Abstract
Some new nonlinear impulsive differential and integral inequalities with nonlocal integral jump conditions are presented in this paper. Using the method of mathematical induction, we obtain a new upper bound estimation of certain differential and integral inequalities; these inequalities have both nonlocal integral jump and weakly singular kernels. Finally, we give two examples of these inequalities in estimating solutions of certain equations with Riemann–Liouville fractional integral conditions.
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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 \(t_{k}\), \(k=1, 2, \ldots\)}.
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 \(p, q\in C[\mathbb{R}_{+}, \mathbb{R}]\), \(d_{k}\geq0\) and \(b_{k}\) (\(k=1, 2, \ldots\)) are constants. Then
$$\begin{aligned} m(t) \leq& m(t_{0})\prod_{t_{0}< t_{k}< t}d_{k}e^{\int^{t}_{t_{0}}p(\xi )\,d\xi }+ \sum_{t_{0}< t_{k}< t}\biggl(\prod_{t_{k}< t_{j}< t}d_{j}e^{\int ^{t}_{t_{k}}p(\xi)\,d\xi} \biggr)b_{k} \\ &{}+ \int^{t}_{t_{0}}\prod_{s< t_{k}< t}d_{k}e^{\int^{t}_{s}p(\xi)\,d\xi }q(s) \,ds,\quad t\geq t_{0}. \end{aligned}$$(1.3)
In [12], Thiramanus and Tariboon developed the impulsive inequalities with the following integral jump conditions:
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:
We note that a weak singular kernel is involved in the nonlocal jump conditions. They gave the estimation of \(m(t)\) as follows.
Theorem 1.2
Let (H0) and (H1) be true. Suppose that \(p, q\in C [\mathbb{R}_{+}, \mathbb{R}]\) and for \(k=1, 2, \ldots\) , \(t\geq t_{0}\),
where \(c_{k}, d_{k}\geq0\), \(\beta_{k}>0\) and \(b_{k}\) (\(k=1, 2, \ldots\)) are constants. Then, for all \(t\geq t_{0}\),
where \(t_{l}=\max\{t_{k}:t\geq t_{k}, k=1, 2, \ldots\}\).
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:
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.
Lemma 1.1
([9])
Assume that \(a, b\in\mathbb{R}\), \(p\geq 0\). Then
where \(C_{p}=1\) for \(0\leq p\leq1\), and \(C_{p}=2^{p-1}\) for \(p>1\).
Lemma 1.2
Let \(\{a_{n}\}\), \(\{b_{n}\}\) be two sequences of numbers. Then we have
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 \(t_{l}=\max\{t_{k}:t\geq t_{k}, k=1, 2, \ldots\}\).
Theorem 2.1
Let (H0) and (H1) hold. Suppose that \(p, q\in C[\mathbb{R}_{+}, \mathbb{R}]\) and for \(k=1, 2, \ldots\) , \(t\geq t_{0}\),
where \(0<\alpha<1\), \(c_{k}, d_{k}\geq0\), \(\beta_{k}>0\) and \(b_{k}\) (\(k=1, 2, \ldots\)) are constants. Then, for \(t\geq t_{0}\), we have
Proof
Let
Then applying (2.1), we can get
The inequality (2.2) can be written as
Next, we set
then (2.3) reduces to
By the definition of \(v(t)\), we just need to prove
We prove it by induction. For \(t\in[t_{0}, t_{1}]\), inequality (2.4) can be written as
Integrating (2.10) from \(t_{0}\) to t, for \(t\in[t_{0}, t_{1}]\), we have
Hence (2.9) is valid on \([t_{0}, t_{1}]\). Assume that (2.9) 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 (2.4) and (2.11) that
Applying (2.5) with (2.12), we get
By induction and (2.13), we get
Hence (2.9) is valid on \([t_{n}, t_{n+1}]\). Therefore, the inequalities (2.9) is valid for \(t_{0}\leq t\leq t_{n+1}\). We know that \(v(t)=m^{1-\alpha}(t)\), this completes the proof of Theorem 2.1. □
If \(d_{k}\equiv1\) in Theorem 2.1, we obtain the following theorem.
Theorem 2.2
Suppose that (H0) and (H1) hold, \(p, q\in C[\mathbb{R}_{+}, \mathbb{R}]\) and for \(k=1, 2, \ldots\) , \(t\geq t_{0}\),
where \(0<\alpha<1\), \(c_{k}, d_{k}\geq0\), \(\beta_{k}>0\) and \(b_{k}\) (\(k=1, 2, \ldots\)) are constants, \(\Delta m^{1-\alpha}(t_{k})=m^{1-\alpha }(t_{k}^{+})-m^{1-\alpha}(t_{k})\). Then we have the following inequality:
where
Proof
As the proof of Theorem 2.1, from (2.11) we have
which means that (2.17) holds for \(t\in[t_{0}, t_{1}]\).
Now we use the method of mathematical induction; suppose that (2.17) holds for \(t\in[t_{0}, t_{n}]\), then
Then, by (2.16),
By Lemma 1.2, we have
For \(t\in[t_{n}, t_{n+1}]\), (2.18) can be replaced by
Therefore, the estimate (2.17) holds for \(t\in[t_{0}, t_{n+1}]\). This completes the proof. □
Using different estimating methods, we have the following results.
Theorem 2.3
Suppose that all the hypotheses of Theorem 2.1 are fulfilled. Then, for \(t\geq{t_{0}}\), we have:
(i) For \(k=1, 2,\ldots \) , the following estimation holds:
where \(\mu_{k}=\beta_{k}+1\) and \(\nu_{k}=1+\frac{1}{\beta_{k}}\).
(ii) If we assume further that \(\beta_{k}>\frac{1}{2}\), then, for \(k=1, 2,\ldots \) , we have the following estimation:
Proof
To prove (i), we use the Hölder inequality. For \(k=1, 2,\ldots \) , we get
In fact, by changing the variable, we get
and
Substituting the above inequalities in (2.3), we obtain the desired inequality in (2.20).
To prove (ii), since in this case, \(2\beta_{k}-1>0\), \(\Gamma(2\beta_{k}-1)\) are well defined for \(k=1, 2,\ldots \) . We use the Cauchy–Schwartz inequality to get
and
Substituting these two inequalities in (2.3), we get the desired results. The proof is completed. □
If \(d_{k}\equiv0\) and \(p(t)\) is constant function in Theorem 2.2, we obtain the following corollary.
Corollary 2.4
Suppose (H0) and (H1) hold, and for \(q\in C[\mathbb{R}_{+}, \mathbb{R}]\), \(k=1, 2, \ldots\) , \(t\geq t_{0}\),
where \(0<\alpha<1\), \(c_{k}, d_{k}\geq0\), \(\beta_{k}>0\) and \(b_{k}\) (\(k=1, 2, \ldots\)) are constants. Then, for \(t\geq t_{0}\), we have the following estimates.
Case I: \(p\neq\frac{1}{1-\alpha}\):
(i) For \(k=1, 2,\ldots \) , the following estimation holds:
where
(ii) If we assume further \(\beta_{k}>\frac{1}{2}\), then, for \(k=1, 2, \ldots \) ,
where
Case II: \(p=\frac{1}{1-\alpha}\):
(i) For \(k=1, 2,\ldots \) , the following estimation holds:
where
(ii) If we assume further \(\beta_{k}>\frac{1}{2}\), then, for \(k=1, 2, \ldots \) ,
where
If \(p(t)\equiv0\) and \(d_{k}\equiv1\) in Theorem 2.3, we obtain the following corollary.
Corollary 2.5
Let (H0) and (H1) hold, and for \(q\in C[\mathbb{R}_{+}, \mathbb{R}]\), \(k=1, 2, \ldots\) , \(t\geq t_{0}\)
where \(0<\alpha<1\), \(c_{k}, d_{k}\geq0\), \(\beta_{k}>0\) and \(b_{k}\) (\(k=1, 2, \ldots\)) are constants, \(\Delta m^{1-\alpha}(t_{k})=m^{1-\alpha }(t_{k}^{+})-m^{1-\alpha}(t_{k})\). Then, for \(t\geq t_{0}\), we have:
(i) For \(k=1, 2,\ldots \) , the following estimation holds:
where
(ii) If we assume further \(\beta_{k}>\frac{1}{2}\), then, for \(k=1, 2, \ldots \) ,
where
Next, we give another kind of nonlinear impulsive differential inequalities.
Theorem 2.6
Suppose that (H0) and (H1) hold, \(p, q\in C[\mathbb{R}_{+}, \mathbb{R}]\) and for \(k=1, 2, \ldots\) , \(t\geq t_{0}\),
where \(0<\alpha<1\), \(c_{k}, d_{k}\geq0\), \(\beta_{k}>0\) and \(b_{k}\) (\(k=1, 2, \ldots\)) are constants, \(\Delta m(t_{k})=m(t_{k}^{+})-m(t_{k})\). Then we have the following estimation:
where
Proof
Obviously, using (2.11), we have (2.34) holds for \(t\in [t_{0}, t_{1}]\). Suppose (2.34) holds for \(t\in[t_{0}, t_{n}]\), then, by mathematical induction, we see that
Since \(\frac{1}{1-\alpha}>1\), by Lemma 1.1,
thus
Then, for \(t\in[t_{n}, t_{n+1}]\), since \(0<1-\alpha<1\), by Lemma 1.1 and (2.11), we obtain
Therefore, the estimation (2.34) is valid on \([t_{n}, t_{n+1}]\). This completes the proof. □
Now, we present and prove a bound for the solutions of nonlinear impulsive integral inequalities with nonlocal jump conditions.
Theorem 2.7
Assume that (H0) and (H1) hold, \(p, q, m \in C[\mathbb{R}_{+}, \mathbb{R}_{+}]\), and, for \(k=1, 2, \ldots\) , \(t\geq t_{0}\),
where \(0<\alpha<1\), \(\gamma_{k}\geq0\), \(\beta_{k}>0\) (\(k=1, 2, \ldots\)) and c are constants. Then, for \(t\geq t_{0}\), the following assertions hold:
where
Proof
We denote by \(g(t)\) the right-side function of (2.35), and \(g(t_{0})=c\). Then we get
Since \(m(t)\leq g(t)\), we have
Applying Theorem 2.6, we deduce the estimation of \(g(t)\) as
Moreover, \(m(t)\leq g(t)\), this completes the proof. □
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.
Definition 3.1
The Riemann–Liouville fractional integral of order \(\alpha>0\) of a function f: \([t_{0}, \infty)\rightarrow\mathbb{R}\) is defined by
where \(\Gamma(\cdot)\) is the Gamma function.
Proposition 3.2
Suppose that \(y\in PC^{1}[J, \mathbb{R}]\) which satisfies
where \(R>0\), \(a\in C[\mathbb{R}_{+}, \mathbb{R}_{+}]\), \(0=t_{0}< t_{1}< t_{2}<\cdots<t_{n}<t_{n+1}=T\), \(0<\alpha<1\), \(c_{k}, b_{k}\geq0\), \(\beta_{k}>0\) (\(k=1, 2, \ldots, n\)) and θ are constants. If either of the following four cases fulfilled:
(i) \(R\neq\frac{1}{1-\alpha}\) and for \(k=1, 2,\ldots, n\), the following hypotheses hold:
where \(\mu_{k}=\beta_{k}+1\) and \(\nu_{k}=1+\frac{1}{\beta_{k}}\);
(ii) \(R\neq\frac{1}{1-\alpha}\), \(\beta_{k}>\frac{1}{2}\), and for \(k=1, 2, \ldots, n\),
and (\(\mathrm{P}_{3}\)) holds;
(iii) \(R=\frac{1}{1-\alpha}\) and for \(k=1, 2, \ldots, n\),
(iv) \(R=\frac{1}{1-\alpha}\), \(\beta_{k}>\frac{1}{2}\), and for \(k=1, 2, \ldots, n\),
and (\(\mathrm{P}_{8}\)) holds. Then we get \(y(t)\leq0\) for \(t\in[0, T]\).
Proof
Firstly, we use Case I(i) of Corollary 2.4 to prove (i). For \(t\in[0, T]\), we have
where
It is obvious that \(\widetilde{A_{k}}\geq0\) for all \(k=1, 2, \ldots, n\). In fact, for the case of \((1-\alpha)R>1\) and \((1-\alpha)R\leq1\), both the denominator and the numerator in \(\widetilde{A_{k}}\) have the same sign, hence we get \(\widetilde{A_{k}}\geq0\). The condition (\(\mathrm{P}_{2}\)) implies that \(\widetilde{B_{k}}\geq0\) for all \(k=1, 2,\ldots n\). Then it is easy to show that \(y(0)\leq0\). In fact, for \(t=T\), we have
that is to say,
Using the hypothesis \(y^{1-\alpha}(0)=y^{1-\alpha}(T)+\theta\) and (\(\mathrm{P}_{1}\)), (\(\mathrm{P}_{2}\)), we get
which implies that \(y^{1-\alpha}(0)\leq0\), since \(0<\alpha<1\), we get \(y(0)\leq0\).
To prove (ii), applying Case I(ii) of Corollary 2.4 for \(t\in[0, T]\), we have
where
Then using a similar method to proof of Case I(i) with conditions (\(\mathrm{P}_{1}\)) and (\(\mathrm{P}_{3}\)), we deduce \(y(0)\leq0\).
Next, we prove (iii). Applying Case II(i) of Corollary 2.4 for \(t\in [0, T]\), we have
where
It is easy to see that \(\widetilde{E_{k}}\geq0\) and \(\widetilde {F_{k}}\leq0\) for all \(k=1, 2, \ldots, n\). Then using a similar method to proof (i) with conditions (\(\mathrm{P}_{6}\)) and (\(\mathrm{P}_{7}\)), it is easy to show that \(y(0)\leq0\).
Similarly, to prove (iv), we apply Case II(ii) of Corollary 2.4 for \(t\in[0, T]\), we get
where
Then using a similar method as that of (iii), it is easy to show that \(y(0)\leq0\). This completes the proof. □
Example 3.3
Let \(x\in PC^{1}[\mathbb{R}_{+}, \mathbb{R}]\), and for \(k=1, 2, \ldots\) , we suppose
where \(f\in C(\mathbb{R}_{+}\times\mathbb{R}, \mathbb{R})\), \(X_{k}\in C(\mathbb{R}, \mathbb{R})\), \(0\leq t_{0}< t_{1}< t_{2}<\cdots\), \(\lim_{k\rightarrow\infty}t_{k}=\infty\), \(\Delta x(t_{k})=x(t_{k}^{+})-x(t_{k})\), \(\beta_{k}>0\) (\(k=1, 2, \ldots\)) and \(x_{0}\) are constants. Assume there exists a constant \(L>0\), such that
and there exists a constant \(M_{k}>0\), such that
Then, for \(t\geq t_{0}\), the following inequalities hold:
where
Proof
Suppose \(x=x(t)\) is a solution of (3.2), we integrate this equation to obtain
then (3.3) and (3.4) imply that
Then Theorem 2.7 yields the estimate of (3.5), and
□
As a special case, we consider the following initial value problem of impulsive differential equation with finite discontinuous points.
Example 3.4
Consider the initial value problem of the form
where \(f(x)= \big\{\scriptsize{\begin{array}{l@{\quad}l} 2\sqrt{|x|}, & |x|<1, \\ \sqrt[3]{x}+1, & |x|\geq1, \end{array}}\) \(t_{k}=k\), \(\beta_{k}=\frac{1}{k+1}\) for \(1\leq k \leq10\). In this case, we see that \(|f(x)|\leq2\sqrt{|x|}\) with \(L=2\) and \(\alpha=\frac{1}{2}\), and \(|X_{k}(x)|=|x|\) with \(M_{k}=1\). By direct calculation, we get
Then the solution of the initial value problem (3.6) can be estimated as
Moreover, for \(t\geq10\), we have
for some constant calculated through (3.7) with \(k=10\).
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Acknowledgements
The authors sincerely thank the referees for constructive suggestions and corrections, which have significantly improved the contents and the exposition of the paper.
Funding
This project is supported by the NNSF of China (Grants 11671227 and 11271225), NSF of Shandong (Grant No. ZR2018LA004), and Science and Technology Project of High Schools of Shandong Province (Grant Nos. J18KA220, J18KB107).
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ZZ came with the main thoughts and helped to draft the manuscript, YZ proved the main theorems, JS revised the paper. All authors read and approved the final manuscript.
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Zheng, Z., Zhang, Y. & Shao, J. Nonlinear impulsive differential and integral inequalities with nonlocal jump conditions. J Inequal Appl 2018, 170 (2018). https://doi.org/10.1186/s13660-018-1762-3
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DOI: https://doi.org/10.1186/s13660-018-1762-3