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

Abstract: In this article, we consider the existence of upper and lower solutions to a second-order random impulsive differential equation. We first study the solution form of the corresponding linear impulsive system of the second-order random impulsive differential equation. Based on the form of the solution, we define the resolvent operator. Then, we prove that the fixed point of this operator is the solution to the equation. Finally, we construct the sum of two monotonic iterative sequences and prove that they are convergent. Thus, we conclude that the system has upper and lower solutions.


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 [3-6, 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 [7-10, 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].

(1.3)
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.
is a stochastic process taking values in the Euclidean space (R, · ). Then, we introduce τ k to be a random variable defined from Ω to E k = (0, d k ), with 0 < d k < 1 for every k ∈ N + . We assume that τ i and τ j are independent of each other when i j for every i, j ∈ N + and b k : E k → R satisfies for every k ∈ N + , b k (τ k ) ≥ 0. Set ξ k = ξ k−1 + τ k . Obviously, {ξ k } is a process with independent increments and the impulsive moments ξ k form a strictly increasing sequence, i.e. 0 = ξ 0 < ξ 1 < ξ 2 < · · · < ξ k < · · · < 1. We hold the opinion that The convergence is under the meaning of the orbit. Since for a realization (sample) of random process, {ξ 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 {N(t) : t ≥ 0} is the simple counting process generated by ξ k . Let J = [0, 1], R + = (0, +∞) and J = J \ {ξ 1 , ξ 2 , · · · }. Here, α 0 , α 1 , β 0 , β 1 , x 0 , x * 0 are constants satisfying α 0 α 1 0, β 0 0, α 0 , α 1 , β 0 , β 1 , x 0 , x * 0 ≥ 0. 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.

Preliminaries
Suppose (Ω, Γ, P) is a probability space. Let L p (Ω, R n ) be the collection of all strongly measurable, pth integrable, and Γ t -measurable with R n -valued random variables x : Ω → R n and norm L p (Ω, R n ) for p ≥ 1. Here, E(x) = Ω xdP < ∞ is the expectation of x, and L p (Ω, R n ) is equipped with its natural ) is a strongly measurable, square integrable, random process from J into L 2 (Ω, R n ), and u(t) is continuous when t ∈ J and left continuous when t ∈ J \ J }. We can prove that PC is a Banach space with norm (2.1) Then, we consider the space PC 1 = PC 1 (J, L 2 (Ω, R n )):={u(t) | u(t) = u(t, ω) is a strongly measurable, square integrable, random process from J into L 2 (Ω, R n ), u(t) is continuously differentiable when t ∈ J and left continuous when t ∈ J \ J , u (ξ − k ) and u (ξ + k ) exist for k = 1, 2, · · · }. It is easy to see that PC 1 is also a Banach space with norm (2. 2) The functions in PC 1 which satisfy the equation ( where h(t) ∈ PC 1 (J, R) and M is a positive constant. and (2.7) (2.8) .
(2.10) 12) and the index function where and (2.16) When t ∈ (ξ 1 , ξ 2 ], we assume that the solution of the equation (2.3) is Plug in the initial conditions we can get (2.20) In the same way, we can get when t ∈ (ξ k , ξ k+1 ], (2.21) Based on the above discussion, mathematical induction can be obtained as (2.25) Therefore So, for every n ∈ N + , we have (2.30) In the same way, we can prove that δ + k is uniformly bounded and 3. For every n ∈ N + , ∆ − n,k and ∆ + n,k are uniformly bounded series.
Proof. We can easily prove that and we have proved the lemma.    (ii) Functions in the set M are equally continuous, that is to say, for all > 0, there exists δ = δ( ) such that when t 1 , t 2 ∈ J and t 1 − t 2 < δ for all u(t) ∈ M, there is u(t 1 ) − u(t 2 ) < . 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 x n = Ax n−1 , y n = Ay n−1 , n = 1, 2, · · · . (2.38) Thus, x 0 ≤ x 1 ≤ · · · ≤ x n ≤ · · · ≤ y n ≤ · · · ≤ y 1 ≤ y 0 , (2.39) and x n → x * , y n → y * . (2.40)

Main result
(H 1 ) The equation (1.4) has the lower solution v 0 (t) and the upper solution ω 0 (t) and they meet the for any t ∈ J.
(H 5 ) There exist a function Ψ(t, x, y) and a constant K such that (i) For each t ∈ J, the function Ψ(t, ·, ·) : R × R → R is continuous and Ψ(t, 0, 0) = 0. For every x, y ∈ R, the function Ψ(·, x, y) : J → R is measurable; ( Then we define the sequences v n and ω n as Proof. We will prove this theorem in five steps. Step (1). We prove that v 0 (t) and ω 0 (t) are the lower and upper solutions of the operator Λ, i.e., we should prove v 0 (t) ≤ Λv 0 (t) and ω 0 (t) ≥ Λω 0 (t).
When there is no random impulsive, we set v 1 (t) = Λv 0 (t). Now, we only need to prove that v 0 (t) ≤ v 1 (t). Here, we use proof by contradiction. If it is not true, then there exist t 0 ∈ J and ε > 0 such that for every t ∈ J. If t 0 ∈ J \ ({0} {1}), it is easy to see that which is a contradiction to the inequality v 0 (t 0 ) − v 1 (t 0 ) ≤ 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) + ε = v 0 (0), it is easy to see that By the boundary value conditions, we can get (3.14) Thus, we have which is a contradiction to the hypothesis. Therefore, we have proved that v(t) ≤ Λv(t).
When the equation has the random pulses, we have (3.16) and v 0 (t) is the lower solution of the equation (1.4). Thus, according to the second inequality of (2.35), Thus, based on our discussion, we conclude that for every t ∈ (ξ k , In the same way, we can prove that ω 0 (t) ≥ Λω 0 (t) for every t ∈ J.
Then hence, Thus, we have proved that Λ is a continuous operator.
Step (4). We prove that the functions in the set u ∈ PC 1 (J, R) | u ∈ Λ(D) are uniformly bounded.
Because u ∈ Λ(D), for any u ∈ u ∈ PC 1 (J, R) | u ∈ Λ(D) , there exists h(t) ∈ D such that u = Λh(t),   (3.50) Using the same way, we can prove that E Λ h(t) 2 is bounded. Thus, we have proved that the functions in the set u(t) ∈ C 2 (J, R) | u(t) ∈ Λ(D) are uniformly bounded.

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.

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 [12-14, 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.