Abstract
In this paper, we introduce the concept of square-mean piecewise almost automorphic function. By using the theory of semigroups of operators and the contraction mapping principle, the existence of square-mean piecewise almost automorphic mild solutions for linear and nonlinear impulsive stochastic evolution equations is investigated. In addition, the exponential stability of square-mean piecewise almost automorphic mild solutions for nonlinear impulsive stochastic evolution equations is obtained by the generalized Gronwall–Bellman inequality. Finally, we provide an illustrative example to justify the results.
Similar content being viewed by others
1 Introduction
The concept of the almost automorphic function is proposed by Bochner in the paper [1], which is an important generalization of the classical almost periodic function and is related to some aspects of differential geometry. Also, the almost automorphic solutions for differential systems have been extensively investigated in [2–9]. Moreover, the square-mean almost automorphic function is defined as almost automorphic function in stochastic process, which has more extensive applications (see [10–15]). On the other hand, the theory of impulsive evolution equations has become an active area of investigation, since it fully considers the impact of instantaneous changes on the whole process, and has the characteristics of differential equations and difference equations. There are several interesting results concerning the existence and stability of solutions, especially for piecewise almost periodic type solutions and piecewise almost automorphic type solutions for impulsive evolution equations (see [16–25]). In [21], the authors introduced a PC-almost automorphic function and investigated the existence of PC-almost automorphic solution to impulsive fractional functional differential equations by α-resolvent family of bounded linear operators, Sadovskii’s fixed point theorem and Schauder’s fixed point theorem. In [25], by introducing the concept of equipotentially almost automorphic sequence, the definition of weighted piecewise pseudo almost automorphic function on time scale is proposed, and the existence and stability of the weighted piecewise pseudo almost automorphic mild solutions to abstract impulsive dynamic equation on time scale is investigated.
However, besides impulsive effects, stochastic effects likewise exist in real systems. Therefore, we must import the stochastic effects into the investigation of impulsive evolution systems. Recently, many authors studied the square-mean piecewise almost periodic solutions for impulsive stochastic evolution equations (see [26–30]). In [28], the authors introduced the concept of square-mean piecewise almost periodic functions for impulsive stochastic processes, and studied the existence and stability of square-mean piecewise almost periodic solutions for linear and nonlinear impulsive stochastic differential equations. Furthermore, Yan et al. [31–34] discussed the square-mean piecewise pseudo almost periodic solutions and the square-mean piecewise weighted pseudo almost periodic solutions for impulsive stochastic evolution equations. However, there are few studies on the square-mean piecewise almost automorphic solutions of impulsive stochastic evolution equations.
Based on this, in this paper, we will construct a square-mean piecewise almost automorphic function and study its composite properties. Further we use these properties to prove the existence of the square-mean piecewise almost automorphic mild solutions for two types of impulsive stochastic evolution equations. Also, the stability of the square-mean piecewise almost automorphic mild solutions for impulsive stochastic evolution equations is studied by the generalized Gronwall–Bellman inequality. In the end, we give an example to illustrate our results.
2 Preliminaries
Throughout this paper, let \((H,\|\cdot \| )\) be a real and separable Hilbert space. Let \((\varOmega ,F,P )\) be a complete probability space. Let \(L^{2}(P,H)\) be a space of the H-valued random variables x such that \(E\|x\|^{2}=\int _{\varOmega }\|x\|^{2}\,dP<\infty \), and \((L^{2}(P,H),\|\cdot \|_{2} )\) is a Hilbert space when it is equipped with the norm \(\|x\|_{2}= (\int _{\varOmega }\|x\|^{2}\,dP )^{1/2}\).
Definition 2.1
A stochastic process \(x:R\rightarrow L^{2}(P,H)\) is said to be stochastically bounded if there exists \(M>0\) such that \(E \|x(t) \|^{2}\leq M\) for all \(t\in R\).
Definition 2.2
A stochastic process \(x:R\rightarrow L^{2}(P,H)\) is said to be stochastically continuous, if
for all \(s\in R\).
Let \(T= \{\{t_{i}\}_{i\in Z}\mid \gamma =\inf_{i\in Z} (t_{i+1}-t_{i} )>0 \}\). For \(\{t_{i} \}_{i\in Z}\in T\), let \(PC (R,L^{2}(P, H) )\) be the space consisting of all stochastically bounded piecewise continuous functions \(f:R\rightarrow L^{2}(P,H)\) such that f is stochastically continuous in \(t\neq t_{i}\), \(i\in Z\), and \(f(t_{i}^{+})\), \(f(t_{i}^{-})\) exist, \(f(t_{i})=f(t_{i}^{-})\). Let \(PC (R\times L^{2}(P,H),L^{2}(P,H) )\) be the space formed by all stochastically uniformly bounded piecewise continuous functions \(f:R\times L^{2}(P,H)\rightarrow L^{2}(P,H)\) such that \(f(\cdot ,x)\in PC (R,L^{2}(P,H) )\) and \(f(t,\cdot )\) is continuous.
Definition 2.3
A function \(\phi \in PC (R,L^{2}(P,H) )\) is said to be square-mean piecewise almost automorphic if the following conditions are fulfilled:
- (i)
\(\{t^{j}_{i}:i\in Z \}_{j\in Z}\) are equipotentially almost automorphic, that is, for any sequence \(\{s_{n} \}\subseteq Z\), there exists a subsequence \(\{\tau _{n} \}\) such that
$$ \lim_{n\rightarrow \infty }t^{\tau _{n}}_{k}=\eta _{k} $$and
$$ \lim_{n\rightarrow \infty }\eta ^{-\tau _{n}}_{k}=t_{k} $$for each \(k\in Z\).
- (ii)
For any sequence \(\{s^{\prime }_{n} \}\subseteq R\), there exist a subsequence \(\{s_{n} \}\subseteq \{s^{\prime }_{n} \}\) and \(\varphi \in PC (R,L^{2}(P,H) )\) such that
$$ \lim_{n\rightarrow \infty }E \bigl\Vert \phi (t+s_{n})-\varphi (t) \bigr\Vert ^{2}=0 $$and
$$ \lim_{n\rightarrow \infty }E \bigl\Vert \varphi (t-s_{n})-\phi (t) \bigr\Vert ^{2}=0 $$for all \(t\in R\) and \(t\neq t_{i}\).
Denote by \(AA_{T} (R,L^{2}(P,H) )\) the set of all square-mean piecewise almost automorphic functions.
Definition 2.4
A function \(f\in PC (R\times L^{2}(P,H),L^{2}(P,H) )\) is said to be square-mean piecewise almost automorphic in \(t\in R\) uniformly in \(x\in L^{2}(P,H)\), if for any \(t\in R\) and \(t\neq t_{i}\), \(x\in L^{2}(P,H)\) such that \(f(\cdot ,x)\in AA_{T} (R,L^{2}(P,H) )\). Denote by \(AA_{T} (R\times L^{2}(P,H),L^{2}(P,H) )\) the set of all such functions.
Theorem 2.1
Let\(\phi \in AA_{T} (R,L^{2}(P,H) )\), then\(R(\phi )\)is a relatively compact set of\(L^{2}(P,H)\).
Proof
Let \(\{\phi (x_{n}) \}\subseteq L^{2}(P,H)\). Since \(\phi \in AA_{T} (R,L^{2}(P,H) )\), by Definition 2.3, there exists a subsequence \(\{x^{\prime }_{n} \}\subseteq \{x_{n} \}\) such that \(\lim_{n\rightarrow \infty }E \|\phi (x^{\prime }_{n} )-\varphi (0) \|^{2}=0\), that is, \(\{\phi (x^{\prime }_{n}) \}\) is the convergent subsequence of \(\{\phi (x_{n}) \}\) in \(L^{2}(P,H)\). Therefore, \(R(\phi )= \{\phi (x):x\in R \}\) is a relatively compact set of \(L^{2}(P,H)\). □
Theorem 2.2
Assume\(f\in AA_{T} (R\times L^{2}(R,H),L^{2}(P,H) )\), and thatfsatisfies the following Lipschitz continuous condition: there exists a number\(L>0\)such that, for any\(x,y\in L^{2}(P,H)\),
If\(g\in AA_{T} (R,L^{2}(P,H) )\), then\(f (\cdot ,g(\cdot ) )\in AA_{T} (R,L^{2}(P,H) )\).
Proof
Since \(g\in AA_{T} (R,L^{2}(P,H) )\) and \(f\in AA_{T} (R\times L^{2}(R,H),L^{2}(P,H) )\), for any sequence \(\{s_{n} \}\subseteq Z\), there exists a subsequence \(\{s^{\prime }_{n} \}\subseteq \{s_{n} \}\) such that
Let \(F(t)=f (t,g(t) )\), \(G(t)=\widetilde{f} (t,\widetilde{g}(t) )\), therefore, we only need to prove
and
Since \(E \|f(t,x)-f(t,y) \|^{2}\leq LE \|x-y \|^{2}\), we have
Combining (1), (3) and (5), we have
Similarly,
Therefore, \(f (\cdot ,g(\cdot ) )\in AA_{T} (R,L^{2}(P,H) )\). □
Lemma 2.1
If\(\phi \in AA_{T} (R,L^{2}(P,H) )\), then\(\{\phi (t_{i}):i\in Z \}\)is a square-mean almost automorphic sequence.
The proof of Lemma 2.1 is similar to the proof of Lemma 3.12 in [25], one may refer to [25] for more details.
Theorem 2.3
Assume\(\phi \in AA_{T} (R,L^{2}(P,H) )\), \(\{I_{i}(\cdot ):i\in Z \}\)is a square-mean almost automorphic function sequence, that is, for any\(x\in L^{2}(P,H)\), \(\{I_{i}(x):i\in Z \}\)is a square-mean almost automorphic sequence, if\(I_{i}\)satisfies the following Lipschitz continuous condition: there exists a number\(L>0\), for any\(x,y\in L^{2}(P,H)\), \(i\in Z\),
then\(\{I_{i} (\phi (t_{i}) ):i\in Z \}\)is a square-mean almost automorphic sequence.
Proof
Since \(\phi \in AA_{T} (R,L^{2}(P,H) )\), by Lemma 2.1, \(\{\phi (t_{i}):i\in Z \}\) is a square-mean almost automorphic sequence. Let
and
Since \(\{I_{i}(x):i\in Z \}\) is a square-mean almost automorphic sequence, by Lemma 2.1, \(I\in AA (R\times L^{2}(P,H),L^{2}(P,H) )\), \(\varPhi \in AA (R,L^{2}(P,H) )\).
For any \(t\in R\), there exists \(i\in Z\) such that \(|t-i|\leq 1\), then
By the composite property of the square-mean almost automorphic function, we have \(I (\cdot ,\varPhi (\cdot ) )\in AA (R,L^{2}(P,H) )\).
Thus, \(\{I (i,\varPhi (i) ):i\in Z \}\) is a square-mean almost automorphic sequence, that is, \(\{I_{i} (\phi (t_{i}) ):i\in Z \}\) is a square-mean almost automorphic sequence. □
We list the following result for a square-mean piecewise almost automorphic function, one may refer to [21] for more details.
Lemma 2.2
Assume\(f,g\in AA_{T} (R,L^{2}(P,H) )\), the sequence\(\{x_{i} \}_{i\in Z}\)is a square-mean almost automorphic, then, for any\(\varepsilon >0\)and\(\{s^{\prime }_{n} \}\subseteq R\), \(\{\tau ^{\prime }_{n} \}\subseteq Z\), there exist subsequences\(\{s_{n} \}\subseteq \{s^{\prime }_{n} \}\), \(\{\tau _{n} \}\subseteq \{\tau ^{\prime }_{n} \}\)and\(\widetilde{f},\widetilde{g}\in PC (R,L^{2}(P,H) )\), \(\{y_{i} \}_{i\in Z}\)such that
- (i)
\(E \|f(t+s_{n})-\widetilde{f}(t) \|^{2}<\varepsilon \)and\(E \|\widetilde{f}(t-s_{n})-f(t) \|^{2}<\varepsilon \)for all\(t\in R\), \(|t-t_{i}|>\varepsilon \), \(\{s_{n} \}\subseteq R\), \(i\in Z\).
- (ii)
\(E \|g(t+s_{n})-\widetilde{g}(t) \|^{2}<\varepsilon \)and\(E \|\widetilde{g}(t-s_{n})-g(t) \|^{2}<\varepsilon \)for all\(t\in R\), \(|t-t_{i}|>\varepsilon \), \(\{s_{n} \}\subseteq R\), \(i\in Z\).
- (iii)
\(E \|x_{i+\tau _{n}}-y_{i} \|^{2}<\varepsilon \)and\(E \|y_{i-\tau _{n}}-x_{i} \|^{2}<\varepsilon \)for all\(\{\tau _{n} \}\subseteq Z\), \(i\in Z\).
- (iv)
\(E \|t_{i+\tau _{n}}-t_{i}-s_{n} \|^{2}<\varepsilon \)for all\(\{\tau _{n} \}\subseteq Z\), \(\{s_{n} \}\subseteq R\), \(i\in Z\).
3 Square-mean piecewise almost automorphic mild solutions for impulsive stochastic evolution equations
In this part, we study the existence and stability of the square-mean piecewise almost automorphic mild solution for impulsive stochastic evolution equations.
3.1 Linear impulsive stochastic evolution equations
Consider the following linear impulsive stochastic evolution equations:
where A is an infinitesimal generator of \(C_{0}\)-semigroup \(\{T(t):t\geq 0 \}\) such that, for all \(t\geq 0\), \(\|T(t) \|\leq Me^{-\delta t}\) with \(M,\delta >0\), and \(w(t)\) is a two-sided standard one-dimensional Brownian motion, which is defined on the filtered probability space \((\varOmega ,F,P,F_{\sigma } )\) with \(F_{t}=\sigma \{w(u)-w(v):u,v\leq t \}\).
Definition 3.1
A function \(x\in PC (R,L^{2}(P,H) )\) is called a mild solution of linear impulsive stochastic evolution equations (6), if
where \(t>\sigma \), \(\sigma \neq t_{i}\), \(i\in Z\). If \(x\in AA_{T} (R,L^{2}(P,H) )\), then x is called the square-mean piecewise almost automorphic mild solution of Eq. (6).
Theorem 3.1
Assume\(f,g\in AA_{T} (R,L^{2}(P,H) )\), \(\{\beta _{i}:i\in Z \}\)is a square-mean almost automorphic sequence, then Eq. (6) has a square-mean piecewise almost automorphic mild solution.
Proof
From semigroup theory, we know
is a mild solution to
For any \(t\in (\sigma ,t_{i}]\),
by using \(\bigtriangleup x(t_{i})=x (t^{+}_{i} )-x (t^{-}_{i} )= \beta _{i}\), we get
If \(t\in (t_{1},t_{2}]\), then
Since
by \(\bigtriangleup x(t_{i})=x (t^{+}_{i} )-x (t^{-}_{i} )= \beta _{i}\), we get
If \(t\in (t_{2},t_{3}]\), then
Therefore, reiterating this procedure, we get
By Definition 3.1, (7) is a mild solution of Eq. (6), therefore, we only need to prove the above (7) is a square-mean piecewise almost automorphic process.
Let \(\sigma \rightarrow -\infty \), then \(\|T(t-\sigma ) \|\leq Me^{-\delta (t-\sigma )}=Me^{-\delta t}e^{ \delta \sigma }\rightarrow 0\), by Definition 2.1, \(x(\sigma )\) is stochastically bounded, so (7) can be defined as
Next we show that \(x\in AA_{T} (R,L^{2}(P,H) )\). The following verification procedure is divided into three steps.
Step 1. \(x_{1}\in AA_{T} (R,L^{2}(P,H) )\)
Since \(f\in AA_{T} (R,L^{2}(P,H) )\), by Definition 2.3, for any sequence \(\{s^{\prime }_{n} \}\subseteq R\), there exist a subsequence \(\{s_{n} \}\subseteq \{s^{\prime }_{n} \}\) and \(\widetilde{f}\in PC (R,L^{2}(P,H) )\) such that
and
for every \(t\in R\) and \(t\neq t_{i}\).
Let \(\widetilde{x_{1}}(t)=\int ^{t}_{-\infty }T(t-s)\widetilde{f}(s)\,ds\), then
Similarly,
So, by Lebesgue’s dominated convergence theorem, we get
and
Since \(\lim_{n\rightarrow \infty }E \|f(t+s_{n})-\widetilde{f}(t) \|^{2}=0\) and \(\lim_{n\rightarrow \infty }E \|\widetilde{f}(t-s_{n})-f(t) \|^{2}=0\), \(x_{1}\in AA_{T} (R, L^{2}(P,H) )\).
Step 2. \(x_{2}\in AA_{T} (R,L^{2}(P,H) )\)
Since \(g\in AA_{T} (R,L^{2}(P,H) )\), by Lemma 2.2, for the above sequence \(\{s^{\prime }_{n} \}\subseteq R\), there exist a subsequence \(\{s_{n} \}\subseteq \{s^{\prime }_{n} \}\) and \(\widetilde{g}\in PC (R,L^{2}(P,H) )\) such that
and
for every \(t\in R\) and \(t\neq t_{i}\).
Let \(\widetilde{x_{2}}(t)=\int ^{t}_{-\infty }T(t-s)\widetilde{g}(s)\,dw(s)\), by the Ito integral, then
Similarly,
So, by Lebesgue’s dominated convergence theorem, we get
and
Since \(\lim_{n\rightarrow \infty }E \|g(t+s_{n})-\widetilde{g}(t) \|^{2}=0\) and \(\lim_{n\rightarrow \infty }E \|\widetilde{g}(t-s_{n})-g(t) \|^{2}=0\), \(x_{2}\in AA_{T} (R, L^{2}(P,H) )\).
Step 3. \(x_{3}\in AA_{T} (R,L^{2}(P,H) )\)
Since \(\beta _{i}\) is a square-mean almost automorphic sequence, by Lemma 2.2, for any sequence \(\{\tau ^{\prime }_{n} \}\subseteq Z\), there exists a subsequence \(\{\tau _{n} \}\subseteq \{\tau ^{\prime }_{n} \}\) and \(\widetilde{\beta _{i}}\) is a stochastically bounded piecewise continuous function sequence such that
and
for every \(i\in Z\).
For \(t_{i}< t\leq t_{i+1}\), \(|t-t_{i}|>\varepsilon \), \(|t-t_{i+1}|>\varepsilon \), \(i\in Z\), by Lemma 2.2, we have
and
Therefore,
Let \(\widetilde{x_{3}}(t)=\sum_{t_{i}< t}T(t-t_{i})\widetilde{\beta _{i}}\), by Lemma 2.2, then
So, by Lebesgue’s dominated convergence theorem, we get
and
Since \(\lim_{n\rightarrow \infty }E \|\beta _{i+\tau _{n}}-\widetilde{ \beta _{i}} \|^{2}=0\) and \(\lim_{n\rightarrow \infty }E \|\widetilde{\beta }_{i-\tau _{n}}- \beta _{i} \|^{2}=0\), \(x_{3}\in AA_{T} (R,L^{2}(P,H) )\).
Thus, \(x\in AA_{T} (R,L^{2}(P,H) )\). □
3.2 Nonlinear impulsive stochastic evolution equations
Consider the following nonlinear impulsive stochastic evolution equation:
where \(f,g:R\times L^{2}(P,H)\rightarrow L^{2}(P,H)\), \(I_{i}:L^{2}(P,H)\rightarrow L^{2}(P,H)\), \(i\in Z\), and \(w(t)\) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space \((\varOmega ,F,P,F_{\sigma } )\) with \(F_{t}=\sigma \{w(u)-w(v):u,v\leq t \}\).
Definition 3.2
A function \(x\in PC (R,L^{2}(P,H) )\) is called a mild solution of Eq. (8), if x satisfies
where \(t>\sigma \), \(\sigma \neq t_{i}\), \(i\in Z\). If \(x\in AA_{T} (R,L^{2}(P,H) )\), then x is called a square-mean piecewise almost automorphic mild solution of Eq. (8).
Theorem 3.2
Suppose Eq. (8) satisfies the following conditions:
- (i)
The operator\(A:D(A)\subseteq L^{2}(P,H)\rightarrow L^{2}(P,H)\)is the infinitesimal generator of a\(C_{0}\)-semigroup\(\{T(t):t\geq 0 \}\), that is, there exist\(M,\delta >0\)such that\(\|T(t) \|\leq Me^{-\delta t}\), \(t\geq 0\).
- (ii)
The functions\(f,g\in AA_{T} (R\times L^{2}(P,H),L^{2}(P,H) )\), \(\{I_{i}(\cdot ):i\in Z \}\)is a square-mean almost automorphic function sequence, and there exist positive numbers\(L_{1}\), \(L_{2}\), Lsuch that
$$\begin{aligned}& E \bigl\Vert f(t,x)-f(t,y) \bigr\Vert ^{2}\leq L_{1}E \Vert x-y \Vert ^{2}, \\& E \bigl\Vert g(t,x)-g(t,y) \bigr\Vert ^{2}\leq L_{2}E \Vert x-y \Vert ^{2} \end{aligned}$$and
$$ E \bigl\Vert I_{i}(x)-I_{i}(y) \bigr\Vert ^{2}\leq LE \Vert x-y \Vert ^{2}. $$
If\(\frac{3M^{2}}{\delta ^{2}}L_{1}+\frac{3M^{2}}{2\delta }L_{2}+ \frac{3M^{2}}{(1-e^{-\delta \gamma })^{2}}L<1\), then Eq. (8) has a square-mean piecewise almost automorphic mild solution.
Proof
Let
For any \(\varphi \in AA_{T} (R,L^{2}(P,H) )\), by (ii) and Theorem 2.2, we have \(f (\cdot ,\varphi (\cdot ) ),g (\cdot , \varphi (\cdot ) )\in AA_{T} (R,L^{2}(P,H) )\), by Theorem 2.3, \(\{I_{i} (\varphi (t_{i}) ):i\in Z \}\) is a square-mean almost automorphic sequence. Similar to the proof of Theorem 3.1, we have \(\varGamma \varphi (t)\in AA_{T} (R,L^{2}(P,H) )\).
For any \(\varphi ,\psi \in AA_{T} (R,L^{2}(P,H) )\), by \((a+b+c )^{2}\leq 3 (a^{2}+b^{2}+c^{2} )\) and Cauchy–Schwarz inequality, we have
Since \(\frac{3M^{2}}{\delta ^{2}}L_{1}+\frac{3M^{2}}{2\delta }L_{2}+ \frac{3M^{2}}{(1-e^{-\delta \gamma })^{2}}L<1\), Γ is a contraction. Therefore, Eq. (8) has a square-mean piecewise almost automorphic mild solution. □
Lemma 3.1
(Generalized Gronwall–Bellman inequality)
Assume\(u\in PC(R,R)\)satisfies the following inequality:
where\(C\geq 0\), \(\beta _{i}\geq 0\), \(\nu (\tau )>0\), then the following estimate holds for the function\(u(t)\):
Theorem 3.3
Suppose the conditions of Theorem 3.2hold, if further that
then Eq. (8) has an exponentially stable square-mean piecewise almost automorphic mild solution.
Proof
By Theorem 3.2, \(\varphi (t)\) is a solution of Eq. (8), then
Let \(\psi (t)\) be a solution of Eq. (8), then
Thus, by Cauchy–Schwarz inequality and the Ito integral, we have
So,
Let \(\varUpsilon (t)=e^{\delta t}E \|\varphi (t)-\psi (t) \|^{2}\), then
By Lemma 3.1, we get
that is,
Therefore
that is,
Since
Equation (8) has an exponentially stable square-mean piecewise almost automorphic mild solution. □
4 Applications
Consider the following impulsive stochastic evolution equation:
where \(w(t)\) is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space \((\varOmega ,F,P,F_{t} )\), \(\beta _{i}=\frac{1}{6}\sin \frac{1}{2+\cos i+\cos \sqrt{2}i}\), \(t_{i}=i+\frac{1}{3} |\sin \frac{1}{2+\cos i+\cos \sqrt{2}i} |\) (\(\varphi (i)=\frac{1}{2+\cos i+\cos \sqrt{2}i}\)).
Let \(X=L^{2}(0,\pi )\), define the operators \(A:D(A)\subseteq X\rightarrow X\) by \(Au=u^{\prime \prime }\). It is well known that A is the infinitesimal generator of a semigroup \(\{T(t):t\geq 0 \}\) on X and \(\|T(t) \|\leq e^{-t}\) for \(t\geq 0\) with \(M=\delta =1\), then condition (i) of Theorem 3.2 is satisfied. By Definition 2.3, \(\{t^{j}_{i}:i\in Z \}_{j\in Z}\) are equipotentially almost automorphic sequence and
Hence, \(\gamma =\inf_{i\in Z}(t_{i+1}-t_{i})>\frac{1}{3}>0\).
Let \(f(t,u)=\frac{1}{6}\sin \frac{1}{2+\cos t+\cos \sqrt{2}t}\sin u\), \(g(t,u)=\frac{1}{3}\sin \frac{1}{2+\cos t+\cos \sqrt{2}t}\sin u\) and \(I_{i}(u)=\beta _{i}\sin u\), then \(f,g\in AA_{T} (R\times L^{2}(P,H),L^{2}(P,H) )\) and \(\{I_{i}(\cdot ):i\in Z \}\) is a square-mean almost automorphic function sequence.
For any u, v, we have
Similarly, \(E \|g(t,u)-g(t,v) \|^{2}\leq \frac{1}{9}E \|u-v \|^{2}\), \(E \|I_{i}(u)-I_{i}(v) \|^{2}\leq \frac{1}{36}E \|u-v \|^{2}\), then \(L_{1}=\frac{1}{36}\), \(L_{2}=\frac{1}{9}\), \(L=\frac{1}{36}\). Therefore, condition (ii) of Theorem 3.2 is satisfied.
Since \(\frac{3M^{2}}{\delta ^{2}}L_{1}+\frac{3M^{2}}{2\delta }L_{2}+ \frac{3M^{2}}{(1-e^{-\delta \gamma })^{2}}L=3\times \frac{1}{36}+ \frac{3}{2}\times \frac{1}{9}+\frac{3}{(1-e^{-1/3})^{2}}\times \frac{1}{36}<1\), by Theorem 3.2, Eq. (9) has a square-mean piecewise almost automorphic mild solution.
Also since
by Theorem 3.3, Eq. (9) has an exponentially stable square-mean piecewise almost automorphic mild solution.
5 Conclusion
In this paper, we mainly construct the square-mean piecewise almost automorphic function, and the existence and exponential stability of square-mean piecewise almost automorphic mild solutions for impulsive stochastic evolution equations is proved by the theory of semigroups of operators, the contraction mapping principle and the generalized Gronwall–Bellman inequality. Finally, an interesting example is given to illustrate our results.
References
Bochner, S.: A new approach to almost-periodicity. Proc. Natl. Acad. Sci. USA 48, 2039–2043 (1962)
N’Guérékata, G.M.: Topics in Almost Automorphy, pp. 41–94. Springer, New York (2005)
Liu, J.H., Song, X.Q., Lu, F.L.: Almost automorphic and pseudo almost automorphic solutions of semilinear differential equations. Acta Anal. Funct. Appl. 11, 294–300 (2009)
Zhao, Z.H., Chang, Y.K., Nieto, J.J.: Almost automorphic and pseudo almost automorphic mild solutions to an abstract differential equation in Banach spaces. Nonlinear Anal. 72, 1886–1894 (2009)
Gal, C.S., Gal, S.G., N’Guérékata, G.M.: Almost automorphic functions in frechet spaces and applications to differential equations. Semigroup Forum 71, 201–230 (2005)
N’Guérékata, G.M.: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Springer, Heidelberg (2001)
Diagana, T.: Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces. Springer, Heidelberg (2013)
Fu, M.M.: Almost automorphic solutions for nonautonomous stochastic differential equations. J. Math. Anal. Appl. 393, 231–238 (2012)
Wang, C., Agarwal, R.P.: Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations. Discrete Contin. Dyn. Syst., Ser. B 25, 781–798 (2020)
Bedouhene, F., Challali, N., Mellah, O., de Fitte, P.R., Smaali, M.: Almost automorphy various extensions for stochastic processes. J. Math. Anal. Appl. 429, 1113–1152 (2015)
Bezandry, P.H., Diagana, T.: Square-mean almost periodic solutions nonautonomous stochastic differential equations. Electron. J. Differ. Equ. 2007, 117, 1–10 (2007)
Diagana, T., Mbaye, M.M.: Square-mean almost periodic solutions to some singular stochastic differential equations. Appl. Math. Lett. 54, 48–53 (2016)
Fu, M.M., Liu, Z.X.: Square-mean almost automorphic solutions for some stochastic differential equations. Proc. Am. Math. Soc. 138, 3689–3701 (2010)
Chang, Y.K., Zhao, Z.H., N’Guérékata, G.M.: Square-mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces. Comput. Math. Appl. 61, 384–391 (2011)
Chang, Y.K., Zhao, Z.H., N’Guérékata, G.M.: A new composition theorem for square-mean almost automorphic functions and applications to stochastic differential equations. Nonlinear Anal. 74, 2210–2219 (2011)
Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations, pp. 12–228. Word Scientific, Singapore (1995)
Lin, Y., Feng, C.: On the existence of almost periodic solutions for a kind of delay differential equations with impulses. Guang Xi Sci. 17, 22–26 (2010)
Henriquez, H.R., de Andrade, B., Rabelo, M.: Existence of almost periodic solutions for a class of abstract impulsive differential equations. ISRN Math. Anal. 2011, 1–21 (2011)
Stamov, G.T., Alzabut, J.O.: Almost periodic solutions for abstract impulsive differential equations. Nonlinear Anal. 72, 2457–2464 (2010)
Stamov, G.T.: Almost Periodic Solutions of Impulsive Differential Equations. Springer, Heidelberg (2012)
Mahto, L., Abbas, S.: Pc-almost automorphic solution of impulsive fractional differential equations. Mediterr. J. Math. 12, 771–790 (2015)
Liu, J.W., Zhang, C.Y.: Existence and stability of almost periodic solutions for impulsive differential equations. Adv. Differ. Equ. 2012, 34, 1–14 (2012)
Aouiti, C., Dridi, F.: Piecewise asymptotically almost automorphic solutions for impulsive non-autonomous high-order Hopfield neural networks with mixed delays. Neural Comput. Appl. 31, 5527–5545 (2019)
Wang, C.: Piecewise pseudo almost periodic solution for impulsive non-autonomous high-order Hopfield neural networks with variable delays. Neurocomputing 171, 1291–1301 (2016)
Wang, C., Agarwal, R.P.: Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive ∇-dynamic equations on time scales. Adv. Differ. Equ. 2014, 153, 1–29 (2014)
Zhang, R.J., Ding, N., Wang, L.S.: Mean square almost periodic solutions for impulsive stochastic differential equations with delays. J. Appl. Math. 2012, 1–14 (2012)
Wang, C.: Existence and exponential stability of piecewise mean-square almost periodic solutions for impulsive stochastic Nicholson’s blowflies model on time scales. Appl. Math. Comput. 248, 101–112 (2014)
Liu, J.W., Zhang, C.Y.: Existence and stability of almost periodic solutions to impulsive stochastic differential equations. CUBO 15, 77–96 (2013)
Zhou, H., Zhou, Z.F., Qiao, Z.M.: Mean-square almost periodic solution for impulsive stochastic Nicholson’s blowflies model with delays. Appl. Math. Comput. 219, 5943–5948 (2013)
Wang, C., Agarwal, R.P.: Almost periodic solution for a new type of neutral impulsive stochastic Lasota–Wazewska timescale model. Appl. Math. Lett. 70, 58–65 (2017)
Yan, Z.M., Lu, F.X.: Existence and exponential stability of pseudo almost periodic solutions for impulsive nonautonomous partial stochastic evolution equations. Adv. Differ. Equ. 2016, 294, 1–37 (2016)
Yan, Z.M., Yan, X.X.: Optimal controls for impulsive partial stochastic differential equations with weighted pseudo almost periodic coefficients. Int. J. Control 2019, 1–38 (2019)
Yan, Z.M., Han, L.: A class of stochastic hyperbolic evolution equations via weighted pseudo almost periodic coefficients and optimal controls. Optim. Control Appl. Methods 40, 819–847 (2019)
Yan, Z.M., Jia, X.M.: Pseudo almost periodicity and its applications to impulsive nonautonomous partial functional stochastic evolution equations. Int. J. Nonlinear Sci. Numer. Simul. 2018, 1–19 (2018)
Acknowledgements
We would like to thank the referees for their valuable comments.
Availability of data and materials
No applicable.
Funding
The work of the first author is supported by the National Natural Science Foundation of China (Grant No. 11526179) and the Natural Science Foundation of Hebei Province (Grant No. A2016203341). The work of the third author is supported by the research project of Tianjin Municipal Education Commission (Grant No. 160022), the Doctoral Scientific Research Foundation of Tianjin University of Commerce (Grant No. R160101) and the National Nurture Fund of Tianjin University of Commerce (Grand No. 170113).
Author information
Authors and Affiliations
Contributions
All authors contributed equally to this manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Liu, J., Ren, R. & Xie, R. Square-mean piecewise almost automorphic mild solutions to a class of impulsive stochastic evolution equations. Adv Differ Equ 2020, 136 (2020). https://doi.org/10.1186/s13662-020-02574-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-020-02574-4