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Some new integral inequalities with mixed nonlinearities for discontinuous functions

Advances in Difference Equations20182018:22

https://doi.org/10.1186/s13662-017-1450-5

  • Received: 20 July 2017
  • Accepted: 13 December 2017
  • Published:

Abstract

In this paper, we establish some new integral inequalities with mixed nonlinearities for discontinuous functions, which provide a handy tool in deriving the explicit bounds for the solutions of impulsive differential equations and differential-integral equations with impulsive conditions.

Keywords

  • integral inequalities
  • discontinuous functions
  • mixed nonlinearities
  • impulsive differential equations

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, [218] 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). □

Declarations

Acknowledgements

The author thanks the reviewers for their helpful and valuable suggestions and comments on this paper. This research was supported by the National Natural Science Foundation of China (No. 11671227) A Project of Shandong Province Higher Educational Science and Technology Program (China) (No. J14LI09).

Authors’ contributions

All authors read and approved the final manuscript.

Competing interests

The author declares that he has no competing interests.

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)
School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, P.R. China

References

  1. Bainov, DD, Simeonov, PS: Impulsive Differential Equations: Periodic Solution and Applications. Longman, Harlow (1993) MATHGoogle Scholar
  2. Mitropolskiy, YuA, Iovane, G, Borysenko, SD: About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their applications. Nonlinear Anal. 66, 2140-2165 (2007) MathSciNetView ArticleMATHGoogle Scholar
  3. Borysenko, SD: About one integral inequality for piece-wise continuous functions. In: Proc. X Int. Kravchuk Conf., Kyiv, p. 323 (2004) Google Scholar
  4. Iovane, G: Some new integral inequalities of Bellman-Bihari type with delay for discontinuous functions. Nonlinear Anal. 66, 498-508 (2007) MathSciNetView ArticleMATHGoogle Scholar
  5. Gallo, A, Piccirillo, AM: About new analogies of Gronwall-Bellman-Bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems. Nonlinear Anal. 67, 1550-1559 (2007) MathSciNetView ArticleMATHGoogle Scholar
  6. Borysenko, SD, Ciarletta, M, Iovane, G: Integro-sum inequalities and motion stability of systems with impulse perturbations. Nonlinear Anal. 62, 417-428 (2005) MathSciNetView ArticleMATHGoogle Scholar
  7. Borysenko, S, Iovane, G: About some new integral inequalities of Wendroff type for discontinuous functions. Nonlinear Anal. 66, 2190-2203 (2007) MathSciNetView ArticleMATHGoogle Scholar
  8. Gallo, A, Piccirillo, AM: On some generalizations Bellman-Bihari result for integro-functional inequalities for discontinuous functions and their applications. Bound. Value Probl. 2009, Article ID 808124 (2009) MathSciNetView ArticleMATHGoogle Scholar
  9. Iovane, G: On Gronwall-Bellman-Bihari type integral inequalities in several variables with retardation for discontinuous functions. Math. Inequal. Appl. 11, 599-606 (2008) MathSciNetMATHGoogle Scholar
  10. Gallo, A, Piccirillo, AM: About some new generalizations of Bellman-Bihari results for integro-functional inequalities with discontinuous functions and applications. Nonlinear Anal. 71, 2276-2287 (2009) MathSciNetView ArticleMATHGoogle Scholar
  11. Deng, SF, Prather, C: Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay. J. Inequal. Pure Appl. Math. 9, Article ID 34 (2008) MathSciNetMATHGoogle Scholar
  12. Wang, WS, Zhou, XL: A generalized Gronwall-Bellman-Ou-Iang type inequality for discontinuous functions and applications to BVP. Appl. Math. Comput. 216, 3335-3342 (2010) MathSciNetMATHGoogle Scholar
  13. Shao, J, Meng, FW: Nonlinear impulsive differential and integral inequalities with integral jump conditions. Adv. Differ. Equ. 2016, 112 (2016) MathSciNetView ArticleGoogle Scholar
  14. Zheng, B: Some generalized Gronwall-Bellman type nonlinear delay integral inequalities for discontinuous functions. J. Inequal. Appl. 2013, 297 (2013) MathSciNetView ArticleMATHGoogle Scholar
  15. Mi, YZ, Deng, SF, Li, XP: Nonlinear integral inequalities with delay for discontinuous functions and their applications. J. Inequal. Appl. 2013, 430 (2013) MathSciNetView ArticleMATHGoogle Scholar
  16. Mi, YZ: Generalized integral inequalities for discontinuous functions with two independent variables and their applications. J. Inequal. Appl. 2014, 524 (2014) MathSciNetView ArticleMATHGoogle Scholar
  17. Wang, WS, Li, ZZ, Tang, AM: Nonlinear retarded integral inequalities for discontinuous functions and their applications. In: Computer, Informatics, Cybernetics and Applications, vol. 107, pp. 149-159 (2012) View ArticleGoogle Scholar
  18. Zheng, ZW, Gao, X, Shao, J: Some new generalized retarded inequalities for discontinuous functions and their applications. J. Inequal. Appl. 2016, 7 (2016) MathSciNetView ArticleMATHGoogle Scholar
  19. Li, YS: The bound, stability and error estimates of the solution of nonlinear different equations. Acta Math. Sin. 12(1), 28-36 (1962) (in Chinese) Google Scholar
  20. Tian, YZ, Cai, YL, Li, LZ, Li, TX: Some dynamic integral inequalities with mixed nonlinearities on time scales. J. Inequal. Appl. 2015, 12 (2015) MathSciNetView ArticleMATHGoogle Scholar

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