The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem

1.   Introduction

Impulsive differential equations have many applications in engineering, science and finance. As a ubiquitous phenomenon, pulses exist in mechanical systems with impacts, optimal control models in economics, and the transfers of satellite orbit. It is difficult to model such phenomena using continuous models or discrete models ^[1,2]. In the 1950s, an impulsive model was developed to describe such specific evolution of a dynamic system ^[1]. Impulsive differential systems describe the dynamic processes with discontinuous jump caused by sudden changes. A variety of impulsive systems were investigated in the literature ^{[3,4,5,6,23,35,38,40]}.

The characteristics of impulsive differential equations have attracted the attention of scholars ^[31,32]. In recent years, many scholars have studied the initial and boundary value problem of fixed impulse differential equations ^[16,17,22]. For example, the boundary value problem of impulsive equations have been examined in the literature ^{[7,8,9,10,15,18,24]}. The existence and uniqueness of solutions to the following impulsive equation with boundary value problems have been investigated in the literature ^[11].

where $a > 0, b\geq0, c > 0, d\geq0$, $0 = t_{0} < t_{1} < t_{2} < \cdots < t_{m} = 1$, $J = [0, 1]$.

Some kinds of stochastic differential equation with fixed impulsive moments and Random impulsive differential equations also obtained considerable attention in the literature ^{[12,13,14,20,21,25,26,36,37]}. The Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays has been studied in ^[19]. The authors considered the following system

where $x_{t}$ is $\mathbb{R}^{d}$ -valued stochastic process such that $x_{t}\in \mathbb{R}^{d}$, $x_{t} = \left\{x(t+\theta):-\tau\leq\theta\leq0\right\}$. Here, $0 = \xi_{0} < \xi_{1} < \xi_{2} < \cdots < \xi_{k} < \cdots < \lim_{k\rightarrow\infty}\xi_{k} = +\infty$, and $x(\xi_{k}^{-}) = \lim_{t\rightarrow\xi_{k}-0}x(t)$. Note that $\left\{N(t), t\geq0\right\}$ is the simple counting process generated by ${\xi_{k}}$, and $\left\{W(t):t\geq0\right\}$ is a given $m$-dimensional Winer process.

The fixed point method ^[28] had been used to study the random impulsive differential equations. Niu et al. ^[27] used the fixed point method to address the existence and Hyers-Ulam stability for the following differential equation

Upper and lower solution method can be used to study fractional evolution equations ^[33] and impulsive differential equations ^[34].

To the best of our knowledge, the boundary value problem of second order random impulsive differential equation has not been studied using the upper and lower solution method in the literature. In this paper, we use the upper and lower solution method to study the following second order random impulsive differential equation with boundary value problem.

Where $f:J\times \mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ is a continuous mapping. $x(t)$ is a stochastic process taking values in the Euclidean space $(\mathbb{R}, \|\cdot\|)$. Then, we introduce $\tau_{k}$ to be a random variable defined from $\Omega$ to $E_{k}\overset{\triangle}{ = }(0, d_{k})$, with $0 < d_{k} < 1$ for every $k\in\mathbb{N}^+$. We assume that $\tau_{i}$ and $\tau_{j}$ are independent of each other when $i\neq j$ for every $i, j\in\mathbb{N}^{+}$ and $b_{k}$: $E_{k}\rightarrow \mathbb{R}$ satisfies for every $k\in\mathbb{N}^{+}$, $b_{k}(\tau_{k})\geq0$. Set $\xi_{k} = \xi_{k-1}+\tau_{k}$. Obviously, $\{{\xi_{k}}\}$ is a process with independent increments and the impulsive moments $\xi_{k}$ form a strictly increasing sequence, i.e. $0 = \xi_{0} < \xi_{1} < \xi_{2} < \cdots < \xi_{k} < \cdots < 1$. We hold the opinion that $x(\xi_{k}^{-}) = \lim_{t\rightarrow\xi_{k}-0}x(t)$, $x(\xi_{k}^{+}) = \lim_{t\rightarrow\xi_{k}+0}x(t)$. The convergence is under the meaning of the orbit. Since for a realization (sample) of random process, $\{\xi_{k}\}$ will become a series of fixed time points. Under that sense, so we can define the limit as we would in general. We suppose that $\left\{N(t):t\geq0\right\}$ is the simple counting process generated by ${\xi_{k}}$. Let $J = [0, 1]$, $\mathbb{R}^{+} = (0, +\infty)$ and $J'\overset{\triangle}{ = }J\setminus\{\xi_{1}, \xi_{2}, \cdots\}$. Here, $\alpha_{0}, \alpha_{1}, \beta_{0}, \beta_{1}, x_{0}, x_{0}^{*}$ are constants satisfying $\alpha_{0}\alpha_{1}\neq0$, $\beta_{0}\neq0$, $\alpha_{0}, \alpha_{1}, \beta_{0}, \beta_{1}, x_{0}, x_{0}^{*}\geq0$.

The rest of the paper is organised as follows: In section 2, we introduce some notations and preliminaries. In section 3, we use the upper and lower solution method to study the existence of solutions to the second order random impulsive differential equations. In section 4, we give an example to show the application of the main result. Finally, conclusions are presented.

2.   Preliminaries

Suppose $(\Omega, \Gamma, P)$ is a probability space. Let $L^{p}(\Omega, \mathbb{R}^{n})$ be the collection of all strongly measurable, $pth$ integrable, and $\Gamma_{t}$-measurable with $\mathbb{R}^{n}$-valued random variables $x: \Omega\rightarrow \mathbb{R}^{n}$ and norm $L^{p}(\Omega, \mathbb{R}^{n})$ for $p\geq1$. Here, $\mathrm{E}(x) = \int_\Omega x\mathrm{d}\mathbb{P} < \infty$ is the expectation of $x$, and $L^{p}(\Omega, \mathbb{R}^{n})$ is equipped with its natural norm $\|x\|_{L^p(\Omega, \mathbb{R}^{n})} = \left(\int_\Omega\|x\|^p\mathrm{d}\mathbb{P}\right)^{\frac{1}{p}} = (\mathrm{E}\|x\|^p)^{\frac{1}{p}}$.

We introduce the space $PC = PC(J, L^{2}(\Omega, \mathbb{R}^n))$: = $\{ u(t)$ $\mid$ $u(t) = u(t, \omega)$ is a strongly measurable, square integrable, random process from $J$ into $L^{2}(\Omega, \mathbb{R}^n)$, and $u(t)$ is continuous when $t\in J'$ and left continuous when $t\in J\setminus J' \}$. We can prove that $PC$ is a Banach space with norm

Then, we consider the space $PC^{1} = PC^{1}(J, L^{2}(\Omega, \mathbb{R}^n))$: = $\{ u(t)$ $\mid$ $u(t) = u(t, \omega)$ is a strongly measurable, square integrable, random process from $J$ into $L^{2}(\Omega, \mathbb{R}^n)$, $u(t)$ is continuously differentiable when $t\in J'$ and left continuous when $t\in J\setminus J'$, $u'(\xi_{k}^{-})$ and $u'(\xi_{k}^{+})$ exist for $k = 1, 2, \cdots \}$. It is easy to see that $PC^{1}$ is also a Banach space with norm

The functions in $PC^{1}$ which satisfy the equation $(1.4)$ are called the solutions of the equation $(1.4)$.

For convince, in the rest of the paper we write $PC(J, L^{2}(\Omega, \mathbb{R}))$ as $PC(J, \mathbb{R})$, write $PC^{1}(J, L^{2}(\Omega, \mathbb{R}))$ as $PC^{1}(J, \mathbb{R})$. In this paper, the upper and lower solutions of equation (1.4) are studied in $PC^1(J, \mathbb{R})$ space.

Now we consider the equation (2.3)

where $h(t)\in PC^{1}(J, \mathbb{R})$ and $M$ is a positive constant.

Lemma 2.1. Equation (2.3) has one solution, given by

and

Denote

and the index function

where

and

Proof. Suppose $\xi_1$, $\xi_2$, $\cdots$ is a sample orbit. Thus, when $t\in[0, \xi_{1}]$, the solution of the equation $(2.3)$ is

When $t\in(\xi_{1}, \xi_{2}]$, we assume that the solution of the equation $(2.3)$ is

Plug in the initial conditions

we can get

In the same way, we can get when $t\in(\xi_{k}, \xi_{k+1}]$,

Based on the above discussion, mathematical induction can be obtained as

Where $\delta_{k}^{+}, \delta_{k}^{-}, \Delta_{k}^{+}, \Delta_{k}^{-}$ are defined as the equations $(2.6)$ and $(2.7)$.

So, the solution of the equation $(2.3)$ is

and

Therefore

Substituting these equations into the boundary value conditions of the equation (2.3) yields and using the Cramer's rule, it follows from (2.26) and (2.27) that (2.5)–(2.16) hold.

Lemma 2.2. $\delta_{k}^{-}$ and $\delta_{k}^{+}$ are uniformly bounded series.

Proof. We firstly consider $\delta_{k}^{-}$,

So, for every $n\in\mathbb{N}^{+}$, we have

In the same way, we can prove that $\delta_{k}^{+}$ is uniformly bounded and

Lemma 2.3. For every $n\in\mathbb{N}^{+}$, $\Delta_{n, k}^{-}$ and $\Delta_{n, k}^{+}$ are uniformly bounded series.

Proof. We can easily prove that

So, we have

and we have proved the lemma.

Definition 2.1. Define the operator $\Lambda: PC^{1}[J, \mathbb{R}]\rightarrow PC^{1}[J, \mathbb{R}]$ such that

Definition 2.2. $v_{0}(t)\in PC^{1}[J, \mathbb{R}]$ is called a lower solution of equation (1.4) if $v_{0}(t)$ satisfies the inequality group

Definition 2.3. $\omega_{0}(t)\in PC^{1}[J, \mathbb{R}]$ is called an upper solution of equation (1.4) if $\omega_{0}(t)$ satisfies the inequality group

Lemma 2.4. $h(t)\in PC^{1}[J, \mathbb{R}]$ is the solution of equation (1.4) if and only if $h(t)\in PC^{1}[J, \mathbb{R}]$ is the fix point of the operator $\Lambda$.

Proof. If $h(t)$ is the fix point of the operator $\Lambda$, it is to say that $h(t)$ satisfies the equation $\Lambda h(t) = h(t)$, then, in the equation (2.3), we have $u(t) = \Lambda h(t) = h(t)$, so, we can replace $u(t)$ with $h(t)$ in the equation (2.3), and we have

and we have proved that $h(t)$ is a solution of the equation (1.4).

If $h(t)$ is the solution of the equation (1.4), then using the same method, we can easily know that it is also the fix point of the operator $\Lambda$, and we have proved the lemma.

Lemma 2.5.(The Arzela-Ascoli Theorem)(^[29]) The set $M\subset C^{2}[J, \mathbb{R}^{n}]$ is column compact tight if and only if

$(\rm{i})$ The functions in the set $M$ are uniformly bounded, that is to say, there exists a fixed constant $K$ for all $u(t)\in M$, where $\|u(t)\|\leq K$.

$(\rm{ii})$ Functions in the set $M$ are equally continuous, that is to say, for all $\epsilon > 0$, there exists $\delta = \delta(\epsilon)$ such that when $t_{1}, t_{2}\in J$ and $\parallel t_{1}-t_{2}\parallel < \delta$ for all $u(t)\in M$, there is $\parallel u(t_{1})-u(t_{2})\parallel < \epsilon$.

Lemma 2.6.(^[30]) Suppose $E$ is a semi-ordered Banach space. For $x_{0}, y_{0}\in E$, $x_{0}\leq y_{0}$, and $D = [x_{0}(t), y_{0}(t)]$, $A:D\rightarrow E$ is an operator. Assuming that the following conditions are satisfied

$(\rm{i})$ $A$ is an increasing operator,

$(\rm{ii})$ $x_{0}$ is the lower solution of $A$ and $y_{0}$ is the upper solution of $A$,

$(\rm{iii})$ $A$ is a continuous operator,

$(\rm{iv})$ $A(D)$ is a relatively compact set of columns in $E$.

Then, $A$ has a maximum fixed point and a minimum fixed point in $D$. Let $x_{0}$ and $y_{0}$ be the initial conditions. We then have the iteration sequences

Thus,

and

3.   Main result

$(H_{1})$ The equation (1.4) has the lower solution $v_{0}(t)$ and the upper solution $\omega_{0}(t)$ and they meet the inequality

for any $t\in J$.

$(H_{2})$ There exists a constant $M > 0$, such that

for any $t\in J$, $y\in PC^{1}(J, \mathbb{R})$ and $v_{0}(t)\leq x_{2}(t)\leq x_{1}(t)\leq \omega_{0}(t)$.

$(H_{3})$ There exist constants $B_{1}$ and $B_{2}$ such that

$(H_{4})$ There exists an increasing continuous function $\Theta(x)$ satisfies that $\frac{\Theta(x)}{x}$ is a decreasing function and

for every $t\in J$, $x_{1}, x_{2}\in D = [v_{0}(t), \omega_{0}(t)]$, and $y\in PC^{1}(J, \mathbb{R})$.

$(H_{5})$ There exist a function $\Psi(t, x, y)$ and a constant $K$ such that

$(\rm{i})$ For each $t\in J$, the function $\Psi(t, \cdot, \cdot):\mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and $\Psi(t, 0, 0) = 0$. For every $x, y\in \mathbb{R}$, the function $\Psi(\cdot, x, y):J\rightarrow \mathbb{R}$ is measurable;

$(\rm{ii})$

for every $t\in J$, $x(t)\in D = [v_{0}(t), \omega_{0}(t)]$.

$(H_{6})$ Define $P = 12\frac{e^{2\sqrt{M}}}{M}$, $P^{*} = 12e^{2\sqrt{M}}+\ln3(m_{1}+m_{2})$, $m_{1} = \sup\limits_{t}\mathrm{E}\|C_{1}^{k}e^{\sqrt{M}t}\|^{2}$, and $m_{2} = \sup\limits_{t}\mathrm{E}\|C_{2}^{k}e^{-\sqrt{M}t}\|^{2}$.

Then we define the sequences $v_{n}$ and $\omega_{n}$ as

Theorem 3.1.

If conditions $(H_{1})\sim(H_{6})$ are met, the equation (1.4) has the maximum solution $x^{*}(t)$ and the minimum solution $x_{*}(t)$ in $[v_{0}(t), \omega_{0}(t)]\bigcap PC^{1}[J, \mathbb{R}]$. And there exist $\omega_{n}(t) = \Lambda\omega_{n-1}(t)$ uniformly convergent to $x^{*}(t)$, $v_{n}(t) = \Lambda v_{n-1}(t)$ uniformly convergent to $x_{*}(t)$, $n = 1, 2, \cdots$. And if $x(t)$ is the solution of the equation (1.4), it satisfies

Proof. We will prove this theorem in five steps.

Step(1). We prove that $v_{0}(t)$ and $\omega_{0}(t)$ are the lower and upper solutions of the operator $\Lambda$, i.e., we should prove $v_{0}(t)\leq \Lambda v_{0}(t)$ and $\omega_{0}(t)\geq \Lambda\omega_{0}(t)$.

When there is no random impulsive, we set $v_{1}(t) = \Lambda v_{0}(t)$. Now, we only need to prove that $v_{0}(t)\leq v_{1}(t)$. Here, we use proof by contradiction. If it is not true, then there exist $t_{0}\in J$ and $\varepsilon > 0$ such that

for every $t\in J$. If $t_{0}\in J\setminus(\left\{0\right\}\bigcup\left\{1\right\})$, it is easy to see that

However,

which is a contradiction to the inequality $v''_{0}(t_{0})-v''_{1}(t_{0})\leq 0$. Thus, our hypothesis does not work.

When $t_{0} = 0$ or $t_{0} = 1$, we assume that $t_{0} = 0$. Therefore,

assuming that $v_{1}(0)+\varepsilon = v_{0}(0)$, it is easy to see that

By the boundary value conditions, we can get

Thus, we have

which is a contradiction to the hypothesis. Therefore, we have proved that $v(t)\leq \Lambda v(t)$.

When the equation has the random pulses, we have

and $v_{0}(t)$ is the lower solution of the equation $(1.4)$. Thus, according to the second inequality of (2.35), for every $t\in\left\{\xi_{k}\right\}_{k\in\mathbb{N}^{+}}$, we have

Thus, based on our discussion, we conclude that for every $t\in(\xi_{k}, \xi_{k+1}]$, $k = 1, 2, \cdots$,

Hence, we have proved that $v_{1}(t)\geq v_{0}(t)$ for every $t\in J$.

In the same way, we can prove that $\omega_{0}(t)\geq \Lambda\omega_{0}(t)$ for every $t\in J$.

Step(2). We prove that $\Lambda$ is an increasing operator.

First of all, we take any $h_{1}(t)$ and $h_{2}(t)$, $h_{1}(t), h_{2}(t)\in PC^{1}[J, \mathbb{R}]$. Suppose $h_{1}(t)\geq h_{2}(t)$ for any $t\in J$. Then, we prove that $\Lambda h_{1}(t)\geq \Lambda h_{2}(t)$. Here, we use proof by contradiction. Let $h^{*}_{1}(t) = \Lambda h_{1}(t)$, $h^{*}_{2}(t) = \Lambda h_{2}(t)$. Then, we need to prove $h^{*}_{1}(t)\geq h^{*}_{2}(t)$.

When there is no random pulse, if the hypothesis is not true, then there must exist $t_{0}\in J$ and $\varepsilon > 0$ such that $h^{*}_{1}(t_{0})+\varepsilon = h^{*}_{2}(t_{0})$ and $h^{*}_{1}(t)+\varepsilon\geq h^{*}_{2}(t)$ for every $t\in J$. If $t_{0}\in J\setminus(\left\{0\right\}\bigcup\left\{1\right\})$, then we have

and

Thus

Which is a contradiction. When $t_{0} = 0$ or $t_{0} = 1$, we assume that $t_{0} = 0$ and $h^{*}_{1}(0)+\varepsilon = h^{*}_{2}(0)$.

Then

Thus

Take the difference of these equation yields

That is to say $h'^{*}_{1}(0)-h'^{*}_{2}(0) < 0$, which is a contradiction.

When there exits random pulses, we have

Then, for every $t\in(\xi_{k}, \xi_{k+1}]$, $k = 1, 2, \cdots$, we have

Thus, for every $t\in J$, the inequality $\Lambda h_{1}(t)\geq \Lambda h_{2}(t)$ is true.

Step(3). We prove that $\Lambda$ is a continuous operator. That is to say, we should prove that for any $\varepsilon > 0$, there exists $\delta(\varepsilon) > 0$ such that when $\parallel h_{1}(t)-h_{2}(t)\parallel_{PC^{1}} < \delta$, $\parallel \Lambda h_{1}(t)-\Lambda h_{2}(t)\parallel_{PC^{1}} < \varepsilon$.

We assume that

Among them,

and we have

and

combing with $(H_{5})$, we have

From (3.31) and (3.32), we can easily know that $|Q|$ is dependent with $h(t)$. So, based on the above discussion, we can know when $\parallel h_{1}(t)-h_{2}(t)\parallel\rightarrow0$, $\|\hat{h}_{1}(t)-\hat{h}_{2}(t)\|\rightarrow0$, $\|C_{1}-\widetilde{C}_{1}\|\rightarrow0$ and $\|C_{2}-\widetilde{C}_{2}\|\rightarrow0$. And then, we can get when $\parallel h_{1}(t)-h_{2}(t)\parallel\rightarrow0$, $\mathrm{E}\parallel \Lambda h_{1}(t)-\Lambda h_{2}(t)\parallel^{2}\rightarrow0$.

Then

hence,

which implies that

Thus, we have proved that $\Lambda$ is a continuous operator.

Step(4). We prove that the functions in the set $\left\{u\in PC^{1}(J, \mathbb{R})\mid u\in \Lambda(D)\right\}$ are uniformly bounded.

Because $u\in \Lambda(D)$, for any $u\in\left\{u\in PC^{1}(J, \mathbb{R})\mid u\in \Lambda(D)\right\}$, there exists $h(t)\in D$ such that $u = \Lambda h(t)$,

suppose $r_{1} = \frac{1}{M}$, $r_{2} = \frac{e^{2\sqrt{M}}}{M}$ and we have

so, if $h(t)$ is the solution of the equation $(1.4)$, we have

Next, define $\phi(t) = \mathrm{E}\|h(t)\|^{2}$ and we have the inequality

Define the right of the inequality (3.43) as the function $\varphi(t)$, we can get

so,

then we can easily get

For $\Delta_{n, k}^{-}$, $\Delta_{n, k}^{+}$, $\delta_{k}^{-}$ and $\delta_{k}^{-}$ are bounded, combining with

and consider that $\Theta(s)$ satisfies the condition $(H_{4})$ and $D = [v_{0}(t), \omega_{0}(t)]$, where $v_{0}(t)$, $\omega_{0}(t)$ are all square integrable, so, $E\|\Lambda h(t)\|^{2}$ is bounded.

Then,

and

Using the same way, we can prove that $E\parallel\Lambda' h(t)\parallel^{2}$ is bounded. Thus, we have proved that the functions in the set $\left\{u(t)\in C^{2}(J, \mathbb{R})\mid u(t)\in \Lambda(D)\right\}$ are uniformly bounded.

Step(5). We prove that the set $\left\{u(t)\mid u(t) = \Lambda h(t)\right\}$ is equicontinuous.

For every $u(t)\in\left\{u(t)\mid u(t) = \Lambda h(t)\right\}$, and every $t_{1}, t_{2}\in J$, $\|t_{1}-t_{2}\| < \delta$,

and,

We suppose $t_{1}\in(\xi_{k_{1}}, \xi_{k_{1}+1})$, $t_{2}\in(\xi_{k_{2}}, \xi_{k_{2}+1})$, so when $\|t_{1}-t_{2}\| < \delta$, $\|\xi_{k_{1}+1}-\xi_{k_{2}}\| < \delta$

So, it is easy to see that when $\delta\rightarrow0$, $E\parallel u(t_{1})-u(t_{2})\parallel^{2}\rightarrow0$. Then, we consider

Therefore,

Using the same method, we can prove that when $\delta\rightarrow0$, $\mathrm{E}\parallel u'(t_{1})-u'(t_{2})\parallel^{2}\rightarrow0$. We have already proved that when $\| t_{1}-t_{2}\|\rightarrow0$, $\mathrm{E}\parallel u(t_{1})-u(t_{2})\parallel^{2}\rightarrow0$. So

That is to say, the set $\left\{u(t)\mid u(t) = \Lambda h(t)\right\}$ is equicontinuous.

Using Lemma (2.5), we know that the set $\left\{u(t)\mid u(t) = \Lambda h(t)\right\}$ is a column compact set. It follows from Lemma (2.6) that the equation $(1.4)$ has a solution in $D = [v_{0}(t), \omega_{0}(t)]$, where $t\in[0, 1]$. Thus, theorem $(3.1)$ is established.

4.   Example

The main result could have many applications, now, we give an example to illustrate this theorem. We consider the following second order random impulsive differential equation with boundary value problems.

Let $\tau_{k}\sim U(0, \frac{1}{2^{k}})$, then the probability density function of $\tau_{k}$ is

Set $\xi_{0} = 0$, $\xi_{k+1} = \xi_{k}+\tau_{k+1}$. Obviously, $\{{\xi_{k}}\}$ is a process with independent increments and the impulsive moments $\xi_{k}$ form a strictly increasing sequence. And for every $k\in\mathbb{N}$,

So in this example, $b_{k}(\tau_{k}) = \frac{k}{3^{k}}\tau_{k}$, $\tau_{k}$ is a random variable defined from $\Omega$ to $E_{k} = (0, d_{k}) = (0, \frac{1}{2^{k}})$. Suppose $\tau_{i}$ and $\tau_{j}$ are independent of each other when $i\neq j$, $x(\xi_{k}^{+}) = \lim\limits_{t \rightarrow \xi_{k}+0}x(t)$ and $x(\xi_{k}^{-}) = \lim\limits_{t \rightarrow \xi_{k}-0}x(t)$.

Taking $v_{0}(t) = 0$, $\omega_{0}(t) = 6\cos t$, we can easily prove that $v_{0}(t)$ is the lower solution and $\omega_{0}(t)$ is the upper solution of the equation (4.1). And for every $v_{0}(t)\leq x_{2}(t) < x_{1}(t)\leq \omega_{0}(t)$, we have

So, we can easily know that $M = 147$.

So, we have proved that $\sum\limits_{k = 0}^{\infty}\Big\{\sum\limits_{n = 1}^{k}\Delta_{n, k}^{-}b_{n}(\tau_{n})\Big\}I_{(\xi_{k}, \xi_{k+1}]}(t)\leq3$. That is to say $B_{1} = 3$. In the same way we can prove that $B_{2} = 3e^{3\sqrt{147}}$.

For every $v_{0}(t)\leq x_{2}(t) < x_{1}(t)\leq\omega_{0}(t)$, we have

and

So, the equation (4.1) meets all the conditions of the theorem (3.1). We can get the solution of the equation of the equation (4.1) between $v_{0}(t) = 0$ and $\omega_{0}(t) = 6\cos t$ by constructing iterative sequences starting from $v_{0}$ an $\omega_{0}$ respectively.

5.   Conclusions

In this article, we study the existence of upper and lower solutions of second order random impulse equation (1.4). First, we study the solution form of the corresponding linear impulsive system (2.3) induced by system (1.4). Based on the form of the solution, we define the solution operator. Secondly, we prove that the fixed point of this operator is the solution of equation (1.4). Finally, we construct two monotone iterative sequences by the solution to (2.3). We then prove that they converge. Thus, it is concluded that there exists upper and lower solution to system (1.4). Impulsive differential equations have been studied in literature ^[7,8,9,10]. Random impulsive differential equations have also been discussed in the literature ^{[12,13,14,19,27,39]}. In this paper, we extend the form of solutions to initial value problems of random impulsive differential equations to more general boundary value problems. The upper and lower methods are applied to Random impulsive differential equations and the related conclusions are generalized.

Acknowledgments

The authors would like to thank the editor and the reviewers for their helpful comments and suggestions.

Conflict of interest

No potential conflict of interest was reported by the authors.