1 Introduction

In recent years, the theory of impulsive differential systems has been attracting the attention of many mathematicians, and the interest in the subject is still growing. This is partly due to broad applications of it in many areas including threshold theory in biology, ecosystems management and orbital transfer of satellite, see [1]. One effective method for investigating the properties of solutions to impulsive differential systems is related to the integral inequalities for discontinuous functions (integro-sum inequalities). Up to now, a lot of integro-sum inequalities (for example, [2–18] and the references therein) have been discovered. For example, in 2003, Borysenko [3] considered the following integro-sum inequality:

$$x(t)\leq a(t)+ \int_{t_{0}}^{t}q(\tau)x^{m}(\tau)\,\mathrm {d}\tau+ \sum _{t_{0}< t_{i}< t}\beta_{i}x^{m}(t_{i}-0),\quad m>0, m\neq1. $$

In 2009, Gallo and Piccirillo [8] further discussed the following nonlinear integro-sum inequality:

$$\begin{aligned} x(t)\leq c(t)+h(t) \int_{t_{0}}^{t}f(s)w\bigl(x\bigl(b(s)\bigr)\bigr) \,\mathrm {d}s+ \sum_{t_{0}< t_{i}< t}\beta_{i}x^{m}(t_{i}-0),\quad m>0. \end{aligned}$$

In 2012, Wang et al. [17] considered the nonlinear integro-sum inequality as follows:

$$\begin{aligned} x^{m}(t) \leq& c(t)+2 \int_{\alpha(t_{0})}^{\alpha(t)}\bigl[M_{1}f_{1}(t,s)u ^{\frac{m}{2}}(s)+N_{1}g_{1}(t,s)u^{m}(s)\bigr] \,\mathrm {d}s \\ &{}+2 \int_{t_{0}}^{t}\bigl[M_{2}f_{2}(t,s)u^{\frac{m}{2}}(s)+N_{2}g_{2}(t,s)u ^{m}(s)\bigr]\,\mathrm {d}s+ \sum_{t_{0}< t_{i}< t} \beta_{i}x(t_{i}-0), \quad m>0. \end{aligned}$$

Very recently, in 2016, Zheng et al. [18] considered the following nonlinear integro-sum inequality under the condition \(p>q>0\):

$$\begin{aligned} x^{p}(t) \leq& a_{0}(t)+\frac{p-q}{p}\sum _{i=1}^{N} \int_{t_{0}}^{t}g _{i}(s) x^{q} \bigl(\phi_{i}(s)\bigr)\,\mathrm {d}s \\ &{}+\sum_{j=1}^{L} \int_{t_{0}}^{t}b_{j}(s) \int_{t_{0}}^{s}c_{j}(\theta ) x^{q}\bigl(w_{j}(s)\bigr)\,\mathrm {d}\theta \,\mathrm {d}s+\sum _{t_{0}< t_{i}< t} \beta_{i}x^{m}(t_{i}-0). \end{aligned}$$

Motivated by [3, 8, 17, 18], in this paper, we investigate some new integro-sum inequality with mixed nonlinearities under the condition \(p>0\), \(q>0\) (\(p\neq q\)):

$$\begin{aligned} x^{p}(t) \leq& a(t)+ \int_{t_{0}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t_{0}}^{t}f_{2}(s) \int_{t_{0}}^{s}g_{1}(\tau)x^{p}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s \\ &{}+ \int_{t_{0}}^{t}f_{3}(s) \int_{t_{0}}^{s}g_{2}(\tau)x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s +c(t)\sum_{t_{0}< t_{i}< t} \beta_{i}x^{m}(t _{i}-0) \end{aligned}$$

and the more general form

$$\begin{aligned} x^{p}(t) \leq& a(t)+ \int_{t_{0}}^{t}f(s)x^{q}(s)\,\mathrm {d}s+\sum _{j=1} ^{L} \int_{t_{0}}^{t}b_{j}(s) \int_{t_{0}}^{s}g_{j}(\tau)x^{p}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s \\ &{}+\sum_{k=1}^{M} \int_{t_{0}}^{t}c_{k}(s) \int_{t_{0}}^{s}\theta_{k}( \tau)x^{q}(\tau)\,\mathrm {d}\tau \,\mathrm {d}s +d(t)\sum _{t_{0}< t_{i}< t} \beta_{i}x^{m}(t_{i}-0). \end{aligned}$$

We also discuss some nonlinear integro-sum inequality with positive and negative coefficients under the condition \(0< q< p< r\):

$$\begin{aligned} x^{p}(t) \leq& a(t)+b(t) \int_{t_{0}}^{t} \bigl[f(s)x^{p}(s)+g(s)x^{q}(s)-h(s)x ^{r}(s) \bigr]\,\mathrm {d}s \\ &{}+c(t)\sum_{t_{0}< t_{i}< t}\beta_{i}x^{m}(t_{i}-0), \end{aligned}$$

and the more general form under the condition \(0< q_{j}< p< r_{j}\) (\(j=1,2,\ldots, L\)):

$$\begin{aligned} x^{p}(t) \leq& a(t)+ \int_{t_{0}}^{t}f(s)x^{p}(s)\,\mathrm {d}s+\sum _{j=1} ^{L} \int_{t_{0}}^{t}g_{j}(s)x^{q_{j}}(s) \,\mathrm {d}s-\sum_{j=1}^{L} \int_{t_{0}}^{t}h_{j}(s)x^{r_{j}}(s) \,\mathrm {d}s \\ &{}+c(t)\sum_{t_{0}< t_{i}< t}\beta_{i}x^{m}(t_{i}-0). \end{aligned}$$

Based on these inequalities, we provide explicit bounds for unknown functions concerned and then apply the results to research the qualitative properties of solutions of certain impulsive differential equations.

2 Preliminaries

Throughout the present paper, R denotes the set of real numbers; \(\mathrm {R}_{+}=[0,+\infty)\) is the subset of R; \(C(D, E)\) denotes the class of all continuous functions defined on the set D with range in the set E.

Lemma 2.1

([19])

Assume that the following conditions for \(t\geq t_{0}\) hold:

  1. (i)

    \(x_{0}\) is a nonnegative constant,

  2. (ii)
    $$x(t)\leq x_{0}+ \int_{t_{0}}^{t} \bigl[e(s) x(s) +l(s) x^{\alpha}(s) \bigr]\,\mathrm {d}s, $$

    where x, e and l are nonnegative continuous functions and \(\alpha\neq1\) is a positive constant.

If

$$1+(1-\alpha)x_{0}^{(\alpha-1)} \int_{t_{0}}^{t} l(s)\exp \biggl(( \alpha-1) \int_{t_{0}}^{s}e(\tau)\,\mathrm {d}\tau \biggr) \,\mathrm {d}s>0 $$

holds, then

$$\begin{aligned} x(t)&\leq x_{0}\exp \biggl( \int_{t_{0}}^{t}e(s)\,\mathrm {d}s \biggr) \\ &\quad{}\times \biggl\{ 1+(1- \alpha)x_{0}^{(\alpha-1)} \int_{t_{0}}^{t} l(s)\exp \biggl((\alpha-1) \int_{t_{0}}^{s}e(\tau)\,\mathrm {d}\tau \biggr)\,\mathrm {d}s \biggr\} ^{\frac{1}{1- \alpha}}, \quad t\geq t_{0}. \end{aligned} $$

Lemma 2.2

([20])

Let x be a nonnegative function, \(0< q< p< r\), \(c_{1}\geq0\), \(k_{2}\geq0\), \(c_{2}>0\) and \(k_{1}>0\). Then

$$c_{1}x^{q}-c_{2}x^{r} \leq(k_{1}-k_{2})x^{p}+\theta_{1}(p,q,c_{1},k _{1})+\theta_{2}(p,r,c_{2},k_{2}), $$

where

$$\theta_{1}(p,q,c_{1},k_{1}):=\frac{p-q}{q} \biggl(\frac{q}{p} \biggr)^{ \frac{p}{p-q}}{c_{1}^{\frac{p}{p-q}}} {k_{1}^{{\frac{-q}{p-q}}}}, \quad\quad \theta _{2}(p,r,c_{2},k_{2}):= \frac{r-p}{r} \biggl(\frac{p}{r} \biggr)^{ \frac{p}{r-p}}{c_{2}^{\frac{-p}{r-p}}} {k_{2}^{{\frac{r}{r-p}}}}. $$

3 Main results

Theorem 3.1

Suppose that x is a nonnegative piecewise continuous function defined on \([t_{0},\infty)\) with discontinuities of the first kind in the points \(t_{i}\) (\(i=1,2,\ldots\)) and satisfies the integro-sum inequality

$$\begin{aligned} \begin{aligned}[b] x^{p}(t) &\leq a(t)+ \int_{t_{0}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t_{0}}^{t}f_{2}(s) \int_{t_{0}}^{s}g_{1}(\tau)x^{p}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{} + \int_{t_{0}}^{t}f_{3}(s) \int_{t_{0}}^{s}g_{2}(\tau )x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s +c(t)\sum_{t_{0}< t_{i}< t} \beta_{i}x^{m}(t _{i}-0), \quad t\geq t_{0}, \end{aligned} \end{aligned}$$
(3.1)

where \(0\leq t_{0}< t_{1} < t_{2}<\cdots\) , \(\lim_{i\rightarrow\infty}t _{i}=\infty\), functions \(a(t)\geq0\) and \(c(t)\geq0\) are defined on \([t_{0},\infty)\), \(f_{1},f_{2},f_{3},g_{1},g_{2}\in C(\mathrm {R}_{+},\mathrm {R}_{+})\), \(\beta_{i}\geq0\) (\(i=1,2,\ldots\)), \(p>0\), \(q>0\), \(p\neq q\) and \(m>0\) are constants. If

$$1+\frac{p-q}{p}r_{i}^{\frac{q-p}{p}}(t) \int_{t_{i-1}}^{t} l(s)\exp \biggl(\frac{q-p}{p} \int_{t_{i-1}}^{s}e(\tau)\,\mathrm {d}\tau \biggr) \,\mathrm {d}s>0, \quad i=1,2,\ldots, $$

then, for \(t\geq t_{0}\), the following estimates hold:

$$\begin{aligned}& x(t)\leq v_{1}(t), \quad t\in[t_{0},t_{1}], \end{aligned}$$
(3.2)
$$\begin{aligned}& x(t)\leq v_{i}(t), \quad t\in(t_{i-1},t_{i}], i=2,3,\ldots, \end{aligned}$$
(3.3)

where

$$\begin{aligned}& \begin{aligned} v_{i}(t) &=r_{i}^{\frac{1}{p}}(t)\exp \biggl( \frac{1}{p} \int_{t_{i-1}} ^{t}e(s)\,\mathrm {d}s \biggr) \\ &\quad {}\times \biggl\{ 1+\frac{p-q}{p}r_{i}^{\frac{q-p}{p}}(t) \int_{t_{i-1}}^{t} l(s) \exp \biggl(\frac{q-p}{p} \int_{t_{i-1}}^{s}e(\tau)\,\mathrm {d}\tau \biggr)\,\mathrm {d}s \biggr\} ^{\frac{1}{p-q}},\quad i=1,2,\ldots, \end{aligned} \\& e(t)=f_{2}(t) \int_{t_{0}}^{t}g_{1}(\tau)\,\mathrm {d}\tau,\quad\quad l(t)= f_{1}(t)+f_{3}(t) \int_{t_{0}}^{t}g_{2}(\tau)\,\mathrm {d}\tau, \end{aligned}$$
(3.4)
$$\begin{aligned}& r_{1}(t)=\max_{t_{0}\leq\tau\leq t}\bigl\vert a( \tau)\bigr\vert ,\quad\quad h(t)= \max_{t_{0}\leq\tau\leq t}\bigl\vert c(\tau)\bigr\vert , \\& \begin{aligned} r_{i+1}(t) &=r_{i}(t) + \int_{t_{i-1}}^{t_{i}}f_{1}(s)v_{i}^{q}(s) \,\mathrm {d}s + \int_{t_{i-1}}^{t} \biggl(f_{2}(s) \int_{t_{i-1}}^{t_{i}}g _{1}(\tau)v_{i}^{p}( \tau)\,\mathrm {d}\tau \biggr)\,\mathrm {d}s \\ &\quad{} + \int_{t_{i-1}}^{t} \biggl(f_{3}(s) \int_{t_{i-1}}^{t_{i}}g_{2}(\tau)v _{i}^{q}(\tau)\,\mathrm {d}\tau \biggr)\,\mathrm {d}s +h(t) \beta_{i}v_{i} ^{m}(t_{i}-0),\quad i=1,2,\ldots. \end{aligned} \end{aligned}$$
(3.5)

Proof

From (3.1) and (3.5), we have, for \(t\in I_{0}=[t_{0},t _{1}]\),

$$\begin{aligned} \begin{aligned}[b] x^{p}(t) &\leq r_{1}(t)+ \int_{t_{0}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t_{0}}^{t}f_{2}(s) \int_{t_{0}}^{s}g_{1}(\tau)x^{p}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{} + \int_{t_{0}}^{t}f_{3}(s) \int_{t_{0}}^{s}g_{2}(\tau )x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s \end{aligned} \end{aligned}$$
(3.6)

and \(r_{1}(t)\) is non-decreasing on \([t_{0},\infty)\). Take any fixed \(T\in[t_{0},t_{1}]\), and for arbitrary \(t\in[t_{0},T]\), we have

$$\begin{aligned} \begin{aligned}[b] x^{p}(t) &\leq r_{1}(T)+ \int_{t_{0}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t_{0}}^{t}f_{2}(s) \int_{t_{0}}^{s}g_{1}(\tau)x^{p}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{} + \int_{t_{0}}^{t}f_{3}(s) \int_{t_{0}}^{s}g_{2}(\tau )x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s. \end{aligned} \end{aligned}$$
(3.7)

Let \(u(t)=x^{p}(t)\). Inequality (3.7) is equivalent to

$$\begin{aligned} \begin{aligned}[b] u(t) &\leq r_{1}(T)+ \int_{t_{0}}^{t}f_{1}(s)u^{\frac{q}{p}}(s) \,\mathrm {d}s+ \int_{t_{0}}^{t}f_{2}(s) \int_{t_{0}}^{s}g_{1}(\tau)u( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{} + \int_{t_{0}}^{t}f_{3}(s) \int_{t_{0}}^{s}g_{2}(\tau)u^{\frac{q}{p}}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s. \end{aligned} \end{aligned}$$
(3.8)

Let

$$\begin{aligned} \begin{aligned}[b] V(t) &=r_{1}(T)+ \int_{t_{0}}^{t}f_{1}(s)u^{\frac{q}{p}}(s) \,\mathrm {d}s+ \int_{t_{0}}^{t}f_{2}(s) \int_{t_{0}}^{s}g_{1}(\tau)u(\tau) \,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{} + \int_{t_{0}}^{t}f_{3}(s) \int_{t_{0}}^{s}g_{2}(\tau)u^{\frac{q}{p}}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s. \end{aligned} \end{aligned}$$
(3.9)

It follows from (3.8) and (3.9) that

$$ u(t)\leq V(t), \quad\quad V(t_{0})=r_{1}(T), $$
(3.10)

\(V(t)\) is non-decreasing and

$$\begin{aligned} V'(t)=f_{1}(t)u^{\frac{q}{p}}(t)+f_{2}(t) \int_{t_{0}}^{t}g_{1}(\tau)u( \tau)\,\mathrm {d}\tau+f_{3}(t) \int_{t_{0}}^{t}g_{2}(\tau)u^{ \frac{q}{p}}( \tau)\,\mathrm {d}\tau. \end{aligned}$$
(3.11)

Since \(V(t)\) is non-decreasing, from (3.11) we have

$$\begin{aligned} V'(t) \leq& f_{1}(t)V^{\frac{q}{p}}(t)+f_{2}(t) \int_{t_{0}}^{t}g_{1}( \tau)V(\tau)\,\mathrm {d}\tau+f_{3}(t) \int_{t_{0}}^{t}g_{2}(\tau)V ^{\frac{q}{p}}( \tau)\,\mathrm {d}\tau \\ \leq& f_{1}(t)V^{\frac{q}{p}}(t)+f_{2}(t) \int_{t_{0}}^{t}g_{1}( \tau)\,\mathrm {d}\tau V(t) +f_{3}(t) \int_{t_{0}}^{t}g_{2}(\tau) \,\mathrm {d}\tau V^{\frac{q}{p}}(t) \\ \leq& e(t) V(t) +l(t) V^{\frac{q}{p}}(t), \end{aligned}$$
(3.12)

where \(e(t)\) and \(l(t)\) are defined as in (3.4). Integrating (3.12) from \(t_{0}\) to t yields

$$V(t)\leq r_{1}(T)+ \int_{t_{0}}^{t}\bigl[e(s) V(s) +l(s) V^{\frac{q}{p}}(s)\bigr] \,\mathrm {d}s. $$

From the above and Lemma 2.1, we get

$$V(t)\leq r_{1}(T)\exp \biggl( \int_{t_{0}}^{t}e(s)\,\mathrm {d}s \biggr) \biggl\{ 1+ \frac{p-q}{p}r_{1}^{\frac{q-p}{p}}(T) \int_{t_{0}}^{t} l(s)\exp \biggl(\frac{q-p}{p} \int_{t_{0}}^{s}e(\tau)\,\mathrm {d}\tau \biggr) \,\mathrm {d}s \biggr\} ^{\frac{p}{p-q}}, $$

and then from (3.10) and the assumption \(u(t)=x^{p}(t)\), we have

$$\begin{aligned} x(t)&\leq r_{1}^{\frac{1}{p}}(T)\exp \biggl(\frac{1}{p} \int_{t_{0}}^{t}e(s) \,\mathrm {d}s \biggr) \\ &\quad{}\times \biggl\{ 1+ \frac{p-q}{p}r_{1}^{\frac{q-p}{p}}(T) \int _{t_{0}}^{t} l(s)\exp \biggl(\frac{q-p}{p} \int_{t_{0}}^{s}e(\tau) \,\mathrm {d}\tau \biggr) \,\mathrm {d}s \biggr\} ^{\frac{1}{p-q}}. \end{aligned} $$

Since the above inequality is true for any \(t\in[t_{0},T]\), we obtain

$$\begin{aligned} x(T)&\leq r_{1}^{\frac{1}{p}}(T)\exp \biggl(\frac{1}{p} \int_{t_{0}}^{T}e(s) \,\mathrm {d}s \biggr) \\ &\quad{}\times \biggl\{ 1+ \frac{p-q}{p}r_{1}^{\frac{q-p}{p}}(T) \int _{t_{0}}^{T} l(s)\exp \biggl(\frac{q-p}{p} \int_{t_{0}}^{s}e(\tau) \,\mathrm {d}\tau \biggr) \,\mathrm {d}s \biggr\} ^{\frac{1}{p-q}}. \end{aligned} $$

Replacing T by t yields

$$\begin{aligned} \begin{aligned}[b] x(t) &\leq r_{1}^{\frac{1}{p}}(t) \exp \biggl(\frac{1}{p} \int_{t_{0}} ^{t}e(s)\,\mathrm {d}s \biggr) \\ &\quad{}\times \biggl\{ 1+ \frac{p-q}{p}r_{1}^{\frac{q-p}{p}}(t) \int_{t_{0}}^{t} l(s)\exp \biggl(\frac{q-p}{p} \int_{t_{0}}^{s}e(\tau) \,\mathrm {d}\tau \biggr) \,\mathrm {d}s \biggr\} ^{\frac{1}{p-q}} \\ &= v_{1}(t), \quad t\in I_{0}=[t_{0},t_{1}]. \end{aligned} \end{aligned}$$
(3.13)

This means that (3.1) is true.

For \(t\in I_{1}=(t_{1},t_{2}]\), from (3.1), (3.2), (3.5) and (3.13), we get

$$\begin{aligned} x^{p}(t) &\leq r_{1}(t)+ \int_{t_{0}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t_{0}}^{t}f_{2}(s) \int_{t_{0}}^{s}g_{1}(\tau)x^{p}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{} + \int_{t_{0}}^{t}f_{3}(s) \int_{t_{0}}^{s}g_{2}(\tau )x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s+h(t)\beta_{1}x^{m}(t_{1}-0) \\ &=r_{1}(t)+ \int_{t_{0}}^{t_{1}}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t _{1}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s + \int_{t_{0}}^{t_{1}}f_{2}(s) \int_{t_{0}}^{s}g_{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{} + \int_{t_{1}}^{t}f_{2}(s) \int_{t_{0}}^{s}g_{1}(\tau )x^{p}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s + \int_{t_{0}}^{t_{1}}f_{3}(s) \int_{t_{0}} ^{s}g_{2}(\tau)x^{q}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{} + \int_{t_{1}}^{t}f_{3}(s) \int_{t_{0}}^{s}g_{2}(\tau )x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s +h(t)\beta_{1}x^{m}(t_{1}-0) \\ &\leq r_{1}(t)+ \int_{t_{0}}^{t_{1}}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t_{1}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s + \int_{t_{0}}^{t_{1}}f _{2}(s) \int_{t_{0}}^{t_{1}}g_{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{} + \int_{t_{1}}^{t}f_{2}(s) \int_{t_{0}}^{t_{1}}g_{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s+ \int_{t_{1}}^{t}f_{2}(s) \int_{t_{1}} ^{s}g_{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{}+ \int_{t_{0}}^{t_{1}}f_{3}(s) \int_{t_{0}}^{t_{1}}g_{2}(\tau)x^{q}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s + \int_{t_{1}}^{t}f_{3}(s) \int_{t_{0}} ^{t_{1}}g_{2}(\tau)x^{q}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{}+ \int_{t_{1}}^{t}f_{3}(s) \int_{t_{1}}^{s}g_{2}(\tau)x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s +h(t)\beta_{1}x^{m}(t_{1}-0) \\ &\leq r_{1}(t)+ \int_{t_{0}}^{t_{1}}f_{1}(s)v_{1}^{q}(s) \,\mathrm {d}s+ \int_{t_{1}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s + \int_{t_{0}}^{t_{1}}f _{2}(s) \int_{t_{0}}^{t_{1}}g_{1}(\tau)v_{1}^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{}+ \int_{t_{1}}^{t}f_{2}(s) \int_{t_{0}}^{t_{1}}g_{1}(\tau)v_{1}^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s+ \int_{t_{1}}^{t}f_{2}(s) \int_{t_{1}} ^{s}g_{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{}+ \int_{t_{0}}^{t_{1}}f_{3}(s) \int_{t_{0}}^{t_{1}}g_{2}(\tau)v_{1} ^{q}(\tau)\,\mathrm {d}\tau \,\mathrm {d}s + \int_{t_{1}}^{t}f_{3}(s) \int_{t_{0}}^{t_{1}}g_{2}(\tau)v_{1}^{q}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{}+ \int_{t_{1}}^{t}f_{3}(s) \int_{t_{1}}^{s}g_{2}(\tau)x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s +h(t)\beta_{1}v_{1}^{m}(t_{1}-0) \\ &=r_{1}(t)+ \int_{t_{0}}^{t_{1}}f_{1}(s)v_{1}^{q}(s) \,\mathrm {d}s+ \int_{t_{1}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s + \int_{t_{0}}^{t}f_{2}(s) \int_{t_{0}}^{t_{1}}g_{1}(\tau)v_{1}^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{}+ \int_{t_{1}}^{t}f_{2}(s) \int_{t_{1}}^{s}g_{1}(\tau)x^{p}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s+ \int_{t_{0}}^{t}f_{3}(s) \int_{t_{0}}^{t _{1}}g_{2}(\tau)v_{1}^{q}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{}+ \int_{t_{1}}^{t}f_{3}(s) \int_{t_{1}}^{s}g_{2}(\tau)x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s +h(t)\beta_{1}v_{1}^{m}(t_{1}-0) \\ &=r_{2}(t)+ \int_{t_{1}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t_{1}} ^{t}f_{2}(s) \int_{t_{1}}^{s}g_{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{}+ \int_{t_{1}}^{t}f_{3}(s) \int_{t_{1}}^{s}g_{2}(\tau)x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s. \end{aligned}$$
(3.14)

Inequality (3.14) is the same as (3.6) if we replace \(r_{1}(t)\) and \(t_{0}\) with \(r_{2}(t)\) and \(t_{1}\) in (3.6), respectively. Thus, by (3.14), we have, for \(t\in I_{1}=(t_{1},t_{2}]\),

$$\begin{aligned} \begin{aligned} x(t)&\leq r_{2}^{\frac{1}{p}}(t)\exp \biggl(\frac{1}{p} \int_{t_{1}}^{t}e(s) \,\mathrm {d}s \biggr) \\ & \quad{}\times \biggl\{ 1+ \frac{p-q}{p}r_{2}^{\frac{q-p}{p}}(t) \int _{t_{1}}^{t} l(s)\exp \biggl(\frac{q-p}{p} \int_{t_{1}}^{s}e(\tau) \,\mathrm {d}\tau \biggr) \,\mathrm {d}s \biggr\} ^{\frac{1}{p-q}} = v_{2}(t). \end{aligned} \end{aligned}$$

Suppose that

$$\begin{aligned} \begin{aligned}[b] x(t) &\leq r_{i}^{\frac{1}{p}}(t) \exp \biggl(\frac{1}{p} \int_{t_{i-1}} ^{t}e(s)\,\mathrm {d}s \biggr) \\ &\quad{}\times \biggl\{ 1+\frac{p-q}{p}r_{i}^{\frac{q-p}{p}}(t) \int_{t_{i-1}}^{t} l(s) \exp \biggl(\frac{q-p}{p} \int_{t_{i-1}}^{s}e(\tau)\,\mathrm {d}\tau \biggr)\,\mathrm {d}s \biggr\} ^{\frac{1}{p-q}}= v_{i}(t) \end{aligned} \end{aligned}$$
(3.15)

holds for \(t\in I_{i-1}=(t_{i-1}, t_{i}]\), \(i=2,3,\ldots\) . Then, for \(t\in I_{i}=(t_{i}, t_{i+1}]\), from (3.1), (3.2), (3.5) and (3.15) we obtain

$$\begin{aligned} x^{p}(t) \leq& r_{1}(t)+ \int_{t_{0}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t_{0}}^{t}f_{2}(s) \int_{t_{0}}^{s}g_{1}(\tau)x^{p}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s \\ &{}+ \int_{t_{0}}^{t}f_{3}(s) \int_{t_{0}}^{s}g_{2}(\tau)x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s +h(t)\sum_{t_{0}< t_{i}< t} \beta_{i}x^{m}(t _{i}-0) \\ =&r_{1}(t)+\sum_{k=0}^{i-1} \int_{t_{k}}^{t_{k+1}}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t_{i}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s \\ &{} +\sum_{k=0}^{i-1} \int_{t_{k}}^{t_{k+1}}f_{2}(s) \int_{t_{0}}^{s}g _{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s + \int_{t_{i}}^{t}f _{2}(s) \int_{t_{0}}^{s}g_{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &{} +\sum_{k=0}^{i-1} \int_{t_{k}}^{t_{k+1}}f_{3}(s) \int_{t_{0}}^{s}g _{2}(\tau)x^{q}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s + \int_{t_{i}}^{t}f _{3}(s) \int_{t_{0}}^{s}g_{2}(\tau)x^{q}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &{} +h(t)\sum_{t_{0}< t_{i}< t}\beta_{i}x^{m}(t_{i}-0) \\ \leq& r_{1}(t)+\sum_{k=0}^{i-1} \int_{t_{k}}^{t_{k+1}}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t_{i}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s \\ &{}+\sum_{k=0}^{i-1} \int_{t_{k}}^{t_{k+1}} \Biggl(f_{2}(s)\sum _{j=0}^{k} \int_{t_{j}}^{t_{j+1}}g_{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \Biggr) \,\mathrm {d}s \\ &{}+ \int_{t_{i}}^{t} \Biggl(f_{2}(s)\sum _{j=0}^{i-1} \int_{t_{j}}^{t_{j+1}}g _{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \Biggr)\,\mathrm {d}s+ \int_{t_{i}} ^{t}f_{2}(s) \int_{t_{i}}^{s}g_{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &{} +\sum_{k=0}^{i-1} \int_{t_{k}}^{t_{k+1}} \Biggl(f_{3}(s)\sum _{j=0}^{k} \int_{t_{j}}^{t_{j+1}}g_{2}(\tau)x^{q}( \tau) \Biggr)\,\mathrm {d}\tau \,\mathrm {d}s \\ &{}+ \int_{t_{i}}^{t} \Biggl(f_{3}(s)\sum _{j=0}^{i-1} \int_{t_{j}}^{t_{j+1}}g _{2}(\tau)x^{q}( \tau)\,\mathrm {d}\tau \Biggr)\,\mathrm {d}s+ \int_{t_{i}} ^{t}f_{3}(s) \int_{t_{i}}^{s}g_{2}(\tau)x^{q}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &{}+h(t)\sum_{t_{0}< t_{i}< t}\beta_{i}x^{m}(t_{i}-0) \\ \leq& r_{1}(t)+\sum_{k=0}^{i-1} \int_{t_{k}}^{t_{k+1}}f_{1}(s)v_{k+1} ^{q}(s)\,\mathrm {d}s+ \int_{t_{i}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s \\ &{}+\sum_{k=0}^{i-1} \int_{t_{k}}^{t_{k+1}} \Biggl(f_{2}(s)\sum _{j=0}^{k} \int_{t_{j}}^{t_{j+1}}g_{1}(\tau)v_{j+1}^{p}( \tau) \Biggr)\,\mathrm {d}\tau \,\mathrm {d}s \\ &{}+ \int_{t_{i}}^{t} \Biggl(f_{2}(s)\sum _{j=0}^{i-1} \int_{t_{j}}^{t_{j+1}}g _{1}(\tau)v_{j+1}^{p}( \tau)\,\mathrm {d}\tau \Biggr)\,\mathrm {d}s+ \int _{t_{i}}^{t}f_{2}(s) \int_{t_{i}}^{s}g_{1}(\tau)x^{p}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s \\ &{} +\sum_{k=0}^{i-1} \int_{t_{k}}^{t_{k+1}} \Biggl(f_{3}(s)\sum _{j=0}^{k} \int_{t_{j}}^{t_{j+1}}g_{2}(\tau)v_{j+1}^{q}( \tau) \Biggr)\,\mathrm {d}\tau \,\mathrm {d}s \\ &{}+ \int_{t_{i}}^{t} \Biggl(f_{3}(s)\sum _{j=0}^{i-1} \int_{t_{j}}^{t_{j+1}}g _{2}(\tau)v_{j+1}^{q}( \tau)\,\mathrm {d}\tau \Biggr)\,\mathrm {d}s+ \int _{t_{i}}^{t}f_{3}(s) \int_{t_{i}}^{s}g_{2}(\tau)x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s \\ &{}+h(t)\sum_{t_{0}< t_{i}< t}\beta_{i}v_{i}^{m}(t_{i}-0) \\ =&r_{i+1}(t)+ \int_{t_{i}}^{t}f_{1}(s)x^{q}(s) \,\mathrm {d}s+ \int_{t_{i}} ^{t}f_{2}(s) \int_{t_{i}}^{s}g_{1}(\tau)x^{p}( \tau)\,\mathrm {d}\tau \,\mathrm {d}s \\ &{}+ \int_{t_{i}}^{t}f_{3}(s) \int_{t_{i}}^{s}g_{2}(\tau)x^{q}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s. \end{aligned}$$
(3.16)

Inequality (3.16) is the same as (3.6) if we replace \(r_{1}(t)\) and \(t_{0}\) with \(r_{i+1}(t)\) and \(t_{i}\) in (3.6), respectively. Thus, by (3.16), we have, for \(t\in I_{i}=(t_{i}, t_{i+1}]\),

$$\begin{aligned} x(t)&\leq r_{i+1}^{\frac{1}{p}}(t)\exp \biggl(\frac{1}{p} \int_{t_{i}} ^{t}e(s)\,\mathrm {d}s \biggr) \\ &\quad{}\times \biggl\{ 1+ \frac{p-q}{p}r_{i+1}^{\frac{q-p}{p}}(t) \int_{t_{i}}^{t} l(s)\exp \biggl(\frac{q-p}{p} \int_{t_{i}}^{s}e(\tau) \,\mathrm {d}\tau \biggr) \,\mathrm {d}s \biggr\} ^{\frac{1}{p-q}}. \end{aligned} $$

By induction, we know that (3.3) holds for \(t\in(t_{i},t_{i+1}]\), for any nonnegative integer i. This completes the proof of Theorem 3.1. □

Theorem 3.2

Suppose that x is a nonnegative piecewise continuous function defined on \([t_{0},\infty)\) with discontinuities of the first kind in the points \(t_{i}\) (\(i=1,2,\ldots\)) and satisfies the integro-sum inequality

$$\begin{aligned} \begin{aligned}[b] x^{p}(t) &\leq a(t)+ \int_{t_{0}}^{t}f(s)x^{q}(s)\,\mathrm {d}s+\sum _{j=1} ^{L} \int_{t_{0}}^{t}b_{j}(s) \int_{t_{0}}^{s}g_{j}(\tau)x^{p}( \tau) \,\mathrm {d}\tau \,\mathrm {d}s \\ &\quad{} +\sum_{k=1}^{M} \int_{t_{0}}^{t}c_{k}(s) \int_{t_{0}}^{s}\theta_{k}( \tau)x^{q}(\tau)\,\mathrm {d}\tau \,\mathrm {d}s +d(t)\sum _{t_{0}< t_{i}< t} \beta_{i}x^{m}(t_{i}-0), \quad t\geq t_{0}, \end{aligned} \end{aligned}$$
(3.17)

where \(0\leq t_{0}< t_{1} < t_{2}<\cdots\) , \(\lim_{i\rightarrow\infty}t _{i}=\infty\), \(a(t)\geq0\) is defined on \([t_{0},\infty)\), \(f\in C(\mathrm {R}_{+},\mathrm {R}_{+})\), \(b_{j},g_{j}\in C( \mathrm {R}_{+},\mathrm {R}_{+})\) (\(j=1,2,\ldots, L\)), \(c_{j},\theta _{j}\in C(\mathrm {R}_{+},\mathrm {R}_{+})\) (\(j=1,2,\ldots, M\)), \(\beta_{i}\geq0\) (\(i=1,2,\ldots\)), \(p>0\), \(q>0\), \(p\neq q\), and \(m>0\) are constants. If

$$1+\frac{p-q}{p}r_{i}^{\frac{q-p}{p}}(t) \int_{t_{i-1}}^{t} l(s)\exp \biggl(\frac{q-p}{p} \int_{t_{i-1}}^{s}e(\tau)\,\mathrm {d}\tau \biggr) \,\mathrm {d}s>0, \quad i=1,2,\ldots, $$

then, for \(t\geq t_{0}\), the following estimates hold:

$$\begin{aligned}& x(t)\leq v_{1}(t),\quad t\in[t_{0},t_{1}], \end{aligned}$$
(3.18)
$$\begin{aligned}& x(t)\leq v_{i}(t),\quad t\in(t_{i-1},t_{i}], i=2,3,\ldots, \end{aligned}$$
(3.19)

where

$$\begin{aligned}& \begin{aligned} v_{i}(t) &=r_{i}^{\frac{1}{p}}(t)\exp \biggl( \frac{1}{p} \int_{t_{i-1}} ^{t}e(s)\,\mathrm {d}s \biggr) \\ &\quad {}\times \biggl\{ 1+\frac{p-q}{p}r_{i}^{\frac{q-p}{p}}(t) \int_{t_{i-1}}^{t} l(s) \exp \biggl(\frac{q-p}{p} \int_{t_{i-1}}^{s}e(\tau)\,\mathrm {d}\tau \biggr)\,\mathrm {d}s \biggr\} ^{\frac{1}{p-q}}, \\ &\quad\ i=1,2,\ldots, \end{aligned} \\& e(t)=\sum_{j=1}^{L}b_{j}(t) \int_{t_{0}}^{t}g_{j}(\tau)\,\mathrm {d}\tau,\quad\quad l(t)= f(t)+\sum_{k=1}^{M}c_{k}(t) \int_{t_{0}}^{t}\theta _{k}(\tau)\,\mathrm {d}\tau, \\& r_{1}(t)=\max_{t_{0}\leq\tau\leq t}\bigl\vert a(\tau)\bigr\vert , \quad\quad h(t)= \max_{t_{0}\leq\tau\leq t}\bigl\vert d(\tau)\bigr\vert , \\& \begin{aligned} r_{i+1}(t) &=r_{i}(t) + \int_{t_{i-1}}^{t_{i}}f(s)v_{i}^{q}(s) \,\mathrm {d}s +\sum_{j=1}^{L} \int_{t_{i-1}}^{t} \biggl(b_{j}(s) \int_{t_{i-1}} ^{t_{i}}g_{j}(\tau)v_{i}^{p}( \tau)\,\mathrm {d}\tau \biggr)\,\mathrm {d}s \\ &\quad{} +\sum_{k=1}^{M} \int_{t_{i-1}}^{t} \biggl(c_{k}(s) \int_{t_{i-1}}^{t_{i}} \theta_{k}( \tau)v_{i}^{q}(\tau)\,\mathrm {d}\tau \biggr)\,\mathrm {d}s +h(t) \beta_{i}v_{i}^{m}(t_{i}-0), \quad i=1,2,\ldots. \end{aligned} \end{aligned}$$

The proof is similar to that of Theorem 3.1, and we omit these details.

Theorem 3.3

Suppose that x is a nonnegative piecewise continuous function defined on \([t_{0},\infty)\) with discontinuities of the first kind in the points \(t_{i}\) (\(i=1,2,\ldots\)) and satisfies the integro-sum inequality:

$$\begin{aligned} \begin{aligned}[b] x^{p}(t) &\leq a(t)+b(t) \int_{t_{0}}^{t} \bigl[f(s)x^{p}(s)+g(s)x^{q}(s)-h(s)x ^{r}(s) \bigr]\,\mathrm {d}s \\ &\quad{} +c(t)\sum_{t_{0}< t_{i}< t}\beta_{i}x^{m}(t_{i}-0), \quad t\geq t_{0}, \end{aligned} \end{aligned}$$
(3.20)

where \(0\leq t_{0}< t_{1} < t_{2}<\cdots\) , \(\lim_{i\rightarrow\infty}t _{i}=\infty\), \(a(t)\) is defined on \([t_{0},\infty)\) and \(a(t_{0}) \neq0\), \(b(t)\geq0\) and \(c(t)\geq0\) are defined on \([t_{0},\infty )\), \(f,g\in C(\mathrm {R}_{+},\mathrm {R}_{+})\), \(h\in C(\mathrm {R}_{+},(0,+\infty))\), \(0< q< p< r\), \(\beta_{i}\geq0\) (\(i=1,2,\ldots\)) and \(m>0\) are constants.

Then, for any continuous functions \(k_{1}(t)>0\) and \(k_{2}(t)\geq0\) on \([t_{0},\infty)\) satisfying \(k(t)= k_{1}(t)-k_{2}(t)\geq0\), the following estimates hold:

$$\begin{aligned}& x(t)\leq v_{1}(t), \quad t\in[t_{0},t_{1}], \end{aligned}$$
(3.21)
$$\begin{aligned}& x(t)\leq v_{i}(t), \quad t\in(t_{i-1},t_{i}], i=2,3,\ldots, \end{aligned}$$
(3.22)

where

$$\begin{aligned}& v_{i}(t)=r_{i}^{\frac{1}{p}}(t)\exp \biggl\{ {\frac{1}{p}}e(t) \int_{t _{i-1}}^{t}\bigl[f(s)+k(s)\bigr]\,\mathrm {d}s \biggr\} ,\quad i=1,2,\ldots, \end{aligned}$$
(3.23)
$$\begin{aligned}& d(t)=\max_{t_{0}\leq\tau\leq t}\bigl\vert a(\tau)\bigr\vert , \quad\quad e(t)= \max_{t_{0}\leq\tau\leq t}\bigl\vert b(\tau)\bigr\vert , \quad\quad l(t)= \max _{t_{0}\leq\tau\leq t}\bigl\vert c(\tau)\bigr\vert , \end{aligned}$$
(3.24)
$$\begin{aligned}& \begin{gathered} r_{1}(t)=d(t)+e(t)w(t), \\ w(t)= \int_{t_{0}}^{t} \bigl[\theta_{1} \bigl(p,q,g(s),k_{1}(s) \bigr)+\theta_{2} \bigl(p,r,h(s),k_{2}(s) \bigr) \bigr]\,\mathrm {d}s, \end{gathered} \end{aligned}$$
(3.25)
$$\begin{aligned}& \theta_{1} \bigl(p,q,g(s),k_{1}(s) \bigr)= \frac{p-q}{q} \biggl(\frac{q}{p} \biggr)^{\frac{p}{p-q}}{g^{\frac{p}{p-q}}(s)} {k_{1}^{{\frac{-q}{p-q}}}(s)}, \end{aligned}$$
(3.26)
$$\begin{aligned}& \theta_{2} \bigl(p,r,h(s),k_{2}(s) \bigr)= \frac{r-p}{r} \biggl(\frac{p}{r} \biggr)^{\frac{p}{r-p}}{h^{\frac{-p}{r-p}}(s)} {k_{2}^{{\frac{r}{r-p}}}(s)}, \end{aligned}$$
(3.27)
$$\begin{aligned}& r_{i+1}(t)=r_{i}(t)+e(t) \int_{t_{i-1}}^{t_{i}}\bigl[f(s)+k(s)\bigr]v_{i}^{p}(s) \,\mathrm {d}s+l(t)\beta_{i}v_{i}^{m}(t_{i}-0),\quad i=1,2,\ldots. \end{aligned}$$
(3.28)

Proof

From (3.20) and (3.24), we obtain, for \(t\in I_{0}=[t _{0},t_{1}]\),

$$\begin{aligned} x^{p}(t)\leq d(t)+e(t) \int_{t_{0}}^{t} \bigl[f(s)x^{p}(s)+g(s)x^{q}(s)-h(s)x ^{r}(s) \bigr]\,\mathrm {d}s. \end{aligned}$$
(3.29)

From Lemma 2.1, (3.24)-(3.27) and (3.29), we have

$$\begin{aligned} x^{p}(t) \leq& d(t)+e(t) \int_{t_{0}}^{t} \bigl[\bigl[f(s)+k(s) \bigr]x^{p}(s)+ \theta_{1}\bigl(p,q,g(s),k_{1}(s) \bigr)+\theta_{2}\bigl(p,r,h(s),k_{2}(s)\bigr) \bigr] \,\mathrm {d}s \\ =&d(t)+e(t) \int_{t_{0}}^{t} \bigl[\theta_{1} \bigl(p,q,g(s),k_{1}(s)\bigr)+\theta _{2} \bigl(p,r,h(s),k_{2}(s)\bigr) \bigr]\,\mathrm {d}s \\ &{}+e(t) \int_{t_{0}}^{t}\bigl[f(s)+k(s)\bigr]x^{p}(s) \,\mathrm {d}s \\ =&d(t)+e(t)w(t)+e(t) \int_{t_{0}}^{t}\bigl[f(s)+k(s)\bigr]x^{p}(s) \,\mathrm {d}s \\ =&r_{1}(t)+e(t) \int_{t_{0}}^{t}\bigl[f(s)+k(s)\bigr]x^{p}(s) \,\mathrm {d}s, \end{aligned}$$
(3.30)

\(r_{1}(t)\) and \(e(t)\) are non-decreasing on \([t_{0},\infty)\). Take any fixed \(T\in[t_{0},t_{1}]\), and for arbitrary \(t\in[t_{0},T]\), we have

$$ x^{p}(t)\leq r_{1}(T)+e(T) \int_{t_{0}}^{t}\bigl[f(s)+k(s)\bigr]x^{p}(s) \,\mathrm {d}s. $$
(3.31)

Let \(u(t)=x^{p}(t)\). Inequality (3.31) is equivalent to

$$ u(t)\leq r_{1}(T)+e(T) \int_{t_{0}}^{t}\bigl[f(s)+k(s)\bigr]u(s)\,\mathrm {d}s. $$
(3.32)

Define a function \(V(t)\) by the right-hand side of (3.32). Then \(V(t)\) is positive and

$$ V(t_{0})=r_{1}(T), \quad\quad u(t)\leq V(t), $$
(3.33)
$$V'(t)=e(T)\bigl[f(t)+k(t)\bigr]u(t)\leq e(T)\bigl[f(t)+k(t) \bigr]V(t), \quad t\in[t_{0},T]. $$

We have

$$\begin{aligned} \begin{aligned}[b] V(t) &\leq V(t_{0})\exp \biggl\{ e(T) \int_{t_{0}}^{t}\bigl[f(s)+k(s)\bigr] \,\mathrm {d}s \biggr\} \\ &=r_{1}(T)\exp \biggl\{ e(T) \int_{t_{0}}^{t}\bigl[f(s)+k(s)\bigr]\,\mathrm {d}s \biggr\} , \end{aligned} \end{aligned}$$
(3.34)

and then, from (3.33), (3.34) and the assumption \(u(t)=x^{p}(t)\), we get

$$x(t)\leq r_{1}^{\frac{1}{p}}(T)\exp \biggl\{ \frac{1}{p}e(T) \int_{t_{0}} ^{t}\bigl[f(s)+k(s)\bigr]\,\mathrm {d}s \biggr\} . $$

Since the above inequality is true for any \(t\in[t_{0},T]\), we obtain

$$x(T)\leq r_{1}^{\frac{1}{p}}(T)\exp \biggl\{ \frac{1}{p}e(T) \int_{t_{0}} ^{T}\bigl[f(s)+k(s)\bigr]\,\mathrm {d}s \biggr\} . $$

Replacing T by t yields

$$ x(t)\leq r_{1}^{\frac{1}{p}}(t)\exp \biggl\{ { \frac{1}{p}}e(t) \int_{t _{0}}^{t}\bigl[f(s)+k(s)\bigr]\,\mathrm {d}s \biggr\} = v_{1}(t), \quad t\in I_{0}=[t _{0},t_{1}]. $$
(3.35)

This means that (3.21) is true for \(t\in[t_{0},t_{1}]\).

For \(t\in I_{1}=(t_{1},t_{2}]\), from Lemma 2.1 and (3.20), (2.24)-(2.27) and (3.35), we obtain

$$\begin{aligned} x^{p}(t) \leq& d(t)+e(t) \int_{t_{0}}^{t} \bigl[f(s)x^{p}(s)+g(s)x^{q}(s)-h(s)x ^{r}(s) \bigr]\,\mathrm {d}s+l(t)\beta_{1}x^{m}(t_{1}-0) \\ \leq& d(t)+e(t) \int_{t_{0}}^{t} \bigl[\bigl[f(s)+k(s) \bigr]x^{p}(s)+\theta_{1}\bigl(p,q,g(s),k _{1}(s) \bigr)+\theta_{2}\bigl(p,r,h(s),k_{2}(s)\bigr) \bigr] \,\mathrm {d}s \\ &{}+l(t)\beta_{1}v_{1}^{m}(t_{1}-0) \\ =&d(t)+e(t) \int_{t_{0}}^{t}\bigl[\theta_{1} \bigl(p,q,g(s),k_{1}(s)\bigr)+\theta_{2}\bigl(p,r,h(s),k _{2}(s)\bigr)\bigr]\,\mathrm {d}s+l(t)\beta_{1}v_{1}^{m}(t_{1}-0) \\ &{}+e(t) \int_{t_{0}}^{t}\bigl[f(s)+k(s)\bigr]x^{p}(s) \,\mathrm {d}s \\ \leq& d(t)+e(t)w(t)+l(t)\beta_{1}v_{1}^{m}(t_{1}-0)+e(t) \int_{t_{0}} ^{t_{1}}\bigl[f(s)+k(s)\bigr]v_{1}^{p}(s) \,\mathrm {d}s \\ &{}+e(t) \int_{t_{1}}^{t}\bigl[f(s)+k(s)\bigr]x^{p}(s) \,\mathrm {d}s \\ =&r_{1}(t)+l(t)\beta_{1}v_{1}^{m}(t_{1}-0) \\ &{}+e(t) \int_{t_{0}}^{t_{1}}\bigl[f(s)+k(s)\bigr]v_{1}^{p}(s) \,\mathrm {d}s+e(t) \int_{t_{1}}^{t}\bigl[f(s)+k(s)\bigr]x^{p}(s) \,\mathrm {d}s \\ =&r_{2}(t)+e(t) \int_{t_{1}}^{t}\bigl[f(s)+k(s)\bigr]x^{p}(s) \,\mathrm {d}s. \end{aligned}$$
(3.36)

Inequality (3.36) is the same as (3.30) if we replace \(r_{1}(t)\) and \(t_{0}\) with \(r_{2}(t)\) and \(t_{1}\) in (3.36), respectively. Thus, by (3.35) and (3.36), we get, for \(t\in I_{1}=(t_{1},t_{2}]\),

$$x(t)\leq r_{2}^{\frac{1}{p}}(t)\exp \biggl\{ \frac{1}{p}e(t) \int_{t_{1}} ^{t}\bigl[f(s)+k(s)\bigr]\,\mathrm {d}s \biggr\} = v_{2}(t). $$

Suppose that

$$\begin{aligned} \begin{aligned}[b] x(t) &\leq r_{i}^{\frac{1}{p}}(t)\exp \biggl\{ { \frac{1}{p}}e(t) \int_{t_{i-1}}^{t}\bigl[f(s)+k(s)\bigr]\,\mathrm {d}s \biggr\} \\ &=v_{i}(t) \quad \text{holds for } t\in I_{i-1}=(t_{i-1}, t_{i}], i=2,3,\ldots. \end{aligned} \end{aligned}$$
(3.37)

Then, for \(t\in I_{i}=(t_{i}, t_{i+1}]\), from Lemma 2.1 and (3.20), (3.24)-(3.27) and (3.37), we have

$$\begin{aligned} x^{p}(t) \leq& d(t)+e(t) \int_{t_{0}}^{t} \bigl[f(s)x^{p}(s)+g(s)x^{q}(s)-h(s)x ^{r}(s) \bigr]\,\mathrm {d}s +l(t)\sum_{t_{0}< t_{i}< t} \beta_{i}x^{m}(t_{i}-0) \\ \leq& d(t)+e(t) \int_{t_{0}}^{t} \bigl[\bigl[f(s)+k(s) \bigr]x^{p}(s)+\theta_{1}\bigl(p,q,g(s),k _{1}(s) \bigr)+\theta_{2}\bigl(p,r,h(s),k_{2}(s)\bigr) \bigr] \,\mathrm {d}s \\ &{}+l(t)\sum_{t_{0}< t_{i}< t}\beta_{i}v_{i}^{m}(t_{i}-0) \\ =&d(t)+e(t)w(t)+e(t)\sum_{k=0}^{i-1} \int_{t_{k}}^{t_{k+1}}\bigl[f(s)+k(s)\bigr]x ^{p}(s)\,\mathrm {d}s+e(t) \int_{t_{i}}^{t}\bigl[f(s)+k(s)\bigr]x^{p}(s) \,\mathrm {d}s \\ &{}+l(t)\sum_{t_{0}< t_{i}< t}\beta_{i}v_{i}^{m}(t_{i}-0) \\ \leq& r_{1}(t)+e(t)\sum_{k=0}^{i-1} \int_{t_{k}}^{t_{k+1}}\bigl[f(s)+k(s)\bigr] v_{k+1}^{p}(s)\,\mathrm {d}s+e(t) \int_{t_{i}}^{t}\bigl[f(s)+k(s)\bigr]x^{p}(s) \,\mathrm {d}s \\ &{}+l(t)\sum_{t_{0}< t_{i}< t}\beta_{i}v_{i}^{m}(t_{i}-0) \\ \leq& r_{i+1}(t)+e(t) \int_{t_{i}}^{t}\bigl[f(s)+k(s)\bigr]x^{p}(s) \,\mathrm {d}s. \end{aligned}$$
(3.38)

Inequality (3.38) is the same as (3.30) if we replace \(r_{1}(t)\) and \(t_{0}\) with \(r_{i+1}(t)\) and \(t_{i}\) in (3.38), respectively. Thus, by (3.35) and (3.38), we have, for \(t\in I_{i}=(t_{i}, t_{i+1}]\),

$$x(t)\leq r_{i+1}^{\frac{1}{p}}(t)\exp \biggl\{ \frac{1}{p}e(t) \int_{t _{i}}^{t}\bigl[f(s)+k(s)\bigr]\,\mathrm {d}s \biggr\} . $$

By induction, we know that (3.30) holds for \(t\in(t_{i},t_{i+1}]\), for any nonnegative integer i. This completes the proof of Theorem 3.3. □

Theorem 3.4

Suppose that x is a nonnegative piecewise continuous function defined on \([t_{0},\infty)\) with discontinuities of the first kind in the points \(t_{i}\) (\(i=1,2,\ldots\)) and satisfies the integro-sum inequality

$$\begin{aligned} x^{p}(t) \leq& a(t)+ \int_{t_{0}}^{t}f(s)x^{p}(s)\,\mathrm {d}s+\sum _{j=1} ^{L} \int_{t_{0}}^{t}g_{j}(s)x^{q_{j}}(s) \,\mathrm {d}s-\sum_{j=1}^{L} \int_{t_{0}}^{t}h_{j}(s)x^{r_{j}}(s) \,\mathrm {d}s \\ &{}+c(t)\sum_{t_{0}< t_{i}< t}\beta_{i}x^{m}(t_{i}-0),\quad t\geq t_{0}, \end{aligned}$$

where \(0\leq t_{0}< t_{1} < t_{2}<\cdots\) , \(\lim_{i\rightarrow\infty}t _{i}=\infty\), \(a(t)\) is defined on \([t_{0},\infty)\) and \(a(t_{0}) \neq0\), \(b(t)\geq0\) and \(c(t)\geq0\) are defined on \([t_{0},\infty )\), \(f,g\in C(\mathrm {R}_{+},\mathrm {R}_{+})\), \(h\in C(\mathrm {R}_{+},(0,+\infty))\), \(0< q_{j}< p< r_{j}\) (\(j=1,2,\ldots, L\)), \(\beta_{i}\geq0\), \(i=1,2,\ldots\) , and \(m>0\) are constants.

Then, for any continuous functions \(k_{1}(t)>0\) and \(k_{2}(t)\geq0\) on \([t_{0},\infty)\) satisfying \(k(t)= k_{1}(t)-k_{2}(t)\geq0\), the following estimates hold:

$$\begin{aligned}& x(t)\leq v_{1}(t), \quad t\in[t_{0},t_{1}], \\& x(t)\leq v_{i}(t), \quad t\in(t_{i-1},t_{i}],i=2,3,\ldots, \end{aligned}$$

where

$$\begin{aligned}& v_{i}(t)=r_{i}^{\frac{1}{p}}(t)\exp \biggl\{ { \frac{1}{p}} \int_{t_{i-1}} ^{t}\bigl[f(s)+Lk(s)\bigr]\,\mathrm {d}s \biggr\} , \quad i=1,2,\ldots, \\ & d(t)=\max_{t_{0}\leq\tau\leq t}\bigl\vert a(\tau)\bigr\vert , \quad\quad l(t)= \max _{t_{0}\leq\tau\leq t}\bigl\vert c(\tau)\bigr\vert , \quad\quad r_{1}(t)=d(t)+w(t), \\ & w(t)=\sum_{j=1}^{L} \int_{t_{0}}^{t} \bigl[\theta_{j} \bigl(p,q_{j},g_{j}(s),k _{1}(s) \bigr)+ \widetilde{\theta}_{j} \bigl(p,r_{j},h_{j}(s),k_{2}(s) \bigr) \bigr]\,\mathrm {d}s, \\ & \theta_{j} \bigl(p,q_{j},g_{j}(s),k_{1}(s) \bigr)=\frac{p-q_{j}}{q_{j}} \biggl(\frac{q_{j}}{p} \biggr)^{\frac{p}{p-q_{j}}}{g_{j}^{ \frac{p}{p-q_{j}}}(s)} {k_{1}^{{\frac{-q}{p-q_{j}}}}(s)}, \quad j=1,2,\ldots,L, \\ & \widetilde{\theta}_{j} \bigl(p,r_{j},h_{j}(s),k_{2}(s) \bigr)= \frac{r _{j}-p}{r_{j}} \biggl(\frac{p}{r_{j}} \biggr)^{\frac{p}{r_{j}-p}}{h_{j}^{\frac{-p}{r _{j}-p}}(s)} {k_{2}^{{\frac{r_{j}}{r_{j}-p}}}(s)}, \quad j=1,2,\ldots,L, \\ & r_{i+1}(t)=r_{i}(t)+ \int_{t_{i-1}}^{t_{i}}\bigl[f(s)+Lk(s)\bigr]v_{i}^{p}(s) \,\mathrm {d}s+l(t)\beta_{i}v_{i}^{m}(t_{i}-0),\quad i=1,2,\ldots. \end{aligned}$$

The proof is similar to that of Theorem 3.3, and we omit these details.

4 Application

In this section, we will apply the results which we have established above to the estimates of solutions of certain impulsive differential equations.

Example 4.1

Consider the following impulsive differential equation:

$$ \textstyle\begin{cases} \frac{\hbox{d}x^{p}(t)}{\hbox{d}t} = F (t,x(t),\int_{t_{0}} ^{t}G(s,t,x(s))\,\mathrm {d}s ), \quad t\neq t_{i}, \\ \bigtriangleup x| _{t=t_{i}} = d(t)\beta_{i}x^{m}(t_{i}-0), \\ x{(t_{0})} = x_{0}, \end{cases} $$
(4.1)

where \(p>0\), \(m>0\) are constants, the functions \(d(t)\geq0\), \(t\in[t _{0},\infty)\), \(F\in C(\mathrm {R}\times \mathrm {R}\times \mathrm {R}, \mathrm {R}_{+})\) and \(G\in C(\mathrm {R}\times \mathrm {R}\times \mathrm {R},\mathrm {R}_{+})\) satisfy the following conditions:

$$\begin{aligned}& \bigl\vert F(t,u,v)\bigr\vert \leq f(t)\vert u\vert ^{q}+\vert v\vert , \end{aligned}$$
(4.2)
$$\begin{aligned}& \bigl\vert G(s,t,w)\bigr\vert \leq\sum _{j=1}^{L}b_{j}(t)g_{j}(s) \vert w\vert ^{p}+\sum_{k=1}^{M}c _{k}(t)\theta_{k}(s)\vert w\vert ^{q}, \end{aligned}$$
(4.3)

where \(q>0\) (\(q\neq p\)) is a constant, and \(f(t)\), \(g_{j}(t)\), \(b_{j}(t)\) (\(j=1,2,\ldots, L\)), \(c_{j}(t)\), \(\theta_{j}(t)\) (\(j=1,2,\ldots, M\)) are defined as in Theorem 3.2. If

$$1+\frac{p-q}{p}r_{i}^{\frac{q-p}{p}}(t) \int_{t_{i-1}}^{t} l(s)\exp \biggl(\frac{q-p}{p} \int_{t_{i-1}}^{s}e(\tau)\,\mathrm {d}\tau \biggr) \,\mathrm {d}s>0, \quad i=1,2,\ldots, $$

then for \(t\geq t_{0}\), every solution \(x(t)\) of Eq. (4.1) satisfies the following estimates:

$$\begin{aligned}& \bigl\vert x(t)\bigr\vert \leq v_{1}(t), \quad t \in[t_{0},t_{1}], \end{aligned}$$
(4.4)
$$\begin{aligned}& \bigl\vert x(t)\bigr\vert \leq v_{i}(t),\quad t \in(t_{i-1},t_{i}], i=2,3,\ldots, \end{aligned}$$
(4.5)

where \(l(t)\), \(e(t)\), \(r_{i}(t)\) and \(v_{i}(t)\) (\(i=1,2,\ldots\)) are defined as in Theorem 3.2.

Proof

The solution \(x(t)\) of Eq. (4.1) satisfies the following equivalent equation:

$$\begin{aligned} x^{p}(t)=x_{0}^{p} + \int_{t_{0}}^{t}F \biggl(\tau,x(\tau), \int_{t_{0}} ^{\tau}G\bigl(s,\tau,x(s)\bigr)\,\mathrm {d}s \biggr)\,\mathrm {d}\tau+d(t) \sum_{t_{0}< t_{i}< t} \beta_{i}x^{m}(t_{i}-0). \end{aligned}$$

From conditions (4.2) and (4.3), it is easy to have

$$\begin{aligned} \bigl\vert x(t)\bigr\vert ^{p} \leq& \vert x_{0} \vert ^{p} + \int_{t_{0}}^{t}\biggl\vert F \biggl(\tau,x( \tau), \int_{t_{0}}^{\tau} G\bigl(s,\tau,x(s)\bigr)\,\mathrm {d}s \biggr)\biggr\vert \,\mathrm {d}\tau \\ &{}+c(t)\sum_{t_{0}< t_{i}< t}\beta_{i}\bigl\vert x(t_{i}-0)\bigr\vert ^{m} \\ \leq&\vert x_{0}\vert ^{p}+ \int_{t_{0}}^{t}f(\tau)\bigl\vert x(\tau)\bigr\vert ^{q}\,\mathrm {d}\tau+\sum_{j=1}^{L} \int_{t_{0}}^{t}b_{j}(\tau) \int_{t_{0}}^{\tau}g _{j}(s)\bigl\vert x(s) \bigr\vert ^{p}\,\mathrm {d}s\,\mathrm {d}\tau \\ &{}+\sum_{k=1}^{M} \int_{t_{0}}^{t}c_{k}(\tau) \int_{t_{0}}^{\tau} \theta_{k}(s)\bigl\vert x(s)\bigr\vert ^{q}\,\mathrm {d}s\,\mathrm {d}\tau +d(t)\sum _{t_{0}< t _{i}< t}\beta_{i}\bigl\vert x(t_{i}-0) \bigr\vert ^{m}, \quad t\geq t_{0}. \end{aligned}$$

By using Theorem 3.2, we easily obtain estimates (4.4) and (4.5) of solutions of Eq. (4.1). □

Example 4.2

Consider the following impulsive differential equation:

$$ \textstyle\begin{cases} \frac{\hbox{d}x(t)}{\hbox{d}t}=f(t)x(t)+g(t)x^{\frac{1}{3}}(t)-h(t)x ^{2}(t), \quad t\neq t_{i}, \\ \bigtriangleup x| _{t=t_{i}}=a(t)\beta_{i}x^{3}(t_{i}-0), \\ x{(t_{0})}=x_{0}, \end{cases} $$
(4.6)

where \(0\leq t_{0}< t_{1} < t_{2}<\cdots\) , \(\lim_{i\rightarrow\infty}t _{i}=\infty\), \(f,g\in C(\mathrm {R}_{+},\mathrm {R}_{+})\), \(h\in C(\mathrm {R}_{+},(0,+\infty))\), \(a(t)\geq0\) is defined on \([t_{0},\infty)\) and \(\beta_{i}\geq0\) (\(i=1,2,\ldots\)) are constants. Then, for any continuous functions \(k_{1}(t)>0\) and \(k_{2}(t)\geq0\) on \([t_{0},\infty)\) satisfying \(k(t)= k_{1}(t)-k_{2}(t)\geq0\), the following estimates hold:

$$\begin{aligned}& \bigl\vert x(t)\bigr\vert \leq v_{1}(t), \quad t \in[t_{0},t_{1}], \end{aligned}$$
(4.7)
$$\begin{aligned}& \bigl\vert x(t)\bigr\vert \leq v_{i}(t), \quad t \in(t_{i-1},t_{i}], i=2,3,\ldots, \end{aligned}$$
(4.8)

where

$$\begin{aligned}& v_{i}(t)=r_{i}(t)\exp \biggl\{ \int_{t_{i-1}}^{t}\bigl[f(s)+k(s)\bigr]\,\mathrm {d}s \biggr\} , \quad i=1,2,\ldots, \end{aligned}$$
(4.9)
$$\begin{aligned}& r_{1}(t)=\vert x_{0}\vert +w(t),\quad\quad w(t)= \int_{t_{0}}^{t} \biggl[\frac{2}{3 \sqrt{3}}{g^{\frac{3}{2}}(s)} {k_{1}^{{-\frac{1}{2}}}(s)}+\frac{1}{4} {h^{-1}(s)} {k_{2}^{{2}}(s)} \biggr]\,\mathrm {d}s, \end{aligned}$$
(4.10)
$$\begin{aligned}& r_{i+1}(t)=r_{i}(t)+ \int_{t_{i-1}}^{t_{i}}\bigl[f(s)+k(s)\bigr]v_{i}(s) \,\mathrm {d}s+l(t)\beta_{i}v_{i}^{3}(t_{i}-0), \quad i=1,2,\ldots, \end{aligned}$$
(4.11)
$$\begin{aligned}& \hbox{and} \quad l(t)=\max_{t_{0}\leq\tau\leq t}\bigl\vert a(t) \bigr\vert . \end{aligned}$$
(4.12)

Proof

The solution \(x(t)\) of Eq. (4.6) satisfies the following equivalent equation:

$$\begin{aligned} x(t)=x_{0} + \int_{t_{0}}^{t} \bigl(f(s)x(s)+g(s)x^{\frac{1}{3}}(s)-h(s)x ^{2}(s) \bigr)\,\mathrm {d}s+a(t)\sum_{t_{0}< t_{i}< t} \beta_{i}x^{3}(t_{i}-0). \end{aligned}$$

From the assumptions of f, g and h, it follows

$$\begin{aligned} \bigl\vert x(t)\bigr\vert \leq& \vert x_{0}\vert + \int_{t_{0}}^{t} \bigl(f(s)\bigl\vert x(s)\bigr\vert +g(s)\bigl\vert x(s)\bigr\vert ^{ \frac{1}{3}}-h(s)\bigl\vert x(s)\bigr\vert ^{2} \bigr)\,\mathrm {d}s \\ &{}+a(t)\sum_{t_{0}< t_{i}< t}\beta_{i}\bigl\vert x(t_{i}-0)\bigr\vert ^{3}, \quad t\geq t_{0}. \end{aligned}$$

By using Theorem 3.3, we easily obtain estimates (4.7) and (4.8) of solutions of Eq. (4.6). □