Abstract
In this paper, the existence and exponential stability of mild solutions of random impulsive fractional differential equations are studied. The results are obtained by using the Leray–Schauder alternative fixed point theorem.
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1 Introduction
For impulsive differential systems, most published papers deal with the problem with fixed time impulses [1–3]. However, actual jumps do not always happen at fixed points but usually at random points. The solutions of random impulsive differential equations is a stochastic process. It is different from deterministic impulsive differential equations and also it is different from stochastic differential equations. At present, the properties of solutions to some integer order differential equations with random impulses have been studied [4–8]. The existence, uniqueness, and stability of fractional differential equations without random impulses have been shown by many authors [9–18]. However, the properties of solutions to fractional delay differential equations with random impulses have not been studied. In [19], the authors study the existence, uniqueness, and stability through continuous dependence on initial conditions and Hyers–Ulam–Rassias stability for random impulsive fractional differential equations, but in that paper, the authors did not consider delay. In some applications, besides impulsive effects, delay effects cannot be ignored.
Motivated by the above consideration, we consider fractional differential equations with random impulses of the form
Let X be a real separable Hilbert space and Ω be a nonempty set. For the sake of simplicity, we denote \(R_{\tau }=[\tau ,+\infty ),R^{+}=[0,+\infty )\). Assume that \(\tau _{k}\) is a random variable defined from Ω to \(D_{k}\stackrel{\mathrm{def.}}{=}(0,d_{k})\) for \(k=1,2,\ldots\) , where \(0< d_{k}<+\infty \). Furthermore, assume that \(\tau _{i}\) and \(\tau _{j}\) are independent of each other as \(i\neq j\) for \(i,j=1,2,\ldots \) . A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators \(S(t)\) with domain \(D(A)\subset X\); the functional \(f:R^{+}\times C\rightarrow X,C=C([-r,0],X)\) is the set of piecewise continuous functions, \(r>0\); \(x_{t}\) is a function when t is fixed, \(x_{t}(\theta ) =x(t+\theta )\), \(\theta \in [-r,0]\); \(\xi _{0}=t_{0}\) and \(\xi _{k}=\xi _{k-1}+\tau _{k}\) for \(k=1,2,\ldots \) , here \(t_{0}\in R_{\tau }\) is an arbitrary given real number. The impulse moments \({\xi _{k}}\) form a strictly increasing sequence, i.e., \(t_{0}=\xi _{0}<\xi _{1}<\xi _{2}<\cdots <\lim_{k\rightarrow \infty }\xi _{k}\); \(b_{k}:D_{k}\rightarrow X\) for each \(k=1,2,\ldots \) ; \(x(\xi _{k}^{-})=\lim_{t\rightarrow \xi _{k}}x(t)\) according to their paths with the norm \(\Vert x \Vert _{t}=\sup_{t-r\leq s\leq t}\vert x(s) \vert \) for each t satisfying \(t\geq t_{0}\); \(\Vert \cdot \Vert \) is any given norm in X; φ is a function defined from \([-r,0]\) to X; \({}^{c}D^{q}\) is the Caputo fractional derivative of order \(0< q<1\).
This paper studies the properties of solutions to system (1.1) by using the theory of functional differential equation, fractional differential equation, and stochastic analysis techniques. Firstly, we investigate the existence of a mild solution for (1.1) by using the Leray–Schauder alternative fixed point theorem. Secondly, we study the exponential stability in the quadratic mean of (1.1). Finally, an example is presented to illustrate our results.
2 Some preliminaries
In this section, we shall recall some basic definitions and lemma which will be used in this paper.
Denote by \(\{B_{t},t\geq 0\}\) the simple counting process generated by \(\{\xi _{n}\}\), that is, \(\{B_{t}\geq n\}=\{\xi _{n}\leq t\}\), and denote by \(F_{t}\) the σ-algebra generated by \(\{B_{t},t\geq 0\}\). Then \((\Omega ,P,\{F_{t}\})\) is a probability space. Let \(L_{2}=L_{2}(\Omega ,F_{t},X)\) denote the Hilbert space of all \(F_{t}\)-measurable square integrable random variables with values in X.
Assume that \(T>t_{0}\) is any fixed time to be determined later and ß denotes the Banach space \(\ss ([t_{0}-r,T],L_{2})\), the family of all \(F_{t}\)- measurable, C-valued random variables φ with the norm
Let \(L_{2}^{0}(\Omega ,\ss )\) denote the family of all \(F_{0}\)-measurable, ß-valued random variables φ.
Definition 2.1
The fractional integral of order q with the lower limit 0 for a function f is defined as
provided the right-hand side is pointwise defined on \([0,\infty )\), where Γ is the gamma function.
Definition 2.2
The Riemann–Liouville derivative of order q with the lower limit 0 for a function \(f:[0,\infty )\rightarrow R\) can be written as
Definition 2.3
The Caputo derivative of order q for a function \(f:[0,\infty )\rightarrow R\) can be written as
Definition 2.4
A semigroup \(\{S(t),t\geq t_{0}\}\) is said to be uniformly bounded if \(\Vert S(t) \Vert \leq M\) for all \(t\geq t_{0}\), where \(M\geq 1\) is some constant. If \(M=1\), then the semigroup is said to be contraction semigroup.
Definition 2.5
A map \(f:[\tau ,T]\times C\rightarrow X\) is said to be \(L^{2}\)-Carathéodory if
-
(i)
\(t\rightarrow f(t,u)\) is measurable for each \(u\in C\);
-
(ii)
\(u\rightarrow f(t,u)\) is continuous for almost all \(t\in [\tau ,T]\);
-
(iii)
for each positive integer \(m>0\), there exists \(\alpha _{m}\in L^{1}([\tau ,T],R^{+})\) such that
$$ \sup_{\Vert u \Vert _{\ss }\leq m} \bigl\Vert f(t,u) \bigr\Vert ^{2}\leq \alpha _{m}(t) \quad \mbox{for } t\in [\tau ,T],\mbox{ a.e.} $$
Definition 2.6
For a given \(T\in (t_{0},+\infty )\), a stochastic process \(\{x(t)\in \ss , t_{0}-r\leq t\leq T\}\) is called a mild solution to Eq. (1.1) in \((\Omega ,P,\{F_{t}\})\) if
-
(i)
\(x(t)\in \ss \) is \(F_{t}\)-adapted for \(t\geq t_{0}\);
-
(ii)
\(x(t_{0}+s)=\varphi (s)\in L_{2}^{0}(\Omega ,\ss )\), when \(s\in [-r,0]\), and
$$\begin{aligned} x(t)& = \sum_{k=0}^{+\infty } \Biggl[ \prod_{i=1}^{k}b_{i}(\tau _{i})S(t-t_{0})\varphi (0) \\ &\quad {}+\frac{1}{\Gamma (q)}\sum_{i=1}^{k} \prod_{j=i}^{k}b_{j}( \tau _{j}) \int _{\xi _{i-1}}^{\xi _{i}}(t-s)^{q-1}S(t-s)f(s,x_{s}) \,ds \\ &\quad {}+\frac{1}{\Gamma (q)} \int _{\xi _{k}}^{t}(t-s)^{q-1}S(t-s)f(s,x_{s}) \,ds \Biggr]I_{[\xi _{k},\xi _{k+1})}(t),\quad t\in [t_{0},T], \end{aligned}$$where \(\prod_{j=m}^{n}(\cdot )=1\) as \(m>n\), \(\prod_{j=i}^{k}b_{j}(\tau _{j})=b_{k}(\tau _{k})b_{k-1}(\tau _{k-1})\cdots b_{i}(\tau _{i})\), and \(I_{A}(\cdot )\) is the index function, i.e.,
$$ I_{A}(t)= \textstyle\begin{cases} 1,&\mbox{if } t\in A , \\ 0,& \mbox{if } t\notin A. \end{cases} $$
Our existence and exponential stability theorems are based on the following theorem, which is a version of the topological transversality theorem.
Lemma 2.1
Let E be a convex subset of a Banach space X, and assume that \(0\in E\). Let \(F:E\rightarrow E\) be a completely continuous operator, and let
then either \(U(F)\) is unbounded or F has a fixed point.
3 Existence results
In this section, we prove the existence of a mild solution of system (1.1) by using the following hypotheses:
- \((H_{1})\) :
-
There exist a continuous non-decreasing function \(H:R^{+}\rightarrow (0,\infty )\) and \(\delta \in L^{1}([\tau ,T],R^{+})\) such that
$$ E \bigl\Vert f(t,\psi _{t}) \bigr\Vert ^{2}\leq \delta (t)H \bigl(E\Vert \psi \Vert _{t}^{2} \bigr) $$for every \(t\in [\tau ,T]\) and for any stochastic process \(\psi \in C\).
- \((H_{2})\) :
-
\(\max_{i,k}\{\prod_{j=i}^{k}\Vert b_{j}(\tau _{j}) \Vert \}\) is uniformly bounded, that is, there is \(B>0\) such that
$$ \max_{i,k} \Biggl\{ \prod_{j=i}^{k} \bigl\Vert b_{j}(\tau _{j}) \bigr\Vert \Biggr\} \leq B $$for all \(\tau _{j}\in D_{j}\), \(j=1,2,\ldots\) .
Theorem 3.1
Assume \((H_{1})\)–\((H_{2})\) hold, then system (1.1) has a mild solution \(x(t)\), defined on \([t_{0},T]\), provided that the following inequality is satisfied:
where \(M_{1}=2M^{2}\max \{1,B^{2}\}\frac{(T-t_{0})^{2q-1}}{(2q-1)\Gamma (q)}\), \(c_{1}=2M^{2}B^{2}E\Vert \varphi \Vert ^{2}\), and \(MC\geq \frac{1}{\sqrt{2}}\).
Proof
Let T be an arbitrary positive number \(t_{0}< T<+\infty \). Define an operator \(\Phi :\ss \rightarrow \ss \) as follows:
First we establish the a priori estimates for the solutions of the integral equation and \(\lambda \in (0,1)\):
Thus by \((H_{1})\)–\((H_{2})\), we have
Let
Then, for any \(t\in [t_{0},T]\), it follows that
Denoting by \(u(t)\) the right-hand side of the above inequality, we obtain that
and
Then
Integrating the above inequality from \(t_{0}\) to t and making use of the change of variable, we obtain
From the above inequality and by the mean value theorem, there is a constant K such that \(u(t)\leq K\), and hence \(\ell (t)\leq K\). Since \(\sup_{t_{0}\leq \upsilon \leq t}E[\Vert x \Vert _{\upsilon }^{2}]=\ell (t)\) holds for every \(t\in [t_{0},T]\), we have \(\sup_{t_{0}\leq \upsilon \leq T}E[\Vert x \Vert _{\upsilon }^{2}]\leq K\), where K only depends on T and the functions δ and H, consequently
In the next steps, we will prove that Φ is continuous and completely continuous.
Step 1. We prove that Φ is continuous.
Let \(\{x_{n}\}\) be a convergent sequence of elements of x in ß. Then, for each \(t\in [t_{0},T]\), we have
Thus
and
Thus Φ is continuous.
Step 2. We prove that Φ is a completely continuous operator.
Denote
for some \(m\geq 0\).
Step 2.1. We show that Φ maps \(B_{m}\) into an equicontinuous family.
Let \(x\in B_{m}\) and \(t_{1},t_{2}\in [t_{0},T]\). If \(t_{0}< t_{1}< t_{2}< T\), then by using \((H_{1})\)–\((H_{2})\) and condition (3.1), we have
Then
where
and
Furthermore,
where \(M^{*}=\sup \{\delta (t):t\in [t_{0},T]\}\).
Thus, Φ maps \(B_{m}\) into an equicontinuous family of functions.
Step 2.2. We show that \(\Phi B_{m}\) is uniformly bounded.
From (3.1), \(\Vert x \Vert _{\ss }^{2}\leq m\), by \((H_{1})\)–\((H_{2})\), we obtain
Thus
This yields that the set \(\{(\Phi x)(t),\Vert x \Vert _{\ss }^{2}\leq m\}\) is uniformly bounded, so \(\{\Phi B_{m}\}\) is uniformly bounded, by the Arzela–Ascoli theorem, Φ maps \(B_{m}\) into a precompact set in X.
Step 2.3. We show that \(\Phi B_{m}\) is compact.
Let \(t_{0}< t\leq T\) be fixed, and let ϵ be a real number satisfying \(\epsilon \in (0,t-t_{0})\), for \(x\in B_{m}\), we define
Since \(S(t)\) is a compact operator, the set
is precompact in X for every \(\epsilon \in (0,t-t_{0})\). Moreover, for every \(x\in B_{m}\), we have
By using \((H_{1})\)–\((H_{2})\), condition (3.1), and \(\Vert x \Vert _{\ss }^{2}\leq m\), we have
Therefore, there are precompact sets arbitrarily close to the set \(\{(\Phi x)(t):x\in B_{m}\}\). Hence the set \(\{(\Phi x)(t):x\in B_{m}\}\) is precompact in X. Therefore, Φ is a completely continuous operator.
Moreover, the set \(U(\Phi )=\{x\in \ss :x=\lambda \Phi x \quad for \quad some \quad 0<\lambda <1\}\) is bounded. Consequently, by Lemma 2.1, the operator Φ has a fixed point in ß. Therefore, system (1.1) has a mild solution. □
4 Exponential stability in the quadratic mean
In this section, we will study the exponential stability of the second moment of a mild solution of system (1.1). For an \(F_{t}\)-adapted process, \(\Phi (t):[-r,\infty )\rightarrow R\) is almost surely continuous in t. For the purposes of stability, we may assume that \(f(t,0)\equiv 0\) for any \(t\geq t_{0}\) so that system (1.1) admits a trivial solution. Moreover, \(\Phi (t)=\varphi (t-t_{0})\) for \(t\in [t_{0}-r,t_{0}]\) and \(E\Vert \Phi \Vert _{t}^{2}\rightarrow 0\) as \(t\rightarrow \infty \).
Definition 4.1
System (1.1) is said to be exponentially stable in the quadratic mean if there exist positive constants \(C_{1}>0\) and \(\lambda >0\) such that
Now we introduce the following hypotheses used in our discussion:
- \((H_{3})\) :
-
\(\mu H(\psi )\leq H(\mu \psi )\) for all \(\psi \in R^{+}\), where \(\mu >1\).
- \((H_{4})\) :
-
\(\Vert S(t) \Vert \leq Me^{-r(t-t_{0})}\), \(t\geq t_{0}\), where \(M\geq 1,r>0\).
Theorem 4.1
Assume that the hypotheses of Theorem 3.1 and \((H_{3})\)–\((H_{4})\) hold. If the following inequality is satisfied, then system (1.1) is exponentially stable in the quadratic mean:
where \(M_{2}=2M^{2}\max \{1,B^{2}\}\frac{(T-t_{0})^{2q-1}}{\Gamma (q)(2q-1)}\), \(c_{2}=2M^{2}B^{2}E[\Vert \varphi \Vert ^{2}]\), and \(MB\geq \frac{1}{\sqrt{2}}\).
Proof
Let Φ be the map defined in Theorem 3.1. Using hypotheses \((H_{1})\)–\((H_{4})\), we have
Let
Then, for any \(t\in [t_{0},T]\), it follows that
Denoting by \(u_{1}(t)\) the right-hand side of the above inequality, we obtain that
and
Then
Integrating the above inequality from \(t_{0}\) to t and making use of the change of variable, we obtain
From the above inequality and by the mean value theorem, there is a constant \(K_{1}\) such that \(u_{1}(t)\leq K_{1}\), and hence \(\ell _{1}(t)\leq K_{1}\). Since \(\sup_{t_{0}\leq \upsilon \leq t}e^{2r(\upsilon -t_{0})}E\Vert x \Vert _{\upsilon }^{2}=\ell _{1}(t)\) holds for every \(t\in [t_{0},T]\), we have \(\sup_{t_{0}\leq \upsilon \leq t}e^{2r(\upsilon -t_{0})}E\Vert x \Vert _{\upsilon }^{2}\leq K_{1}\), where \(K_{1}\) depends only on δ and H. Consequently,
In the following, we proceed as in the previous theorem. Now, we will show that Φ is a completely continuous operator using a two-step proof.
Step 1. We prove that Φ is continuous.
Let \(\{x_{n}\}\) be a convergent sequence of elements of x in ß. Then, for each \(t\in [t_{0},T]\), we have
Thus Φ is continuous.
Step 2. We prove that Φ is a completely continuous operator.
Denote
for some \(m_{1}\geq 0\).
Step 2.1. We show that Φ maps \(B_{m_{1}}\) into an equicontinuous family.
Let \(x\in B_{m}\) and let \(t_{1},t_{2}\in [t_{0},T]\). If \(t_{0}< t_{1}< t_{2}< T\), then by using \((H_{1})\)–\((H_{4})\) and condition (4.1) and then by following a process similar to Step 2.1 of Theorem 3.1, we get
Thus, Φ maps \(B_{m_{1}}\) into an equicontinuous family of functions.
Step 2.2. We show that \(\Phi B_{m_{1}}\) is uniformly bounded.
From condition (4.1) and \((H_{1})\)–\((H_{4})\), we obtain
Thus
where \(M^{*}=\sup \{\delta (t):t\in [t_{0},T]\}\). Since \(e^{-2r(t-t_{0})}\rightarrow 0\), the right-hand side of the above inequality tends to 0 as \(t\rightarrow \infty \). That is,
This yields that the set \(\{(\Phi x)(t),\Vert x \Vert _{\ss }^{2}\leq m_{1}\}\) is uniformly bounded, so \(\{\Phi B_{m_{1}}\}\) is uniformly bounded.
Step 2.3. We show that \(\Phi B_{m_{1}}\) is compact.
Let \(t_{0}< t\leq T\) be fixed and let ϵ be a real number satisfying \(\epsilon \in (0,t-t_{0})\), for \(x\in B_{m_{1}}\), we define
Since \(S(t)\) is a compact operator, the set
is precompact in X for every \(\epsilon \in (0,t-t_{0})\).
Making use of hypotheses \((H_{1})\)–\((H_{4})\), condition (4.1), and the fact that \(\Vert x \Vert _{\ss }^{2}\leq m_{1}\), we have
Therefore, there are precompact sets arbitrarily close to the set \(\{(\Phi x)(t):x\in B_{m_{1}}\}\). Hence the set \(\{(\Phi x)(t):x\in B_{m_{1}}\}\) is precompact in X. Therefore, Φ is a completely continuous operator.
Moreover, the set \(U(\Phi )=\{x\in \ss :x=\lambda \Phi x \mbox{ for some } 0<\lambda <1\}\) is bounded. Consequently, by Lemma 2.1, the operator Φ has a fixed point in ß. Therefore, system (1.1) has a mild solution with \(\Phi (t)=\varphi (t-t_{0})\) when \(t\in [t_{0}-r,t_{0}]\) and \(E\Vert \Phi \Vert _{t}^{2}\rightarrow 0\) as \(t\rightarrow \infty \). This completes the proof. □
5 Example
In this section, we provide an example to illustrate our main results. Consider the following random impulsive fractional differential system:
Let \(X=L^{2}([0,\pi ])\) and \(\tau _{k}\) be a random variable defined on \(D_{k}\equiv (0,d_{k})\) for \(k=1,2,\ldots\) , where \(0< d_{k}<+\infty\). Furthermore, assume that \(\tau _{k}\) follows the Erlang distribution, where \(k=1,2,\ldots \) and \(\tau _{i}\) and \(\tau _{j}\) are independent of each other as \(i\neq j\) for \(i,j=1,2,\ldots \) ; q is a function of k; \(\xi _{0}=t_{0}\) and \(\xi _{k}=\xi _{k-1}+\tau _{k}\) for \(k=1,2,\ldots \) , here \(t_{0}\in R_{\tau }\) is an arbitrary given real number.
Define A an operator on X by \(Au=\frac{\partial ^{2}u}{\partial x^{2}}\) with the domain \(D(A)=\{u\in X\mid u\mbox{ }\mbox{ and }\frac{\partial u}{\partial x}\mbox{ }\mbox{ are } \mbox{absolutely continuous, }\frac{\partial ^{2}u}{\partial x^{2}}\in X,u(0)=0=u(\pi )\}\).
It it well known that A generates a strongly continuous semigroup \(S(t)\) which is compact, analytic, and self-adjoint. Moreover, the operator A can be expressed as
where \(u_{n}(\zeta )=\sqrt{\frac{2}{\pi }}\sin n\zeta \), \(n\in N\) is the orthogonal set of eigenvectors of A, and for every \(u\in X, S(t)u=\sum_{n=1}^{\infty }\exp(-n^{2}t)(u,u_{n})u_{n}\), which satisfies
Hence \(S(t)\) is a contraction semigroup.
Equation (5.1) can be reformulated as the following abstract equation in \(X=L^{2}([0,\pi ])\):
where \(z(t)=u(\cdot,t)\), i.e., \(z(t)(x)=u(x,t),z_{t}(\theta )(x)=u(x,t+\theta ),t\in [t_{0},T],\theta \in [-r,0],z(t,x)=\varphi (t,x),t\in [-r,0],x\in [0,\pi ]\). The operator A is defined as above. The functions \(f:[\tau,T]\times C\rightarrow X\) and \(b_{k}:D_{k}\rightarrow X\) are given by
where \(a>0\).
It is easy to check that system (5.1) has a mild solution if hypotheses \((H_{1}),(H_{2})\) and the inequality
hold, where \(c_{1}=2B^{2}E\Vert \varphi \Vert ^{2}\).
Furthermore, by Theorem 4.1, if the following conditions hold
where \(c_{2}=2B^{2}E[\Vert \varphi \Vert ^{2}]\), then system (5.1) is exponentially stable in the quadratic mean.
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This research was supported by the National Nature Science Foundation of China (No. 11371027), the Research Fund for the Doctoral Program (No. 2016QD110).
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Zhang, S., Jiang, W. The existence and exponential stability of random impulsive fractional differential equations. Adv Differ Equ 2018, 404 (2018). https://doi.org/10.1186/s13662-018-1779-4
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DOI: https://doi.org/10.1186/s13662-018-1779-4