In this section, to establish sufficient conditions ensuring the exponential stability in p-moment () for a mild solution to Eq. (1), we firstly establish a new integral inequality to overcome the difficulty when the neutral term and impulsive effects are present.
Lemma 3.1 For any , assume that there exist some positive constants (), () and a function such that
for each . If
where is the unique solution to the equation: and .
Proof Let , then by (6) and the existence theorem of the root, there exists a positive constant such that .
For any , let
To now prove the result, we only claim that (4) and (5) imply
Clearly, for any , (9) holds. By the contradiction, assume that there is a positive constant such that
This, together with (5), yields (note that )
By (8), we have
Hence, by (11), we obtain , which contradicts (10). Therefore (9) holds.
Since ε is arbitrarily small, so (7) holds. This completes the proof. □
We can now state our main result of this paper.
Theorem 3.1 If (H1)-(H4) hold for some , , then the mild solution of Eq. (1) is exponentially stable in the pth moment, provided
where , and is defined in Lemma 3.1.
Proof From the condition (13), we can always find a number small enough such that
On the other hand, recall the inequalities and for , . Then, for any ,
From (2) and (14),
Now we compute the right-hand terms of (15). Firstly, by (H1) and (H3), we can easily obtain
By (H4) and the Hölder inequality, for , , , we have
By (H3), Lemma 3.1 and the Hölder inequality,
Similar to (20), by (H2) and the Hölder inequality, we have
By Da Prato and Zabczyk [, Lemma 7.7, p.194], similar to (20), (H2) and the Hölder inequality, we have
Substituting (16)-(22) into (15) yields
This, together with Lemma 3.1 and (13), gives that there exist two positive constants and such that for any . This completes the proof. □
If , then we get the following corollary from Theorem 3.1.
Corollary 3.1 If (H1)-(H4) hold for some , then the mild solution of Eq. (1) is mean-square exponentially stable, provided
Remark 3.1 Unlike earlier studies, ours does not make use of general methods such as Lyapunov methods, fixed point theory and so forth. As we know, in general, it is impossible to construct a suitable Lyapunov function (functional) and to find an appropriate fixed point theorem for stochastic partial differential equations with memory, even for constant delays, to deal with stability. In this work, we use the new impulsive integral inequality to derive the sufficient conditions for stability.
Remark 3.2 Without delay and impulsive effect, Eq. (1) becomes stochastic neutral partial differential equations, which is investigated in . Without the neutral term and impulsive effect, Eq. (1) reduces to stochastic partial differential delay equations, which is studied in [6, 17]. Therefore, we generalize by the integral inequality the results to cover a class of more general impulsive stochastic neutral partial differential equations with memory. Moreover, unlike , we need not require the functions , , to be differentiable.
Remark 3.3 In Eq. (1), provided , Eq. (1) becomes stochastic neutral partial differential equations without impulsive effects, that is to say, our Theorem 3.1 is effective for it.