Boundedness analysis of neutral stochastic differential systems with mixed delays

doi:10.1016/j.aml.2021.107545

Applied Mathematics Letters

Volume 122, December 2021, 107545

https://doi.org/10.1016/j.aml.2021.107545 Get rights and content

Abstract

This article aims to investigate the boundedness of neutral stochastic differential systems with mixed delays. By virtue of Itô formula and some important inequalities, several $p$ th moment exponential ultimate boundedness conditions are obtained for the considered systems.

Introduction

Boundedness is one of the important fundamental concepts in the theory of differential equations and plays a critical role in the study of the stability, synchronization, invariant set, attracting set, periodic solution and so on. The problem of boundedness can be briefly described as the estimate of the range in which the solutions stay as time tends to infinity. In recent years, increasing efforts have been devoted to study the boundedness problem of various stochastic systems, for example, the exponential ultimate boundedness of impulsive stochastic differential systems with or without delay has been discussed in [1], [2], [3], the moment and almost sure asymptotic boundedness of stochastic delay differential systems has been studied in [4], some boundedness conditions for stochastic differential systems driven by $G$ -Brownian motion have been obtained in [5], [6], the boundedness for stochastic differential systems with Lévy noise and with distributed delay has been discussed in [7], the boundedness of stochastic pantograph differential systems has been investigated in [8].

But so far, the boundedness analysis of neutral stochastic differential systems with mixed delays has not been concerned. With the above motivations, the present article aims to investigate the boundedness of neutral stochastic differential systems with mixed delays. By virtue of Itô formula and some important inequalities, several sufficient criteria are formulated for the $p$ th moment exponential ultimate boundedness of the considered systems.

Section snippets

Preliminaries

The notation $(Ω, ℱ, {ℱ_{t}}_{t \geq 0}, P)$ stands for a complete probability space with a filtration ${ℱ_{t}}_{t \geq 0}$ which satisfies the usual conditions. Let $ω (t) = {(ω_{1} (t), \dots, ω_{m} (t))}^{T}$ stand for an $m$ -dimensional Brownian motion defined on the probability space. The symbol $C [[- μ, 0], R^{n}]$ with $μ > 0$ stands for the family of continuous functions $ψ : [- μ, 0] \to R^{n}$ with the norm $‖ ψ ‖ = {sup}_{- μ \leq θ \leq 0} | ψ (θ) |$ , where $| \cdot |$ denotes the Euclidean norm. The notation $C_{ℱ_{0}}^{b} [[- μ, 0], R^{n}]$ denotes the set of all bounded $ℱ_{0}$ -measurable, $C [[- μ, 0], R^{n}]$ -valued

Main results

Theorem 3.1

Let $p \geq 2$ and conditions H1 to H3 hold. If there exists a positive constant $ϑ$ such that $C_{1} (ϑ) ≔ 2^{p - 1} (1 + δ^{p}) + 2^{p - 1} δ^{p} μ e^{ϑ μ} \{ϑ + (p - 2) (ς_{1} + ς_{2} + ς_{3} + ς_{4}) + \frac{(p - 2) (p - 4) (ς_{5} + ς_{6} + ς_{7} + ς_{8})}{2}\} + 2 \{ς_{2} + (p - 2) ς_{6}\} μ e^{ϑ μ} + 2 \{ς_{3} + (p - 2) ς_{7}\} Q^{p} μ^{p} \frac{e^{ϑ μ} - 1}{ϑ} > 0$ and $C_{2} (ϑ) ≔ 2^{p - 1} (1 + δ^{p} e^{ϑ μ}) \{ϑ + (p - 2) (ς_{1} + ς_{2} + ς_{3} + ς_{4}) + \frac{(p - 2) (p - 4) (ς_{5} + ς_{6} + ς_{7} + ς_{8})}{2}\} + 2 [ς_{1} + (p - 2) ς_{5}] + 2 [ς_{2} + (p - 2) ς_{6}] e^{ϑ μ} + 2 [ς_{3} + (p - 2) ς_{7}] Q^{p} μ^{p - 1} \frac{e^{ϑ μ} - 1}{ϑ} < 0 .$ Then there holds for the solution $y (t)$ of system (1) that $E {| y (t) |}^{p} \leq \frac{1}{{(1 - δ)}^{p}} C_{1} (ϑ) E {‖ ζ ‖}^{p} e^{- ϑ t} + \frac{1}{ϑ {(1 - δ)}^{p}} 2 [ς_{4} + (p - 2) ς_{8}], t \geq 0 .$ That is, system (1) is $p$ th moment exponentially

Example

Example 4.1

Consider the one-dimensional neutral stochastic differential systems with mixed delays $d [y (t) - 0.125 y (t - 2)] = [- 10 y (t) + 0.5 y (t - 2) + 0.25 \int_{t - 2}^{t} y (s) d s + 1] d t + [0.5 y (t) + 0.5 y (t - 2) + 0.25 \int_{t - 2}^{t} y (s) d s] d ω (t), t \geq 0 .$ A direct check shows that conditions H1 to H3 hold with $δ = 0.125, Q = 1, ς_{1} = - 8.625, ς_{2} = 1.2656, ς_{3} = 0.2344, ς_{4} = 0.5, ς_{5} = 2.4492, ς_{6} = 0.2139, ς_{7} = 0.1904, ς_{8} = 0$ . Taking $ϑ = 0.5$ , we have $- \frac{(1 + δ^{2})}{μ} - δ^{2} e^{ϑ μ} ϑ = - 0.5290 < ς_{2} e^{ϑ μ} + ς_{3} Q^{2} μ \frac{e^{ϑ μ} - 1}{ϑ} = 5.0513 < - (1 + δ^{2} e^{ϑ μ}) ϑ - ς_{1} = 8.1038$ and $C_{1} (ϑ) = 2 (1 + δ^{2}) + 2 δ^{2} μ e^{ϑ μ} ϑ + 2 ς_{2} μ e^{ϑ μ} + 2 ς_{3} Q^{2} μ^{2} \frac{e^{ϑ μ} - 1}{ϑ} = 22.3215 > 0 .$ According to

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^☆: The work is supported by the NNSF of China under Grant 11701512.

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Boundedness analysis of neutral stochastic differential systems with mixed delays☆

Abstract

Introduction

Section snippets

Preliminaries

Main results

Example

Appl. Math. Lett.

Appl. Math. Lett.

Appl. Math. Lett.

Appl. Math. Comput.

J. Math. Anal. Appl.