General Decay Stability for Stochastic Functional Differential Equations with Infinite Delay

Abstract

So far there are not many results on the stability for stochastic functional differential equations with infinite delay. The main aim of this paper is to establish some new criteria on the stability with general decay rate for stochastic functional differential equations with infinite delay. To illustrate the applications of our theories clearly, this paper also examines a scalar infinite delay stochastic functional differential equations with polynomial coefficients.

1. Introduction

Stability is one of the central problems for both deterministic and stochastic dynamic systems. Due to introduction of stochastic factors, stochastic stability mainly includes almost sure stability and the moment stability. In a series of papers (see [1–5]), Mao et al. examined the moment exponential stability and almost sure exponential stability for various stochastic systems.

In many cases we may find that the Lyapunov exponent equals zero, namely, the equation is not exponentially stable, but the solution does tend to zero asymptotically. By this phenomenon, Mao [6] considered polynomial stability of stochastic system, which shows that solution tends to zero polynomially. Then in [7], he extended these two classes of stability into the general decay stability.

In general, time delay and system uncertainty are commonly encountered and are often the source of instability (see [8]). Many studies focused on stochastic systems with delay. Especially, infinite delay systems have received the increasing attention in the recent years since they play important roles in many applied fields (cf. [7, 9–13]). Under the Lipschitz condition and the linear growth condition, Wei and Wang [14] built the existence-and-uniqueness theorem of global solutions to stochastic functional differential equations with infinite delay. There is also some other literature to consider stochastic functional differential equations with infinite delay and we here only mention [15–17].

However, to the best knowledge of the authors, there are not many results on the stability with general decay rate for stochastic functional equations with infinite delay. It is therefore interesting to consider the stability of infinite delay stochastic systems. The main aim of this paper is to establish some new criteria for 𝑝th moment stability and almost surely asymptotic stability with general decay rate of the global solution to stochastic functional differential equations with infinite delay

d𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥_𝑡)d𝑡+𝑔(𝑡,𝑥(𝑡),𝑥_𝑡)d𝑤(𝑡), (1.1) where 𝑓=(𝑓₁,…,𝑓_𝑑)^T∶ℝ₊×ℝ^𝑑×𝐶^𝑏((−∞,0];ℝ^𝑑)→ℝ^𝑑, and 𝑔=[𝑔_𝑖𝑗]_𝑑×𝑟∶ℝ₊×ℝ^𝑑×𝐶^𝑏((−∞,0];ℝ^𝑑)→ℝ^𝑑×𝑟 are Borel measurable functionals, and 𝑤(𝑡) is an 𝑟-dimensional Brownian motion. Without the linear growth condition, we will show that (1.1) has the following properties.

(i)This equation almost surely admits a global solution on [0,∞).(ii)There exists a pair of positive constants 𝑝 and 𝑞 such that this global solution has properties limsup_𝑡→∞lnE|𝑥(𝑡,𝜉)|^𝑝ln𝜓(𝑡)⩽−𝑞,limsup_𝑡→∞ln|𝑥(𝑡,𝜉)|𝑞ln𝜓(𝑡)⩽−𝑝,a.s., (1.2) where 𝜓(𝑡) is a general decay function defined in the next section, namely, this solution is 𝑝th moment and almost surely asymptotically stable with general decay rate.

In the next section, we introduce some necessary notation and definitions. Section 3 gives the main result of this paper by establishing a new criteria for 𝑝th moment stability and almost surely asymptotic stability with general decay rate for the global solution of (1.1). To make our results more applicable, Section 4 gives the further result. To illustrate the application of our result, Section 5 considers a scalar stochastic functional differential equation with infinite delay in detail.

2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notation. Let (Ω,ℱ,{ℱ_𝑡}_𝑡⩾0,ℙ) be a complete probability space with the filtration {ℱ_𝑡}_𝑡⩾0 satisfying the usual conditions, that is, it is right continuous and increasing while ℱ₀ contains all ℙ-null sets. 𝑤(𝑡) is an 𝑟-dimensional Brownian motion defined on this probability space.

Let ℝ₊=[0,+∞), ℝ₊₊=(0,+∞), and ℝ₋=(−∞,0]. Let |𝑥| be the Euclidean norm of vector 𝑥∈ℝ^𝑛. If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴^T. For a matrix 𝐴, its trace norm is denoted by |𝐴|=trace(𝐴^T𝐴). Denote by 𝐶_𝑏=𝐶^𝑏(ℝ₋;ℝ^𝑑) the family of all bounded continuous functions 𝜑 from ℝ₋ to ℝ^𝑑 with the norm ‖𝜑‖=sup_{−∞<𝜃⩽0}|𝜑(𝜃)|, which forms a Banach space. In this paper, 𝑐𝑜𝑛𝑠𝑡 always represents some positive constants whose precise value is not important. If 𝑥(𝑡) is an ℝ^𝑑-valued stochastic process on ℝ, for any 𝑡⩾0, define 𝑥_𝑡=𝑥_𝑡(𝜃)={𝑥(𝑡+𝜃)∶𝜃∈ℝ₋}. 𝐶²(ℝ^𝑑,ℝ) denotes the family of continuously twice differentiable ℝ-valued functions defined on ℝ^𝑑. For any 𝑉(𝑥)∈𝐶²(ℝ^𝑑,ℝ₊), define an operator ℒ𝑉∶ℝ₊×ℝ^𝑑×𝐶_𝑏→ℝ by

ℒ𝑉(𝑡,𝑥,𝜑)=𝑉_𝑥1(𝑥)𝑓(𝑡,𝑥,𝜑)+2trace[𝑔^T(𝑡,𝑥,𝜑)𝑉_𝑥𝑥(𝑥)𝑔(𝑡,𝑥,𝜑)], (2.1) where

𝑉_𝑥(𝑥)=(𝜕𝑉(𝑥)𝜕𝑥₁,𝜕𝑉(𝑥)𝜕𝑥₂,…,𝜕𝑉(𝑥)𝜕𝑥_𝑑),𝑉_𝑥𝑥𝜕(𝑥)=[²𝑉(𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗]_𝑑×𝑑. (2.2)

If 𝑥(𝑡) is a solution of (1.1), for any 𝑉(𝑥)∈𝐶²(ℝ^𝑑,ℝ), applying the Itô formula yields

d𝑉(𝑥(𝑡))=𝐿𝑉(𝑥(𝑡))d𝑡+𝑉_𝑥(𝑥(𝑡))𝑔(𝑡,𝑥(𝑡),𝑥_𝑡)d𝑤(𝑡), (2.3)

where 𝐿𝑉(𝑥(𝑡))=ℒ𝑉(𝑡,𝑥(𝑡),𝑥_𝑡).

Let us introduce the following 𝜓-type function, which will be used as the decay function.

Definition 2.1. The function 𝜓∶ℝ→(0,∞) is said to be the 𝜓-type function if it satisfies the following conditions:(i)it is continuous and nondecreasing in ℝ and differentiable in ℝ₊,(ii)𝜓(0)=1 and 𝜓(∞)=∞,(iii)𝜙∶=sup_𝑡⩾0𝜓₁(𝑡)<∞, where 𝜓₁(𝑡)=𝜓′(𝑡)/𝜓(𝑡),(iv) for any 𝜃⩽0 and 𝑡⩾0, 𝜓(𝑡)⩽𝜓(−𝜃)𝜓(𝑡+𝜃).

It is is easy to find that functions 𝜓(𝑡)=𝑒^𝛾𝑡 and 𝜓(𝑡)=(1+𝑡⁺)^𝛾 for any 𝛾,𝛾>0 are 𝜓-type functions.

For any 𝑝,𝑞⩾0 and 𝜑∈𝐶_𝑏, define

𝒯_𝑝,𝑞(𝜑)=⁰_−∞𝜓^𝑞(𝜃)|𝜑(𝜃)|^𝑝d𝜃 (2.4) and 𝐶(𝑝,𝑞)={𝜑∈𝐶_𝑏∶𝒯_𝑝,𝑞(𝜑)<∞}. Denote by 𝑀₀ the family of all probability measures on ℝ₋. For any 𝜇∈𝑀₀ and 𝜀⩾0, define

𝑀_𝜀={𝜇∈𝑀₀∶𝜇_𝜀∶=⁰_−∞𝜓^𝜀(−𝜃)d𝜇(𝜃)<∞}. (2.5) We also impose the following standard assumption on coefficients 𝑓 and 𝑔.

Assumption 2.2. Let 𝑓 and 𝑔 satisfy the Local Lipschitz condition. That is, for every integer 𝑛⩾1, there is 𝑘_𝑛>0 such that |𝑓(𝑡,𝑥,𝜑)−𝑓(𝑡,𝑥,𝜑)|∨|𝑔(𝑡,𝑥,𝜑)−𝑔(𝑡,𝑥,𝜑)|⩽𝑘_𝑛(|𝑥−𝑥|+‖𝜑−𝜑‖), (2.6)for all 𝑡⩾0 and those 𝑥,𝑥∈ℝ^𝑛, 𝜑,𝜑∈𝐶_𝑏 with |𝑥|∨|𝑥|∨‖𝜑‖∨‖𝜑‖⩽𝑛.

Let us present the continuous semimartingale convergence theory (cf. [18]).

Lemma 2.3. Let 𝑀(𝑡) be a real-value local martingale with 𝑀(0)=0 a.s. Let 𝜁 be a nonnegative ℱ₀-measurable random variable. If 𝑋(𝑡) is a nonnegative continuous ℱ_𝑡-adapted process and satisfies 𝑋(𝑡)≤𝜁+𝑀(𝑡)for𝑡≥0, (2.7)then 𝔼𝑋(𝑡)≤𝜁 and 𝑋(𝑡) is almost surely bounded, namely, lim_𝑡→∞𝑋(𝑡)<∞, a.s.

3. Main Results

In this section, we establish the stability result with general decay rate for (1.1). This result includes the global existence and uniqueness of the solution, the 𝑝th moment stability, and almost surely asymptotic stability with general decay rate.

In order for a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of this equation are generally required to satisfy the linear growth condition and the local Lipschitz condition (see [18, 19]) or a given non-Lipschitz condition and the linear growth condition (cf. [20, 21]). These show that the linear growth condition plays an important role to suppress the potential explosion of solutions and guarantee existence of global solutions. References [16, 22] extended these two classes conditions to infinite delay cases. However, many well-known infinite delay systems such that the Lotka-Volterra (see [13]) do not satisfy the linear growth condition. It is therefore necessary to examine the global existence of the solution for (1.1).

It is well known for stochastic differential equations that the linear growth condition for global solutions may be replaced by the use of the Lyapunov functions [23, 24]. By this idea, this paper establishes the existence-and-uniqueness theorem for (1.1).

For 𝑖=1,2,…,𝑘, let 𝜁_𝑖,𝛼_𝑖∈ℝ₊ and probability measures 𝜇_𝑖∈𝑀_𝜀. Define Γ_𝜀∶ℝ^𝑛×𝐶_𝑏→ℝ as

Γ_𝜀(𝑥,𝜑)=^𝑘_𝑖=0𝜁_𝑖(⁰_−∞|𝜑(𝜃)|^𝛼𝑖d𝜇_𝑖(𝜃)−𝜇_𝑖𝜀|𝑥|^𝛼𝑖), (3.1) where 𝜇_𝑖𝜀 is defined by (2.5). Then the following theorem follows.

Theorem 3.1. Assume that there exist positive constants 𝑎, 𝑝, 𝜀, 𝜁_𝑖, 𝛼_𝑖 and probability measures 𝜇_𝑖∈𝑀_𝜀, where 𝑖=1,2,…,𝑘, such that for any 𝑥∈ℝ^𝑑 and 𝜑∈𝐶_𝑏, the function 𝑉(𝑥)=|𝑥|^𝑝 satisfies ℒ𝑉(𝑡,𝑥,𝜑)⩽Γ_𝜀(𝑥,𝜑)−𝑎|𝑥|^𝑝. (3.2) Under Assumption 2.2, there exists a constant 𝑞>0 such that for any 𝜉∈𝐶(𝛼,𝑞), where 𝛼=min_{0⩽𝑖⩽𝑘}{𝛼_𝑖}, (1.1) almost surely admits a unique global solution 𝑥(𝑡) on [0,∞) and this solution has the properties (1.2).

Proof. For sufficiently small 𝑞∈(0,𝜀), fix the initial data 𝜉∈𝐶(𝛼,𝑞). We divide this proof into the two steps.
Step 1 (existence and uniqueness of the global solution). Under Assumption 2.2, (1.1) has a unique maximal local solution 𝑥(𝑡) on [0,𝜌_𝑒) (see [21]), where 𝜌_𝑒 is the explosion time. If we can show 𝜌_𝑒=∞, a.s., then 𝑥(𝑡) is actually a global solution. Let 𝑛₀ be a positive integer such that sup_𝜃⩽0|𝜉(𝜃)|<𝑛₀. For each integer 𝑛⩾𝑛₀, define the stopping time 𝜎_𝑛=inf{𝑡∈[0,𝜌_𝑒)∶|𝑥(𝑡)|^𝑝⩾𝑛}. (3.3)Obviously, 𝜎_𝑛 is increasing and 𝜎_𝑛→𝜎_∞⩽𝜌_𝑒 as 𝑛→∞. Thus, to prove 𝜌_𝑒=∞ a.s., it is sufficient to show that 𝜎_∞=∞ a.s., which is equivalent to the statement that for any 𝑡>0, ℙ(𝜎_𝑛⩽𝑡)→0 as 𝑛→∞.
For any 𝑡⩾0, define 𝑡_𝑛=𝑡⋀𝜎_𝑛. Applying the Ito formula to 𝜓^𝑞(𝑡)𝑉(𝑥(𝑡)) yields 𝑛ℙ(𝜎_𝑛⩽𝑡)=E[𝐼_{{𝜎𝑛⩽𝑡}}𝑉(𝑥(𝑡_𝑛))]⩽E𝑉(𝑥(𝑡_𝑛))⩽E[𝜓^𝑞(𝑡_𝑛)𝑉(𝑥(𝑡_𝑛))]=const+E^𝑡𝑛₀𝐿[𝜓^𝑞(𝑠)𝑉(𝑥(𝑠))]d𝑠=const+E^𝑡𝑛₀𝜓^𝑞(𝑠)[𝐿𝑉(𝑥(𝑠))+𝑞𝜓₁(𝑠)𝑉(𝑥(𝑠))]d𝑠⩽const+E^𝑡𝑛₀𝜓^𝑞(𝑠)[𝐿𝑉(𝑥(𝑠))+𝑞𝜙𝑉(𝑥(𝑠))]d𝑠⩽const+E^𝑡𝑛₀𝜓^𝑞(𝑠)[Γ_𝜀(𝑥(𝑠),𝑥_𝑠)−𝑎|𝑥(𝑠)|^𝑝+𝑞𝜙𝑉(𝑥(𝑠))]d𝑠. (3.4) Note that by (2.5), 𝜇_𝑖𝜀≥𝜇_𝑖𝑞 for 𝑞≤𝜀. By the Fubini theorem and a substitution technique, we have ^𝑡𝑛₀𝜓^𝑞(𝑠)Γ_𝜀(𝑥(𝑠),𝑥_𝑠=)d𝑠_𝑘𝑖=0𝜁_𝑖[⁰_−∞d𝜇_𝑖(𝜃)^𝑡𝑛₀𝜓^𝑞(𝑠)|𝑥(𝑠+𝜃)|^𝛼𝑖d𝑠−𝜇_𝑖𝜀^𝑡𝑛₀𝜓^𝑞(𝑠)|𝑥(𝑠)|^𝛼𝑖⩽d𝑠]_𝑘𝑖=0𝜁_𝑖[⁰_−∞d𝜇_𝑖(𝜃)^𝑡𝑛_𝜃^+𝜃𝜓^𝑞(𝑠−𝜃)|𝑥(𝑠)|^𝛼𝑖d𝑠−𝜇_𝑖𝑞^𝑡𝑛₀𝜓^𝑞(𝑠)|𝑥(𝑠)|^𝛼𝑖⩽d𝑠]_𝑘𝑖=0𝜁_𝑖[⁰_−∞𝜓^𝑞(−𝜃)d𝜇_𝑖(𝜃)^𝑡𝑛_𝜃^+𝜃𝜓^𝑞(𝑠)|𝑥(𝑠)|^𝛼𝑖d𝑠−𝜇_𝑖𝑞^𝑡𝑛₀𝜓^𝑞(𝑠)|𝑥(𝑠)|^𝛼𝑖⩽d𝑠]_𝑘𝑖=0𝜁_𝑖[𝜇_𝑖𝑞^𝑡𝑛_−∞𝜓^𝑞(𝑠)|𝑥(𝑠)|^𝛼𝑖d𝑠−𝜇_𝑖𝑞^𝑡𝑛₀𝜓^𝑞(𝑠)|𝑥(𝑠)|^𝛼𝑖=d𝑠]_𝑘𝑖=0𝜁_𝑖𝜇_𝑖𝑞⁰_−∞𝜓^𝑞(𝜃)|𝜉(𝜃)|^𝛼𝑖d𝜃. (3.5)(1)(2)(3)(4)(5)Noting that 𝜉∈𝐶(𝛼,𝑞), we have 𝜉∈𝐶(𝛼_𝑖,𝑞), which implies that for all 𝑖=1,…,𝑘, ⁰_−∞𝜓^𝑞(𝜃)|𝜉(𝜃)|^𝛼𝑖d𝜃<∞. (3.6)Hence, there exists ^𝑡𝑛₀𝜓^𝑞(𝑠)Γ_𝜀(𝑥(𝑠),𝑥_𝑠)d𝑠<∞. (3.7) By (3.4) and (3.7), we have 𝑛ℙ(𝜎_𝑛⩽𝑡)⩽const+E^𝑡𝑛₀𝜓^𝑞(𝑠)[𝑞𝜙𝑉(𝑥(𝑠))−𝑎|𝑥(𝑠)|^𝑝]d𝑠. (3.8) Choosing 𝑞 sufficiently small such that 𝑞𝜙⩽𝑎, by (3.8) we have 𝑛ℙ(𝜎_𝑛⩽𝑡)⩽const, which implies that ℙ(𝜎_𝑛⩽𝑡)→0 as 𝑛→∞.
Step 2 (Proof of (1.2)). Define ℎ(𝑡)=𝜓^𝑞(𝑡)𝑉(𝑥(𝑡)). (3.9) By the Itô formula and (3.2), ℎ(𝑡)=ℎ(0)+^𝑡₀𝜓^𝑞(𝑠)[𝐿𝑉(𝑥(𝑠))+𝑞𝜓₁(𝑠)𝑉(𝑥(𝑠))]d𝑠+𝑀(𝑡)⩽ℎ(0)+^𝑡₀𝜓^𝑞(𝑠)[Γ_𝜀(𝑥(𝑠),𝑥_𝑠)−𝑎|𝑥|^𝑝+𝑞𝜙𝑉(𝑥(𝑠))]d𝑠+𝑀(𝑡), (3.10) where 𝑀(𝑡)=^𝑡₀𝜓^𝑞(𝑠)𝑉_𝑥(𝑥(𝑠))𝑔(𝑠,𝑥(𝑠),𝑥_𝑠)d𝑤(𝑠) (3.11) is a continuous local martingale with 𝑀(0)=0. Similar to (3.7), there exists ^𝑡₀𝜓^𝑞(𝑠)Γ_𝜀(𝑥(𝑠),𝑥_𝑠)d𝑠<∞. (3.12) By (3.10), (3.12), noting that 𝑞𝜙⩽𝑎, ℎ(𝑡)⩽const+^𝑡₀𝜓^𝑞(𝑠)(𝑞𝜙−𝑎)|𝑥(𝑠)|^𝑝d𝑠+𝑀(𝑡)⩽const+𝑀(𝑡). (3.13)By Lemma 2.3, we have limsup_𝑡→∞Eℎ(𝑡)<∞,limsup_𝑡→∞ℎ(𝑡)<∞,a.s., (3.14)which implies the required assertions.

4. Further Result

In Theorem 3.1, it is not convenient to check condition (3.2) since it is not related to coefficients 𝑓 and 𝑔 explicitly. To make our theory more applicable, let us impose the following assumption on coefficients 𝑓 and 𝑔.

Assumption 4.1. There exist positive constants 𝜎, 𝜎, 𝛼, 𝛽, 𝜀, and nonnegative constants 𝜎, 𝜆, 𝜆, 𝜆, 𝜎_𝑖, 𝜎_𝑖, 𝜆_𝑗, 𝜆_𝑗, 𝛼_𝑖, 𝛽_𝑗, such that for any 𝑥∈ℝ^𝑛, 𝜑∈𝐶_𝑏, 𝑥^T𝑓(𝑡,𝑥,𝜑)⩽−𝜎|𝑥|^𝛼+2+𝜎⁰_−∞|𝜑(𝜃)|^𝛼+2d𝜇(𝜃)−𝜎|𝑥|²+^𝑘_𝑖=0(𝜎_𝑖|𝑥|^𝛼𝑖+2+𝜎_𝑖⁰_−∞|𝜑(𝜃)|^𝛼𝑖+2d𝜇_𝑖(𝜃)),|𝑔(𝑡,𝑥,𝜑)|⩽𝜆|𝑥|^𝛽+1+𝜆⁰_−∞|𝜑(𝜃)|^𝛽+1+d𝜈(𝜃)+𝜆|𝑥|_𝑙𝑗=0(𝜆_𝑗|𝑥|^𝛽𝑗+1+𝜆_𝑗⁰_−∞|𝜑(𝜃)|^𝛽𝑗+1d𝜈_𝑗(𝜃)), (4.1)(4.2) where 0⩽𝛼₀<𝛼₁<⋯<𝛼_𝑘<𝛼, 0⩽𝛽₀<𝛽₁<⋯<𝛽_𝑙<𝛽, 2𝛽⩽𝛼, and 𝜇,𝜇_𝑖,𝜈,𝜈_𝑗∈𝑀_𝜀.

We also need the following lemma.

Lemma 4.2. Let 𝛼,𝑝>0. Assume that 𝛼₀,𝛼₁,…,𝛼_𝑘, 𝑐₀,𝑐₁,…,𝑐_𝑘 are nonnegative constants such that 0⩽𝛼₀⩽𝛼₁⩽⋯⩽𝛼_𝑘⩽𝛼, 𝑏>𝑐=∶∑^𝑘_𝑖=0𝑐_𝑘 and 𝑎>𝑐𝜌, where 𝜌=1,if𝛼₀=0,0,if𝛼₀=𝛼,(𝛼−𝛼₀𝛼)(^𝛼00𝛼^𝛼)^{1/(𝛼−𝛼0)},if𝛼₀∈(0,𝛼), (4.3) then, there is 𝑎∈(0,𝑎) such that for all 𝑡⩾0, 𝑎𝑡^𝑝+𝑏𝑡^𝛼+𝑝−^𝑘_𝑖=0𝑐_𝑖𝑡^{𝛼𝑖+𝑝}⩾𝑎𝑡^𝑝. (4.4)

Proof. Noting that 𝑎>𝑐𝜌, choose the constant 𝑎 such that 𝑐𝜌<𝑎<𝑎. (4.5) If we can show that for any 𝑡∈[0,∞), 𝐹(𝑡)=∶𝑎+𝑏𝑡^𝛼−∑^𝑘_𝑖=0𝑐_𝑖𝑡^𝛼𝑖⩾0, then the inequality 𝑎+𝑏𝑡^𝛼−^𝑘_𝑖=0𝑐_𝑖𝑡^𝛼𝑖⩾𝑎−𝑎 (4.6)holds. Let 𝑎=𝑎−𝑎. This is equivalent to prove that 𝑎𝑡^𝑝+𝑏𝑡^𝛼+𝑝−^𝑘_𝑖=0𝑐_𝑖𝑡^{𝛼𝑖+𝑝}⩾𝑎𝑡^𝑝. (4.7)
For all 𝑡∈(1,+∞), there exists 𝐹(𝑡)⩾𝑎+𝑏𝑡^𝛼−𝑐𝑡^𝛼. By 𝑎>𝑐𝜌⩾0 and 𝑏>𝑐, we have 𝐹(𝑡)⩾𝑎+𝑏𝑡^𝛼−𝑐𝑡^𝛼>0.
For all 𝑡∈[0,1], there exists 𝐹(𝑡)⩾𝐹_∗(𝑡)=∶𝑎+𝑏𝑡^𝛼−𝑐𝑡^𝛼0. To prove 𝐹(𝑡)⩾0, we consider three cases of 𝛼₀, respectively.
Case 1. 𝛼₀=0. By 𝛼₀=0, we have 𝐹_∗(𝑡)=𝑎+𝑏𝑡^𝛼−𝑐 and 𝑎∈(𝑐,𝑎). Then there exists 𝐹(𝑡)⩾𝐹_∗(𝑡)>0.Case 2. 𝛼₀=𝛼. By 𝛼₀=𝛼, we have 𝐹_∗(𝑡)=𝑎+𝑏𝑡^𝛼−𝑐𝑡^𝛼 and 𝑎∈(0,𝑎). Noting 𝑏>𝑐, we obtain 𝐹(𝑡)⩾𝐹_∗(𝑡)>0.Case 3. 𝛼₀∈(0,𝛼). Without the loss of generality, we assume that 𝑐>0. Obviously, on (0,+∞) the derivative function 𝐹^′_∗(𝑡)=𝑏𝛼𝑡^𝛼−1−𝑐𝛼₀𝑡^𝛼0−1 has a unique null point 𝑡₀=∶(𝛼₀𝑐/𝛼𝑏)^{1/(𝛼−𝛼0)}<1. We can compute that 𝐹_∗(𝑡₀𝛼)=𝑎+𝑏(⁰𝑐)𝛼𝑏^{𝛼/(𝛼−𝛼0)}𝛼−𝑐(⁰𝑐)𝛼𝑏^{𝛼0/(𝛼−𝛼0)}=𝑐𝑎−𝑐(𝑏)^{𝛼0/(𝛼−𝛼0)}(𝛼−𝛼₀𝛼)(^𝛼00𝛼^𝛼)^{1/(𝛼−𝛼0)}. (4.8) Since 0<𝛼₀<𝛼 and 𝑏>𝑐, we know that 𝑐0<(𝑏)^{𝛼0/(𝛼−𝛼0)}<1. (4.9) By (4.8) and (4.9), we obtain that 𝐹_∗(𝑡₀)>𝑎−𝑐𝜌>0. Then we have that for any 𝑡∈[0,1], 𝐹(𝑡)⩾𝐹_∗(𝑡)⩾𝐹_∗(𝑡₀)>0. The proof is completed.

For the purpose of simplicity, we introduce the following notations:

𝜎_⋅=^𝑘_𝑖=0𝜎_𝑖,𝜎_⋅=^𝑘_𝑖=0𝜎_𝑖,𝜆_⋅=^𝑙_𝑗=0𝜆_𝑗,𝜆_⋅=^𝑙_𝑗=0𝜆_𝑗,𝑄=𝜎−𝜎_⋅−𝜎_⋅−𝜎,𝑆=𝜆+𝜆+𝜆+𝜆_⋅+𝜆_⋅ (4.10)

Then the following theorem follows.

Theorem 4.3. Let Assumptions 2.2 and 4.1 hold. Assume that 22𝑄>𝑆(𝑆−𝜆),𝜎−2𝜌(𝜎_⋅+𝜎_⋅)>𝑆[𝜆+𝜌(𝑆−𝜆)], (4.11)(4.12) where 𝜌 is defined by Lemma 4.2 except that 𝛼₀ is replaced by 𝛼₀⋀2𝛽. For any 𝑝∈(2,𝑝₁𝑝₂), (4.13) where 𝑝₁=1+2𝑄𝑆(𝑆−𝜆),𝑝₂2=1+𝜎−2𝜌(𝜎_⋅+𝜎_⋅)𝑆[𝜆+𝜌(𝑆−𝜆)], (4.14) there exists a positive constant 𝑞 such that for any initial data 𝜉∈𝐶((𝛼₀⋀2𝛽₀)+𝑝,𝑞), (1.1) admits a unique global solution 𝑥(𝑡) on [0,∞) and this solution has the properties (1.2).

Proof. Define 𝑉(𝑥)=|𝑥|^𝑝 for 𝑝>2. Applying (2.1) gives ℒ𝑉(𝑡,𝑥,𝜑)=𝑝|𝑥|^𝑝−2𝑥^T𝑝𝑓+2(𝑝−2)|𝑥|^𝑝−4|𝑔^T𝑥|²+𝑝2|𝑥|^𝑝−2|𝑔|²⩽𝑝|𝑥|^𝑝−2𝑥^T𝑝𝑓+2(𝑝−1)|𝑥|^𝑝−2|𝑔|²=∶𝐼₁+𝐼₂. (4.15) By (4.1) and the Young inequality, 𝐼₁=𝑝|𝑥|^𝑝−2𝑥^T𝑓⩽𝑝|𝑥|^𝑝−2[−𝜎|𝑥|^𝛼+2+𝜎⁰_−∞|𝜑(𝜃)|^𝛼+2d𝜇(𝜃)−𝜎|𝑥|²+^𝑘_𝑖=0(𝜎_𝑖|𝑥|^𝛼𝑖+2+𝜎_𝑖⁰_−∞|𝜑(𝜃)|^𝛼𝑖+2d𝜇_𝑖(𝜃))]⩽−𝑝(𝜎−𝜎𝑝−2𝛼+𝑝)|𝑥|^𝛼+𝑝−𝑝𝜎|𝑥|^𝑝+𝑝𝜎𝛼+2𝛼+𝑝⁰_−∞|𝜑(𝜃)|^𝛼+𝑝d𝜇(𝜃)+𝑝^𝑘_𝑖=0(𝜎_𝑖+𝜎_𝑖𝑝−2𝛼_𝑖+𝑝)|𝑥|^{𝛼𝑖+𝑝}+𝑝^𝑘_𝑖=0𝜎_𝑖𝛼_𝑖+2𝛼_𝑖+𝑝⁰_−∞|𝜑(𝜃)|^{𝛼𝑖+𝑝}d𝜇_𝑖(𝜃). (4.16) Recall the following elementary inequality: for any 𝜆_𝑗⩾0 and 𝑥_𝑗∈ℝ, 𝑗=0,1,…,𝑛, applying the Hölder inequality yields (^𝑛_𝑗=0𝜆_𝑗𝑥_𝑗)²⩽^𝑛_𝑗=0𝜆_𝑗^𝑛_𝑗=0𝜆_𝑗𝑥²_𝑗. (4.17) By (4.2) and (4.17), applying the Young inequality and the Hölder inequality, we have 𝐼₂=𝑝2(𝑝−1)|𝑥|^𝑝−2|𝑔|²⩽𝑝(𝑝−1)2|𝑥|^𝑝−2[𝜆|𝑥|^𝛽+1+𝜆⁰_−∞|𝜑(𝜃)|^𝛽+1+d𝜈(𝜃)+𝜆|𝑥|_𝑙𝑗=0(𝜆_𝑗|𝑥|^𝛽𝑗+1+𝜆_𝑗⁰_−∞|𝜑(𝜃)|^𝛽𝑗+1d𝜈_𝑗(𝜃))]²⩽𝑆𝑝(𝑝−1)2[(𝜆+𝜆𝑝−22𝛽+𝑝)|𝑥|^2𝛽+𝑝+𝜆2𝛽+22𝛽+𝑝⁰_−∞|𝜑(𝜃)|^2𝛽+𝑝d𝜈(𝜃)+𝜆|𝑥|^𝑝+^𝑙_𝑗=0(𝜆_𝑗+𝜆_𝑗𝑝−22𝛽_𝑗+𝑝)|𝑥|^{2𝛽𝑗+𝑝}+^𝑙_𝑗=0𝜆_𝑗2𝛽_𝑗+22𝛽_𝑗+𝑝⁰_−∞|𝜑(𝜃)|^{2𝛽𝑗+𝑝}d𝜈_𝑗(𝜃)]. (4.18) Substituting (4.16) and (4.18) into (4.15) yields ℒ𝑉(𝑡,𝑥,𝜑)⩽Γ_𝜀𝑝(𝑥,𝜑)−2𝐻(𝑥), (4.19) where Γ_𝜀(𝑥,𝜑)=^𝑘_𝑖=0𝑝𝜎_𝑖𝛼_𝑖+2𝛼_𝑖(+𝑝⁰_−∞|𝜑(𝜃)|^{𝛼𝑖+𝑝}d𝜇_𝑖(𝜃)−𝜇_𝑖𝜀|𝑥|^{𝛼𝑖+𝑝})+𝑝𝜎𝛼+2𝛼+𝑝(⁰_−∞|𝜑(𝜃)|^𝛼+𝑝d𝜇(𝜃)−𝜇_𝜀|𝑥|^𝛼+𝑝)+^𝑙_𝑗=0𝑆𝑝(𝑝−1)2𝜆_𝑗2𝛽_𝑗+22𝛽_𝑗+𝑝(⁰_−∞|𝜑(𝜃)|^{2𝛽𝑗+𝑝}d𝜈_𝑗(𝜃)−𝜈_𝑗𝜀|𝑥|^{2𝛽𝑗+𝑝})+𝑆𝑝(𝑝−1)2𝜆2𝛽+2(2𝛽+𝑝⁰_−∞|𝜑(𝜃)|^2𝛽+𝑝d𝜈(𝜃)−𝜈_𝜀|𝑥|^2𝛽+𝑝), (4.20) whose expression is similar to (3.1) and 𝐻(𝑥)=𝑎|𝑥|^𝑝+𝑏(𝜀)|𝑥|^𝛼+𝑝−𝑐(̃𝜀)|𝑥|^2𝛽+𝑝−^𝑘_𝑖=0𝑐_𝑖(𝜀)|𝑥|^{𝛼𝑖+𝑝}−^𝑙_𝑗=0𝑐_𝑗̃(𝜀)|𝑥|^{2𝛽𝑗+𝑝}, (4.21) in which 𝑎=2𝜎−𝑆(𝑝−1)𝜆,𝑏(𝜀)=2𝜎−2𝜎𝑝−2𝛼+𝑝−2𝜎𝛼+2𝜇𝛼+𝑝^𝜀,𝑐(̃𝜀)=𝑆(𝑝−1)(𝜆+𝜆𝑝−2+2𝛽+𝑝𝜆2𝛽+2𝜇2𝛽+𝑝^𝜀𝑐),_𝑖(𝜀)=2𝜎_𝑖+2𝜎_𝑖𝑝−2𝛼_𝑖+𝑝+2𝜎_𝑖𝛼_𝑖+2𝛼_𝑖𝜇+𝑝^𝑖𝜀,𝑐_𝑗̃(𝜀)=𝑆(𝑝−1)(𝜆_𝑗+𝜆_𝑗𝑝−22𝛽_𝑗++𝑝𝜆_𝑗2𝛽_𝑗+22𝛽_𝑗𝜈+𝑝^𝑗𝜀). (4.22) Let 𝑐(𝜀)=𝑐(̃𝜀)+∑^𝑘_𝑖=0𝑐_𝑖(𝜀)+∑^𝑙_𝑗=0𝑐_𝑗̃(𝜀). Note that 𝑏(0)=2(𝜎−𝜎), 𝑐(̃0)=𝑆(𝑝−1)(𝜆+𝜆), 𝑐_𝑖(0)=2(𝜎_𝑖+𝜎_𝑖), 𝑐_𝑗̃(0)=𝑆(𝑝−1)(𝜆_𝑗+𝜆_𝑗), 𝑐(0)=𝑆(𝑝−1)(𝑆−𝜆)+2(𝜎_.+𝜎_.). By (4.13), we obtain that 𝑐(̃0)>0, 𝑐(0)>0, and 𝑐_𝑗̃(0)>0 for all 0⩽𝑗⩽𝑙. By (4.11) and (4.13), we have 𝑏(0)>𝑐(0). By (4.12) and (4.13), we obtain 𝑎(0)>𝜌𝑐(0). Choose sufficiently small 𝜀 such that 𝑎>𝜌𝑐(𝜀),𝑏(𝜀)>𝑐(𝜀). (4.23) By (4.23) and Lemma 4.2, there exists a constant 𝑎∈(0,𝑎) such that 𝑎|𝑥|^𝑝⩽𝑎|𝑥|^𝑝+𝑏(𝜀)|𝑥|^𝛼+𝑝−𝑐(̃𝜀)|𝑥|^2𝛽+𝑝−^𝑘_𝑖=0𝑐_𝑖(𝜀)|𝑥|^{𝛼𝑖+𝑝}−^𝑙_𝑗=0𝑐_𝑗̃(𝜀)|𝑥|^{2𝛽𝑗+𝑝}. (4.24) By (4.19), (4.21), and (4.24), we therefore have ℒ𝑉(𝑡,𝑥,𝜑)⩽Γ_𝜀𝑝(𝑥,𝜑)−2𝑎|𝑥|^𝑝, (4.25) which implies that condition (3.2) is satisfied. By (4.20), (4.25), and the fact that 0⩽𝛼₀<𝛼₁<⋯<𝛼_𝑘<𝛼 and 0⩽𝛽₀<𝛽₁<⋯<𝛽_𝑙<𝛽, applying Theorem 3.1 yields that there exists 𝑞>0, such that for any 𝜉∈𝐶((𝛼₀⋀2𝛽₀)+𝑝,𝑞), the desired assertions hold. The proof is completed.

5. A Scalar Case

To illustrate the application of our result, this section considers a scalar stochastic functional differential equations

d𝑥(𝑡)=[^𝑛_𝑟=1𝑥^𝑟(𝑡)𝑢_𝑟(𝑡)+_{0⩽𝑟<𝑟+𝑠⩽𝑛}𝑥^𝑟(𝑡)⁰_−∞𝑥^𝑠(𝑡+𝜃)𝑢_𝑟𝑠(𝑡,𝜃)d𝜃]d𝑡+[^𝑚_𝑘=1𝑥^𝑘(𝑡)𝑣_𝑘(𝑡)+_{0⩽𝑘<𝑘+𝑙⩽𝑚}𝑥^𝑘(𝑡)⁰_−∞𝑥^𝑙(𝑡+𝜃)𝑣_𝑘𝑙(𝑡,𝜃)d𝜃]d𝑤(𝑡), (5.1) where for 𝑟=1,2,…,𝑛 and 𝑘=1,2,…,𝑚, 𝑢_𝑟(𝑡),𝑣_𝑘(𝑡)∈𝐶(ℝ₊), for 0⩽𝑟<𝑟+𝑠⩽𝑛 and 0⩽𝑘<𝑘+𝑙⩽𝑚,𝑢_𝑟𝑠(𝑡,𝜃),𝑣_𝑘𝑙(𝑡,𝜃)∈𝐶(ℝ₊×ℝ₋),𝑛⩾3 is an odd number, 𝑚⩾2, and 2𝑚⩽𝑛+1. In this section, ∑_{0≤𝑟<𝑟+𝑠≤𝑛}∶=∑^𝑛_𝑟=0∑^𝑛_𝑠=0 with 𝑟+𝑠≤𝑛 and ∑_{0≤𝑘<𝑟+𝑙≤𝑚} has similar explanation. Assume

𝑢₁(𝑡)⩽−𝑎₁𝑢<0,_𝑛(𝑡)⩽−𝑎_𝑛<0,|𝑢_𝑟(𝑡)|⩽𝑎_𝑟,where2⩽𝑟⩽𝑛−1,|𝑢_𝑟𝑠(𝑡,𝜃)|⩽𝑎_𝑟𝑠2𝜀(1−𝜃)^−1−2𝜀,where0⩽𝑟<𝑟+𝑠⩽𝑛,|𝑣_𝑘(𝑡)|⩽𝑏_𝑘,where1⩽𝑘⩽𝑚,|𝑣_𝑘𝑙(𝑡,𝜃)|⩽𝑏_𝑘𝑙2𝜀(1−𝜃)^−1−2𝜀,where0⩽𝑘<𝑘+𝑙⩽𝑚, (5.2)(5.3) in which 𝑎_𝑟, 𝑎_𝑟𝑠, 𝑏_𝑘, 𝑏_𝑘𝑙 are nonnegative constants and 𝜀>0. Define

𝑓(𝑡,𝑥,𝜑)=^𝑛_𝑟=1𝑥^𝑟(𝑡)𝑢_𝑟(𝑡)+_{0⩽𝑟<𝑟+𝑠⩽𝑛}𝑥^𝑟(𝑡)⁰_−∞𝜑^𝑠(𝜃)𝑢_𝑟𝑠(𝑡,𝜃)d𝜃,𝑔(𝑡,𝑥,𝜑)=^𝑚_𝑘=1𝑥^𝑘(𝑡)𝑣_𝑘(𝑡)+_{0⩽𝑘<𝑘+𝑙⩽𝑚}𝑥^𝑘(𝑡)⁰_−∞𝜑^𝑙(𝜃)𝑣_𝑘𝑙(𝑡,𝜃)d𝜃. (5.4) It is obvious that 𝑓(𝑡,𝑥,𝜑) and 𝑔(𝑡,𝑥,𝜑) satisfy the local Lipschtiz condition. By (5.4), (5.1) can be rewritten as (1.1).

Choose the Ψ-type function 𝜓(𝑡)=1+𝑡⁺. Let d𝜇(𝜃)=2𝜀(1−𝜃)^−1−2𝜀d𝜃. It is obvious that ∫⁰_−∞d𝜇(𝜃)=1 and

⁰_−∞𝜓^𝜀(−𝜃)d𝜇(𝜃)=⁰_−∞[1+(−𝜃)⁺]^𝜀2𝜀(1−𝜃)^−1−2𝜀d𝜃=⁰_−∞2𝜀(1−𝜃)^−1−𝜀d𝜃=2<∞, (5.5)

which shows that 𝜇∈𝑀_𝜀.

By (5.2) and the Young inequality, we have that

𝑥^T𝑓(𝑡,𝑥,𝜑)⩽−𝑎₁|𝑥|²+^𝑛−2_𝑖=1𝑎_𝑖+1|𝑥|^𝑖+2−𝑎_𝑛|𝑥|^𝑛+1+_{0⩽𝑟<𝑟+𝑠⩽𝑛}𝑎_𝑟𝑠|𝑥|^𝑟+10_−∞|𝜑(𝜃)|^𝑠d𝜃⩽−𝑎₁|𝑥|²+^𝑛−2_𝑖=1𝑎_𝑖+1|𝑥|^𝑖+2−𝑎_𝑛|𝑥|^𝑛+1+_{0⩽𝑟<𝑟+𝑠⩽𝑛}𝑎_𝑟𝑠(𝑟+1𝑟+𝑠+1|𝑥|^𝑟+𝑠+1+𝑠𝑟+𝑠+1⁰_−∞|𝜑(𝜃)|^𝑟+𝑠+1d𝜇(𝜃))=−(𝑎_𝑛−_{0⩽𝑟<𝑟+𝑠=𝑛}(𝑟+1)𝑎_𝑟𝑠𝑛+1)|𝑥|^𝑛−1+2+_{0⩽𝑟<𝑟+𝑠=𝑛}𝑠𝑎_𝑟𝑠𝑛+1⁰_−∞|𝜑(𝜃)|^𝑛−1+2d𝜇(𝜃)−𝑎₁|𝑥|²+12𝑎₀₁|𝑥|²+^𝑛−2_𝑖=1(𝑎_𝑖+1+_{0⩽𝑟<𝑟+𝑠=𝑖+1}(𝑟+1)𝑎_𝑟𝑠𝑖+2)|𝑥|^𝑖+2+^𝑛−2_{𝑖=00⩽𝑟<𝑟+𝑠=𝑖+1}𝑠𝑎_𝑟𝑠𝑖+2⁰_−∞|𝜑(𝜃)|^𝑖+2d𝜇(𝜃)=∶−𝜎|𝑥|^𝛼+2+𝜎⁰_−∞|𝜑(𝜃)|^𝛼+2d𝜇(𝜃)−𝜎|𝑥|²+^𝑛−2_𝑖=0(𝜎_𝑖|𝑥|^𝛼𝑖+2+𝜎_𝑖⁰_−∞|𝜑(𝜃)|^𝛼𝑖+2d𝜇(𝜃)), (5.6)

which shows that condition (4.1) holds with

𝜎=𝑎_𝑛−_{0⩽𝑟<𝑟+𝑠=𝑛}(𝑟+1)𝑎_𝑟𝑠,𝑛+1𝜎=_{0⩽𝑟<𝑟+𝑠=𝑛}𝑠𝑎_𝑟𝑠,𝑛+1𝜎=𝑎₁,𝜎_𝑖=𝑎_𝑖+1+_{1⩽𝑟<𝑟+𝑠=𝑖+1}(𝑟+1)𝑎_𝑟𝑠,𝑖+2𝜎_𝑖=_{1⩽𝑟<𝑟+𝑠=𝑖+1}𝑠𝑎_𝑟𝑠,𝑖+2𝛼=𝑛−1,𝛼_𝑖=𝑖. (5.7)

By (5.3) and the Young inequality, we get that

|𝑔(𝑡,𝑥,𝜑)|⩽𝑏₁|𝑥|+^𝑚−1_𝑗=1𝑏_𝑗|𝑥|^𝑗+𝑏_𝑚|𝑥|^𝑚+_{0⩽𝑘<𝑘+𝑙⩽𝑚}𝑏_𝑘𝑙|𝑥|^𝑘0_−∞|𝜑(𝜃)|^𝑙d𝜃⩽𝑏₁|𝑥|+^𝑚−1_𝑗=1𝑏_𝑗|𝑥|^𝑗+𝑏_𝑚|𝑥|^𝑚+_{0⩽𝑘<𝑘+𝑙⩽𝑚}𝑏_𝑘𝑙(𝑘𝑘+𝑙|𝑥|^𝑘+𝑙+𝑙𝑘+𝑙⁰_−∞|𝜑(𝜃)|^𝑘+𝑙d𝜇(𝜃))=(𝑏_𝑚+_{0⩽𝑘<𝑘+𝑙=𝑚}𝑘𝑏_𝑘𝑙𝑚)|𝑥|^𝑚−1+1+_{0⩽𝑘<𝑘+𝑙=𝑚}𝑙𝑏_𝑘𝑙𝑚⁰_−∞|𝜑(𝜃)|^𝑚−1+1d𝜇(𝜃)+𝑏₁+|𝑥|_{𝑚−2𝑗=1}(𝑏_𝑗+1+_{0⩽𝑘<𝑘+𝑙=𝑗+1}𝑘𝑏_𝑘𝑙𝑗+1)|𝑥|^𝑗+1+^𝑚−2_{𝑗=00⩽𝑘<𝑘+𝑙=𝑗+1}𝑙𝑏_𝑘𝑙𝑗+1⁰_−∞|𝜑(𝜃)|^𝑗+1d𝜇(𝜃)=∶𝜆|𝑥|^𝛽+1+𝜆⁰_−∞|𝜑(𝜃)|^𝛽+1+d𝜇(𝜃)−𝜆|𝑥|_{𝑚−2𝑗=0}(𝜆_𝑗|𝑥|^𝛽𝑗+1+𝜆_𝑗⁰_−∞|𝜑(𝜃)|^𝛽𝑗+1d𝜇(𝜃)), (5.8)

which shows that condition (4.2) holds with

𝜆=𝑏_𝑚+_{0⩽𝑘<𝑘+𝑙=𝑚}𝑘𝑏_𝑘𝑙𝑚,𝜆=_{0⩽𝑘<𝑘+𝑙=𝑚}𝑙𝑏_𝑘𝑙𝑚,𝜆=𝑏₁,𝜆_𝑗=_{0⩽𝑘<𝑘+𝑙=𝑗+1}𝑙𝑏_𝑘𝑙𝑗+1,𝜆_𝑗=𝑏0,if𝑗=0,_𝑗+1+_{0⩽𝑘<𝑘+𝑙=𝑗+1}𝑘𝑏_𝑘𝑙𝑗+1,if1⩽𝑗⩽𝑚−2,𝛽=𝑚−1,𝛽_𝑗=𝑗. (5.9)

By 2𝑚⩽𝑛+1, we have 2(𝑚−1)⩽𝑛−1, which implies 2𝛽⩽𝛼. It is easy to see that 𝜎, 𝜎, 𝜆, 𝜆, and 𝜆 are positive, and 𝜎_𝑖, 𝜎_𝑖, 𝜆_𝑗, 𝜆_𝑗, are nonnegative, where 0⩽𝑖⩽𝑛−2,0⩽𝑗⩽𝑚−2.

By the parameters in Theorem 4.3, we can compute

𝜌=1,𝜎_⋅+𝜎_⋅=^𝑛−1_𝑖=2𝑎_𝑖+_{0⩽𝑟<𝑟+𝑠⩽𝑛−1}𝑎_𝑟𝑠,𝑆=𝑏_⋅+_{0⩽𝑘<𝑘+𝑙⩽𝑚}𝑏_𝑘𝑙,𝑄=𝑎_𝑛−(^𝑛−1_𝑖=2𝑎_𝑖+_{0⩽𝑟<𝑟+𝑠⩽𝑛}𝑎_𝑟𝑠),𝑆−𝜆₀=^𝑚_𝑗=2𝑏_𝑗+_{0⩽𝑘<𝑘+𝑙⩽𝑚}𝑏_𝑘𝑙. (5.10)

In Assumption 4.1, the parameter 𝜎 is positive, so it is required that

𝑎_𝑛>_{0⩽𝑟<𝑟+𝑠=𝑛}(𝑟+1)𝑎_𝑟𝑠𝑛+1. (5.11) Let

𝑊₁=^𝑛−1_𝑖=2𝑎_𝑖+_{0⩽𝑟<𝑟+𝑠⩽𝑛−1}𝑎_𝑟𝑠,𝑊₂=𝑏_⋅+_{0⩽𝑘<𝑘+𝑙⩽𝑚}𝑏_𝑘𝑙,𝑊₃=^𝑛−1_𝑖=2𝑎_𝑖+_{0⩽𝑟<𝑟+𝑠⩽𝑛}𝑎_𝑟𝑠. (5.12) To apply Theorem 4.3, it is necessary to test that (4.11)–(4.13) are satisfied. This requires that

𝑎₁>𝑊₁+12𝑊²₂,𝑎_𝑛>𝑊₃+12𝑊₂(𝑊₂−𝑏₁). (5.13)(5.14) Obviously, (5.11) can be obtained from (5.14). By (4.14),

𝑝₁=1+2(𝑎_𝑛−𝑊₃)𝑊₂(𝑊₂−𝑏₁),𝑝₂=1+2(𝑎₁−𝑊₁)𝑊²₂. (5.15) Thus, we have the following corollary from Theorem 4.3.

Corollary. Let conditions (5.2), (5.3), (5.13), and (5.14) be satisfied, where 𝑊₁, 𝑊₂, and 𝑊₃ are given in (5.12). For any 𝑝∈(2,𝑝₁⋀𝑝₂), where 𝑝₁ and 𝑝₂ are given in (5.15), there exist 𝑞>0, for any 𝜉∈𝐶(𝑝,𝑞), (5.1) has a unique global solution 𝑥(𝑡)=𝑥(𝑡,𝜉), and this solution has properties limsup_𝑡→∞lnE|𝑥(𝑡,𝜉)|^𝑝ln(1+𝑡⁺)⩽−𝑞,limsup_𝑡→∞ln|𝑥(𝑡,𝜉)|ln(1+𝑡⁺)𝑞⩽−𝑝,a.s. (5.16)