Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays

1 Introduction

As stochastic modeling is used in the fields such as physics, economics, chemistry, and scholars have paid more and more attention to stochastic differential equations. Therefore, the existence and uniqueness of solutions of the equation have become a hot topic in recent years. The Volterra equation is a significant differential equation, which has been applied to the circulating fuel nuclear reactor, the neural networks, the population projection and others. In 1928, Volterra [1] first proposed the Volterra equation, i.e.,

and Levin [2] obtained the asymptotical stability of (1). Burton investigated the stability of equation (1) by the contraction mapping principle in [3]. Zhao and Yuan [4] considered 3/2-stability of a generalized Volterra-Levin equation. The discrete Volterra equation describing the evolutionary process of the population was recently investigated in [5].

To analyze the Volterra equation, Levin [2] used the limited condition that is pretty hard to be checked in practical application, lim ∣ x ∣ → ∞ ∫ 0 x f ( x ) d x = ∞ , and the author also required that the function p ( t ) has good properties, such as d p d t ≤ 0 , d 2 p d t 2 ≥ 0 and d 3 p ( t ) d t 3 ≤ 0 for any t ∈ ( 0 , + ∞ ) . Although the conditions of f ( x ) were simplified by averages in [3], there were still more requirements for the function f ( x ) . In this paper, the constrains of f ( x ) and p ( t ) will be weaken in the stochastic Volterra-Levin equation with variable delays.

Let { Ω , ℱ , P } be a complete probability space equipped with some filtration { ℱ t } t ≥ 0 satisfying the usual conditions, that the filtration is right continuous and { ℱ 0 } contains all P-null sets. Let { W ( t ) , t ≥ 0 } denote a standard Brownian motion defined on { Ω , ℱ , P } . We investigate the existence and uniqueness of solutions for the stochastic Volterra-Levin equations with variable delays, i.e.,

where f ( x ) and g ( x ) are known functions satisfying certain conditions, the constant L > 0 , p ( s ) ∈ C ( [ − L , T ] ; R ) , and R = ( − ∞ , + ∞ ) , α ( t ) and β ( t ) are the variable delays, satisfying α ( t ) , β ( t ) ∈ [ 0 , τ ] .

Scholars have become increasingly interested in the stochastic Volterra-Levin equations. The equation has been applied to many special research fields, such as the population model of spatial heterogeneity [6], the predator-prey model [7], and the nonautonomous competitive model [8]. After reviewing and sorting out the literature, it is found that most scholars currently use the principle of contraction mapping to explore the equation. For example, Luo [9] analyzed the exponential stability of the classical stochastic Volterra-Levin equations. Zhao et al. [10] investigated the mean square asymptotic stability of the generalized stochastic Volterra-Levin equations, which improved the results in [9]. Li and Xu [11] demonstrated the existence and global attractiveness of periodic solutions for impulsive stochastic Volterra-Levin equations. In this paper, the Picard iteration method is directly used to prove the existence and uniqueness of solutions of the stochastic Volterra-Levin equations with variable delays, which can give a more intuitive approximate solution. Recently, for the case without delay, Jaber [12] proved the weak existence and uniqueness of affine stochastic Volterra equations. Dung [13] revealed Itô differential representation of the stochastic Volterra integral equations. For the case of constant delay, Guo and Zhu [14] used this approximate method to prove the existence of solutions of stochastic Volterra-Levin equations. Some delay Volterra integral problems on a half-line were analyzed in [15]. The qualitative properties of solutions of nonlinear Volterra equations without random disturbance were investigated in [16]. However, there are only a few results of the stochastic Volterra equations with variable delay.

Generally, a time delay is inevitable and variable in practical application, and the future state of an existing system depends not only on the current state of the systems but also on the past [17,18,19]. When the function f ( x ) = x in equation (2), Benhadri and Zeghdoudi [20] applied the variable delays to the Volterra-Levin equation with Poisson jump and obtained the mean square stability by the fixed-point theory. The authors in [5] discussed the linear discrete Volterra equation with infinite delay when the function g ( x ) = 0 , which means there are not any random noises. In this paper, we will investigate the Volterra-Levin equation with the variable delays and the standard Brownian motion in more general conditions. Moreover, the Picard successive approximation method is used to prove the existence and uniqueness of the solution in some sufficient conditions. Compared with [2] and [3], these conditions are easier to be verified.

The rest of this paper is organized as follows. In Section 2, some necessary conditions and lemmas are established. In Section 3, the existence and uniqueness of solutions are proved. In Section 4, two examples are given to demonstrate the validity of the main results.

2 Assumptions and lemmas

To obtain the existence and uniqueness of the solutions for equation (2), the following assumptions are given in this paper.

1. lim x → 0 f ( x ) x = β > 0 .

2. g ( 0 ) = f ( 0 ) = 0 , and there exists a constant μ > 0 , such that f ( x ) x > μ .

3. ∫ − L 0 p ( s ) d s = m > 0 , ∫ − L 0 ∣ p ( s ) s ∣ d s = m 1 > 0 and max − L ≤ s ≤ 0 ∣ p ( s ) ∣ = m 2 .

4. There is a positive constant K > 0 , such that ∣ f ( x ) − f ( y ) ∣ ∨ ∣ g ( x ) − g ( y ) ∣ ≤ K ∣ x − y ∣ for all x , y ∈ R .

5. 5 K 2 2 μ K ( 1 − e − m μ T ) + 2 m 1 2 + 1 2 m μ ( 1 − e − 2 m μ T ) ∨ K 2 L 4 m 2 2 + 1 2 K 2 + 9 10 e m K T < 1 .

Now, we transform (2) into the following form by using the properties of integrals.

3 Existence and uniqueness

Picard iteration is the most commonly used method in the proof of the existence of solutions to the stochastic equations [20,21, 22,23]. In this paper, the existence and uniqueness of solutions for equation (2) are proved by the Picard iteration method. An important characteristic of this method is that it is constructive, and the bounds on the difference between iterates and the solutions are easily available. Such bounds are not only useful for the approximation of solutions but also necessary in the study of qualitative properties of solutions.

Now, let’s briefly summarize this idea of the Picard iteration method. To obtain the solution for a class of integral equation y ( t ) = y 0 + ∫ 0 t f ( τ , y ( τ ) ) d τ , Picard successive approximation sequences are constructed as follows.

If the sequences { y m ( t ) } converge uniformly to a continuous function y ( t ) in some interval J , then we may past to the limit in both sides of the aforementioned equation to obtain

So that y ( t ) is the desired solution.