- Research
- Open Access
Convergence and stability of stochastic parabolic functional differential equations
- Zhao Li^{1} and
- Shuyong Li^{2}Email author
https://doi.org/10.1186/s13662-018-1703-y
© The Author(s) 2018
- Received: 15 April 2018
- Accepted: 16 June 2018
- Published: 9 August 2018
Abstract
The main purpose of this paper is to investigate the convergence and stability of stochastic parabolic functional differential equations. Firstly, a comparison theorem in the context of Lyapunov-like function together with differential inequality is established. Secondly, various criteria for the convergence and stability are obtained on the basis of the comparison theorem and stochastic analysis techniques. Finally, two examples are provided to illustrate the significance of the theoretical results.
Keywords
- Stochastic parabolic functional differential equations
- Comparison theorem
- Functional differential equation
- Convergence
- Stability
1 Introduction
In recent years, stochastic parabolic partial differential equations have received a great deal of attention due to their important applications in biological, control engineering, economics, physics, social sciences, and so on. Hence, the theory of stochastic parabolic partial differential equations has developed very quickly [1–3]. On the other hand, in many realistic models, their future states depend not only on the present states but also on the past ones. Therefore, time delay should be taken into consideration in modeling. As in [4], stochastic parabolic partial differential equations with time-delay are called stochastic parabolic functional differential equations (also called SPFDEs for short). The study of SPFDEs is very complex and difficult since most of them do not have an explicit solution. Thus, many researchers have been interested in investigating the stability; all kinds of ways to study stability of the solutions have been put out in [5–10] and the references cited therein. For example, Caraballo and Liu [5] have analyzed the exponential stability for mild solution to stochastic partial differential equations with delays by utilizing the well-known Gronwall inequality, and they imposed the requirement of the monotone decreasing behavior of the delays. Liu [6] has considered the exponential stability for SPFDEs by means of the Razumikhin-type theorem. Luo [7] has established some sufficient conditions ensuring the exponential stability and asymptotic stability for mild solution of stochastic partial differential equations with delays basing on the fixed point theorem. Taniguchi [8] has proved the almost sure exponential stability of mild solution for stochastic partial functional differential equations by using the analytic technique. Taniguchi [9] has investigated the exponential stability for stochastic delay partial differential equations by energy inequality. Wang and Yang [10] have given some sufficient conditions for determining the domain of pth-moment stability region for stochastic partial differential equation with delays on the basis of the properties of nonnegative matrices, stochastic convolution, and the inequality technique.
It is well known that in previous work by Lakshmikatham, Leela and Ladde [11, 12], comparison principle as an important method has been successfully used in investigating the convergence and stability of differential equations. The main characteristic of this principle in stability analysis is that, by considering the relations which are based on a suitable Lyapunov-like function (or Lyapunov-like functional) together with the techniques of differential inequality between the original system and a comparison system, the stability of the solutions of the comparison system will imply the corresponding stability properties of the solutions of the original system. Usually, the stability of the comparison system (e.g., ordinary differential equations and linear differential difference equation) is obtained easily. Recently, Anabtawi and Sathananthan [13] employed the comparison principle and Lyapunov-like functional [14] techniques to study the convergence and stability behavior of stochastic parabolic partial differential equations.
Motivated by the above, we have the main aim in this paper to investigate the convergence and stability for SPFDEs. A comparison theorem in the context of Lyapunov-like function will be established. By the comparison theorem, some sufficient conditions for the convergence and stability of SPFDEs can be obtained by considering the convergence and stability of deterministic functional differential equation.
This paper is organized as follows. In Sect. 2, we introduce the basic notations and set up the comparison theorem for SPFDEs. In Sect. 3 and Sect. 4, we get our main results on the convergence and stability of solution process. In Sect. 5, we illustrate the significance of our results by two examples. Finally, conclusions are drawn in Sect. 6.
2 Preliminaries
Let \((\Omega,\mathscr{F},\{\mathscr{F}_{t}\}_{t \geq t_{0}},\mathbb {P})\) be a complete probability space with a filtration \(\{\mathscr{F}_{t}\}_{t \geq t_{0}}\) satisfying the usual conditions, \(W(t)\) is an m-dimensional normalized Brownian motion defined on the complete probability space, and the increment \(W(t+h)-W(t)\) is independent of every event in \(\mathscr{F}_{t}\). Let \(\mathbb{R}^{n}\) and \(\mathbb{R}^{n\times m}\) denote the n-dimensional real Euclidean space and the set of all \(n\times m\) real matrices, respectively. Let H be a bounded domain in \(\mathbb{R}^{d}\) with a smooth boundary ∂H. Let \(\tau\geq0\) be a bounded constant, \(t_{0}\in\mathbb {R}_{+}=[0,+\infty)\), \(J=[t_{0},+\infty)\), \(J_{\tau}=[t_{0}-\tau,+\infty )\), \(Q_{t_{0}}=J\times H\), and \(\Gamma_{t_{0}}=J\times\partial H\). Let \(C(X,Y)\) be the space of continuous mappings from the topological space X to the topological space Y. Especially, let \(C_{1}\triangleq C([-\tau,0]\times H;\mathbb{R}^{n})\) denote the Banach space of all continuous functions \(\varphi(s,x)\) from \([-\tau,0]\times H \) to \(\mathbb{R}^{n}\) with the norm \(\|\varphi\| _{0}=\sup_{-\tau\leq s\leq0}\|\varphi(s)\|\), where \(\|\cdot\|\) is some norm in H; let \(C_{2}\triangleq C([-\tau,0];\mathbb{R}_{+})\) denote the family of all continuous functions ϕ from \([-\tau,0]\) to \(\mathbb{R}_{+}\) with the norm \(|\phi|_{0}=\sup_{-\tau\leq s \leq0}\phi(s)\). Denote by \(L_{\mathscr{F}_{t}}^{p}(\Omega,C_{1})\) the family of all \(\mathscr{F}_{t}\) measurable, \(C_{1}\)-valued random variables \(\varphi (s,x)\) with \(\sup_{-\tau\leq s \leq0}\mathbb{E}\|\varphi(s)\| ^{p}<\infty\), where \(\{\mathbb{E}(\cdot)\}\) stands for the mathematical expectation operator with respect to the given probability measure \(\mathbb{P}\), the superscript “T” denotes the transpose of a matrix or vector, \(\operatorname{Tr}(\cdot)\) denotes the trace of the corresponding matrix.
In the paper, we always assume that F, G, and the domain are smooth enough so that Eq. (2.1) has a solution process on \(J\times H\). Further, assume that \(F(t,x,0,0,0,0)\equiv0\) and \(G(t,x,0,0)\equiv0\). Then Eq. (2.1) obviously has a trivial solution \(u(t,x)\equiv0\).
Denote by \(C^{1,2}(\mathbb{R}\times S_{\rho};\mathbb{R}_{+})\) the space of all nonnegative functions \(V(t,u)\) defined on \(\mathbb{R}\times S_{\rho}\) which are once continuously differential in t and twice in u, where \(S_{\rho}\equiv\{(t,x)\in\mathbb{R}\times H:\|u(t,x)\|<\rho\}\).
Theorem 2.1
- (i)\(g(t,y,\sigma)\in C(J\times\mathbb{R}_{+}\times C_{2};\mathbb{R})\), \(g(t,y,\sigma)\) is concave and nondecreasing in σ for each \((t,y)\) and satisfieswhere \(V_{t}=V(t+s,u(t+s,x))\), \(s\in[-\tau,0]\), \(x\in H \), and$$ \mathcal{L}V\bigl(t,u(t,x)\bigr)\leq g(t,V,V_{t}), $$(2.2)where \(\frac{\partial V(t,u(t,x))}{\partial u}=(\frac{\partial V(t,u(t,x))}{\partial u_{1}},\ldots,\frac{\partial V(t,u(t,x))}{\partial u_{n}})\), \(\frac{\partial^{2}V(t,u(t,x))}{\partial u^{2}}=(\frac{\partial ^{2}V(t,u(t,x))}{\partial u_{i}\partial u_{j}})_{n\times n}\);$$\begin{aligned} &\mathcal{L}V\bigl(t,u(t,x)\bigr) \\ &\quad =\frac{\partial V(t,u(t,x))}{\partial t} \\ &\qquad {}+\frac{\partial V(t,u(t,x))}{\partial u}F\biggl(t,x,u(t,x),u_{t}(s,x), \frac {\partial u(t,x)}{\partial u}, \frac{\partial^{2}u(t,x)}{\partial u^{2}}\biggr) \\ &\qquad {}+\frac{1}{2}\operatorname{Tr}\biggl[\bigl(G\bigl(t,x,u(t,x),u_{t}(s,x) \bigr)\bigr)^{T}\frac{\partial^{2} V(t,u(t,x))}{\partial u^{2}}G\bigl(t,x,u(t,x),u_{t}(s,x) \bigr)\biggr], \end{aligned}$$(2.3)
- (ii)\(r(t;t_{0},\xi(s))\) is the maximal solution of the functional differential equationwhere \(y_{t}=y(t+s)\);$$ \textstyle\begin{cases} \frac{dy}{dt}=g(t,y,y_{t}),\quad t\in J, \\ y_{t_{0}}=\xi(s)\in C_{2},\quad s\in[-\tau,0], \end{cases} $$(2.4)
- (iii)for the solution process \(u(t,x)\) of SPFDEs (2.1), \(\mathbb{E}[V(t,u(t,x))]\) exists for \(t\in J_{\tau}\), and$$ \textstyle\begin{cases} \mathbb{E}V(t_{0}+s,u(t_{0}+s,x))\leq\xi(s),\quad s\in[-\tau,0], x\in H, \\ \mathbb{E}V(t,\psi(t,x))\leq r(t;t_{0},\xi(s)),\quad (t,x)\in\Gamma_{t_{0}}. \end{cases} $$(2.5)
Proof
This completes the proof of Theorem 2.1. □
3 Convergence analysis
In this section, we utilize the comparison theorem in the context of Lyapunov-like function established in Sect. 2 to investigate the convergence of the solution process of SPFDEs (2.1). Let \(u(t,x,\omega)\) be any solution process of SPFDEs (2.1).
Definition 3.1
- (i)converge in the pth moment if, for every \(\varepsilon >0\) and \(t_{0}\in J\), there exist \(\delta>0\) and a positive number \(T=T(\varepsilon,\delta ,t_{0})>0\) such thatwhenever \(\sup_{-\tau\leq s\leq0}\mathbb{E}\|\varphi(s,\omega)\| ^{p}\leq\delta\) and \(\mathbb{E}(\|\psi(t,\omega)\| ^{p})<\varepsilon\), \(t\geq T+t_{0}\);$$ \mathbb{E}\bigl( \bigl\Vert u(t,\omega) \bigr\Vert ^{p}\bigr)< \varepsilon, \quad \mbox{for } t\geq T+t_{0}, $$(3.1)
- (ii)converge in probability if, for every \(\varepsilon>0\), \(\mu>0\), and \(t_{0}\in J\), there exist \(\delta>0\) and a positive number \(T=T(t_{0},\varepsilon ,\delta,\mu)\) such thatwhenever \(\sup_{-\tau\leq s\leq0}\mathbb{E}\|\varphi(s,\omega)\| ^{p}\leq\delta\) and \(P\{\omega:\|\psi(t,\omega)\| \geq\varepsilon\}<\mu\), \(t\geq T+t_{0}\).$$ P\bigl\{ \omega: \bigl\Vert u(t,\omega) \bigr\Vert \geq \varepsilon\bigr\} < \mu,\quad t\geq T+t_{0}, $$(3.2)
Definition 3.2
Theorem 3.3
- (i)
all the hypotheses of Theorem 2.1 hold;
- (ii)
\(V(t,u)\) has the unique zero at \(u\equiv0\);
- (iii)for \((t,u)\in J_{\tau}\times\mathbb{R}^{n}\), \(V(t,u)\) satisfies the following inequality:where \(b\in\mathcal{VK}\), \(\mathcal{VK}\) is the collection of all continuous, convex, and increasing functions defined on \(\mathbb {R}^{+}\) into itself with \(b(0)=0\) and \(b(r)\rightarrow\infty\) as \(r\rightarrow\infty\);$$ b\bigl( \bigl\Vert u(t) \bigr\Vert ^{p}\bigr)\leq V \bigl(t,u(t,x)\bigr), $$(3.4)
- (iv)
the maximal solution \(r(t;t_{0},\xi(s))\) of (2.4) through \((t_{0},\xi(s))\) converges to zero as \(t\rightarrow \infty\).
Then the solution process of SPFDEs (2.1) converges in the pth moment to the zero vector.
Proof
From the conclusion of Theorem 2.1, we have \(\mathbb {E}V(t,u(t,x,\omega))< r(t;t_{0},\xi(s))\).
Now we claim that \(\sup_{-\tau\leq s\leq0}\mathbb{E}\|\varphi(s,\omega)\| ^{p}\leq\delta \) and \(\mathbb{E}\|\psi(t,\omega)\| ^{p}<\varepsilon\) for \(t\geq T+t_{0}\) imply \(\mathbb{E}(\|u(t,\omega)\| ^{p})<\varepsilon\), \(t\geq T+t_{0}\). Suppose it is not true, there exists a sequence \(\{t_{j}\}\) such that \(t_{j}\geq T+t_{0}\), \(t_{j}\rightarrow\infty\) as \(j\rightarrow\infty\), and \(\mathbb{E}(\|u(t_{j},\omega)\| ^{p})=\varepsilon\).
Theorem 3.4
- (i)
all the hypotheses of Theorem 2.1 hold;
- (ii)
\(V(t,u)\) has the unique zero at \(u\equiv0\);
- (iii)for \((t,u)\in J_{\tau}\times\mathbb{R}^{n}\), \(V(t,u)\) satisfies the following inequality:$$ b\bigl( \bigl\Vert u(t) \bigr\Vert \bigr)\leq V\bigl(t,u(t,x) \bigr); $$(3.9)
- (iv)
the maximal solution \(r(t;t_{0},\xi(s))\) of (2.4) through \((t_{0},\xi(s))\) converges to zero as \(t\rightarrow \infty\).
Then the solution process of SPFDEs (2.1) converges in probability to the zero vector.
Proof
Now we claim that \(\sup_{-\tau\leq s\leq0}E\|\varphi(s,\omega)\|<\delta \) and \(P\{\omega:\|\psi(t,\omega)\| \geq\varepsilon\}<\mu\) for \(t\geq T+t_{0}\) imply \(P\{\omega:\|u(t,\omega)\| \geq\varepsilon\}<\mu\), \(t\geq T+t_{0}\). Suppose it is not true, there exists a sequence \(\{t_{j}\}\) such that \(t_{j}\geq T+t_{0}\), \(t_{j}\rightarrow\infty\) as \(j\rightarrow\infty\), and \(P\{\omega: \|u(t_{j},\omega)\| \geq\varepsilon\}\geq\mu\).
Therefore our claim is true, this completes the proof of Theorem 3.4. □
4 Stability analysis
In this section, we consider the stability of the trivial solution process of SPFDEs (2.1) by means of the comparison theorem.
Definition 4.1
- (i)Stable in the pth moment if, for each \(\varepsilon >0\) and for all \(t\in J\), there exists a constant \(\delta=\delta(t_{0},\varepsilon)>0\) such thatwhenever \(\sup_{-\tau\leq s \leq0}\mathbb{E}\|\varphi(s,\omega)\| ^{p}\leq\delta \) and \(\mathbb{E}(\|\psi(t,\omega)\| ^{p})<\varepsilon\) for \(t\in J\).$$ \mathbb{E}\bigl( \bigl\Vert u(t,\omega) \bigr\Vert ^{p}\bigr)< \varepsilon,\quad t \in J, $$(4.1)
- (ii)
Asymptotically stable in the pth moment if it is stable in the pth moment, and if, for \(\varepsilon>0\), \(t_{0}\in J\), there exist positive numbers \(\delta_{0}=\delta_{0}(t_{0})\) and \(T=T(t_{0},\varepsilon)\) such that \(\sup_{-\tau\leq s\leq0}\mathbb{E}\| \varphi(s,\omega)\| ^{p}\leq\delta\) and \(\mathbb{E}(\|\psi(t,\omega)\| ^{p})<\varepsilon\) for \(t\geq T+t_{0}\) imply \(\mathbb{E}(\|u(t,\omega)\| ^{p})<\varepsilon\), \(t \geq T+t_{0}\).
- (iii)Stable in probability if, for each \(\varepsilon>0\), \(\mu>0\), \(t_{0}\in J\), there exists a positive constant \(\delta=\delta (t_{0},\varepsilon,\mu)\) such thatwhenever \(\sup_{-\tau\leq s \leq0}\mathbb{E}\|\varphi(s,\omega)\| ^{p}\leq\delta \) and \(P\{\omega:\|\psi(t,\omega)\| \geq\varepsilon\}<\mu\).$$ P\bigl\{ \omega: \bigl\Vert u(t,\omega) \bigr\Vert \geq\varepsilon\bigr\} < \mu,\quad t\in J, $$(4.2)
- (iv)
Asymptotically stable in probability if it is stable in probability and, for any \(\varepsilon>0\), \(\mu>0\), \(t_{0}\in J\), there exist numbers \(\delta_{0}=\delta_{0}(t_{0})\) and \(T=T(t_{0},\varepsilon ,\mu)>0\) such that \(\sup_{-\tau\leq s\leq0}\mathbb{E}\|\varphi(s,\omega )\| ^{p}\leq\delta\) and \(P\{\omega:\|\psi(t,\omega)\| \geq\varepsilon\} <\mu\), \(t\geq T+ t_{0}\) imply \(P\{\omega:\|u(t,\omega)\| \geq\varepsilon\}<\mu\), \(t\geq T+t_{0}\).
Definition 4.2
- (i)Stable if, for each \(\varepsilon>0\), \(t_{0}\in J \), there exists a positive constant \(\delta(t_{0},\varepsilon)\) such that when \(|\xi|_{0}\leq\delta\) it implieswhere \(y(t;t_{0},\xi(s))\) is the solution of Eq. (2.4) through \((t_{0},\xi(s))\).$$ y\bigl(t;t_{0},\xi(s)\bigr)< \varepsilon, \quad t\in J, $$(4.3)
- (ii)
Asymptotically stable if it is stable and converges to zero.
Theorem 4.3
Then the trivial solution of (2.4) is stable implies the trivial solution of SPFDEs (2.1) is stable in the pth moment; the trivial solution of (2.4) is asymptotically stable implies the trivial solution of SPFDEs (2.1) is asymptotically stable in the pth moment.
Proof
So, we have \(\mathbb{E}\|u(t,\omega)\| ^{p}<\varepsilon\), \(t\in J\).
Similar to the proof of Theorem 3.3, we can prove that the trivial solution of (2.4) is asymptotically stable implies the trivial solution of SPFDEs (2.1) is asymptotically stable in the pth moment. The details are omitted.
This completes the proof of Theorem 4.3. □
Theorem 4.4
Then the trivial solution of (2.4) is stable implies the trivial solution of SPFDEs (2.1) is stable in probability; the trivial solution of (2.4) is asymptotically stable implies the trivial solution of SPFDEs (2.1) is asymptotically stable in probability.
Proof
Therefore our claim is true. Similar to the proof of Theorem 3.4, we also prove that the trivial solution of (2.4) is asymptotically stable implies the trivial solution of SPFDEs (2.1) is asymptotically stable in probability.
This completes the proof of Theorem 4.4. □
5 Example
In this section we present two examples in order to illustrate our theory. Our aim is to examine the asymptotic stability of one-dimensional SPFDEs and two-dimensional SPFDEs.
Example 5.1
So all the hypotheses of Theorem 4.3 are satisfied. From [15], we know that the trivial solution of system (5.4) is asymptotically stable when \(\varsigma^{2}+2\gamma<4\pi^{2}\alpha\). By Theorem 4.3, the solution process of SPFDEs (5.1) is asymptotically stable in the mean square in \(L^{2}\) when \(\varsigma ^{2}+2\gamma<4\pi^{2}\alpha\).
Example 5.2
From [15], system (5.8) is asymptotically stable if \(\rho (t)+\theta(t)<\frac{2}{M}\).
It is obvious that all the hypotheses of Theorem 4.3 are satisfied. Therefore, the solution process of SPFDEs (5.5) is asymptotically stable in the mean square in \(L^{2}\) if \(\rho(t)+\theta(t)<\frac{2}{M}\).
6 Conclusion
In this paper, some sufficient conditions for the convergence and stability of the solution process of SPFDEs are established by developing the comparison principle in the context of Lyapunov-like function together with differential inequality. We believe that our approach and results are very efficient and comparable to the existing research in this area for stochastic parabolic partial differential equations [13]. We plan to extend this research result into stochastic parabolic functional differential equations with Markovian switching.
Declarations
Acknowledgements
The authors would like to thank the reviewers for their constructive suggestions and helpful comments.
Funding
The research was supported by the National Natural Science Foundation of China (grant number 11571245).
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interest regarding the publication of this paper. The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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