Open Access

Impulsive inequalities with nonlocal jumps and their applications to impulsive fractional integral conditions

  • Walwadee Liengtragulngam1,
  • Phollakrit Thiramanus2,
  • Sotiris K Ntouyas3, 4 and
  • Jessada Tariboon2Email author
Journal of Inequalities and Applications20152015:189

https://doi.org/10.1186/s13660-015-0706-4

Received: 11 December 2014

Accepted: 20 May 2015

Published: 12 June 2015

Abstract

In this paper we establish some impulsive differential and integral inequalities with nonlocal jumps. Two applications to impulsive differential and integral inequalities with Riemann-Liouville fractional integral jump conditions are given.

Keywords

impulsive differential inequality impulsive integral inequality nonlocal jump condition Riemann-Liouville fractional integral

MSC

34A37 34A40 34A08

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 [127] 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 [2836]). 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. □

Declarations

Acknowledgements

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-GEN-57-44.

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

Authors’ Affiliations

(1)
Department of Social and Applied Science, College of Industrial Technology, King Mongkut’s University of Technology North Bangkok
(2)
Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok
(3)
Department of Mathematics, University of Ioannina
(4)
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University

References

  1. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) MATHView ArticleGoogle Scholar
  2. Bainov, DD, Simeonov, PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow (1993) MATHGoogle Scholar
  3. Bainov, DD, Simeonov, PS: Impulsive Differential Equations: Asymptotic Properties of the Solutions. World Scientific, Singapore (1995) MATHGoogle Scholar
  4. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) MATHGoogle Scholar
  5. Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions, vol. 2. Hindawi Publishing Corporation, New York (2006) MATHView ArticleGoogle Scholar
  6. Bainov, DD, Hristova, SG: The method of quasilinearization for the periodic boundary value problem for systems of impulsive differential equations. Appl. Math. Comput. 117, 73-85 (2001) MATHMathSciNetView ArticleGoogle Scholar
  7. Bonotto, EM, Gimenes, LP, Federson, M: Oscillation for a second-order neutral equation with impulses. Appl. Math. Comput. 215, 1-15 (2009) MATHMathSciNetView ArticleGoogle Scholar
  8. Cui, BT, Han, M, Yang, H: Some sufficient conditions for oscillation of impulsive delay hyperbolic systems with Robin boundary conditions. J. Comput. Appl. Math. 180, 365-375 (2005) MATHMathSciNetView ArticleGoogle Scholar
  9. Ding, W, Xing, Y, Han, M: Anti-periodic boundary value problems for first order impulsive functional differential equations. Appl. Math. Comput. 186, 45-53 (2007) MATHMathSciNetView ArticleGoogle Scholar
  10. Franco, D, Nieto, JJ: Maximum principles for periodic impulsive first order problems. J. Comput. Appl. Math. 88, 149-159 (1998) MATHMathSciNetView ArticleGoogle Scholar
  11. Fu, X, Zhang, L: Forced oscillation for impulsive hyperbolic boundary value problems with delay. Appl. Math. Comput. 158, 761-780 (2004) MATHMathSciNetView ArticleGoogle Scholar
  12. Gimenes, LP, Federson, M: Oscillation by impulses for a second-order delay differential equation. Comput. Math. Appl. 52, 819-828 (2006) MATHMathSciNetView ArticleGoogle Scholar
  13. He, Z, Ge, W: Oscillations of second-order nonlinear impulsive ordinary differential equations. J. Comput. Appl. Math. 158, 397-406 (2003) MATHMathSciNetView ArticleGoogle Scholar
  14. He, Z, He, X: Periodic boundary value problems for first order impulsive integro-differential equations of mixed type. J. Math. Anal. Appl. 296, 8-20 (2004) MATHMathSciNetView ArticleGoogle Scholar
  15. Huang, M: Oscillation criteria for second order nonlinear dynamic equations with impulses. Comput. Math. Appl. 59, 31-41 (2010) MATHMathSciNetView ArticleGoogle Scholar
  16. Jiao, J, Chen, L, Li, L: Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations. J. Math. Anal. Appl. 337, 458-463 (2008) MATHMathSciNetView ArticleGoogle Scholar
  17. Li, J: Periodic boundary value problems for second-order impulsive integro-differential equations. Appl. Math. Comput. 198, 317-325 (2008) MATHMathSciNetView ArticleGoogle Scholar
  18. Li, Q, Liang, H, Zhang, Z, Yu, Y: Oscillation of second order self-conjugate differential equation with impulses. J. Comput. Appl. Math. 197, 78-88 (2006) MATHMathSciNetView ArticleGoogle Scholar
  19. Li, J, Shen, J: Periodic boundary value problems for delay differential equations with impulses. J. Comput. Appl. Math. 193, 563-573 (2006) MATHMathSciNetView ArticleGoogle Scholar
  20. Liu, H, Li, Q: Asymptotic behavior of second-order impulsive differential equations. Electron. J. Differ. Equ. 2011, 33 (2011) View ArticleGoogle Scholar
  21. Luo, J: Oscillation of hyperbolic partial differential equations with impulses. Appl. Math. Comput. 133, 309-318 (2002) MATHMathSciNetView ArticleGoogle Scholar
  22. Nieto, JJ, Rodriguez-Lopez, R: New comparison results for impulsive integro-differential equations and applications. J. Math. Anal. Appl. 328, 1343-1368 (2007) MATHMathSciNetView ArticleGoogle Scholar
  23. Peng, M: Oscillation theorems of second-order nonlinear neutral delay difference equations with impulses. Comput. Math. Appl. 44, 741-748 (2002) MATHMathSciNetView ArticleGoogle Scholar
  24. Shen, J: New maximum principles for first-order impulsive boundary value problems. Appl. Math. Lett. 16, 105-112 (2003) MATHMathSciNetView ArticleGoogle Scholar
  25. Stamova, IM: Lyapunov method for boundedness of solutions of nonlinear impulsive functional differential equations. Appl. Math. Comput. 177, 714-719 (2006) MATHMathSciNetView ArticleGoogle Scholar
  26. Zhang, C, Feng, W, Yang, J, Huang, M: Oscillations of second order impulses nonlinear FDE with forcing term. Appl. Math. Comput. 198, 271-279 (2008) MATHMathSciNetView ArticleGoogle Scholar
  27. Akça, H, Covachev, V, Covacheva, Z: Improved stability estimates for impulsive delay reaction-diffusion Cohen-Grossberg neural networks. Tatra Mt. Math. Publ. 54, 1-18 (2013) MATHMathSciNetGoogle Scholar
  28. Deng, S, Prather, C: Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay. J. Inequal. Pure Appl. Math. 9(2), Article ID 34 (2008) MathSciNetGoogle Scholar
  29. Gallo, A, Piccirillo, AM: Multidimensional impulse inequalities and general Bahari type inequalities for discontinuous functions with delay. Nonlinear Stud. 19(1), 13-24 (2012) Google Scholar
  30. Hristova, SG: Nonlinear delay integral inequalities for piecewise continuous functions and applications. J. Inequal. Pure Appl. Math. 5(4), Article ID 88 (2004) MathSciNetGoogle Scholar
  31. Li, J: On some new impulsive integral inequalities. J. Inequal. Appl. 2008, Article ID 312395 (2008) Google Scholar
  32. Mi, Y: Some generalized Gronwall-Bellman type impulsive integral inequalities and their applications. J. Appl. Math. 2014, Article ID 353210 (2014) MathSciNetView ArticleGoogle Scholar
  33. Peng, Y, Kang, Y, Yuan, M, Huang, R, Yang, L: Gronwall-type integral inequalities with impulses on time scales. Adv. Differ. Equ. 2011, 26 (2011) MathSciNetView ArticleGoogle Scholar
  34. Tatar, NE: An impulsive nonlinear singular version of the Gronwall-Bihari inequality. J. Inequal. Appl. 2006, Article ID 84561 (2006) MathSciNetView ArticleGoogle Scholar
  35. Wang, H, Ding, C: A new nonlinear impulsive delay differential inequality and its applications. J. Inequal. Appl. 2011, 11 (2011) MathSciNetView ArticleGoogle Scholar
  36. Wang, X: A generalized Halanay inequality with impulse and delay. Adv. Devel. Technol. Inter. 1(2), 9-23 (2012) Google Scholar
  37. Thiramanus, P, Tariboon, J: Impulsive differential and impulsive integral inequalities with integral jump conditions. J. Inequal. Appl. 2012, 25 (2012) MathSciNetView ArticleGoogle Scholar
  38. Herrmann, R: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011) View ArticleGoogle Scholar

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