}}^{t_{k}}q(s)\,ds+ \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int _{t_{k-1}}^{t_{k}} e^{-\nu_{k}s} \biggl\{ \int _{t_{k-1}}^{s}q(v)\,dv \biggr\} ^{\nu_{k}}\,ds \biggr)^{\frac{1}{\nu_{k}}} \\& \hphantom{H_{k} =}{}+b_{k}, \end{aligned}$$
  2. (ii)

    \(\beta_{k}>\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$ m(t) \leq m(t_{0}) \biggl(\prod _{t_{0}< t_{k}< t}M_{k} \biggr)+\sum _{t_{0}< t_{k}< t} \biggl(\prod_{t_{k}< t_{j}< t}M_{j}N_{k} \biggr)+\int_{t_{l}}^{t}q(s)\,ds, $$
    (2.20)

    where

    $$\begin{aligned}& M_{k} = 1+\frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{\Gamma(2\beta _{k}-1)}{2^{2\beta_{k}}} \bigl[ e^{2(t_{k}-t_{k-1})}-1 \bigr] \biggr)^{\frac{1}{2}}, \\& N_{k} = \int_{t_{k-1}}^{t_{k}}q(s)\,ds+ \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{2t_{k}}\Gamma(2\beta_{k}-1)}{2^{2\beta_{k}-1}} \biggr)^{\frac {1}{2}} \biggl(\int _{t_{k-1}}^{t_{k}} e^{-2s} \biggl\{ \int _{t_{k-1}}^{s}q(v)\,dv \biggr\} ^{2}\,ds \biggr)^{\frac {1}{2}}\\& \hphantom{N_{k} =}{}+b_{k}. \end{aligned}$$

Now we state and prove impulsive integral inequalities with nonlocal jump conditions.

Theorem 2.5

Assume that (H0) and (H1) hold. Suppose that \(p\in C[\mathbb{R}_{+},\mathbb{R}_{+}]\) and, for \(k=1,2,\ldots\) , \(t\geq t_{0}\),

$$\begin{aligned} m(t) \leq& C+\int_{t_{0}}^{t}p(s)m(s)\,ds+ \sum_{t_{0}< t_{k}< t}\gamma _{k}m(t_{k}) \\ &{}+ \sum_{t_{0}< t_{k}< t}\frac{\alpha_{k}}{\Gamma(\beta_{k})} \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}m(s)\,ds, \end{aligned}$$
(2.21)

where \(\alpha_{k}\geq0\), \(\gamma_{k}\geq-1\), \(\beta_{k}>0\), \(k=1,2,\ldots \) , and C are constants. Then, for \(t\geq t_{0}\), the following assertions hold:

  1. (i)

    \(0<\beta_{k}\leq\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} m(t) \leq& C\prod_{t_{0}< t_{k}< t} \biggl\{ (1+ \gamma_{k})e^{\int _{t_{k-1}}^{t_{k}}p(\xi)\,d\xi}+\frac{\alpha_{k}}{\Gamma(\beta_{k})} \biggl( \frac{e^{\mu_{k}t_{k}}}{ \mu_{k}^{\beta_{k}^{2}}}\Gamma\bigl(\beta_{k}^{2}\bigr) \biggr)^{\frac{1}{\mu _{k}}} \\ &{}\times \biggl(\int_{t_{k-1}}^{t_{k}}e^{\nu_{k} (\int _{t_{k-1}}^{s}p(\xi)\,d\xi-s )}\,ds \biggr)^{\frac{1}{\nu_{k}}} \biggr\} e^{\int_{t_{l}}^{t}p(\xi)\,d\xi}, \end{aligned}$$
    (2.22)

    where \(\mu_{k}=\beta_{k}+1\) and \(\nu_{k}=1+\frac{1}{\beta _{k}}\),

  2. (ii)

    \(\beta_{k}>\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} m(t) \leq& C\prod_{t_{0}< t_{k}< t} \biggl\{ (1+ \gamma_{k})e^{\int _{t_{k-1}}^{t_{k}}p(\xi)\,d\xi}+\frac{\alpha_{k}}{\Gamma(\beta_{k})} \biggl( \frac{e^{2t_{k}}}{ 2^{2\beta_{k}-1}}\Gamma(2\beta_{k}-1) \biggr)^{\frac{1}{2}} \\ &{}\times \biggl(\int_{t_{k-1}}^{t_{k}}e^{2 (\int _{t_{k-1}}^{s}p(\xi)\,d\xi-s )}\,ds \biggr)^{\frac{1}{2}} \biggr\} e^{\int_{t_{l}}^{t}p(\xi)\,d\xi}. \end{aligned}$$
    (2.23)

Proof

Define a function \(g(t)\) by the right-hand side of (2.21). Then we have

$$ \textstyle\begin{cases} g'(t)= p(t)m(t),\quad t\neq t_{k}, \quad\quad g(t_{0})=C,\\ g(t_{k}^{+})=g(t_{k})+\gamma_{k}m(t_{k})+\frac{\alpha_{k}}{\Gamma (\beta_{k})} \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}m(s)\,ds. \end{cases} $$

Since \(m(t)\leq g(t)\), we obtain

$$ \textstyle\begin{cases} g'(t)\leq p(t)g(t), \quad t\neq t_{k}, \quad\quad g(t_{0})=C,\\ g(t_{k}^{+})=(1+\gamma_{k})g(t_{k})+\frac{\alpha_{k}}{\Gamma (\beta_{k})} \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}g(s)\,ds. \end{cases} $$

Applying Theorem 2.2, we deduce that:

  1. (i)

    \(0<\beta_{k}\leq\frac{1}{2}\), \(k=1,2,\ldots\) , for \(t\geq t_{0}\),

    $$\begin{aligned} g(t) \leq& C\prod_{t_{0}< t_{k}< t} \biggl\{ (1+ \gamma_{k})e^{\int _{t_{k-1}}^{t_{k}}p(\xi)\,d\xi}+\frac{\alpha_{k}}{\Gamma(\beta_{k})} \biggl( \frac{e^{\mu_{k}t_{k}}}{ \mu_{k}^{\beta_{k}^{2}}}\Gamma\bigl(\beta_{k}^{2}\bigr) \biggr)^{\frac{1}{\mu _{k}}} \\ &{}\times \biggl(\int_{t_{k-1}}^{t_{k}}e^{\nu_{k} (\int _{t_{k-1}}^{s}p(\xi)\,d\xi-s )}\,ds \biggr)^{\frac{1}{\nu_{k}}} \biggr\} e^{\int_{t_{l}}^{t}p(\xi)\,d\xi}, \end{aligned}$$
  2. (ii)

    \(\beta_{k}>\frac{1}{2}\), \(k=1,2,\ldots\) , for \(t\geq t_{0}\),

    $$\begin{aligned} g(t) \leq& C\prod_{t_{0}< t_{k}< t} \biggl\{ (1+ \gamma_{k})e^{\int _{t_{k-1}}^{t_{k}}p(\xi)\,d\xi}+\frac{\alpha_{k}}{\Gamma(\beta_{k})} \biggl( \frac{e^{2t_{k}}}{ 2^{2\beta_{k}-1}}\Gamma(2\beta_{k}-1) \biggr)^{\frac{1}{2}} \\ &{}\times \biggl(\int_{t_{k-1}}^{t_{k}}e^{2 (\int _{t_{k-1}}^{s}p(\xi)\,d\xi-s )}\,ds \biggr)^{\frac{1}{2}} \biggr\} e^{\int_{t_{l}}^{t}p(\xi)\,d\xi}, \end{aligned}$$

    which are the results in (2.22) and (2.23), respectively.

 □

In the case when in place of the constant C involved in Theorem 2.5 we have a function \(h(t)\), we obtain the following result.

Theorem 2.6

Assume that (H0) and (H1) hold. Suppose that \(p\in C[\mathbb{R}_{+},\mathbb{R}_{+}]\), \(h\in PC[\mathbb{R}_{+},\mathbb{R}]\), and, for \(k=1,2,\ldots\) , \(t\geq t_{0}\),

$$\begin{aligned} m(t) \leq& h(t)+\int_{t_{0}}^{t}p(s)m(s)\,ds+ \sum_{t_{0}< t_{k}< t}\gamma _{k}m(t_{k}) \\ &{}+ \sum_{t_{0}< t_{k}< t}\frac{\alpha_{k}}{\Gamma(\beta_{k})}\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}m(s)\,ds, \end{aligned}$$
(2.24)

where \(\alpha_{k}\geq0\), \(\gamma_{k}\geq-1\) and \(\beta_{k}>0\), \(k=1,2,\ldots\) , are constants. Then, for \(t\geq t_{0}\), the following assertions hold:

  1. (i)

    \(0<\beta_{k}\leq\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} m(t) \leq& h(t)+ \biggl\{ \sum_{t_{0}< t_{k}< t} \biggl[\prod_{t_{k}< t_{j}< t} \biggl\{ (1+\gamma_{j})e^{\int_{t_{j-1}}^{t_{j}}p(\xi)\,d\xi }+ \frac{\alpha_{j}}{\Gamma(\beta_{j})} \biggl(\frac{e^{\mu_{j}t_{j}}}{ \mu_{j}^{\beta_{j}^{2}}}\Gamma\bigl(\beta_{j}^{2} \bigr) \biggr)^{\frac{1}{\mu _{j}}} \\ &{}\times \biggl(\int_{t_{j-1}}^{t_{j}}e^{\nu_{j} (\int _{t_{j-1}}^{s}p(\xi)\,d\xi-s )}\,ds \biggr)^{\frac{1}{\nu_{j}}} \biggr\} \biggl\{ (1+\gamma_{k})\int _{t_{k-1}}^{t_{k}}p(s)h(s) e^{\int_{s}^{t_{k}}p(\xi)\,d\xi}\,ds \\ &{}+ \frac{\alpha_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{\mu_{k}t_{k}}}{ \mu_{k}^{\beta_{k}^{2}}}\Gamma\bigl( \beta_{k}^{2}\bigr) \biggr)^{\frac{1}{\mu _{k}}} \\ &{}\times\biggl(\int _{t_{k-1}}^{t_{k}}e^{-\nu_{k}s} \biggl\{ \int _{t_{k-1}}^{s}p(v)h(v)e^{\int_{v}^{s}p(\xi)\,d\xi}\,dv \biggr\} ^{\nu _{k}}\,ds \biggr)^{\frac{1}{\nu_{k}}} \\ &{}+ \gamma_{k}h(t_{k})+\frac{\alpha_{k}}{\Gamma(\beta_{k})} \biggl( \frac {e^{\mu_{k}t_{k}}}{ \mu_{k}^{\beta_{k}^{2}}}\Gamma\bigl(\beta_{k}^{2}\bigr) \biggr)^{\frac{1}{\mu _{k}}} \biggl(\int_{t_{k-1}}^{t_{k}}e^{-\nu_{k}s}h^{\nu_{k}}(s)\,ds \biggr)^{\frac{1}{\nu_{k}}} \biggr\} \biggr] \biggr\} e^{\int_{t_{l}}^{t} p(\xi)\,d\xi} \\ &{}+ \int_{t_{l}}^{t}p(s)h(s)e^{\int_{s}^{t}p(\xi)\,d\xi}, \end{aligned}$$
    (2.25)

    where \(\mu_{k}=\beta_{k}+1\) and \(\nu_{k}=1+\frac{1}{\beta _{k}}\),

  2. (ii)

    \(\beta_{k}>\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} m(t) \leq& h(t)+ \biggl\{ \sum_{t_{0}< t_{k}< t} \biggl[\prod_{t_{k}< t_{j}< t} \biggl\{ (1+\gamma_{j})e^{\int_{t_{j-1}}^{t_{j}}p(\xi)\,d\xi }+ \frac{\alpha_{j}}{\Gamma(\beta_{j})} \biggl(\frac{e^{2t_{j}}}{ 2^{2\beta_{j}-1}}\Gamma(2\beta_{j}-1) \biggr)^{\frac{1}{2}} \\ &{}\times \biggl(\int_{t_{j-1}}^{t_{j}}e^{2 (\int _{t_{j-1}}^{s}p(\xi)\,d\xi-s )}\,ds \biggr)^{\frac{1}{2}} \biggr\} \biggl\{ (1+\gamma_{k})\int _{t_{k-1}}^{t_{k}}p(s)h(s) e^{\int_{s}^{t_{k}}p(\xi)\,d\xi}\,ds \\ &{}+ \frac{\alpha_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{2t_{k}}}{ 2^{2\beta_{k}-1}}\Gamma(2\beta_{k}-1) \biggr)^{\frac{1}{2}} \\ &{}\times \biggl(\int_{t_{k-1}}^{t_{k}}e^{-2s} \biggl\{ \int_{t_{k-1}}^{s}p(v)h(v)e^{\int _{v}^{s}p(\xi)\,d\xi}\,dv \biggr\} ^{2}\,ds \biggr)^{\frac{1}{2}}+ \gamma_{k}h(t_{k}) \\ &{}+\frac{\alpha_{k}}{\Gamma(\beta_{k})} \biggl( \frac{e^{2t_{k}}}{ 2^{2\beta_{k}-1}}\Gamma(2\beta_{k}-1) \biggr)^{\frac{1}{2}} \biggl( \int_{t_{k-1}}^{t_{k}}e^{-2s}h^{2}(s)\,ds \biggr)^{\frac{1}{2}} \biggr\} \biggr] \biggr\} e^{\int_{t_{l}}^{t} p(\xi)\,d\xi} \\ &{}+ \int_{t_{l}}^{t}p(s)h(s)e^{\int_{s}^{t}p(\xi)\,d\xi}. \end{aligned}$$
    (2.26)

Proof

Setting

$$g(t)=\int_{t_{0}}^{t}p(s)m(s)\,ds+\sum _{t_{0}< t_{k}< t}\gamma_{k}m(t_{k})+\sum _{t_{0}< t_{k}< t}\frac{\alpha_{k}}{\Gamma(\beta_{k})}\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}m(s)\,ds, $$

and using the fact that \(m(t)\leq h(t)+g(t)\), we have

$$ \textstyle\begin{cases} g'(t)\leq p(t)g(t)+p(t)h(t), \quad t\neq t_{k},\quad\quad g(t_{0})=0,\\ g(t_{k}^{+})=(1+\gamma_{k})g(t_{k})+\frac{\alpha_{k}}{\Gamma (\beta_{k})} \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}g(s)\,ds+\gamma _{k}h(t_{k})\\ \hphantom{g(t_{k}^{+})=}{} +\frac{\alpha_{k}}{\Gamma(\beta_{k})} \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\beta_{k}-1}h(s)\,ds. \end{cases} $$

Applying Theorem 2.2 for \(0<\beta_{k}\leq 1/2\) and \(\beta_{k}>1/2\) together with \(m(t)\leq h(t)+g(t)\), we then obtain the estimates in (2.25) and (2.26), respectively. □

3 Applications to impulsive fractional integral jump conditions

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

Definition 3.1

The Riemann-Liouville fractional integral of order \(\beta>0\) of a function \(f: (t_{0},\infty)\rightarrow{\mathbb{R}}\) is defined by

$$I^{\beta}_{t_{0}}f(t)=\frac{1}{\Gamma(\beta)}\int _{t_{0}}^{t}(t-s)^{\beta-1}f(s)\,ds, $$

provided the right-hand side is point-wise defined on \((t_{0},\infty)\), where Γ is the Gamma function.

We apply our results to work out the maximum principle of the impulsive differential inequality.

Proposition 3.2

Assume that \(x\in PC^{1}[J,\mathbb{R}]\) satisfies

$$ \textstyle\begin{cases} x'(t)-Mx(t)+a(t)\leq0, & t\neq t_{k}, t\in J=[0, T],\\ x(t_{k}^{+})\leq c_{k}I_{t_{k-1}}^{\beta_{k}}x(t_{k})-b_{k},& k=1,2,\ldots,n,\\ x(0)=x(T)+\lambda, \end{cases} $$
(3.1)

where \(M>0\), \(a\in C[\mathbb{R}_{+},\mathbb{R}_{+}]\), \(0=t_{0}< t_{1}< t_{2}<\cdots<t_{n}<t_{n+1}=T\), \(b_{k},c_{k}\geq0\), \(\beta_{k}>0\), \(k=1,2,\ldots,n\), and λ are constants.

Suppose in addition that:

Case I: \(M\neq1\).

  1. (i)

    \(0<\beta_{k}\leq\frac{1}{2}\) for \(k=1,2,\ldots,n\),

    $$\begin{aligned} \mbox{(Q$_{1}$)} \quad& \prod_{k=1}^{n} \frac{c_{k}}{\Gamma(\beta _{k})} \biggl(\frac{\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl( \frac{1-e^{\nu_{k}(t_{k}-t_{k-1})(1-M)}}{ \nu_{k}(M-1)} \biggr)^{\frac{1}{\nu_{k}}}< e^{-MT}, \\ \mbox{(Q$_{2}$)} \quad& \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int_{t_{k-1}}^{t_{k}}e^{\nu_{k}s(M-1)} \biggl\{ -\int_{t_{k-1}}^{s}a(v)e^{-Mv}\,dv \biggr\} ^{\nu_{k}}\,ds \biggr)^{\frac{1}{\nu_{k}}}\\ &\quad\leq b_{k}, \\ \mbox{(Q$_{3}$)} \quad& \lambda\leq\int_{t_{n}}^{T}a(s)e^{M(T-s)}\,ds, \end{aligned}$$

    where \(\mu_{k}=\beta_{k}+1\) and \(\nu_{k}=1+\frac{1}{\beta _{k}}\),

  2. (ii)

    \(\beta_{k}>\frac{1}{2}\) for \(k=1,2,\ldots,n\),

    $$\begin{aligned} \mbox{(Q$_{4}$)} \quad& \prod_{k=1}^{n} \frac{c_{k}}{2^{\beta _{k}}\Gamma(\beta_{k})} \biggl(\frac{\Gamma(2\beta_{k}-1)}{M-1} \bigl[1-e^{2(t_{k}-t_{k-1})(1-M)} \bigr] \biggr)^{\frac{1}{2}}< e^{-MT}, \\ \mbox{(Q$_{5}$)} \quad& \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{2t_{k}}\Gamma(2\beta_{k}-1)}{2^{2\beta_{k}-1}} \biggr)^{\frac {1}{2}} \biggl(\int_{t_{k-1}}^{t_{k}}e^{2s(M-1)} \biggl\{ -\int_{t_{k-1}}^{s}a(v)e^{-Mv}\,dv \biggr\} ^{2}\,ds \biggr)^{\frac {1}{2}} \\ &\quad\leq b_{k}, \end{aligned}$$

    and (Q3) holds.

Case II: \(M=1\).

  1. (i)

    \(0<\beta_{k}\leq\frac{1}{2}\) for \(k=1,2,\ldots,n\),

    $$\begin{aligned} \mbox{(Q$_{6}$)} \quad& \prod_{k=1}^{n} \frac{c_{k}}{\Gamma(\beta _{k})} \biggl(\frac{\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} (t_{k}-t_{k-1} )^{\frac{1}{\nu_{k}}}< e^{-T} , \\ \mbox{(Q$_{7}$)} \quad&\frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int_{t_{k-1}}^{t_{k}} \biggl\{ -\int_{t_{k-1}}^{s}a(v)e^{-v}\,dv \biggr\} ^{\nu_{k}}\,ds \biggr)^{\frac{1}{\nu_{k}}}\leq b_{k} , \\ \mbox{(Q$_{8}$)} \quad& \lambda\leq\int_{t_{n}}^{T}a(s)e^{(T-s)}\,ds , \end{aligned}$$
  2. (ii)

    \(\beta_{k}>\frac{1}{2}\) for \(k=1,2,\ldots,n\),

    $$\begin{aligned} \mbox{(Q$_{9}$)} \quad& \prod_{k=1}^{n} \frac{c_{k}}{\Gamma(\beta _{k})} \biggl(\frac{\Gamma(2\beta_{k}-1)(t_{k}-t_{k-1})}{2^{2\beta _{k}-1}} \biggr)^{\frac{1}{2}}< e^{-T} , \\ \mbox{(Q$_{10}$)} \quad&\frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{2t_{k}}\Gamma(2\beta_{k}-1)}{2^{2\beta_{k}-1}} \biggr)^{\frac {1}{2}} \biggl(\int_{t_{k-1}}^{t_{k}} \biggl\{ -\int_{t_{k-1}}^{s}a(v)e^{-v}\,dv \biggr\} ^{2}\,ds \biggr)^{\frac {1}{2}}\leq b_{k}, \end{aligned}$$

    and (Q 8) holds.

Then \(x(t)\leq0\) for \(t\in[0, T]\).

Proof

To prove Case I(i), applying Corollary 2.3 for \(t\in[0, T]\), we have

$$x(t) \leq x(0) \biggl(\prod_{t_{0}< t_{k}< t}A_{k}^{*} \biggr)e^{Mt}+\sum_{t_{0}< t_{k}< t} \biggl(\prod _{t_{k}< t_{j}< t}A_{j}^{*}B_{k}^{*}e^{M(t-t_{k})} \biggr) - \int_{t_{l}}^{t}a(s)e^{M(t-s)}\,ds, $$

where

$$\begin{aligned}& A_{k}^{*} = \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{\Gamma(\beta _{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\frac{1-e^{\nu_{k}(t_{k}-t_{k-1})(1-M)}}{\nu_{k}(M-1)} \biggr)^{\frac{1}{\nu_{k}}}, \\& B_{k}^{*} = \frac{c_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma (\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\int_{t_{k-1}}^{t_{k}} e^{\nu_{k}(M-1)s} \biggl\{ -\int_{t_{k-1}}^{s}a(v)e^{-M v}\,dv \biggr\} ^{\nu _{k}}\,ds \biggr)^{\frac{1}{\nu_{k}}}-b_{k}. \end{aligned}$$

It is easy to see that \(A_{k}^{*}\geq0\) for all \(k=1,2,\ldots, n\). The condition (Q2) implies that \(B_{k}^{*}\leq0\) for all \(k=1,2,\ldots, n\). Then it is sufficient to show that \(x(0)\leq0\). For \(t=T\), we have

$$x(T) \leq x(0) \Biggl(\prod_{k=1}^{n}A_{k}^{*} \Biggr)e^{MT}+\sum_{t_{0}< t_{k}< T} \biggl(\prod _{t_{k}< t_{j}< T}A_{j}^{*}B_{k}^{*}e^{M(T-t_{k})} \biggr) - \int_{t_{n}}^{T}a(s)e^{M(T-s)}\,ds. $$

Using the conditions (Q1) and (Q3), we see that

$$\begin{aligned} x(0) \Biggl[1- \Biggl(\prod_{k=1}^{n}A_{k}^{*} \Biggr)e^{MT} \Biggr] \leq& \lambda+\sum _{t_{0}< t_{k}< T} \biggl(\prod_{t_{k}< t_{j}< T}A_{j}^{*}B_{k}^{*}e^{M(T-t_{k})} \biggr)-\int_{t_{n}}^{T}a(s)e^{M(T-s)}\,ds \\ \leq& 0, \end{aligned}$$

which implies that \(x(0)\leq0\).

Using a similar method to prove Case I(i) with suitable conditions, we deduce that \(x(0)\leq0\). This completes the proof. □

The last application shows the boundedness of solution of impulsive differential equation with Riemann-Liouville fractional integral jump conditions.

Proposition 3.3

Let \(x\in PC^{1}[\mathbb{R}_{+},\mathbb{R}]\) such that

$$ \textstyle\begin{cases} x'(t)=f(t,x(t)),& t\neq t_{k}, t\in[t_{0}, \infty),\\ \Delta x(t_{k})= Z_{k} (I_{t_{k-1}}^{\beta _{k}}x(t_{k}) ), & k=1,2,\ldots,\\ x(t_{0})=x_{0}, \end{cases} $$
(3.2)

where \(f\in C(\mathbb{R}_{+}\times\mathbb{R},\mathbb{R})\), \(Z_{k}\in C(\mathbb{R},\mathbb{R})\), \(0\leq t_{0}< t_{1}< t_{2}<\cdots\) , \(\lim_{k\rightarrow\infty}t_{k}=\infty\), \(\Delta x(t_{k})=x(t_{k}^{+})-x(t_{k})\), \(\beta_{k}>0\), \(k=1,2,\ldots\) , and \(x_{0}\) are constants.

Assume that:

(Q11):

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

$$\bigl|f\bigl(t,x(t)\bigr)\bigr|\leq N \bigl|x(t)\bigr|\quad \textit{for }t\geq t_{0}, $$
(Q12):

there exist constants \(L_{k}\geq0\) such that

$$\bigl|Z_{k}(x)\bigr|\leq L_{k}|x|,\quad x\in\mathbb{R}, k=1,2, \ldots. $$

Then, for \(t\geq t_{0}\), the following inequalities hold:

Case I: \(N\neq1\).

  1. (i)

    \(0<\beta_{k}\leq\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} \bigl|x(t)\bigr| \leq&|x_{0}|\prod_{t_{0}< t_{k}< t} \biggl\{ 1+\frac{L_{k}}{\Gamma(\beta _{k})} \biggl(\frac{\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \biggl(\frac{1-e^{\nu_{k}(t_{k}-t_{k-1})(1-N)}}{\nu_{k}(N-1)} \biggr)^{\frac{1}{\nu_{k}}} \biggr\} \\ &{}\times e^{N(t-t_{0})}, \end{aligned}$$
    (3.3)

    where \(\mu_{k}=\beta_{k}+1\) and \(\nu_{k}=1+\frac{1}{\beta _{k}}\),

  2. (ii)

    \(\beta_{k}>\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$\begin{aligned} \bigl|x(t)\bigr| \leq&|x_{0}|\prod_{t_{0}< t_{k}< t} \biggl\{ 1+\frac{L_{k}}{2^{\beta _{k}}\Gamma(\beta_{k})} \biggl(\frac{\Gamma(2\beta_{k}-1)}{N-1} \bigl[1-e^{2(t_{k}-t_{k-1})(1-N)} \bigr] \biggr)^{\frac{1}{2}} \biggr\} \\ &{}\times e^{N(t-t_{0})}. \end{aligned}$$
    (3.4)

Case II: \(N= 1\).

  1. (i)

    \(0<\beta_{k}\leq\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$ \bigl|x(t)\bigr|\leq|x_{0}|\prod_{t_{0}< t_{k}< t} \biggl\{ 1+\frac{L_{k}}{\Gamma(\beta _{k})} \biggl(\frac{\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} (t_{k}-t_{k-1} )^{\frac{1}{\nu_{k}}} \biggr\} e^{(t-t_{0})}, $$
    (3.5)
  2. (ii)

    \(\beta_{k}>\frac{1}{2}\) for \(k=1,2,\ldots\) ,

    $$ \bigl|x(t)\bigr|\leq|x_{0}|\prod_{t_{0}< t_{k}< t} \biggl\{ 1+\frac{L_{k}}{\Gamma(\beta _{k})} \biggl(\frac{\Gamma(2\beta_{k}-1)(t_{k}-t_{k-1})}{2^{2\beta_{k}-1}} \biggr)^{\frac{1}{2}} \biggr\} e^{(t-t_{0})}. $$
    (3.6)

Proof

The solution \(x(t)\) of problem (3.2) satisfies the impulsive integral equation

$$x(t)=x(t_{0})+\int_{t_{0}}^{t}f \bigl(s,x(s)\bigr)\,ds+\sum_{t_{0}< t_{k}< t}Z_{k} \bigl(I_{t_{k-1}}^{\beta_{k}}x(t_{k}) \bigr). $$

From conditions (Q11)-(Q12), it follows for \(t\geq t_{0}\) that

$$\begin{aligned} \bigl|x(t)\bigr| \leq&|x_{0}|+\int_{t_{0}}^{t}\bigl|f \bigl(s,x(s)\bigr)\bigr|\,ds+\sum_{t_{0}< t_{k}< t}\bigl\vert Z_{k} \bigl(I_{t_{k-1}}^{\beta_{k}}x(t_{k}) \bigr)\bigr\vert \\ \leq&|x_{0}|+\int_{t_{0}}^{t}N\bigl|x(s)\bigr|\,ds+ \sum_{t_{0}< t_{k}< t}L_{k}I_{t_{k-1}}^{\beta_{k}}|x|(t_{k}). \end{aligned}$$

Hence Theorem 2.5 yields the estimate

$$\begin{aligned} \bigl|x(t)\bigr| \leq& |x_{0}|\prod_{t_{0}< t_{k}< t} \biggl\{ e^{N(t_{k}-t_{k-1})}+\frac {L_{k}}{\Gamma(\beta_{k})} \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma(\beta _{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \\ &{}\times \biggl(\int_{t_{k-1}}^{t_{k}}e^{\nu_{k} (N(s-t_{k-1})-s )}\,ds \biggr)^{\frac{1}{\nu_{k}}} \biggr\} e^{N(t-t_{l})} \\ =&|x_{0}|\prod_{k=1}^{l} \biggl\{ e^{N(t_{k}-t_{k-1})}+\frac{L_{k}}{\Gamma (\beta_{k})} \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma(\beta_{k}^{2})}{\mu _{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \\ &{}\times \biggl(\frac{e^{\nu_{k}(N(t_{k}-t_{k-1})-t_{k})}-e^{-\nu _{k}t_{k-1}}}{\nu_{k}(N-1)} \biggr)^{\frac{1}{\nu_{k}}} \biggr\} e^{N(t-t_{l})} \\ =&|x_{0}|\prod_{k=1}^{l} \biggl\{ e^{N(t_{k}-t_{k-1})}+\frac{L_{k}}{\Gamma (\beta_{k})} \biggl(\frac{e^{\mu_{k}t_{k}}\Gamma(\beta_{k}^{2})}{\mu _{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \\ &{}\times \biggl(\frac{e^{\nu_{k}(N(t_{k}-t_{k-1})-t_{k})} (1-e^{\nu _{k}(t_{k}-t_{k-1})(1-N)} )}{\nu_{k}(N-1)} \biggr)^{\frac{1}{\nu _{k}}} \biggr\} e^{N(t-t_{l})} \\ =&|x_{0}|\prod_{k=1}^{l} \biggl\{ e^{N(t_{k}-t_{k-1})}+e^{N(t_{k}-t_{k-1})}\frac{L_{k}}{\Gamma(\beta _{k})} \biggl( \frac{\Gamma(\beta_{k}^{2})}{\mu_{k}^{\beta_{k}^{2}}} \biggr)^{\frac{1}{\mu_{k}}} \\ &{}\times \biggl(\frac{1-e^{\nu_{k}(t_{k}-t_{k-1})(1-N)}}{\nu _{k}(N-1)} \biggr)^{\frac{1}{\nu_{k}}} \biggr\} e^{N(t-t_{l})},\quad t\geq t_{0}. \end{aligned}$$

Therefore, inequality (3.3) holds for \(t\geq t_{0}\). For the other cases the proofs are similar and thus omitted. The proof is completed. □