Gronwall-type integral inequalities with impulses on time scales

Abstract

In this article, some Gronwall-type integral inequalities with impulses on time scales are investigated. Our results extend some known dynamic inequalities on time scales, unify and extend some continuous inequalities and their corresponding discrete analogues. Some applications of the main results are given in the end of this article.

AMS (MOS) Subject Classification 34D09; 34D99, 37M10, 35D05, 49K25, 90C46.

Introduction

The theory of time scales, which has recently received a lot of attention, was initiated by Hilger [1] in his Ph.D. thesis in 1988 to contain both difference and differential calculus in a consistent way. Since then many authors have investigated the dynamic equations, the calculus of variations and the optimal control problem on time scales (see [2–11]). At the same time, a few papers have studied the theory of integral inequalities on time scales (see [12–14]).

In this article, we study some Gronwall-type integral inequalities on time scales, which extend some known dynamic inequalities on time scales, unify and extend some continuous inequalities and their corresponding discrete analogues. It is helpful in our result to study dynamic systems and optimal control problem on time scales.

Preliminaries

For convenience, we present some important theorem on time scales in this section. The approach is based on the ideas in [9] and will be of fundamental importance in following results.

A time scale $T$ is a nonempty closed subset of R. The two most popular examples are $T=R$ and $T=Z$. Define the forward and backward jump operators $\sigma ,\rho :T\to T$ by

$$\sigma \left(t\right)=inf\left\{s\in T|s>t\right\},\rho \left(t\right)=sup\left\{s\in T|s<t\right\},t\in T,$$

where, in this definition, we write $sup\varnothing =infT\equiv a$ and $inf\varnothing =supT\equiv b$. A point $t\in T$ is said to be left-dense, left-scattered, right-dense, right-scattered if ρ(t) = t, ρ (t) < t, σ (t) = t, σ (t) > t, respectively. The forward (backward) graininess $\mu :T\to \left[0,+\infty \right)\left(\nu :T\to \left[0,+\infty \right)\right)$ is defined by μ(t) = σ(t) - t(v(t) = t - ρ(t)). Define $\Lambda =\left\{t_i\in T|i=1,2,\cdots ,n\right\}$, $P{C}_{lrd}\left(T,R\right)\left(P{C}_{rrd}\left(T,R\right)\right)=\{x:T\to R|x$ is continuous at right-dense point $t\in T\backslash \Lambda$, x exists right limit at t ∈ Λ or left-dense point $t\in T$, x is left (right) continuous and exists right (left) limit at t ∈ Λ}. Endowed with norm

$$\left|\right|x|{|}_{PC}=max\left\{\underset{t\in T}{sup}\left|\right|x\left(t+\right)\left|\right|,\underset{t\in T}{sup}\left|\right|x\left(t-\right)\left|\right|\right\}.$$

$\left\{P{C}_{lrd}\left(T,X\right),\left|\right|\cdot |{|}_{PC}\right\}$ and $\left\{P{C}_{rrd}\left(T,X\right),\left|\right|\cdot |{|}_{PC}\right\}$ are Banach spaces.

Theorem 2.1. (1) Let $f\in L^1\left(T,R\right)$. Then

$$F\left(t\right)=\underset{\left[a,t\right)}{\int }f\left(\tau \right)\Delta \tau ,t\in T$$

implies $F\in C_{rd}\left(T,R\right)$ is Δ-differentiable Δ-a.e. on $T$ and

$$F\left(t\right)-F\left(s\right)=\underset{\left[s,t\right)}{\int }f\left(\tau \right)\Delta \tau fors,t\in T.$$

• (2) If f and g are Δ-differentiable Δ-a.e. on $T$, then

  $$\underset{\left[s,t\right)}{\int }\left[f^{\Delta }\left(\tau \right)g\left(\tau \right)+f^{\sigma }\left(\tau \right)g^{\Delta }\left(\tau \right)\right]\Delta \tau =f\left(t\right)g\left(t\right)-f\left(s\right)g\left(s\right)fors,t\in T.$$

The exponential function e _p on time scale plays a very important role for discussing dynamic equations on time scales. Define ${\Gamma }_1\left(T\right)=\left\{p\in L_{\mathsf{\text{loc}}}^1\left(T,R\right)|1+\mu \left(t\right)p\left(t\right)\ne 0\right\}$. For any $p,q\in {\Gamma }_1\left(T\right)$, define p ⊕ q = p + q + μpq, $\ominus p=-\frac{p}{1+\mu p}$, $p\ominus q=\frac{p-q}{1+\mu q}$. Further, we can show that $p\oplus q,p\ominus q,\ominus p\in {\Gamma }_1\left(T\right)$. Define the generalized exponential function as follows:

$$e_p\left(t,s\right)=exp\left\{\underset{\left[s,t\right)}{\int }{\xi }_{\mu \left(\tau \right)}\left(p\left(\tau \right)\right)\Delta \tau \right\}.$$

Theorem 2.2. Assume that $p,q\in {\Gamma }_1\left(T\right)$, then the following hold:

• (1) e ₀(t, s) ≡ 1, e _p (t, t) ≡ 1, e _p (t, s)e _p (s, r) = e _p (t, r), e _p (σ(t), s) = [1 + μ(t)p(t)]e _p (t, s);

• (2) $e_p\left(t,s\right)=\frac{1}{e_p\left(s,t\right)}=e_{\ominus p}\left(s,t\right)$, e _p (t, s)e _q (t, s) = e _p⊕q (t, s), $\frac{e_p\left(t,s\right)}{e_q\left(t,s\right)}=e_{p\ominus q}\left(t,s\right)$;

• (3) $e_p\left(\cdot ,s\right)\in C_{rd}\left(T,R\right)$, (e _p (·, s))^Δ = p(·)e _p (·, s),e _p (s, ·))^Δ = -p(·)e _p (s, σ(·) Δ-a.e. on $T$.

Main results

In this section, we deal with Gronwall-type integral inequalities with impulses on time scales. For convenience, we always assume that $f\in L^1\left(T,R\right)$, p _i , $q_i\in L^1\left(T,R^{+}\right)$ with R ⁺ = [0, +∞), 0 < λ _i < 1 (i = 1, 2, 3, 4), α ≥ 0, β _k ≥ 0 (k = 1, 2, · · ·, n), $x_t=\underset{a\le s\le t}{sup}\left|x\left(s\right)\right|$, $x_{{\tau }_b}\underset{\tau \le s\le b}{sup}\left|x\left(s\right)\right|$ in the section.

Theorem A. (1) If $x\in C_{rd}\left(T,R\right)$ satisfies the following inequality

$$x^{\Delta }\left(t\right)\le p_1\left(t\right)x\left(t\right)+f\left(t\right)\Delta -a.e.onT$$

then

$$x\left(t\right)\le e_{p_1}\left(t,a\right)x\left(a\right)+\underset{\left[a,t\right)}{\int }{e_p}_1\left(t,\sigma \left(\tau \right)\right)f\left(\tau \right)\Delta \tau forallt\in T.$$

• (2) If $x\in P{C}_{lrd}\left(T,R^{+}\right)$ satisfies the following inequality

  $$x\left(t\right)\le \alpha +\underset{\left[a,t\right)}{\int }{p}_1\left(\tau \right)x\left(\tau \right)\Delta \tau +\sum _{t_k<t}{\beta }_{k}x\left(t_k\right),\forall t\in T,$$

then

$$x\left(t\right)\le \alpha \prod _{t_k<t}\left(1+{\beta }_k\right)e_{p_1}\left(t,a\right),\forall t\in T.$$

• (3) If $x\in P{C}_{lrd}\left(T,R^{+}\right)$ satisfies the following inequality

  $$x\left(t\right)\le \alpha +\underset{\left[a,t\right)}{\int }{p}_1\left(\tau \right)x\left(\tau \right)\Delta \tau +\underset{\left[a,t\right)}{\int }{q}_1\left(\tau \right)x^{{\lambda }_1}\left(\sigma \left(\tau \right)\right)\Delta \tau +\sum _{t_k<t}{\beta }_{k}x\left(t_k\right),\forall t\in T,$$

then there is a constant M > 0 such that

$$x\left(t\right)\le M,\forall t\in T.$$

• (4) If $x\in P{C}_{lrd}\left(T,R^{+}\right)$ satisfies the following inequality

  $$\begin{array}{lll}\hfill x\left(t\right)\le & \alpha +\underset{\left[a,t\right)}{\int }{p}_1\left(\tau \right)x\left(\tau \right)\Delta \tau +\underset{\left[a,t\right)}{\int }{p}_2\left(\tau \right)x_{\tau }\Delta \tau +\underset{\left[a,t\right)}{\int }{q}_1\left(\tau \right)x^{{\lambda }_1}\left(\sigma \left(\tau \right)\right)\Delta \tau & \hfill \text{(1)}\\ +\underset{\left[a,t\right)}{\int }{q}_2\left(\tau \right)x_{\sigma \left(\tau \right)}^{{\lambda }_2}\Delta \tau +\sum _{t_k<t}{\beta }_{k}x\left(t_k\right),\forall t\in T,& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

then there is a constant M > 0 such that

$$x\left(t\right)\le M,\forall t\in T.$$

• (5) If $x\in P{C}_{lrd}\left(T,R^{+}\right)$ satisfies the following inequality

  $$\begin{array}{lll}\hfill x\left(t\right)\le & \underset{\left[a,t\right)}{\int }{p}_1\left(\tau \right)x\left(\tau \right)\Delta \tau +\underset{\left[a,t\right)}{\int }{p}_2\left(\tau \right)x_{\tau }\Delta \tau +\underset{\left[a,t\right)}{\int }{q}_1\left(\tau \right)x^{{\lambda }_1}\left(\sigma \left(\tau \right)\right)\Delta \tau +\underset{\left[a,t\right)}{\int }{q}_2\left(\tau \right)x_{\sigma \left(\tau \right)}^{{\lambda }_2}\Delta \tau & \hfill \text{(1)}\\ +\alpha +\underset{\left[a,b\right)}{\int }{q}_3\left(\tau \right)x^{{\lambda }_3}\left(\tau \right)\Delta \tau +\underset{\left[a,b\right)}{\int }{q}_4\left(\tau \right)x_{\sigma \left(\tau \right)}^{{\lambda }_4}\Delta \tau +\sum _{t_k<t}{\beta }_{k}x\left(t_k\right),\forall t\in T,& \hfill \text{(2)}\\ \hfill \text{(3) }\end{array}$$

then there is a constant M > 0 such that

$$x\left(t\right)\le M,\forall t\in T.$$

Proof. (1) Note that $p_1\in L^1\left(T,R^{+}\right)$ implies $p_1\in {\Gamma }_1\left(T\right)$ and 1 + μ(t)p ₁(t) > 0 for all $t\in T$. Now

$${\left[x\left(t\right)e_{\ominus p_1}\left(t,a\right)\right]}^{\Delta }=\left[x^{\Delta }\left(t\right)-p_1\left(t\right)x\left(t\right)\right]e_{\ominus p_1}\left(\sigma \left(t\right),a\right)\Delta -\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.}}\mathsf{\text{on}}T.$$

Therefore,

$$x\left(t\right)e_{\ominus p_1}\left(t,a\right)-x\left(a\right)=\underset{\left[a,t\right)}{\int }\left[x^{\Delta }\left(\tau \right)-p_1\left(\tau \right)x\left(\tau \right)\right]e_{\ominus p_1}\left(\sigma \left(\tau \right),a\right)\Delta \tau \le \underset{\left[a,t\right)}{\int }f\left(\tau \right)e_{\ominus p_1}\left(\sigma \left(\tau \right),a\right)\Delta \tau ,$$

that is,

$$x\left(t\right)\le e_{p_1}\left(t,a\right)x\left(a\right)+\underset{\left[a,t\right)}{\int }{e}_{p_1}\left(t,\sigma \left(\tau \right)\right)f\left(\tau \right)\Delta \tau \mathsf{\text{for}}\mathsf{\text{all}}t\in T.$$

• (2) Define

  $$y\left(t\right)=\alpha +\underset{\left[a,t\right)}{\int }{p}_1\left(\tau \right)x\left(\tau \right)\Delta \tau +\sum _{t_k<t}{\beta }_{k}x\left(t_k\right),\forall t\in T.$$

By Theorem 2.1, y is Δ-differential Δ-a.e. on $T$ and

$$\left\{\begin{array}{cc}\hfill y^{\Delta }\left(t\right)=p_1\left(t\right)x\left(t\right)\le p_1\left(t\right)y\left(t\right),\hfill & \hfill t\ne t_k,\hfill \\ \hfill y\left(a\right)=\alpha ,y\left(t_k+\right)=\left(1+{\beta }_k\right)y\left(t_k\right).\hfill \end{array}\right.$$

For t ∈ [a, t ₁], it is obvious to

$$y\left(t\right)\le \alpha e_{p_1}\left(t,a\right).$$

Further, we have

$$y\left(t\right)\le y\left(t_i+\right)e_{p_1}\left(t,t_i\right)\le \alpha \left(1+{\beta }_i\right)\prod _{t_k<t_i}\left(1+{\beta }_k\right)e_{p_1}\left(t_i,a\right)e_{p_1}\left(t,t_i\right)=\alpha \prod _{t_k<t}\left(1+{\beta }_k\right)e_{p_1}\left(t,a\right),\forall t\in \left(t_i,t_{i+1}\right].$$

Thus,

$$x\left(t\right)\le \alpha \prod _{t_k<t}\left(1+{\beta }_k\right)e_p\left(t,a\right),\forall t\in T.$$

• (3) Setting

  $$y\left(t\right)=\alpha +\underset{\left[a,t\right)}{\int }{p}_1\left(\tau \right)x\left(\tau \right)\Delta \tau +\underset{\left[a,t\right)}{\int }{g}_1\left(\tau \right)x^{{\lambda }_1}\left(\sigma \left(\tau \right)\right)\Delta \tau +\sum _{t_k<t}{\beta }_{k}x\left(t_k\right),$$

then

$$\left\{\begin{array}{cc}\hfill y^{\Delta }\left(t\right)=p_1\left(t\right)x\left(t\right)+g_1\left(t\right)x^{{\lambda }_1}\left(\sigma \left(t\right)\right)\le p_1\left(t\right)y\left(t\right)+g_1\left(t\right)y^{{\lambda }_1}\left(\sigma \left(t\right)\right)\Delta -\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.}}\mathsf{\text{on}}T,\hfill & \hfill t\ne t_k,\hfill \\ \hfill y\left(a\right)=\alpha ,y\left(t_k+\right)=y\left(t_k\right)+{\beta }_{k}x\left(t_k\right)=\left(1+{\beta }_k\right)y\left(t_k\right).\hfill \end{array}\right.$$

Using the conclusion (1), we have

$$\begin{array}{lll}\hfill y\left(t\right)& \le \alpha \prod _{t_k<t}\left(1+{\beta }_k\right)e_{p_1}\left(t,a\right)+\prod _{t_k<t}\left(2+{\beta }_k\right)\underset{\left[a,t\right)}{\int }{e}_{p_1}\left(t,\sigma \left(\tau \right)\right)g_1\left(\tau \right)y^{{\lambda }_1}\left(\sigma \left(\tau \right)\right)\Delta \tau & \hfill \text{(1)}\\ \le \prod _{t_k<t}\left(2+{\beta }_k\right)\left[\alpha e_{p_1}\left(t,a\right)+\underset{\left[a,t\right)}{\int }{e}_{p_1}\left(t,\sigma \left(\tau \right)\right)g_1\left(\tau \right)y^{{\lambda }_1}\left(\sigma \left(\tau \right)\right)\Delta \tau \right].& \hfill \text{(2)}\\ \hfill \text{(3) }\end{array}$$

For $t\in T$, let

$$h\left(t\right)=\alpha e_{p_1}\left(b,a\right)+e_{p_1}\left(b,a\right)\underset{\left[a,t\right)}{\int }{g}_1\left(\tau \right)y^{{\lambda }_1}\left(\sigma \left(\tau \right)\right)\Delta \tau +e_{p_1}\left(b,a\right)\underset{\left[a,b\right)}{\int }{g}_1\left(\tau \right)y^{{\lambda }_1}\left(\sigma \left(\tau \right)\right)\Delta \tau ,$$

then h is monotone increasing function and $h\left(b\right)=2h\left(a\right)-\alpha e_{p_1}\left(b,a\right)$,

$$h^{\Delta }\left(t\right)=e_{p_1}\left(b,a\right)g_1\left(t\right)y^{{\lambda }_1}\left(\sigma \left(t\right)\right)\le \prod _{k=1}^n{\left(2+{\beta }_k\right)}^{{\lambda }_1}{e}_{p_1}\left(b,a\right)g_1\left(t\right)q^{{\lambda }_1}\left(t\right),\forall t\in T.$$

Δ-integrating from a to t, we obtain

$$h^{1-{\lambda }_1}\left(t\right)-h^{1-{\lambda }_1}\left(a\right)\le \rho \underset{\left[a,t\right)}{\int }{g}_1\left(\tau \right)\Delta \tau ,\forall t\in T,$$

where

$$\rho =\left(1-{\lambda }_1\right)e_{p_1}\left(b,a\right)\prod _{k=1}^n{\left(2+{\beta }_k\right)}^{{\lambda }_1}.$$

Now, we observe that

$${\left(2h\left(a\right)-\alpha e_{p_1}\left(b,a\right)\right)}^{1-{\lambda }_1}-h^{1-{\lambda }_1}\left(a\right)\le \rho \underset{\left[a,b\right)}{\int }{g}_1\left(\tau \right)\Delta \tau .$$

Letting

$$\Gamma \left(z\right)={\left(2z-\alpha e_{p_1}\left(b,a\right)\right)}^{1-{\lambda }_1}-z^{1-{\lambda }_1},$$

then $\Gamma \in C\left(\left[\frac{\alpha e_{p_1}\left(b,a\right)}{2},+\infty \right),R\right)$ and $\Gamma \left(\frac{\alpha e_{p_1}\left(b,a\right)}{2}\right)=-{\left(\frac{\alpha e_{p_1}\left(b,a\right)}{2}\right)}^{1-{\lambda }_1}<0$,

$$\underset{z\to +\infty }{lim}\Gamma \left(z\right)=\underset{z\to +\infty }{lim}\frac{\Gamma \left(z\right)}{z^{1-{\lambda }_1}}{z}^{1-{\lambda }_1}=\underset{z\to +\infty }{lim}\left[{\left(2-\frac{\alpha e_{p_1}\left(b,a\right)}{z}\right)}^{1-{\lambda }_1}-1\right]z^{1-{\lambda }_1}=+\infty .$$

Using the proof by contraction, one can show that there exists a constant M > 0 such that q(a) < M. Thus,

$$x\left(t\right)\le y\left(t\right)\le h\left(a\right)\le M,\forall t\in T.$$

• (4) Setting λ = max{λ ₁, λ ₂},

  $$z\left(t\right)=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill x\left(t\right)<1,\hfill \\ \hfill x\left(t\right),\hfill & \hfill x\left(t\right)\ge 1,\hfill \end{array}\right.p\left(t\right)=p_1\left(t\right)+p_2\left(t\right),g\left(t\right)=g_1\left(t\right)+g_2\left(t\right),\forall t\in T,$$

we have

$$x\left(t\right)\le \alpha +\underset{\left[a,t\right)}{\int }p\left(\tau \right)z_{\tau }\Delta \tau +\underset{\left[a,t\right)}{\int }g\left(\tau \right)z_{\sigma \left(\tau \right)}^{\lambda }\Delta \tau +\sum _{t_k<t}{\beta }_k{z}_{t_k},\forall t\in T.$$

Furthermore,

$$z_t\le \alpha +\underset{\left[a,t\right)}{\int }p\left(\tau \right)z_{\tau }\Delta \tau +\underset{\left[a,t\right)}{\int }g\left(\tau \right)z_{\sigma \left(\tau \right)}^{\lambda }\Delta \tau +\sum _{t_k<t}{\beta }_k{z}_{t_k},\forall t\in T,$$

By the conclusion (3), there is a constant M > 0 such that

$$x\left(t\right)\le M,\forall t\in T.$$

• (5) Setting λ = max{λ ₁, λ ₂, λ ₃, λ ₄},

  $$y\left(t\right)=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill x_t<1,\hfill \\ \hfill x_t,\hfill & \hfill x_t\ge 1,\hfill \end{array}\right.p\left(t\right)=p_1\left(t\right)+p_2\left(t\right),g\left(t\right)=\sum _{i=1}^4{g}_i\left(t\right),\forall t\in T.$$

we have

$$y\left(t\right)\le \alpha +\underset{\left[a,t\right)}{\int }p\left(\tau \right)y\left(\tau \right)\Delta \tau +\underset{\left[a,b\right)}{\int }g\left(\tau \right)y^{\lambda }\left(\tau \right)\Delta \tau +\sum _{t_k<t}{\beta }_{k}y\left(t_k\right).$$

By the conclusions (3) and (4), we can show that the conclusion (5) is true. The proof is completed. □

In Theorem A, we give some Gronwall-type generalized integral inequalities on time scales. Next, we give some backward Gronwall-type generalized integral inequalities on time scales which can not be directly obtained from Gronwall inequalities.

Theorem B. (1) If $x\in C_{rd}\left(T,R^{+}\right)$ satisfies the following inequality

$$x\left(t\right)\le f\left(t\right)+\underset{\left[t,b\right)}{\int }{p}_1\left(\tau \right)x^{\sigma }\left(\tau \right)\Delta \tau ,$$

then

$$x\left(t\right)\le f\left(t\right)+\underset{\left[t,b\right)}{\int }{e}_{p_1}\left(\tau ,t\right)p_1\left(\tau \right)f^{\sigma }\left(\tau \right)\Delta \tau fort\in T.$$

• (2) If $x\in P{C}_{rrd}\left(T,R^{+}\right)$ satisfies the following inequality

  $$x\left(t\right)\le \alpha +\underset{\left[t,b\right)}{\int }{p}_1\left(\tau \right)x^{\sigma }\left(\tau \right)\Delta \tau +\sum _{t_k>t}{\beta }_{k}x\left(t_k\right)$$

then

$$x\left(t\right)\le \alpha \prod _{t_k>t}\left(1+{\beta }_k\right)e_{p_1}\left(b,t\right),\forall t\in T.$$

(3) If $x\in P{C}_{rrd}\left(T,R^{+}\right)$ satisfies the following inequality

$$x\left(t\right)\le \alpha +\underset{\left[t,b\right)}{\int }{p}_1\left(\tau \right)x^{{\lambda }_1}\left(\tau \right)\Delta \tau +\underset{\left[t,b\right)}{\int }{g}_1\left(\tau \right)x^{\sigma }\left(\tau \right)\Delta \tau ,$$

then there is a constant M > 0 such that

$$x\left(t\right)\le M,\forall t\in T.$$

(4) If $x\in P{C}_{rrd}\left(T,R^{+}\right)$ satisfies the following inequality

$$\begin{array}{lll}\hfill x\left(t\right)\le & \alpha +\underset{\left[t,b\right)}{\int }{p}_1\left(\tau \right)x^{{\lambda }_1}\left(\tau \right)\Delta \tau +\underset{\left[t,b\right)}{\int }{p}_2\left(\tau \right)x_{{\tau }_b}^{{\lambda }_2}\Delta \tau +\underset{\left[t,b\right)}{\int }{g}_1\left(\tau \right)x\left(\sigma \left(\tau \right)\right)\Delta \tau & \hfill \text{(1)}\\ +\underset{\left[t,b\right)}{\int }{g}_2\left(\tau \right)x_{\sigma {\left(\tau \right)}_b}\Delta \tau +\sum _{t_k>t}{\beta }_{k}x\left(t_k\right),\forall t\in T,& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

then there is a constant M > 0 such that

$$x\left(t\right)\le M,\forall t\in T.$$

(5) If $x\in P{C}_{rrd}\left(T,R^{+}\right)$ satisfies the following inequality

$$\begin{array}{lll}\hfill x\left(t\right)\le & \underset{\left[t,b\right)}{\int }{p}_1\left(\tau \right)x^{{\lambda }_1}\left(\tau \right)\Delta \tau +\underset{\left[t,b\right)}{\int }{p}_2\left(\tau \right)x_{{\tau }_b}^{{\lambda }_2}\Delta \tau +\underset{\left[t,b\right)}{\int }{g}_1\left(\tau \right)x^{\sigma }\left(\tau \right)\Delta \tau +\underset{\left[t,b\right)}{\int }{g}_2\left(\tau \right)x_{\sigma {\left(\tau \right)}_b}\Delta \tau & \hfill \text{(1)}\\ +\alpha +\underset{\left[a,b\right)}{\int }{g}_3\left(\tau \right)x^{{\lambda }_3}\left(\tau \right)\Delta \tau +\underset{\left[a,b\right)}{\int }{g}_4\left(\tau \right)x_{\sigma {\left(\tau \right)}_b}^{{\lambda }_4}\Delta \tau +\sum _{t_k>t}{\beta }_{k}x\left(t_k\right),& \hfill \text{(2)}\\ \hfill \text{(3) }\end{array}$$

then there is a constant M > 0 such that

$$x\left(t\right)\le M,\forall t\in T.$$

Proof. (1) Define

$$y\left(t\right)=\underset{\left[t,b\right)}{\int }{p}_1\left(\tau \right)x^{\sigma }\left(\tau \right)\Delta \tau \mathsf{\text{for}}t\in T.$$

Then y(b) = 0 and

$$y^{\Delta }\left(t\right)=-p_1\left(t\right)x^{\sigma }\left(t\right)\ge -p_1\left(t\right)y^{\sigma }\left(t\right)-p_1\left(t\right)f^{\sigma }\left(t\right)\Delta -\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.}}\mathsf{\text{on}}T.$$

Note that,

$${\left[y\left(t\right)e_{p_1}\left(t,b\right)\right]}^{\Delta }=\left[y^{\Delta }\left(t\right)+p_1\left(t\right)y^{\sigma }\left(t\right)\right]e_{p_1}\left(t,b\right)\Delta -\mathsf{\text{a}}\mathsf{\text{.e}}\mathsf{\text{.}}\mathsf{\text{on}}T,$$

therefore,

$$-y\left(t\right)e_{p_1}\left(t,b\right)\ge -\underset{\left[t,b\right)}{\int }{p}_1\left(\tau \right)f^{\sigma }\left(\tau \right)e_{p_1}\left(\tau ,b\right)\Delta \tau .$$

Moreover, we obtain

$$x\left(t\right)\le f\left(t\right)+\underset{\left[t,b\right)}{\int }{e}_{p_1}\left(\tau ,t\right)p_1\left(\tau \right)f^{\sigma }\left(\tau \right)\Delta \tau \mathsf{\text{for}}t\in T.$$

• (2) Setting

  $$y\left(t\right)=\alpha +\underset{\left[t,b\right)}{\int }{p}_1\left(\tau \right)x^{\sigma }\left(\tau \right)\Delta \tau +\sum _{t_k>t}{\beta }_{k}x\left(t_k\right),\forall t\in T.$$

then y is Δ-differential Δ-a.e. on $T$ and

$$\left\{\begin{array}{cc}\hfill y^{\Delta }\left(t\right)=-p_1\left(t\right)x^{\sigma }\left(t\right)\ge -p_1\left(t\right)y^{\sigma }\left(t\right),\hfill & \hfill t\ne t_k,\hfill \\ \hfill y\left(t_k-\right)=y\left(t_k\right)+{\beta }_{k}x\left(t_k\right)\le \left(1+{\beta }_k\right)y\left(t_k\right),\hfill \\ \hfill y\left(b\right)=\alpha .\hfill \end{array}\right.$$

For t ∈ [t _n , b], by the conclusion (1) we have

$$y\left(t\right)\le \alpha e_{p_1}\left(b,t\right).$$

When t ∈ (t _n , t _n+1], By Theorem 2.2 and the conclusion (1) we also obtain

$$y\left(t\right)\le y\left(t_n-\right)e_{p_1}\left(t_n,t\right)\le \alpha \left(1+{\beta }_n\right)e_{p_1}\left(b,t\right).$$

Thus,

$$x\left(t\right)\le \alpha \prod _{t_k>t}\left(1+{\beta }_k\right)e_{p_1}\left(b,t\right),\forall t\in T.$$

• (3) Setting γ = (α + 1)(β + 1), β = ∫_[a,b) e _g (τ, a)g(τ)Δτ, then

  $$\begin{array}{lll}\hfill x\left(t\right)& \le \alpha +\underset{\left[t,b\right)}{\int }{p}_1\left(\tau \right)x^{{\lambda }_1}\left(\tau \right)\Delta \tau +\alpha \underset{\left[t,b\right)}{\int }{e}_{g_1}\left(\tau ,t\right)g_1\left(\tau \right)\Delta \tau +\underset{\left[t,b\right)}{\int }{e}_{g_1}\left(\tau ,t\right)g_1\left(\tau \right)\underset{\left[\tau ,b\right)}{\int }{p}_1\left(\nu \right)x^{{\lambda }_1}\left(\nu \right)\Delta \nu \Delta \tau & \hfill \text{(1)}\\ \le \gamma +\gamma \underset{\left[a,b\right)}{\int }{p}_1\left(\tau \right)x^{{\lambda }_1}\left(\tau \right)\Delta \tau .& \hfill \text{(2)}\\ \hfill \text{(3) }\end{array}$$

Letting

$$h\left(t\right)=\gamma +\gamma \underset{\left[t,b\right)}{\int }{p}_1\left(\tau \right)x^{\lambda }\left(\tau \right)\Delta \tau +\gamma \underset{\left[a,b\right)}{\int }{p}_1\left(\tau \right)x^{{\lambda }_1}\left(\tau \right)\Delta \tau ,$$

then h is monotone descending function and

$$h^{\Delta }\left(t\right)\ge -\gamma p_1\left(t\right)h^{\lambda }\left(t\right).$$

Δ-integrating from t to b, we obtain

$$h^{1-{\lambda }_1}\left(t\right)-h^{1-{\lambda }_1}\left(b\right)\le \left(1-{\lambda }_1\right)\gamma \underset{\left[t,b\right)}{\int }{p}_1\left(\tau \right)\Delta \tau .$$

Therefore,

$${\left(2h\left(b\right)-\gamma \right)}^{1-{\lambda }_1}-h^{1-{\lambda }_1}\left(b\right)\le \left(1-{\lambda }_1\right)\gamma \underset{\left[a,b\right)}{\int }{p}_1\left(\tau \right)\Delta \tau .$$

Using the method of the conclusion (3) in Theorem A, one can show that there is a constant M > 0 such that

$$x\left(t\right)\le M,\forall t\in T.$$

• (4) For $t\in T$, define

  $$z\left(t\right)=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill x\left(t\right)<1,\hfill \\ \hfill x\left(t\right),\hfill & \hfill x\left(t\right)\ge 1,\hfill \end{array}\right.$$

we have

$$x\left(t\right)\le \alpha +\underset{\left[t,b\right)}{\int }p\left(\tau \right)z_{{\tau }_b}\Delta \tau +\underset{\left[t,b\right)}{\int }g\left(\tau \right)z_{\sigma {\left(\tau \right)}_b}^{\lambda }\Delta \tau +\sum _{t_k>t}{\beta }_k{z}_{t_{kb}},\forall t\in T,$$

where λ = max{λ ₁, λ ₂}, p(t) = p ₁(t) + p ₂(t)g(t) = g ₁(t) + g ₂(t). Hence

$$z_{t_b}\le \alpha +\underset{\left[t,b\right)}{\int }p\left(\tau \right)z_{{\tau }_b}\Delta \tau +\underset{\left[t,b\right)}{\int }g\left(\tau \right)z_{\sigma {\left(\tau \right)}_b}^{\lambda }\Delta \tau +\sum _{t_k>t}{\beta }_k{z}_{t_{kb}},\forall t\in T,$$

By the conclusion (3), there exists a constant M > 0 such that

$$x\left(t\right)\le M,\forall t\in T.$$

• (5) Setting λ = max{λ ₁, λ ₂, λ ₃, λ ₄},

  $$y\left(t\right)=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill x_t<1,\hfill \\ \hfill x_t,\hfill & \hfill x_t\ge 1,\hfill \end{array}\right.p\left(t\right)=p_1\left(t\right)+p_2\left(t\right),g\left(t\right)=\sum _{i=1}^4{g}_i\left(t\right),\forall t\in T.$$

then we have

$$y\left(t\right)\le \alpha +\underset{\left[t,b\right)}{\int }p\left(\tau \right)y\left(\tau \right)\Delta \tau +\underset{\left[a,b\right)}{\int }g\left(\tau \right)y^{\lambda }\left(\tau \right)\Delta \tau +\sum _{t_k>t}{\beta }_{k}y\left(t_k\right).$$

Further, we also can prove that the conclusion (5) is hold. This completes the proof.

Remark 3.1: (1) If $T=R$, then the inequality established in Theorem A reduces to the inequality established by Peng and Wei in [15].

• (2) Using our main results, we can obtain many dynamic inequalities for some peculiar time scales. Due to limited space, their statements are omitted here.

Application

In this section, we present some applications of Theorems A and B to investigate certain properties of solutions of the following impulsive dynamic integral equation

$$\left\{\begin{array}{cc}\hfill x^{\Delta }\left(t\right)+p\left(t\right)x^{\sigma }\left(t\right)=f\left(t,x\left(t\right),x^{\sigma }\left(t\right),\left(Sx\right)\left(t\right),\left(Wx\right)\left(t\right)\right),\hfill & \hfill t\notin \Lambda ,\hfill \\ \hfill \left(Sx\right)\left(t\right)=\underset{\left[a,t\right)}{\int }k\left(t,\tau \right)s\left(\tau ,x\left(\tau \right),x^{\sigma }\left(\tau \right)\right)\Delta \tau ,\hfill & \hfill t\in T,\hfill \\ \hfill \left(Wx\right)\left(t\right)=\underset{\left[a,b\right)}{\int }m\left(t,\tau \right)w\left(\tau ,x\left(\tau \right),x^{\sigma }\left(\tau \right)\right)\Delta \tau ,\hfill & \hfill t\in T\hfill \\ \hfill x\left(t_k+\right)=x\left(t_k\right)+J_k\left(x\left(t_k\right)\right),\hfill & \hfill t_k\in \Lambda ,\hfill \\ \hfill x\left(a\right)=x_0,\hfill \end{array}\right.$$

(4.1)

where $p\in {\Gamma }_1\left(T\right)$.

Definition 4.1: A function $x\in P{C}_{lrd}\left(T,R\right)$ is said to be a weak solution of (4.1), if x satisfies the impulsive integral equation

$$x\left(t\right)=e_{\ominus p}\left(t,a\right)x_0+\underset{\left[a,t\right)}{\int }{e}_{\ominus p}\left(t,\tau \right)f\left(\tau ,x\left(\tau \right),x^{\sigma }\left(\tau \right),\left(Sx\right)\left(\tau \right),\left(Wx\right)\left(\tau \right)\right)\Delta \tau +\sum _{t_k<t}{e}_{\ominus p}\left(t,t_k\right)J_k\left(x\left(t_k\right)\right),t\in T.$$

Suppose that:

[F] (1) k, $m\in C_{rd}\left(T\times T,R\right)$, the functions $f:T\times R\times R\times R\times R\to R$, $s:T\times R\times R\to R$, $w:T\times R\times R\to R$ are measurable in $t\in T$ and locally Lipschitz continuous, that is, for any ρ > 0, there exists a constant L(ρ) > 0, for all x _i , y _i , z _i , w _i ∈ X, satisfying ||x _i ||, ||y _i ||, ||z _i ||, ||w _i || ≤ ρ (i = 1, 2), we have

$$\begin{array}{lll}\hfill \parallel f\left(t,x_1,y_1,z_1,w_1\right)-f\left(t,x_2,y_2,z_2,w_2\right)\parallel & \le L\left(\rho \right)\left(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel +\parallel z_1-z_2\parallel +\parallel w_1-w_2\parallel \right),& \hfill \text{(1)}\\ \hfill \parallel s\left(t,x_1,y_1\right)-s\left(t,x_2,y_2\right)\parallel & \le L\left(\rho \right)\left(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel \right),& \hfill \text{(2)}\\ \hfill \parallel w\left(t,x_1,y_1\right)-w\left(t,x_2,y_2\right)\parallel & \le L\left(\rho \right)\left(\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel \right).& \hfill \text{(3)}\\ \hfill \text{(4) }\end{array}$$

• (2) There are constants 0 < λ _i < 0 (i = 1, · · ·, 4) and function q ₁, q ₂, $q_3\in L^1\left(T,R^{+}\right)$ such that

  $$\begin{array}{lll}\hfill \left|f\left(t,x,y,w,z\right)\right|& \le q_1\left(t\right)\left(1+\left|x\right|+|y{|}^{{\lambda }_1}+|w|+|z|\right)\mathsf{\text{,}}\forall x,y,w,z\in R,t\in T,& \hfill \text{(1) }\\ \hfill \left|s\left(t,x,y\right)\right|& \le q_2\left(t\right)\left(1+\left|x\right|+|y{|}^{{\lambda }_2}\right)\mathsf{\text{,}}\forall x,y\in R,t\in T,& \hfill \text{(2) }\\ \hfill \left|w\left(t,x,y\right)\right|& \le q_3\left(t\right)\left(1+|x{|}^{{\lambda }_3}+|y{|}^{{\lambda }_4}\right)\mathsf{\text{,}}\forall x,y\in R,t\in T.& \hfill \text{(3) }\\ \hfill \text{(4) }\end{array}$$

• (3) There are constants β _k ≥ 0 such that the mapping J _k : R → R (k = 12 · · · n) satisfies

  $$\parallel J_k\left(x\right)-J_k\left(y\right)\parallel \le {\beta }_k\parallel x-y\parallel ,\forall x,y\in X.$$

Theorem C. Under assumption [F], if $p\in {\Gamma }_1\left(T\right)$, then the system (4.1) has a weak solution $x\in P{C}_{lrd}\left(T,R\right)$.

Proof. Define the operator H on $P{C}_{lrd}\left(T,R\right)$ given by

$$\left(Hx\right)\left(t\right)=\underset{\left[a,t\right)}{\int }{e}_{\ominus p}\left(t,\tau \right)f\left(\tau ,x\left(\tau \right),x^{\sigma }\left(\tau \right),\left(Sx\right)\left(\tau \right),\left(Wx\right)\left(\tau \right)\right)\Delta \tau +e_{\ominus p}\left(t,a\right)x_0+\sum _{t_k<t}{e}_{\ominus p}\left(t,t_k\right)J_k\left(x\left(t_k\right)\right).$$

We can first prove that $H:P{C}_{lrd}\left(T,R\right)\to P{C}_{lrd}\left(T,R\right)$ is continuous and compact.

Let $x\in \Upsilon \equiv \left\{x\in P{C}_{lrd}\left(T,R\right)|x=\delta \left(Hx\right),\delta \in \left[0,1\right]\right\}$. When δ ≠ 0, set $y=\frac{1}{\delta }x$, if not y = 0. Note that

$$\begin{array}{lll}\hfill \left|y\left(t\right)\right|\le & M\left[\left|x_0\right|+\parallel q_1{\parallel }_{L^1}\left(1+\parallel k\parallel \parallel q_2{\parallel }_{L^1}+\parallel m\parallel \parallel q_3{\parallel }_{L^1}\right)+\sum _{k=1}^n\left|J_k\left(0\right)\right|\right]& \hfill \text{(1)}\\ +M\delta \underset{\left[a,t\right)}{\int }{q}_1\left(\tau \right)\left|y\left(\tau \right)\right|\Delta \tau +M\delta \parallel k\parallel \parallel q_2{\parallel }_{L^1}\underset{\left[a,t\right)}{\int }{q}_1\left(\tau \right)\left|y_{\tau }\right|\Delta \tau & \hfill \text{(2)}\\ +M\delta \underset{\left[a,t\right)}{\int }{q}_1\left(\tau \right)|y^{\sigma }\left(\tau \right){|}^{{\lambda }_1}\Delta \tau +M\delta \parallel k\parallel \parallel q_2{\parallel }_{L_1}\underset{\left[a,t\right)}{\int }{q}_1\left(\tau \right)|y_{\sigma \left(\tau \right)}{|}^{{\lambda }_2}\Delta \tau ,& \hfill \text{(3)}\\ +M\delta \parallel m\parallel \parallel q_1{\parallel }_{L^1}\underset{\left[a,b\right)}{\int }{q}_3\left(\tau \right)|y\left(\tau \right){|}^{{\lambda }_3}\Delta \tau & \hfill \text{(4)}\\ +M\delta \parallel m\parallel \parallel q_1{\parallel }_{L^1}\underset{\left[a,b\right)}{\int }{q}_3\left(\tau \right)|y^{\sigma }\left(\tau \right){|}^{{\lambda }_4}\Delta \tau +M\sum _{t_k<t}{\beta }_k|y\left(t_k\right)|.& \hfill \text{(5)}\\ \hfill \text{(6) }\end{array}$$

By Gronwall inequality (5) in Theorem A, there is a constant M > 0 such that

$$\left|y\left(t\right)\right|\le M,\forall t\in T.$$

It follows by Leray-Schauder fixed point theorem, H has a fixed point in $P{C}_{rd}\left(T,R\right)$, that is, the impulsive integro-differential equation (4.1) has a weak solution $x\in P{C}_{rd}\left(T,R\right)$. □

For the following backward problem

$$\left\{\begin{array}{c}\hfill {\phi }^{\Delta }\left(t\right)+p\left(t\right){\phi }^{\sigma }\left(t\right)=w\left(t,\phi \left(t\right),{\phi }^{\sigma }\left(t\right)\right),a\le t<b,\hfill \\ \hfill \phi \left(b\right)={\phi }_1,\hfill \end{array}\right.$$

(4.2)

we introduce the following assumption:

[W] (1) The function $w:T\times R\times R\to R$ is Δ-measurable in $t\in T$ and locally Lipschitz continuous, i.e. for all φ ₁, φ ₂, ψ ₁, ψ ₂ ∈ R, satisfying |φ ₁|, |φ ₂|, |ψ ₁|, |ψ ₂| ≤ ρ, we have

$$|w\left(t,{\phi }_1,{\psi }_1\right)-w\left(t,{\phi }_2,{\psi }_2\right)|\le L\left(\rho \right)\left(|{\phi }_1-{\phi }_2|+|{\psi }_1-{\psi }_2|\right)\mathsf{\text{for}}\mathsf{\text{all}}t\in T.$$

• (2) There exist a constant 0 < λ < 1 and a function $q\in L^1\left(T,\left[0,+\infty \right)\right)$ such that

  $$\left|w\left(t,\phi ,\psi \right)\right|\le q\left(t\right)\left(1+|\phi {|}^{\lambda }+|\psi |\right)\mathsf{\text{for}}\mathsf{\text{all}}\phi ,\psi \in R.$$

We can prove the following result.

Theorem D. Let $p\in {\Gamma }_1\left(T\right)$. Under the assumption [W], the backward problem of the nonlinear dynamical equation (4.2) has a unique weak solution $\phi \in C_{rd}\left(T,R\right)$.