1 Introduction
As is well known, impulsive differential and impulsive integral inequalities play a fundamental part in the study of theory of impulsive equations (see [
1‐
4]). Recently, a lot of experts studied the global existence, uniqueness, bounded-ness, stability, oscillation and other properties of different impulsive inequalities (see [
5‐
18]). For example, in [
1], Lakshmikanthan investigated an impulsive differential inequality given as Theorem
1.1.
Let \(0\leq t_{0}< t_{1}< t_{2}<\cdots\) be a sequence, \(\lim_{k\rightarrow \infty}t_{k}=\infty\), \(\mathbb{R}_{+}=[0, +\infty)\). For \(I\subset\mathbb {R}\), we define the following set of functions:
\(PC(\mathbb{R}_{+}, I)\) = {\(u:\mathbb{R}_{+}\rightarrow I\); \(u(t)\) is continuous for \(t\neq t_{k}\), \(u(0^{+})\), \(u(t_{k}^{+})\), \(u(t_{k}^{-})\) exist, and \(u(t)\) is left-continuous at \(t_{k}\), \(k=1, 2, \ldots\)};
\(PC^{1}(\mathbb{R}_{+}, I)\) = {\(u\in PC(\mathbb{R}_{+}, I)\); \(u'(t)\) is continuous for \(t\neq t_{k}\), \(u'(0^{+})\), \(u'(t_{k}^{+})\), \(u'(t_{k}^{-})\) exist, and \(u'(t)\) is left-continuous at \(t_{k}\), \(k=1, 2, \ldots\)}.
In [
12], Thiramanus and Tariboon developed the impulsive inequalities with the following integral jump conditions:
$$ m\bigl(t_{k}^{+}\bigr)\leq d_{k}m(t_{k})+c_{k} \int_{t_{k}-\tau_{k}}^{t_{k}-\sigma _{k}}m(s)\,ds+b_{k},\quad k=1, 2, \ldots, $$
(1.4)
where
\(0\leq\sigma_{k}\leq\tau_{k}\leq t_{k}-t_{k-1}\). In [
11], Liengtragulngam et al. generalized further results by replacing the integral jump conditions (
1.2) by the following nonlocal jump conditions:
$$ m\bigl(t_{k}^{+}\bigr)\leq\frac{c_{k}}{\Gamma(\beta_{k})} \int ^{t_{k}}_{t_{k-1}}(t_{k}-s)^{\beta_{k}-1} m(s)\,ds+d_{k}m(t_{k})+b_{k}. $$
(1.5)
We note that a weak singular kernel is involved in the nonlocal jump conditions. They gave the estimation of
\(m(t)\) as follows.
These results play fundamental roles in the global existence, uniqueness, stability and other properties of various linear impulsive differential and integral equations.
A lot of authors just study the qualitative properties of linear impulsive inequalities. However, most of the phenomena in the world do not change linearly, such as heart beat, blood pressure, and so on. Hence the nonlinear impulsive differential and integral theories are more accurate than linear impulsive theories in various aspects.
In this paper, we extend the theories of linear impulsive system to nonlinear impulsive inequalities with nonlocal jump conditions. We consider the following nonlinear inequality:
$$ m'(t)\leq p(t)m(t)+q(t)m^{\alpha}(t),\quad t\neq t_{k}, $$
with different nonlocal jump conditions, we give the upper bound estimation of the inequality, and an estimation of solutions of certain nonlinear equations is also involved.
For convenience, we give the following lemmas.
2 Nonlinear impulsive inequalities with nonlocal jump conditions
In this section, we present and prove some new nonlinear impulsive differential and integral inequalities with nonlocal jump conditions. Let \(t_{l}=\max\{t_{k}:t\geq t_{k}, k=1, 2, \ldots\}\).
If
\(d_{k}\equiv1\) in Theorem
2.1, we obtain the following theorem.
Using different estimating methods, we have the following results.
If
\(d_{k}\equiv0\) and
\(p(t)\) is constant function in Theorem
2.2, we obtain the following corollary.
If
\(p(t)\equiv0\) and
\(d_{k}\equiv1\) in Theorem
2.3, we obtain the following corollary.
Next, we give another kind of nonlinear impulsive differential inequalities.
Now, we present and prove a bound for the solutions of nonlinear impulsive integral inequalities with nonlocal jump conditions.
3 Impulsive fractional differential and integral equations with integral jump conditions
In this section, we give some examples about impulsive nonlinear differential and integral inequalities with Riemann–Liouville fractional integral jump conditions.
As a special case, we consider the following initial value problem of impulsive differential equation with finite discontinuous points.
Acknowledgements
The authors sincerely thank the referees for constructive suggestions and corrections, which have significantly improved the contents and the exposition of the paper.
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