The Asymptotic Behavior for Second-Order Neutral Stochastic Partial Differential Equations with Infinite Delay

Abstract

By establishing two Lemmas, the exponential stability and the asymptotical stability for mild solution to the second-order neutral stochastic partial differential equations with infinite delay are obtained, respectively. Our results can generalize and improve some existing ones. Finally, an illustrative example is given to show the effectiveness of the obtained results.

1. Introduction

The neutral stochastic differential equations can play an important role in describing many sophisticated dynamical systems in physical, biological, medical, chemical engineering, aero-elasticity, and social sciences [1–3], and the qualitative dynamics such as the existence and uniqueness, stability, and controllability for first-order neutral stochastic partial differential equations with delays have been extensively studied by many authors; see, for example, the existence and uniqueness for neutral stochastic partial differential equations under the non-Lipschitz conditions was investigated by using the successive approximation [4–6]; in [7], Caraballo et al. have considered the exponential stability of neutral stochastic delay partial differential equations by the Lyapunov functional approach; in [8], Dauer and Mahmudov have analyzed the existence of mild solutions to semilinear neutral evolution equations with nonlocal conditions by using the fractional power of operators and Krasnoselski-Schaefer-type fixed point theorem; in [9], Hu and Ren have established the existence results for impulsive neutral stochastic functional integrodifferential equations with infinite delays by means of the Krasnoselskii-Schaefer-type fixed point theorem; some sufficient conditions ensuring the controllability for neutral stochastic functional differential inclusions with infinite delay in the abstract space with the help of the Leray-Schauder nonlinear alterative have been given by Balasubramaniam and Muthukumar in [10]; Luo and Taniguchi, in [11], have studied the asymptotic stability for neutral stochastic partial differential equations with infinite delay by using the fixed point theorem. Very recently, in [12], the author has discussed the exponential stability for mild solution to neutral stochastic partial differential equations with delays by establishing an integral inequality.

Although there are many valuable results about neutral stochastic partial differential equations, they are mainly concerned with the first-order case. In many cases, it is advantageous to treat the second-order stochastic differential equations directly rather than to convert them to first-order systems. The second-order stochastic differential equations are the right model in continuous time to account for integrated processes that can be made stationary. For instance, it is useful for engineers to model mechanical vibrations or charge on a capacitor or condenser subjected to white noise excitation through a second-order stochastic differential equations. The studies of the qualitative properties about abstract deterministic second-order evolution equation governed by the generator of a strongly continuous cosine family was proposed in [13–15]. Recently, Mahmudov and McKibben, in [16], have established the approximate controllability of second-order neutral stochastic evolution equations; the existence and uniqueness for mild solution to second-order neutral impulsive stochastic evolution equations with delay under the non-Lipschitz condition was considered by the successive approximation [17]; Balasubramaniam and Muthukumar in [10] have also discussed the approximate controllability of second-order neutral stochastic distributed implicit functional differential equations with infinite delay; Sakthivel et al. in [18] have studied the asymptotic stability of second-order neutral stochastic differential equations by the fixed point theorem.

However, the work done by Sakthivel et al. [18] is mainly in connection with no heredity case. Since many systems arising from realistic models heavily depend on histories [19] (i.e., there is the effect of infinite delay on state equations), there is a real need to proceed with studying the second-order neutral stochastic partial differential equations with infinite delay. Although Sakthivel et al. [18] have applied the fixed point theorem to discuss the asymptotic stability for mild solution to the second-order neutral stochastic partial differential equations, the method proposed by Sakthivel et al. [18] is not suitable for such equations with infinite delay. Obviously, the Lyapunov functional method utilized by Caraballo et al. [7] fails to deal with the asymptotic behavior for mild solution to the second-order neutral stochastic partial differential equations with infinite delay since the mild solutions do not have stochastic differentials. Besides, to the best of author’s knowledge, there is no paper which is involved with the exponential stability and the asymptotic stability for mild solution to second-order neutral stochastic partial differential equations with infinite delay. So, in this paper, we will make the first attempt to close this gap.

The format of this work is organized as follows. In Section 2, some necessary definitions, notations, and Lemmas used in this paper are introduced; in Section 3, the main results in this paper are given. Finally, an illustrative example is provided to demonstrate the effectiveness of our obtained results.

2. Preliminaries

Let 𝑋 and 𝑌 be two real, separable Hilbert spaces and 𝐿(𝑌,𝑋) the space of bounded linear operators from 𝑌 to 𝑋. For the sake of convenience, we will use the same notation ‖⋅‖ to denote the norms in 𝑋,𝑌 and 𝐿(𝑌,𝑋) when no confusion possibly arises. Let (Ω,ℑ,{ℑ_𝑡}_𝑡≥0,𝑃) be a complete probability space equipped with some filtration ℑ_𝑡(𝑡≥0) satisfying the usual conditions, that is, the filtration is right continuous and ℑ₀ contains all 𝑃-null sets. Let 𝐶((−∞,0],𝑋) be the space of all bounded and continuous functions 𝜑 from (−∞,0] to 𝑋 with the sup-norm ‖⋅‖_𝐶=sup_{−∞<𝜃≤0}‖𝜑(𝜃)‖, and the space 𝔅 present the family of all ℑ_𝑡(𝑡≥0)-measurable and 𝐶((−∞,0],𝑋)-valued random variables.

Let 𝛽_𝑛(𝑡)(𝑛=1,2,…) be a sequence of real-valued one-dimensional standard Brownian motions mutually independent over (Ω,ℑ,{ℑ_𝑡}_𝑡≥0,𝑃). Set 𝑤(𝑡)=∑^+∞_𝑛=1𝜆_𝑛𝛽_𝑛(𝑡)𝑒_𝑛,𝑡≥0, where 𝜆_𝑛≥0(𝑛=1,2,…) are nonnegative real numbers and {𝑒_𝑛}(𝑛=1,2,…) is a complete orthonormal basis in 𝑌. Let 𝑄∈𝐿(𝑌,𝑌) be an operator defined by 𝑄𝑒_𝑛=𝜆_𝑛𝑒_𝑛 with finite trace tr𝑄=∑^+∞_𝑛=1𝜆_𝑛<+∞. Then, the above 𝑌-valued stochastic process 𝑤(𝑡) is called a 𝑄-Wiener process.

Definition 2.1 (see [20]). Let 𝜎∈𝐿(𝑌,𝑋) and define ‖𝜎‖²_𝐿02∶=tr𝜎𝑄𝜎^∗=^+∞_𝑛=1𝜆_𝑛𝜎𝑒_𝑛². (2.1) If ‖𝜎‖²_𝐿02<+∞, then 𝜎 is called a 𝑄-Hilbert—Schmidt operator, and let 𝐿⁰₂(𝑌,𝑋) denote the space of all 𝑄-Hilbert—Schmidt operators 𝜎∶𝑌→𝑋.
Now, for the definition of an 𝑋-valued stochastic integral of an 𝐿⁰₂(𝑌,𝑋)-valued and ℑ_𝑡-adapted predictable process Φ(𝑡) with respect to the 𝑄-Wiener process 𝑤(𝑡), the readers can refer to [20].
In this paper, we consider the following second-order neutral stochastic partial differential equations with infinite delay: 𝑑𝑥^′(𝑡)−𝑓₀𝑡,𝑥_𝑡=𝐴𝑥(𝑡)+𝑓₁𝑡,𝑥_𝑡𝑑𝑡+𝑓₂𝑡,𝑥_𝑡(𝑡)[),𝑥𝑑𝑤,𝑡∈0,+∞₀(⋅)𝑥=𝜑,_′(0)=𝜉, (2.2) where 𝜑∈𝔅 and 𝜉 is also an ℑ₀-measurable 𝑋-valued random variable independent of the Wiener process 𝑤(𝑡). 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is the infinitesimal generator of a strongly continuous cosine family on 𝑋; 𝑓_𝑖∶[0,+∞)×𝔅→𝑋(𝑖=0,1), 𝑓₂∶[0,+∞)×𝔅→𝐿⁰₂(𝑌,𝑋) are three approximate mappings. In this sequel, the history 𝑥_𝑡∶(−∞,0]→𝑋, 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃)(𝑡≥0) belongs to the space 𝔅.
At the end of this section, let us introduce the following Lemmas and definitions that are useful for the development of our results. The one parameter cosine family {𝐶(𝑡)∶𝑡∈𝑅}⊂𝐿(𝑋,𝑋) satisfying(i)𝐶(0)=𝐼,(ii)𝐶(𝑡)𝑥 is in continuous in 𝑡 on 𝑅 for all 𝑥∈𝑅,(iii)𝐶(𝑡+𝑠)+𝐶(𝑡−𝑠)=2𝐶(𝑡)𝐶(𝑠) for all 𝑡,𝑠∈𝑅is called a strongly continuous cosine family.
The corresponding strongly continuous sine family {𝑆(𝑡)∶𝑡∈𝑅}⊂𝐿(𝑋) is defined by 𝑆(𝑡)𝑥=∫^𝑡₀𝐶(𝑠)𝑥𝑑𝑠,𝑡∈𝑅,𝑥∈𝑋. The generator 𝐴∶𝑋→𝑋 of {𝐶(𝑡)∶𝑡∈𝑅} is given by 𝐴𝑥=(𝑑²/𝑑𝑡²)𝐶(𝑡)𝑥|_𝑡=0 for all 𝑥∈𝐷(𝐴)={𝑥∈𝑋∶𝐶(⋅)𝑥∈𝐶²(𝑅,𝑋)}. It is well known that the infinitesimal generator 𝐴 is a closed, densely defined operator on 𝑋. Such cosine and the corresponding sine families and their generators satisfy the following properties.

Lemma 2.2 (see [21]). Suppose that 𝐴 is the infinitesimal generator of a cosine family of operators {𝐶(𝑡)∶𝑡∈𝑅}. Then, the following holds:(i)there exists 𝑀^∗≥1 and 𝛼≥0 such that ‖C(𝑡)‖≤𝑀^∗𝑒^𝛼𝑡 and hence ‖𝑆(𝑡)‖≤𝑀^∗𝑒^𝛼𝑡,(ii)𝐴∫^𝑟_𝑠𝑆(𝑢)𝑥𝑑𝑢=[𝐶(𝑟)−𝐶(𝑢)]𝑥 for all 0≤𝑠≤𝑟<+∞,(iii)there exists 𝑁^∗≥1 such that ‖𝑆(𝑠)−𝑆(𝑟)‖≤𝑁^∗|∫^𝑠_𝑟𝑒^𝛼|𝜃|𝑑𝜃| for all 0≤𝑟≤𝑠<+∞.

Lemma 2.3 (see [20]). For any 𝑟≥1 and for arbitrary 𝐿⁰₂(𝑌,𝑋)-valued predictable process 𝜙(⋅) such that sup_{𝑠∈[0,𝑡]}𝐸^𝑠₀𝜙(𝑢)𝑑𝑤(𝑢)^2𝑟≤𝐶_𝑟^𝑡₀𝐸𝜙(𝑠)_𝐿^2𝑟₀₂^1/𝑟𝑑𝑠^𝑟,𝑡∈[0,+∞), (2.3) where 𝐶_𝑟=(𝑟(2𝑟−1))^𝑟.

Definition 2.4. An 𝑋-value stochastic process 𝑥(𝑡)(𝑡∈𝑅) is called a mild solution of the system (2.2) if(i)𝑥(𝑡) is adapted to ℑ_𝑡(𝑡≥0) and has càdlàg path on 𝑡≥0 almost surely,(ii)for arbitrary 𝑡∈[0,+∞), 𝑃{𝜔∶∫^𝑡₀‖𝑥(𝑡)‖²𝑑𝑡<+∞}=1 and almost surely 𝑥(𝑡)=𝐶(𝑡)𝜑+𝑆(𝑡)𝜉−𝑓₀0,𝑥₀+^𝑡₀𝐶(𝑡−𝑠)𝑓₀𝑠,𝑥_𝑠+𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑓₁𝑠,𝑥_𝑠𝑑𝑠+^𝑡₀𝑆(𝑡−𝑠)𝑓₂𝑠,𝑥_𝑠𝑑𝑤(𝑠), (2.4) where 𝑥₀(⋅)=𝜑∈𝔅.

Definition 2.5. The solution of integral equation (2.4) is said to be exponentially stable in 𝑝(𝑝≥2) moment, if there exists a pair of positive constants 𝛾>0 and 𝑀₁>0 such that 𝐸‖𝑥(𝑡)‖^𝑝≤𝑀₁𝑒^−𝛾𝑡,𝑡≥0,𝑝≥2, (2.5) for any initial value 𝜑∈𝔅.

Definition 2.6. The solution of integral equation (2.4) is said to be stable in 𝑝(𝑝≥2) moment, if for arbitrarily given 𝜀>0, there exists a 𝛿>0 such that 𝐸‖𝜉‖²_𝐶<𝛿 guarantees that 𝐸sup_𝑡≥0‖𝑥(𝑡)‖^𝑝<𝜀,𝑝≥2. (2.6)

Definition 2.7. The solution of integral equation (2.4) is said to be asymptotically stable in 𝑝(𝑝≥2) moment, if it is stable in mean square and for any 𝜑∈𝔅, a.s., we have lim_𝑇→+∞𝐸sup_𝑡≥𝑇‖𝑥(𝑡)‖^𝑝=0,𝑝≥2. (2.7)

3. Main Results

In order to obtain our main results, we need the following assumptions. (H₁)The cosine family of operators {𝐶(𝑡)∶𝑡≥0} on 𝑋 and the corresponding sine family {𝑆(𝑡)∶𝑡≥0} satisfy the conditions ‖𝐶(𝑡)‖≤𝑀𝑒^−𝑏𝑡 and ‖𝑆(𝑡)‖≤𝑀𝑒^−𝑎𝑡, 𝑡≥0 for some constants 𝑀≥1, 𝑎>0 and 𝑏>0.(H₂)The mappings 𝑓_𝑖(𝑖=0,1,2) satisfy the following conditions: there exist three positive constants 𝐶_𝑖>0(𝑖=0,1,2) and a function: 𝑘∶(−∞,0]→[0,+∞) with two important properties: ∫⁰_−∞𝑘(𝑡)𝑑𝑡=1 and ∫⁰_−∞𝑘(𝑡)𝑒^−𝜈𝑡𝑑𝑡<+∞(𝜈>0), such that 𝑓_𝑖(𝑡,𝑥)−𝑓_𝑖𝑡,𝑦≤𝐶_𝑖⁰_−∞𝑘(𝜃)𝑥(𝑡+𝜃)−𝑦(𝑡+𝜃)𝑑𝜃,𝑓_𝑖()𝑓𝑡,0=0,𝑖=0,1,₂(𝑡,𝑥)−𝑓₂𝑡,𝑦_𝔏02≤𝐶₂⁰_−∞𝑘(𝜃)𝑥(𝑡+𝜃)−𝑦(𝑡+𝜃)𝑑𝜃,𝑓₂(𝑡,0)=0, (3.1) for any 𝑥,𝑦∈𝔅 and 𝑡≥0.(H₃)5^𝑝−1𝑀^𝑝𝑏^−𝑝𝐶^𝑝₀+𝑎^−𝑝𝐶^𝑝₁+𝐶^𝑝₂𝑎^−(𝑝/2)2(𝑝−1)𝑝−2^1−(𝑝/2)𝑝(𝑝−1)2^𝑝/2<1,(𝑝≥2).

Remark 3.1. Obviously, under the conditions: (H₁)-(H₂), the existence and uniqueness of mild solution to the system (2.2) can be shown by using the Picard iterative method, and the proof is very similar to that proposed in [4, 17]. Here, we omit it. In particular, the system (2.2) has one unique trivial mild solution when the initial value 𝜑=0.

Lemma 3.2. For 𝛾₁,𝛾₂∈(0,𝜈], there exist some positive constants: 𝜆_𝑖>0(𝑖=1,2,3,4) and a function 𝑦∶(−∞,+∞)→[0,+∞). If (𝜆₃/𝛾₁)+(𝜆₄/𝛾₂)<1, the following inequality: 𝑦(𝑡)≤𝜆₁𝑒^−𝛾1𝑡+𝜆₂𝑒^−𝛾2𝑡+𝜆₃^𝑡₀𝑒^{−𝛾1(𝑡−𝑠)0}_−∞𝑘(𝜃)𝑦(𝑠+𝜃)𝑑𝜃𝑑𝑠+𝜆₄^𝑡₀𝑒^{−𝛾2(𝑡−𝑠)0}_−∞𝑘(𝜃)𝑦()𝜆𝑠+𝜃𝑑𝜃𝑑𝑠,𝑡≥0,₁𝑒^−𝛾1𝑡+𝜆₂𝑒^−𝛾2𝑡,𝑡∈(−∞,0], (3.2) holds. Then, one has 𝑦(𝑡)≤𝑀₂𝑒^−𝜇𝑡,𝑡∈(−∞,+∞), where 𝜇∈(0,𝛾₁∧𝛾₂) is a positive root of the algebra equation: ((𝜆₃/(𝛾₁−𝜇))+(𝜆₄/(𝛾₂−𝜇)))∫⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃=1 and 𝑀₂=max{𝜆₁+𝜆₂,(𝜆₁(𝛾₁−𝜇)/(𝜆₃∫⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃)),(𝜆₂(𝛾₂−𝜇)/(𝜆₄∫⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃))}>0.

Proof. Letting 𝐹(𝜆)=((𝜆₃/(𝛾₁−𝜆))+(𝜆₄/(𝛾₂−𝜆)))∫⁰_−∞𝑘(𝜃)𝑒^−𝜆𝜃𝑑𝜃−1, we have 𝐹(0)𝐹(𝛾−)<0 holds, that is, there exists a positive constant 𝜇∈(0,𝛾₁∧𝛾₂), such that 𝐹(𝜇)=0.
For any 𝜀>0 and letting 𝐶_𝜀=max𝜆₁+𝜆₂+𝜀,𝜆₁𝛾+𝜀¹−𝜇𝜆₃∫⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃,𝜆𝑑𝜃²𝛾+𝜀²−𝜇𝜆₄∫⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃>0. (3.3)
Now, in order to show this Lemma, we only claim that (3.2) implies 𝑦(𝑡)≤𝐶_𝜀𝑒^−𝜇𝑡,𝑡∈(−∞,+∞). (3.4) It is easily seen that (3.4) holds for any 𝑡∈(−∞,0]. Assume, for the sake of contradiction, that there exists a 𝑡₁>0 such that 𝑦(𝑡)<𝐶_𝜀𝑒^−𝜇𝑡,𝑡∈−∞,𝑡₁,𝑦𝑡₁=𝐶_𝜀𝑒^−𝜇𝑡1. (3.5)
Then, it from (3.2) follows that 𝑦𝑡₁≤𝜆₁𝑒^{−𝛾1𝑡1}+𝜆₂𝑒^{−𝛾2𝑡1}+𝜆₃𝐶_𝜀^𝑡1₀𝑒^{−𝛾1(𝑡1−𝑠)0}_−∞𝑘(𝜃)𝑒^{−𝜇(𝑠+𝜃)}𝑑𝑠+𝜆₄𝐶_𝜀^𝑡1₀𝑒^{−𝛾2(𝑡1−𝑠)0}_−∞𝑘(𝜃)𝑒^{−𝜇(𝑠+𝜃)}=𝜆𝑑𝜃𝑑𝑠₁−𝐶_𝜀𝜆₃𝛾₁−𝜇⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃𝑒^{−𝛾1𝑡1}+𝜆₂−𝐶_𝜀𝜆₄𝛾₂−𝜇⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃𝑒^{−𝛾2𝑡1}+𝜆₃𝛾₁−𝜇⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝜆𝑑𝜃+⁴𝛾₂−𝜇⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃𝐶_𝜀𝑒^−𝜇𝑡1. (3.6) From the definitions of 𝜇 and 𝐶_𝜀, we obtain 𝜆₃𝛾₁−𝜇⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝜆𝑑𝜃+⁴𝛾₂−𝜇⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝜆𝑑𝜃=1,₁−𝜆₃𝐶_𝜀𝛾₁−𝜇⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃=𝜆₁−𝜆₃𝛾₁−𝜇⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃𝜆₁𝛾+𝜀¹−𝜇𝜆₃∫⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝜆𝑑𝜃<0,₂−𝐶_𝜀𝜆₄𝛾₂−𝜇⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃=𝜆₂−𝜆₄𝛾₂−𝜇⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃𝜆₂𝛾+𝜀²−𝜇𝜆₄∫⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃<0. (3.7)
Thus, (3.6) yields 𝑦𝑡₁<𝐶_𝜀𝑒^−𝜇𝑡1, (3.8) which contradicts (3.5), that is, (3.4) holds.
As 𝜀>0 is arbitrarily small, in view of (3.4), it follows 𝑦(𝑡)≤𝑀₂𝑒^−𝜇𝑡,𝑡≥0, (3.9) where 𝑀₂=max{𝜆₁+𝜆₂,(𝜆₁(𝛾₁−𝜇)/(𝜆₃∫⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃)), (𝜆₂(𝛾₂−𝜇)/(𝜆₄∫⁰_−∞𝑘(𝜃)𝑒^−𝜇𝜃𝑑𝜃))}>0. The proof of this Lemma is completed.

Lemma 3.3. For 𝛾₁,𝛾₂>0, there exist some positive constants: 𝜆_𝑖>0(𝑖=1,2,3,4) and a function 𝑦∶(−∞,+∞)→[0,+∞). If (𝜆₃/𝛾₁)+(𝜆₄/𝛾₂)<1, the following inequality: 𝑦(𝑡)≤𝜆₁𝑒^−𝛾1𝑡+𝜆₂𝑒^−𝛾2𝑡+𝜆₃^𝑡₀𝑒^{−𝛾1(𝑡−𝑠)0}_−∞𝑘(𝜃)𝑦(𝑠+𝜃)𝑑𝜃𝑑𝑠+𝜆₄^𝑡₀𝑒^{−𝛾2(𝑡−𝑠)0}_−∞𝑘(𝜃)𝑦()𝜆𝑠+𝜃𝑑𝜃𝑑𝑠,𝑡≥0,₁+𝜆₂,𝑡∈(−∞,0], (3.10) holds. Then, one has lim_𝑡→+∞𝑦(𝑡)=0.

Proof. In order to show the conclusion of this Lemma, the proof is divided into two steps as follows.Step 1. We show that there exists a positive constant 𝑑_𝜀>0 such that 𝑦(𝑡)≤𝑑_𝜀, (3.11) for any 𝑡∈(−∞,+∞). Firstly, for for all 𝜀>0, letting 𝑑_𝜀=max𝜆₁+𝜆₂𝛾+𝜀,¹𝜆₃𝜆₁,𝛾+𝜀²𝜆₄𝜆₂+𝜀>0. (3.12) It is obviously seen that 𝑦(𝑡)≤𝑑_𝜀 for any 𝑡∈(−∞,0]. Assumed that there exists a 𝑡₁>0 such that 𝑦(𝑡)<𝑑_𝜀,𝑡∈−∞,𝑡₁,𝑦𝑡₁=𝑑_𝜀. (3.13)
Then, it from (3.10) implies that 𝑦𝑡₁≤𝜆₁𝑒^{−𝛾1𝑡1}+𝜆₂𝑒^{−𝛾2𝑡1}+𝜆₃𝑑_𝜀^𝑡1₀𝑒^{−𝛾1(𝑡1−𝑠)}𝑑𝑠+𝜆₄𝑑_𝜀^𝑡1₀𝑒^{−𝛾2(𝑡1−𝑠)}=𝜆𝑑𝑠₁−𝜆₃𝑑_𝜀𝛾₁𝑒^{−𝛾1𝑡1}+𝜆₂−𝜆₄𝑑_𝜀𝛾₂𝑒^{−𝛾2𝑡1}+𝜆₃𝛾₁+𝜆₄𝛾₂𝑑_𝜀. (3.14) From the definition of 𝑑_𝜀, we have 𝜆₁−𝜆₃𝑑_𝜀𝛾₁=𝜆₁−𝜆₃𝛾₁𝛾₁𝜆₃𝜆₁𝜆+𝜀<0,₂−𝜆₄𝑑_𝜀𝛾₂=𝜆₂−𝜆₄𝛾₂𝛾₂𝜆₄𝜆₂+𝜀<0. (3.15)
Thus, (3.14) yields 𝑦𝑡₁<𝑑_𝜀, (3.16) which contradicts (3.13), that is, (3.11) holds.
As 𝜀>0 is arbitrarily small, in view of (3.11), it follows 𝑦(𝑡)≤𝑑,𝑡∈(−∞,+∞), (3.17) where 𝑑=max{𝜆₁+𝜆₂,(𝛾₁/𝜆₃)𝜆₁,(𝛾₂/𝜆₄)𝜆₂}.Step 2. We prove that lim_𝑡→+∞𝑦(𝑡)=0.
From the inequality (3.17), it has shown that 𝑦(𝑡) is a bounded function defined on the interval (−∞,+∞). Thus, as 𝑡→+∞, the upper limit (denoted by 𝑙≥0) of 𝑦(𝑡) exists, namely, lim_𝑡→+∞𝑦(𝑡)=𝑙, (3.18) the remaining work is to prove 𝑙=0.
Supposed that 𝑙>0. From (3.18), there must exist arbitrary positive scalar 𝜀>0 and constant 𝑇₁>0 such that 𝑦(𝑡)<𝑙+𝜀,∀𝑡≥𝑇₁. (3.19) On the other hand, since ∫⁰_−∞𝑘(𝑠)𝑑𝑠=1, there must exist 𝑇₂>0 such that ^−𝑇2_−∞𝑘(𝜃)𝑑𝜃<𝜀,∀𝑡≥𝑇₂. (3.20)
Letting 𝑇=max{𝑇₁,𝑇₂}, (3.19) and (3.20) hold for 𝑡>𝑇. Thus, it from (3.10) follows that 𝑦(𝑡)≤𝜆₁𝑒^−𝛾𝑡+𝜆₂𝑒^−𝛾2𝑡+𝜆₃^𝑡₀𝑒^{−𝛾1(𝑡−𝑠)𝑠−𝑇}_−∞𝑘(𝑢−𝑠)𝑦(𝑢)𝑑𝑢𝑑𝑠+𝜆₃^𝑡₀𝑒^{−𝛾1(𝑡−𝑠)𝑠}_𝑠−𝑇𝑘(𝑢−𝑠)𝑦(𝑢)𝑑𝑢𝑑𝑠+𝜆₄^𝑡₀𝑒^{−𝛾2(𝑡−𝑠)𝑠−𝑇}_−∞𝑘(𝑢−𝑠)𝑦(𝑢)𝑑𝑢𝑑𝑠+𝜆₄^𝑡₀𝑒^{−𝛾2(𝑡−𝑠)𝑠}_𝑠−𝑇𝑘()𝑦(𝑢)≤𝜆𝑢−𝑠𝑑𝑢𝑑𝑠₁+𝜆₃₀^2𝑇𝑒^{𝛾1𝑠𝑠}_𝑠−𝑇𝑘(𝑢−𝑠)𝑦(𝑢)𝑑𝑢𝑑𝑠𝑒^−𝛾1𝑡+𝜆₂+𝜆₄₀^2𝑇𝑒^{𝛾2𝑠𝑠}_𝑠−𝑇𝑘(𝑢−𝑠)𝑦(𝑢)𝑑𝑢𝑑𝑠𝑒^−𝛾2𝑡+𝜆₃𝑑𝛾₁+𝜆₄𝑑𝛾₂𝜆𝜀+³𝛾₁+𝜆₄𝛾₂(𝑙+𝜀). (3.21)
By virtue of (3.18), we have 𝜆𝑙≤³𝑑𝛾₁+𝜆₄𝑑𝛾₂𝜆𝜀+³𝛾₁+𝜆₄𝛾₂(𝑙+𝜀). (3.22)
From the arbitrary property of 𝜀, it follows 𝑙≤((𝜆₃/𝛾₁)+(𝜆₄/𝛾₂))𝑙, that is,(𝜆₃/𝛾₁)+(𝜆₄/𝛾₂)≥1, which contradicts the condition: (𝜆₃/𝛾₁)+(𝜆₄/𝛾₂)<1. Thus, 𝑙=0. The proof of this Lemma is completed.

Theorem 3.4. Suppose that the conditions: (H₁)–(H₃) are satisfied and 𝑎,𝑏∈(0,𝜈], then the mild solution to system (2.2) is exponentially stable in 𝑝(𝑝≥2) moment.

Proof. In view of (2.4) and the elementary inequality, we have ‖𝑥(𝑡)‖^𝑝=𝐶(𝑡)𝜑+𝑆(𝑡)𝜉−𝑓₀0,𝑥₀+^𝑡₀𝐶(𝑡−𝑠)𝑓₀𝑠,𝑥_𝑠+𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑓₁𝑠,𝑥_𝑠𝑑𝑠+^𝑡₀𝑆(𝑡−𝑠)𝑓₂(𝑠,𝑥_𝑠)𝑑𝑤(𝑠)^𝑝≤5^𝑝−1𝑀^𝑝𝜑^𝑝𝑒^−𝑏𝑡+5^𝑝−1𝑀^𝑝𝜉−𝑓₀0,𝑥₀^𝑝𝑒^−𝑎𝑡+5^𝑝−1𝑀^𝑝𝑏^1−𝑝𝑡₀𝑒^{−𝑏(𝑡−𝑠)}𝑓₀(𝑠,𝑥_𝑠)^𝑝𝑑𝑠+5^𝑝−1𝑀^𝑝𝑎^1−𝑝𝑡₀𝑒^{−𝑎(𝑡−𝑠)}𝑓₁(𝑠,𝑥_𝑠)^𝑝𝑑𝑠+5^𝑝−1𝑡₀𝑆(𝑡−𝑠)𝑓₂𝑠,𝑥_𝑠𝑑𝑤(𝑠)^𝑝. (3.23)
From the condition (H₂), it from (3.23) concludes that 𝐸‖𝑥(𝑡)‖^𝑝≤5^𝑝−1𝑀^𝑝𝐸𝜑^𝑝𝑒^−𝑏𝑡+5^𝑝−1𝑀^𝑝𝐸𝜉−𝑓₀0,𝑥₀^𝑝𝑒^−𝑎𝑡+5^𝑝−1𝑀^𝑝𝑏^1−𝑝𝐶^𝑝₀^𝑡₀𝑒^{−𝑏(𝑡−𝑠)}𝐸⁰_−∞𝑘(𝜃)‖𝑥(𝑠+𝜃)‖𝑑𝜃^𝑝𝑑𝑠+5^𝑝−1𝑀^𝑝𝑎^1−𝑝𝐶^𝑝₁^𝑡₀𝑒^{−𝑎(𝑡−𝑠)}𝐸⁰_−∞𝑘(𝜃)‖𝑥(𝑠+𝜃)‖𝑑𝜃^𝑝𝑑𝑠+5^𝑝−1𝐸^𝑡₀𝑆(𝑡−𝑠)𝑓₂𝑠,𝑥_𝑠𝑑𝑤(𝑠)^𝑝. (3.24)
From Lemma 2.3, we obtain 𝐸^𝑡₀𝑆(𝑡−𝑠)𝑓₂𝑠,𝑥_𝑠𝑑𝑤(𝑠)^𝑝≤𝑀^𝑝𝑡₀𝑒^{−𝑎𝑝(𝑡−𝑠)}𝐸𝑓₂𝑠,𝑥_𝑠^𝑝_𝐿02^2/𝑝𝑑𝑠^𝑝/2𝑝𝑝−12^𝑝/2=𝑀^𝑝𝑡₀𝑒^{−2𝑎(𝑡−𝑠)}𝐸𝑓₂𝑠,𝑥_𝑠^𝑝_𝐿02^2/𝑝𝑑𝑠^𝑝/2𝑝𝑝−12^𝑝/2≤𝑀^𝑝𝑡₀𝑒^{−(2𝑎(𝑝−1)/(𝑝−2))(𝑡−𝑠)}𝑑𝑠^{(𝑝/2)−1𝑡}₀𝑒^{−𝑎(𝑡−𝑠)}𝐸𝑓₂𝑠,𝑥_𝑠^𝑝_𝐿02𝑝𝑑s𝑝−12^𝑝/2≤𝑀^𝑝𝐶^𝑝₂2𝑎𝑝−1𝑝−2^1−(𝑝/2)𝑝𝑝−12^𝑝/2𝑡₀𝑒^{−𝑎(𝑡−𝑠)}𝐸⁰_−∞𝑘(𝜃)‖𝑥(𝑠+𝜃)‖𝑑𝜃^𝑝𝑑𝑠. (3.25)
Substituting (3.25) into (3.24), it follows 𝐸‖𝑥(𝑡)‖^𝑝≤5^𝑝−1𝑀^𝑝𝐸𝜑^𝑝𝑒^−𝑏𝑡+5^𝑝−1𝑀^𝑝𝐸𝜉−𝑓₀0,𝑥₀^𝑝𝑒^−𝑎𝑡+5^𝑝−1𝑀^𝑝𝑏^1−𝑝𝐶^𝑝₀^𝑡₀𝑒^{−𝑏(𝑡−𝑠)0}_−∞𝑘(𝜃)𝐸‖𝑥(𝑠+𝜃)‖^𝑝𝑑𝜃𝑑𝑠+5^𝑝−1𝑀^𝑝𝑎^1−𝑝𝐶^𝑝₁^𝑡₀𝑒^{−𝑎(𝑡−𝑠)0}_−∞𝑘(𝜃)𝐸‖𝑥(𝑠+𝜃)‖^𝑝𝑑𝜃𝑑𝑠+5^𝑝−1𝑀^𝑝𝐶^𝑝₂2𝑎𝑝−1𝑝−2^1−(𝑝/2)𝑝𝑝−12^𝑝/2𝑡₀𝑒^{−𝑎(t−𝑠)0}_−∞𝑘(𝜃)𝐸‖𝑥(𝑠+𝜃)‖^𝑝𝑑𝜃𝑑𝑠. (3.26)
And it is easily verified that there exists two positive number 𝑀^′>0 and 𝑀^′′>0 such that 𝐸‖𝑥(𝑡)‖²≤𝑀^′𝑒^−𝑏𝑡+𝑀^′′𝑒^−𝑎𝑡, for any 𝑡∈(−∞,0].
By Lemma 3.2, we can derive that 𝐸‖𝑥(𝑡)‖²≤𝑀₁𝑒^−𝜇𝑡,𝑡∈[0,+∞)(𝜇∈(0,𝛾₁∧𝛾₂)), where 𝑀₁=5^𝑝−1𝑀^𝑝𝐸𝜑^𝑝+𝐸𝜉−𝑓₀0,𝑥₀^𝑝,5^𝑝−1𝑀^𝑝𝑏^−𝑝𝐶^𝑝₀,5^𝑝−1𝑀^𝑝𝑎^−𝑝𝐶^𝑝₁+𝐶^𝑝₂𝑎^−𝑝/2×2𝑝−1𝑝−2^1−(𝑝/2)𝑝𝑝−12^𝑝/2>0. (3.27) The proof of this Theorem is completed.

Theorem 3.5. Suppose that the conditions: (H₁)–(H₃) are satisfied, then the mild solution to system (2.2) is asymptotically stable in 𝑝(𝑝≥2) moment.

Proof. Similarly, we can obtain the conclusion as follows: 𝐸‖𝑥(𝑡)‖^𝑝≤5^𝑝−1𝑀^𝑝𝐸𝜑^𝑝𝑒^−𝑏𝑡+5^𝑝−1𝑀^𝑝𝐸𝜉−𝑓₀0,𝑥₀^𝑝𝑒^−𝑎𝑡+5^𝑝−1𝑀^𝑝𝑏^1−𝑝𝐶^𝑝₀^𝑡₀𝑒^{−𝑏(𝑡−𝑠)0}_−∞𝑘(𝜃)𝐸‖𝑥(𝑠+𝜃)‖^𝑝𝑑𝜃𝑑𝑠+5^𝑝−1𝑀^𝑝𝑎^1−𝑝𝐶^𝑝₁^𝑡₀𝑒^{−𝑎(𝑡−𝑠)0}_−∞𝑘(𝜃)𝐸‖𝑥(𝑠+𝜃)‖^𝑝𝑑𝜃𝑑𝑠+5^𝑝−1𝑀^𝑝𝐶^𝑝₂2𝑎𝑝−1𝑝−2^1−(𝑝/2)𝑝𝑝−12^𝑝/2𝑡₀𝑒^{−𝑎(𝑡−s)0}_−∞𝑘(𝜃)𝐸‖𝑥(𝑠+𝜃)‖^𝑝𝑑𝜃𝑑𝑠, (3.28) and it is easily verified that there exists two positive number 𝑀^′_∗>0 and 𝑀_∗^′′>0 such that 𝐸‖𝑥(𝑡)‖²≤𝑀^′_∗+𝑀_∗^′′, for any 𝑡∈(−∞,0].
By Lemma 3.3, we can derive that lim_𝑡→+∞𝐸‖𝑥(𝑡)‖^𝑝=0. (3.29)
To obtain the asymptotical stability in 𝑝(𝑝≥2)-moment, we need to prove that mild solution of system (2.2) is stable in 𝑝(𝑝≥2)-moment. Let 𝜀>0 be given and choose 𝛿>0(𝛿<𝜀) such that 𝐶𝑒^{−(𝑎∧𝑏)𝑡∗}+Δ𝜀<𝜀, (3.30) where 𝐶=5^𝑝−1𝑀^𝑝𝐸𝜑^𝑝+5^𝑝−1𝑀^𝑝𝐸𝜉−𝑓₀(0,𝑥₀)^𝑝,Δ=5^𝑝−1𝑀^𝑝𝑏^−𝑝𝐶^𝑝₀+𝑎^−𝑝𝐶^𝑝₁+𝐶^𝑝₂𝑎^−𝑝/22𝑝−1(𝑝−2)^1−(𝑝/2)𝑝𝑝−12^𝑝/2. (3.31)
If 𝑥(𝑡,0,𝜑) is a mild solution of system (2.2) with sup_{𝜃∈(−∞,0]}𝐸‖𝜑(𝜃)‖^𝑝<𝛿, then 𝑥(𝑡) is defined in (2.4). Now, we claim that 𝐸‖𝑥(𝑡)‖^𝑝<𝜀 for all 𝑡≥0. Notice that sup_{𝜃∈(−∞,0]}𝐸‖𝜑(𝜃)‖^𝑝<𝜀. If there exists 𝑡^∗>0 such that 𝐸‖𝑥(𝑡^∗)‖^𝑝=𝜀 and 𝐸‖𝑥(𝑡)‖^𝑝<𝜀, for all 𝑡∈(−∞,𝑡^∗), then it follows from (2.4) that 𝐸𝑥(𝑡^∗)^𝑝≤5^𝑝−1𝑀^𝑝𝐸𝜑^𝑝+𝐸𝜉−𝑓₀(0,𝑥₀)^𝑝𝑒^{−(𝑎∧𝑏)𝑡∗}+5^𝑝−1𝑀^𝑝𝑏^1−𝑝𝐶^𝑝₀^𝑡∗₀𝑒^{−𝑏(𝑡∗−𝑠)0}_−∞𝑘(𝜃)𝐸‖𝑥(𝑠+𝜃)‖^𝑝𝑑𝜃𝑑𝑠+5^𝑝−1𝑀^𝑝𝑎^1−𝑝𝐶^𝑝₁^𝑡∗₀𝑒^{−𝑎(𝑡∗−𝑠)0}_−∞𝑘(𝜃)𝐸‖𝑥(𝑠+𝜃)‖^𝑝𝑑𝜃𝑑𝑠+5^𝑝−1𝑀^𝑝𝐶^𝑝₂2𝑎𝑝−1𝑝−2^1−(𝑝/2)𝑝𝑝−12^{𝑝/2𝑡∗}₀𝑒^{−𝑎(𝑡∗−𝑠)0}_−∞𝑘(𝜃)𝐸‖𝑥(𝑠+𝜃)‖^𝑝𝑑𝜃𝑑𝑠<𝐶𝑒^{−(𝑎∧𝑏)𝑡∗}+Δ𝜀<𝜀, (3.32) which contradicts the definition of 𝑡^∗. This shows that the mild solution of system (2.2) is asymptotically stable in 𝑝(𝑝≥2)-moment. The proof of this Theorem is completed.

4. An Illustrative Example

In this section, we provide an example to illustrate the obtained results above. Let 𝑋=𝐿²[0,𝜋] and 𝑌=𝑅¹ with the norm ‖⋅‖. And let 𝑒_𝑛∶=2/𝜋sin(𝑛𝜉)(𝑛=1,2,…) denote the completed orthonormal basis in 𝑋. Let 𝑤(𝑡)∶=∑^+∞_𝑛=1𝜆_𝑛𝛽_𝑛(𝑡)𝑒_𝑛,(𝜆_𝑛>0), where {𝛽_𝑛(𝑡)} are one-dimensional standard Brownian motions mutually independent on a usual complete probability space (Ω,ℑ,ℑ_𝑡,𝑃). Define 𝐴∶𝑋→𝑋 by 𝐴=(𝜕²/𝜕𝜉²) with the domain 𝐷(𝐴)={ℎ∈𝑋∶ℎ,(𝜕/𝜕𝜉)ℎ are absolutely continuous, (𝜕²/𝜕𝜉²)ℎ∈𝑋,ℎ(0)=ℎ(𝜋)=0}. Then, 𝐴ℎ=^+∞_𝑛=1𝑛²ℎ,𝑒_𝑛𝑒_𝑛,ℎ∈𝐷(𝐴), (4.1) where 𝑒_𝑛,𝑛=1,2,3,…, is also the orthonormal set of eigenvector of 𝐴. It is well known that ‖𝐶(𝑡)‖≤exp(−𝜋²𝑡) and ‖𝑆(𝑡)‖≤exp(−𝜋²𝑡),𝑡≥0.

Now, we consider the following second-order neutral stochastic partial differential equations with infinite delays: 𝑑𝜕𝑧−𝑎𝜕𝑡𝑡,𝑦⁰𝜋√𝜋⁰_−∞(−𝜃)^−1/2𝑒^𝜋2𝜃𝑧()=𝜕𝑡+𝜃,𝜉𝑑𝜃₂𝜕𝜉²𝑧()+𝛼𝑡,𝜉¹𝜋√𝜋⁰_−∞(−𝜃)^−1/2𝑒^𝜋2𝜃𝑧()+𝛼𝑡+𝜃,𝜉𝑑𝜃𝑑𝑡₂𝜋√𝜋⁰_−∞(−𝜃)^−1/2𝑒^𝜋2𝜃𝑧()(𝑡)[],𝑥()()𝑧()()(][],𝜕𝑡+𝜃,𝜉𝑑𝜃𝑑𝑤,𝑡≥0,𝜉∈0,𝜋𝑡,0=𝑥𝑡,𝜋=0,𝑡≥0,𝜃,𝜉=𝜑𝜃,𝜉,𝜃∈−∞,0,𝜉∈0,𝜋𝑧()(𝜉)[].𝜕𝑡0,𝜉=𝜁,𝜉∈0,𝜋 (4.2)

Define 𝑓₀𝑡,𝑧_𝑡=𝛼₀𝜋√𝜋5⁰_−∞(−𝜃)^−1/2𝑒^𝜋2𝜃𝑧(𝑡+𝜃,𝜉)𝑑𝜃,𝑓₀()𝑓𝑡,0=0,₁𝑡,𝑧_𝑡=𝛼₁𝜋√𝜋5⁰_−∞(−𝜃)^−1/2𝑒^𝜋2𝜃𝑧(𝑡+𝜃,𝜉)𝑑𝜃,𝑓₁()𝑓𝑡,0=0,₂𝑡,𝑧_𝑡=𝛼₂𝜋5⁰_−∞(−𝜃)^−1/2𝑒^𝜋2𝜃𝑧((𝑡+𝜃),𝜉)𝑑𝜃,𝑓₂(𝑡,0)=0, (4.3) for any 𝑧_𝑡∈𝔅.

It is easily verified that 𝑓₀𝑡,𝑧¹_𝑡−𝑓₀𝑡,𝑧²_𝑡≤𝛼₀𝜋√𝜋5⁰_−∞(−𝜃)^−1/2𝑒^𝜋2𝜃𝑧¹(𝑡+𝜃,𝜉)−𝑧²(𝑡+𝜃,𝜉)𝑑𝜃,𝑓₀()𝑓𝑡,0=0,₁𝑡,𝑧¹_𝑡−𝑓₁𝑡,𝑧²_𝑡≤𝛼₁𝜋√𝜋5⁰_−∞(−𝜃)^−1/2𝑒^𝜋2𝜃𝑧¹(𝑡+𝜃,𝜉)−𝑧²(𝑡+𝜃,𝜉)𝑑𝜃,𝑓₁()𝑓𝑡,0=0,₂𝑡,𝑧¹_𝑡−𝑓₂𝑡,𝑧²_𝑡≤𝛼₂𝜋5⁰_−∞(−𝜃)^−1/2𝑒^𝜋2𝜃𝑧¹(𝑡+𝜃,𝜉)−𝑧²(𝑡+𝜃,𝜉)𝑑𝜃,𝑓₂(𝑡,0)=0, (4.4) for any 𝑧¹_𝑡,𝑧²_𝑡∈𝐵.

By virtue of Theorems 3.4 and 3.5, the exponential stability in 𝑝(𝑝≥2)-moment and the asymptotical stability in 𝑝(𝑝≥2)-moment for mild solution to system (4.2) are obtained, provided that the following inequality: 𝛼^𝑝₀+𝛼^𝑝₁+𝛼₂2𝑝−1𝑝−2^1−(𝑝/2)𝑝𝑝−12^𝑝/2<5,𝑝≥2, (4.5) holds.

Remark 4.1. Obviously, the result in [18] is ineffective in dealing with this example, and our results are more general than those proposed in [18]. Besides, our results can be easily extended to investigate two cases: (1) the exponential stability and the asymptotic stability for the second-order neutral stochastic partial differential equations with infinite delay and impulses and (2) the exponential stability for the second-order neutral stochastic partial differential equations with time-varying delays; the readers can refer to [12, 22]. Here, we omit them.