Abstract
The current article presents the study of neutral stochastic functional differential equations driven by G-Brownian motion in the phase space \(C_{q}((-\infty ,0];\mathbb{R}^{n})\). The mean-square boundedness of solutions has been derived. The convergence of solutions with different initial data has been investigated. The boundedness and convergence of solution maps have been obtained. In addition, the \(L^{2}_{G}\) and exponential estimates of solutions have been determined.
Similar content being viewed by others
1 Introduction
Several stochastic dynamical systems not only rely on current and past values but also include derivatives with delays. Neutral stochastic functional differential equations (NSFDEs) are employed to express such type of systems. These equations and their applications in aeroelasticity and chemical engineering were introduced by Kolmanovskii, Nosov and Myshkis [10, 11]. Thenceforth, the theory of NSFDEs has attracted the attention of many authors [15, 16, 29, 32]. The existence-uniqueness and stability of solutions for neutral stochastic functional differential equations driven by G-Brownian motion (G-NSFDEs) with Lipschitz and non-Lipschitz conditions was, respectively, studied by Faiz [4] and Faiz et al. [7]. The pth moment exponential stability for solutions to G-NSFDEs with Markovian switching [13] and the asymptotic stability of Euler–Maruyama numerical solutions for G-NSFDEs [14] was given by Li and Yang. The quasi sure exponential stability for solutions to the stated equations was established by Zhu et al. [33]. The mean-square stability of delayed stochastic neural networks driven by G-Brownian motion and stabilization of SDEs driven by G-Brownian motion can be found in [20, 28]. For the text on stochastic functional differential equations driven by G-Brownian motion we refer the reader to see [3, 5, 18]. The existence theory and estimates for the difference between exact and approximate solutions of stochastic differential equations driven by G-Brownian motion can be found in [2, 8, 9]. Also see [21,22,23,24,25,26]. Unlike to the above briefly discussed literature, this article presents the study of G-NSFDEs with some suitable monotone type conditions in the phase space \(C_{q}\) defined below. We investigate the boundedness and convergence of solutions. We derive the convergence of any two solution maps with distinct initial conditions. Furthermore, the \(L^{2}_{G}\) and exponential estimates for solutions to G-NSFGEs are determined. Let \(\mathbb{R}^{n}\) be an n-dimensional Euclidean space and \(C((- \infty ,0];\mathbb{R}^{n})\) be the collection of continuous functions from \((-\infty ,0]\) to \(\mathbb{R}^{n}\). For a given number \(q>0\) the phase space with the fading memory \(C_{q}((-\infty ,0];\mathbb{R}^{n})\) is defined by
The space \(C_{q}((-\infty ,0];\mathbb{R}^{n})\) endowed with the norm \(\|\psi \|_{q}=\sup_{-\infty <\alpha \leq 0}e^{q\alpha }|\psi (\alpha )|<\infty \) is a Banach space of continuous and bounded functions and for any \(0\leq q_{1}\leq q_{2}<\infty \), \(C_{q_{1}}\subseteq C_{q_{2}}\) [12, 31]. Let \(\mathcal{B}(C_{q})\) be the σ-algebra generated by \(C_{q}\) and \(C_{q}^{0}=\{\psi \in C_{q}: \lim_{\theta \rightarrow -\infty }e^{q\theta }\psi (\theta )=0\}\). Denote by \(C^{2}(C_{q})\) (resp. \(C^{2}(C^{0}_{q})\)) the space of all \(\mathcal{F}\)-measurable \(C_{q}\)-valued (resp. \(C^{0}_{q}\)-valued) stochastic processes ψ such that \(E\|\psi \|_{q}^{2}<\infty \). Let \((\varOmega ,\mathcal{F},\mathbb{P})\) be a complete probability space, \(B(t)\) be a n-dimensional G-Brownian motion and \(\mathcal{F}_{t}= \sigma \{B(s):0\leq s\leq t\}\) be the natural filtration. Let the filtration \(\{\mathcal{F};t\geq 0\}\) satisfies the usual conditions. Let \(\mathcal{P}\) be the collection of all probability measures on \((C_{q},\mathcal{B}(C_{q}))\) and \(C_{b}(C_{q})\) be the set of all bounded continuous functionals. Let \(N_{0}\) be the set of probability measures on \((-\infty ,0]\) such that, for any \(\mu \in N_{0}\), \(\int _{-\infty }^{0}\mu (d\theta )=1\). For any \(m>0\) we define \(N_{m}\) by
where for any \(k\in (0,k_{0})\), \(N_{k_{0}}\subset N_{k}\subset N_{0}\) [31]. We study the G-NSFDE with infinite delay
on \(t\geq 0\) with the given initial data \(z_{0}=\zeta \in C_{q}((- \infty ,0];\mathbb{R}^{{n}})\) and \(z_{t} = \{ z(t + \theta ) : -\infty < \theta \leq 0\}\). The remaining article is divided in three sections. The basic notions and definitions can be found in Sect. 2. Section 3 includes some useful lemmas, boundedness and convergence of solutions and solution maps. The \(L^{2}_{G}\) and exponential estimates are placed in the last Sect. 4.
2 Preliminaries
This section contains some basic notions and results required for our further study in the subsequent sections of this article [2, 6, 17,18,19, 27, 30]. Assume a sublinear expectation space \((S,\mathcal{W},\hat{\mathbb{E}})\) where \(\mathcal{W}\) is a space of real mappings defined on a given non-empty set S. Assume that S denotes the collection of all \(\mathbb{R}^{n}\)-valued continuous trajectories \((z(t))_{t\geq 0}\) with \(z(0)=0\) endowed with the distance
Let \(B(t)=B(t,z)=z(t)\) [18] for any \(z\in S\) and \(t\geq 0\), be the canonical process. For a selected \(T\in [0,\infty )\), we define
where \(C_{b.\mathrm{Lip}}(\mathbb{R}^{n\times m})\) is a space of bounded Lipschitz mappings. For \(t\leq T\), \(\mathcal{L}_{ip}(S_{t})\subseteq \mathcal{L}_{ip}(S_{T})\) and \(\mathcal{L}_{ip}(S)=\bigcup _{n=1}^{\infty }\mathcal{L}_{ip}(S_{n})\). Let \(\mathcal{L}^{p}_{G}(S)\) denote the completion of \(\mathcal{L}_{ip}(S)\) equipped with the Banach norm \(\hat{\mathbb{E}}[|\cdot|^{p}]^{\frac{1}{p}}\), \(p\geq 1\) and \(\mathcal{L} _{G}^{p}(S_{t})\subseteq \mathcal{L}_{G}^{p}(S_{T})\subseteq \mathcal{L}_{G}^{p}(S)\) for \(0\leq t\leq T <\infty \). Let \(\mathcal{F}_{t}=\sigma \{B(v), 0\leq v \leq t\}\) indicate the filtration produced by the stated canonical process and \(\mathcal{F}=\{ \mathcal{F}_{t}\}_{t\geq 0}\). Let \(\pi _{T}=\{t_{0},t_{1},\ldots,t_{ \mathbb{Z}^{+}}\}\), \(0\leq t_{0}\leq t_{1}\leq \cdots\leq t_{\mathbb{Z} ^{+}}\leq \infty \) be a partition of \([0,T]\). For every \({\mathbb{Z} ^{+}}\geq 1\), \(0=t_{0}< t_{1}<\cdots<t_{\mathbb{Z}^{+}}=T\) and \(i=0,1,\ldots, {\mathbb{Z}^{+}}-1\), define the space \(\mathcal{M}^{p,0}_{G}([0,T])\), \(p\geq 1\) of simple processes as
The space (2.1) is complete with the norm \(\|\eta \|= \{\int _{0}^{T}\hat{\mathbb{E}}[|\eta (s)|^{p}]\,ds \}^{1/p}\) and is indicated by \(\mathcal{M}_{G}^{p}(0,T)\), \(p\geq 1\).
Definition 2.1
Let \(\eta _{t}\in \mathcal{M}_{G}^{2,0}(0,T)\). Then the G-Itô integral, say \(J(\eta )\), is given by
We can continuously extend the mapping \(J:\mathcal{M}^{2,0}_{G}(0,T) \mapsto \mathcal{L}^{2}_{G}(\mathcal{F}_{T})\) to \(J:\mathcal{M}^{2} _{G}(0,T)\mapsto \mathcal{L}^{2}_{G}(\mathcal{F}_{T})\). For \(\eta \in \mathcal{M}^{2}_{G}(0,T)\) we can still give the G-Itô integral as
Definition 2.2
The quadratic variation process \(\{\langle B^{\theta }\rangle (t)\} _{t\geq 0}\) of G-Brownian motion is given by
The stated process is increasing, \(\langle B^{\theta }\rangle (0)=0\) and for any \(0\leq s\leq t\), \(\langle B^{\theta }\rangle (t)-\langle B ^{\theta }\rangle (s)\leq \sigma _{\theta \theta ^{\tau }} (t-s)\).
Assume that \(\theta , \hat{\theta } \in \mathbb{R}^{n}\) be given vectors. The mutual variation process of \(B^{\hat{\theta }}\) and \(B^{\theta }\) is given by \(\langle B^{\theta },B^{\hat{\theta }} \rangle =\frac{1}{4}[\langle B^{\theta }+B^{\hat{\theta }}\rangle (t)- \langle B^{\theta }-B^{\hat{\theta }}\rangle (t)]\). A mapping \(\mathcal{W}_{0,T}:\mathcal{M}^{0,1}_{G}(0,T)\mapsto \mathcal{L}^{2} _{G}(\mathcal{F}_{T})\) is defined by
We can continuously extend it to \(\mathcal{M}^{1}_{G}(0,T)\) and for \(\eta \in \mathcal{M}^{1}_{G}(0,T)\) this is still given by
The concept of G-capacity and Lemma 2.3 can be found in [1]. Let \(\mathcal{B}(S)\) be a Borel σ-algebra of S. Let \(\mathcal{Q}\) represent the group of all probability measures on \((S, \mathcal{B}(S)\). The G-capacity ν̂ is given as follows:
where set \(C\in \mathcal{B}(S)\). If \(\hat{\nu }(C)=0\) then the set \(C\in \mathcal{B}(S)\) is called polar. A characteristic holds quasi-surely when it holds external of the set C.
Lemma 2.3
Let \(z\in {{\mathcal{L}_{G}^{p}}}\) and \(\hat{\mathbb{E}}|z|^{p}< \infty \). Then, for each \(c>0\), the G-Markov inequality is given by
For the proof of Lemmas 2.4 and 2.5 see [9].
Lemma 2.4
Let \(\theta \in \mathbb{R}^{n}\), \(\eta \in \mathcal{M}_{G}^{2}(0,T)\), \(p\geq 2\) and \(z(t)=\int _{0}^{t} \eta (s)\,dB^{\theta }(s)\). Then on some \(\bar{S}\subset S\) with \(\nu (\bar{S}^{c})=0\) and \(\forall t\in [0,T]\), \(\hat{\nu }(|z(t)-\bar{z}|\neq 0)=0\) so that
where \(0<\hat{K}<\infty \) is a positive constant and \(\bar{z}(t)\) is a modification of \(z(t)\).
Lemma 2.5
Let \(\theta ,\hat{\theta }\in \mathbb{R}^{n}\), \(p\geq 1\) and \(\eta \in \mathcal{M}_{G}^{p}(0,T)\). A continuous modification \(\bar{z}^{\theta ,\hat{\theta }}(t)\) of \(z^{\theta ,\hat{\theta }}(t)= \int _{0}^{t} \eta (s)\,d \langle B^{\theta }, B^{\hat{\theta }} \rangle (s)\) exists and for \(0\leq s\leq v\leq t\leq T\),
Lemma 2.6
Let \(\lambda < 2q\) and \(\mu _{i}\in N_{m}\), for any \(i\in \mathbb{Z} ^{+}\). Then, for any \(\zeta \in C_{q}((-\infty ,0];\mathbb{R}^{n})\),
Proof
Let \(\zeta \in C_{q}((-\infty ,0];\mathbb{R}^{n})\) and \(\mu _{i}\in N _{2q}\) for any \(i\in \mathbb{Z}^{+}\). By using the definition of norm and the Fubini theorem, we derive
by noticing that \(\int _{-\infty }^{0}\mu _{i}(d\theta )=1\) and \(\int _{-\infty }^{0}e^{-2q\theta }\mu _{i}(d\theta )=\mu _{i}^{(2q)}\), \(i\in \mathbb{Z}^{+}\), we derive
The proof of (2.2) is complete. The assertion (2.3) can be proved in a similar fashion as above. □
The book [16] is a good reference for the following three lemmas.
Lemma 2.7
Let \(a,b\geq 0\) and \(\epsilon \in (0,1)\). Then
Lemma 2.8
Assume \(p\geq 2\) and \(\hat{\epsilon },a,b>0\). Then the following two inequalities hold:
-
(i)
\(a^{p-1}b\leq \frac{(p-1)\hat{\epsilon }a^{p}}{p}+ \frac{b^{p}}{p\hat{\epsilon }^{p-1}}\).
-
(ii)
\(a^{p-2}b^{2}\leq \frac{(p-2)\hat{\epsilon }a ^{p}}{p}+\frac{2b^{p}}{p\hat{\epsilon }^{\frac{p-2}{2}}}\).
Lemma 2.9
Let \(a_{1},a_{2}\in \mathbb{R}\) and \(\delta \in (0,1)\). Then for any \(p>1\)
3 Boundedness and convergence of solutions
Consider that problem (1.1) has a solution \(z(t)\). All through this article we take \(\lambda < pq\) for any \(p\geq 1\). We assume the following two hypotheses:
- \((\mathrm{A}_{1})\) :
-
Let \(y,z\in C_{q}((-\infty ,0];\mathbb{R} ^{n})\) and \(\mu _{1}, \mu _{2},\mu _{3}\in N_{2q}\). Then there are positive constants \(\lambda _{i}\), \(i=1,2,\ldots,5\) so that
$$\begin{aligned}& \bigl[z(0)-y(0)-\bigl(u(z)-u(y)\bigr)\bigr]^{T} \bigl[g(z)-g(y)\bigr]\\& \quad \leq -\lambda _{1} \bigl\vert z(0)-y(0) \bigr\vert ^{2}+ \lambda _{2} \int _{-\infty }^{0} \bigl\vert z(\theta )-y(\theta ) \bigr\vert ^{2}\mu _{1}(d \theta ), \\& \bigl[z(0)-y(0)-\bigl(u(z)-u(y)\bigr)\bigr]^{T} \bigl[h(z)-h(y)\bigr]\\& \quad \leq -\lambda _{3} \bigl\vert z(0)-y(0) \bigr\vert ^{2}+ \lambda _{4} \int _{-\infty }^{0} \bigl\vert z(\theta )-y(\theta ) \bigr\vert ^{2}\mu _{2}(d \theta ), \end{aligned}$$and
$$ \bigl\vert \gamma (z)-\gamma (y) \bigr\vert ^{2} \leq \lambda _{5} \int _{-\infty }^{0} \bigl\vert z( \theta )-y(\theta ) \bigr\vert ^{2}\mu _{3}(d\theta ). $$ - \((\mathrm{A}_{2})\) :
-
Let \(z\in C_{q}((-\infty ,0];\mathbb{R}^{n})\) and \(\mu _{4}\in N_{2q}\) with \(\mu ^{(2q)}_{4}<1\). Then there is a constant \(0<\hat{b}<1\) so that
$$ \bigl\vert u(z) \bigr\vert ^{2}\leq \hat{b} \int _{-\infty }^{0} \bigl\vert z(\theta ) \bigr\vert ^{2}\mu _{4}(d \theta ). $$(3.1)
Let \(p>1\), \(z\in C_{q}((-\infty ,0];\mathbb{R}^{n})\) and \(\mu _{4} \in N_{pq}\) with \(\mu ^{(pq)}_{4}<1\). Then there is a constant \(0< b<1\) so that
Obviously, if \(p=2\) and letting \(b^{2}=\hat{b}\) then (3.2) is the same as (3.1). Firstly, we give the prove of some important lemmas.
Lemma 3.1
Let \(p> 1\), \(q>0\) and \(\zeta \in C_{q}((-\infty ,0];\mathbb{R}^{n})\). Let condition 3.2 hold. Then
where \(0< b<1\).
Proof
By using Lemma 2.9 and condition (3.2) for any \(\delta >0\), it follows that
Taking \(\delta =(\frac{b}{1-b})^{p-1}\) and using the definition of norm, we obtain
simplification yields the desired assertion. The proof is complete. □
Lemma 3.2
Let \(p> 1\), \(q>0\) and \(\zeta \in C_{q}((-\infty ,0];\mathbb{R}^{n})\). Let condition 3.2 hold. Then there exists a constant \(0< b<1\) such that
Proof
In view of Lemma 2.9 and condition (3.2), we obtain
Observing \(|\zeta (0)|^{p}\leq \sup_{-\infty < \alpha \leq 0}e^{pq \theta }|\zeta (\theta )|^{p}=\|\zeta \|_{q}^{p}\) and substituting \(\delta =b^{p-1}\) we have
The proof is completed. □
Lemma 3.3
Let \(p> 1\), \(q>0\) and \(\zeta \in C_{q}((-\infty ,0];\mathbb{R}^{n})\). Let condition 3.2 hold. Then
where \(0< b<1\).
We omit the proof. It can be proved in a similar procedure to the above last lemma. Now let us see one of the main results.
Theorem 3.4
Under assumptions \(\mathrm{A}_{1}\) and \(\mathrm{A}_{2}\), if for any \(\zeta \in C_{q}\), \(\lambda _{i}\), \(i=1,2,\ldots,5\) satisfy \(\lambda _{1}>\mu _{1}^{(2q)}\lambda _{2}-k_{2}\lambda _{3}+k_{2}\mu _{2}^{(2q)}\lambda _{4} +(2k^{2}_{1}+k _{2})\mu _{3}^{(2q)}\lambda _{5}\) then there exists \(\lambda \in (0,\frac{1}{(1+b _{1}\mu _{4}^{(2q)})}(\lambda _{1}+k_{2}\lambda _{3}- \mu _{1}^{(2q)} \lambda _{2}-k_{2}\mu _{2}^{(2q)}\lambda _{4} -(2k_{1}^{2}+k_{2})\mu _{3} ^{(2q)}\lambda _{5})\wedge 2q)\) such that
where \(C=b_{3}c_{1}\), \(K=(b_{2}+b_{3}c_{2})\hat{\mathbb{E}}\|\zeta \| ^{2}_{q}\), \(b_{1}=1+b^{-1}\), \(b_{2}=b\mu _{4}^{(2q)}(1-b)^{-1}\), \(b_{3}=(1-b)^{-2}\), \(c_{1}=\frac{1}{\lambda }[\frac{1}{\epsilon _{1}}|g(0)|^{2}+k _{2}\frac{1}{\epsilon _{1}}|h(0)|^{2}+(k_{2}+k_{3})\frac{1}{\epsilon _{3}}|\gamma (0)|^{2}]\), \(c_{2}=\frac{2}{2q-\lambda }(4(2q-\lambda )+2 \lambda _{2}\mu _{1}^{(2q)}+2k_{2}\lambda _{4}\mu _{2}^{(2q)}+ (2k_{1} ^{2}+k_{2})\mu _{3}^{(2q)}+b_{1}(\lambda +\epsilon _{1}+\epsilon _{1}k _{2})\mu _{4}^{(2q)})\), \(k_{1}\), \(k_{2}\) are positive constants and \(\epsilon _{1}\), \(\epsilon _{2}\) are sufficiently small constants such that
Proof
By virtue of the G-Itô formula, for any \(t\in [0, T]\), it follows that
Applying the G-expectation on both sides, utilizing Lemma 2.5, Lemma 2.4 and Lemma 3.2, there exist \(k_{1}>0\) and \(k_{2}>0\) so that
By using assumption \(A_{1}\) and Lemma 2.8 we derive
similar arguments give
In view of assumption \(A_{1}\) and Lemma 2.7 we derive
By substituting (3.4), (3.5) and (3.6) in (3.3) and using Lemma 3.3 we get
where \(c_{1}=\frac{2}{\lambda }[\frac{1}{\epsilon _{1}}|g(0)|^{2}+k _{2}\frac{1}{\epsilon _{1}}|h(0)|^{2}+(k_{2}+k_{3})\frac{1}{\epsilon _{2}}|\gamma (0)|^{2}]\) and \(b_{1}=1+b^{-1}\). By virtue of Lemma 2.6, it follows that
where \(c_{2}=\frac{2}{2q-\lambda }(4(2q-\lambda )+2\lambda _{2}\mu _{1} ^{(2q)}+2k_{2}\lambda _{4}\mu _{2}^{(2q)}+ (2k_{1}^{2}+k_{2})\mu _{3} ^{(2q)}+b_{1}(\lambda +\epsilon _{1}+\epsilon _{1}k_{2})\mu _{4}^{(2q)})\). Next, we use Lemma 3.1 to derive
where \(b_{2}=b\mu _{4}^{(2q)}(1-b)^{-1}\), \(b_{3}=(1-b)^{-2}\) and \(e^{-(2q-\lambda )s}<1\). From the assumptions, we notice that \(\lambda _{1}>\mu _{1}^{(2q)}\lambda _{2}-k_{2}\lambda _{3}+k_{2}\lambda _{4}\mu _{2}^{(2q)} +(2k_{1}^{2}+k_{2})\mu _{3}^{(2q)}\lambda _{5}\) and \(\lambda \in (0,\frac{1}{(1+b_{1}\mu _{4}^{(2q)})}(\lambda _{1}+k_{2} \lambda _{3}- \mu _{1}^{(2q)}\lambda _{2}-k_{2}\mu _{2}^{(2q)}\lambda _{4} -(2k_{1}^{2}+k_{2})\mu _{3}^{(2q)}\lambda _{5})\wedge 2q)\). Choosing \(\epsilon _{1}\) and \(\epsilon _{2}\) sufficiently small such that
we get the desired result. The proof is completed. □
Next, let us see the convergence of any two solutions of G-NSFDEs with different initial data.
Theorem 3.5
Let the assumptions of Theorem 3.4 hold. Let \(y(t)\) and \(z(t)\) be any two solutions of problem (1.1) with the respective initial conditions ζ and ξ. Then
where \(L=b_{2}+b_{3}c_{3}\), \(b_{2}=b\mu _{4}^{(2q)}(1-b)^{-1}\), \(b_{3}=(1-b)^{-2}\), \(c_{3}= \frac{2}{2q-\lambda }[{{4(2q- \lambda )}}+2\lambda _{2}\mu _{1}^{(2q)} +2k_{2}\lambda _{4}\mu _{2}^{(2q)}+(k _{2}+2k_{1}^{2})\lambda _{5}\mu _{3}^{(2q)}+\lambda b_{1}\mu _{4}^{(2q)}]\), \(b_{1}=1+b^{-1}\), \(k_{1}\) and \(k_{2}\) are positive constants.
Proof
Define \(\varLambda (t)=z(t)-y(t)\), \(\hat{u}(t)=u(z_{t})-u(y_{t})\), \(\hat{g}(t)=g(z_{t})-g(y_{t})\), \(\hat{h}(t)=h(z_{t})-h(y_{t})\), \(\hat{\gamma }(t)=\gamma (z_{t})-\gamma (y_{t})\). By the G-Itó formula and similar arguments to Theorem 3.4 it follows that
By virtue of assumptions \(A_{1}\), we have
Substituting this in (3.8) and using Lemma 3.3 we obtain
where \(b_{1}=1+b^{-1}\). By using Lemma 2.6, it follows that
where \(c_{3}=\frac{2}{2q-\lambda }(4(2q-\lambda )+2\lambda _{2}\mu _{1} ^{(2q)} +2k_{2}\lambda _{4}\mu _{2}^{(2q)}+(k_{2}+2k^{2}_{1})\lambda _{5}\mu _{3}^{(2q)}+\lambda b_{1}\mu _{4}^{(2q)})\). By using Lemma 3.1, we have
where \(b_{2}=b\mu _{4}^{(2q)}(1-b)^{-1}\), \(b_{3}=(1-b)^{-2}\) and \(e^{-(2q-\lambda )s}<1\). By using the assumptions \(\lambda _{1}>\mu _{1}^{(2q)}\lambda _{2}-k_{2}\lambda _{3}+k_{2}\mu _{2}^{(2q)}\lambda _{4} +(2k^{2}_{1}+k_{2})\mu _{3}^{(2q)}\lambda _{5}\) and \(\lambda \in (0,\frac{1}{(1+b_{1}\mu _{4}^{(2q)})}(\lambda _{1}+k_{2}\lambda _{3}- \mu _{1}^{(2q)}\lambda _{2}-k_{2}\mu _{2}^{(2q)}\lambda _{4} -(2k_{1}^{2}+k _{2})\mu _{3}^{(2q)}\lambda _{5})\wedge 2q)\), we derive the desired assertion. The proof is completed. □
The following two theorems show that the solutions maps of G-NSFDEs are bounded and any two solutions maps with different initial data are convergent, respectively.
Theorem 3.6
Let all the assumptions of Theorem 3.5 hold. Then for any initial data \(\zeta \in C_{q}\)
where \(C_{1}=b_{3}c_{1}\), \(K_{1}=(1+b_{2}+b_{3}c_{2})E\|\zeta \|^{2} _{q}\) and \(b_{2}\), \(b_{3}\), \(c_{1}\), \(c_{2}\) are defined in Theorem 3.4.
Proof
By virtue of the definition of norm \(\|\cdot\|\) and observing that \(2q>\lambda \) we have
consequently,
But from (3.7) we have
using the assumptions of Theorem 3.5, it gives
By substituting (3.11) in (3.10), we derive
which yields the desired assertion. The proof is completed. □
Theorem 3.7
Under the assumptions of Theorem 3.5, different solution maps \(z_{t}\) and \(y_{t}\) of problem (1.1) with respective different initial data ζ and ξ converge, i.e.,
where \(L_{1}=(1+b_{2}+b_{3}c_{3})\) and \(b_{2}\), \(b_{3}\), \(c_{3}\) are defined in Theorem 3.5.
Proof
Using similar arguments of Theorem 3.6 we derive
But from (3.9), using the assertion of Theorem 3.5, we derive
which on substituting in (3.12) yields the desired assertion. The proof is completed. □
4 Exponential estimate
For the purpose of exponential estimate one needs to assume that problem (1.1) admits a unique solution \(z(t)\) on \(t\geq 0\). In the following theorem, we give the \(L^{2}_{G}\) and exponential estimates for the solutions of neutral stochastic functional differential equation driven by G-Brownian motion.
Theorem 4.1
Let problem (1.1) has a unique solution \(z(t)\) on \(t\geq 0\) and \(\hat{\mathbb{E}}\|\zeta \|_{q}^{2}<\infty \). Assume that \(A_{1}\) and \(A_{2}\) are satisfied. Then the following results hold:
where \(C_{2}=2b_{3}[4q+\mu _{1}^{2q}+k_{2}\lambda _{4}\mu _{2}^{2q}+ \lambda _{5}(k_{2}+2k_{1}^{2})\mu _{3}^{2q}+(1+k_{2})(1+b^{-1})\mu _{4} ^{2q}]\frac{1}{q}\hat{\mathbb{E}}\|\zeta \|_{q}^{2}+(1+b_{2}) \hat{\mathbb{E}}\|\zeta \|_{q}^{2}+b_{3}c^{*}\), \(K_{2}=4b_{3}(-\lambda _{1}-k_{2}\lambda _{3}+\lambda _{2}+\lambda _{4}k_{2}+\lambda _{5}(k_{2}+2k _{1}^{2})+(2+k_{2})(1+b^{-1}))\), \(b_{2}=b \mu ^{(2q)}_{4}(1-b)^{-1}\) and \(b_{3}=(1-b)^{-2}\) are positive constants. Furthermore,
where \(M=2b_{3}(-\lambda _{1}-k_{2}\lambda _{3}+\lambda _{2}+\lambda _{4}k _{2}+\lambda _{5}(k_{2}+2k_{1}^{2})+(2+k_{2})(1+b^{-1}))\).
Proof
To prove (4.1) in a similar fashion to Theorem 3.5 we derive
which yields
By using assumption \(A_{1}\) and the basic inequality \({ {2a_{1}a_{2}\leq \sum_{i=1}^{2}a_{i}^{2}}}\) we derive
Similar arguments to above give the following:
In view of assumption \(A_{1}\) and the basic inequality \(|\sum_{i=1} ^{2}a_{i}|^{2}\leq \sum_{i=1}^{2}|a_{i}|^{2}\), we obtain
Substituting (4.4), (4.5) and (4.6) in (4.3) and using Lemma 3.3 we have
where \(c^{*}=2(|g(0)|^{2}+k_{2}|h(0)|^{2}+2(k_{2}+2k_{1}^{2})|\gamma (0)|^{2})T\). By using Lemma 2.6, it follows that
By using Lemma 3.1, we derive
where \(b_{2}=b \mu ^{(2q)}_{4}(1-b)^{-1}\), \(b_{3}=(1-b)^{-2}\) and \(e^{-2qs}<1\). Noticing that \(\hat{\mathbb{E}}[\sup_{-\infty < s\leq t}|z(s)|^{2}] \leq \hat{\mathbb{E}}\|\zeta \|_{q}^{2}+\hat{\mathbb{E}}[\sup_{0< s \leq t}|z(s)|^{2}]\) and using the Gronwall inequality, we get the desired assertion. The proof is complete.
To prove (4.2) by the Gronwall inequality from (4.7) we have
where \(\alpha =2b_{3}[4q+\mu _{1}^{2q}+k_{2}\lambda _{4}\mu _{2}^{2q}+ \lambda _{5}(k_{2}+2k_{1}^{2})\mu _{3}^{2q}+(1+k_{2})(1+b^{-1})\mu _{4} ^{2q}]\frac{1}{q}E\|\zeta \|_{q}^{2}+b_{2}\hat{\mathbb{E}}\|\zeta \| _{q}^{2}+b_{3}c^{*}\) and \(\beta =4b_{3}(-\lambda _{1}-k_{2}\lambda _{3}+ \lambda _{2}+\lambda _{4}k_{2}+\lambda _{5}(k_{2}+2k_{1}^{2})+(2+k_{2})(1+b ^{-1}))\). By virtue of the above result (4.8), for each \(k=1,2,3,\ldots\) , we have
For any \(\epsilon >0\), by using Lemma 2.3 we get
But, for almost all \(\delta \in \varOmega \), the Borel–Cantelli lemma shows that there is \(k_{0}=k_{0}(\delta )\) so that
and consequently we derive the desired assertion. The proof is completed. □
References
Denis, L., Hu, M., Peng, S.: Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths. Potential Anal. 34, 139 (2010)
Faizullah, F.: Existence of solutions for stochastic differential equations under G-Brownian motion with discontinuous coefficients. Z. Naturforsch. A 67a, 692–698 (2012)
Faizullah, F.: On the pth moment estimates of solutions to stochastic functional differential equations in the G-framework. SpringerPlus 5(872), 1–11 (2016)
Faizullah, F.: Existence results and moment estimates for NSFDEs driven by G-Brownian motion. J. Comput. Theor. Nanosci. 7(13), 4679–4686 (2016)
Faizullah, F.: Existence and uniqueness of solutions to SFDEs driven by G-Brownian motion with non-Lipschitz conditions. J. Comput. Anal. Appl. 2(23), 344–354 (2017)
Faizullah, F.: A note on p-th moment estimates for stochastic functional differential equations in the framework of G-Brownian motion. Iran. J. Sci. Technol., Trans. A, Sci. 41, 1131–1138 (2017)
Faizullah, F., Bux, M., Rana, M.A., Rahman, G.: Existence and stability of solutions to non-linear neutral stochastic functional differential equations in the framework of G-Brownian motion. Adv. Differ. Equ. 2017, 350 (2017)
Faizullah, F., Khan, I., Salah, M.M., Alhussain, Z.A.: Estimates for the difference between approximate and exact solutions to stochastic differential equations in the G-framework. J. Taibah Univ. Sci. 13(1), 20–26 (2018)
Gao, F.: Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion. Stoch. Process. Appl. 2, 3356–3382 (2009)
Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer Academic Publishers, Norwell (1992)
Kolmanovskii, V., Nosov, V.: Stability of Functional Differential Equations. Academic Press, New York (1986)
Kuang, Y., Smith, H.L.: Global stability for infinite delay Lotka–Volterra type system. J. Differ. Equ. 103, 221–246 (1993)
Li, Q., Yang, Q.: Stability of neutral stochastic functional differential equations with Markovian switching driven by G-Brownian motion. Appl. Anal. 15(97), 2555–2572 (2018)
Li, Q., Yang, Q.: Convergence and asymptotical stability of numerical solutions for neutral stochastic delay differential equations driven by G-Brownian motion. Comput. Appl. Math. 4(37), 4301–4320 (2018)
Liu, K., Xia, X.: On the exponential stability in mean square of neutral stochastic functional differential equations. Syst. Control Lett. 37, 207–215 (1999)
Mao, X.: Stochastic Differential Equations and Their Applications. Horwood Publishing Chichester, Coll House, England (1997)
Peng, S.: Multi-dimentional G-Brownian motion and related stochastic calculus under G-expectation. Stoch. Process. Appl. 12, 2223 (2008)
Ren, Y., Bi, Q., Sakthivel, R.: Stochastic functional differential equations with infinite delay driven by G-Brownian motion. Math. Methods Appl. Sci. 36(13), 1746 (2013)
Ren, Y., Gu, Y., Zhou, Q.: Stability analysis of impulsive stochastic Cohen–Grossberg neural networks driven by G-Brownian motion. Int. J. Control 91, 1745–1756 (2018)
Ren, Y., He, Q., Gu, Y., Sakthivel, R.: Mean-square stability of delayed stochastic neural networks with impulsive effects driven by G-Brownian motion. Stat. Probab. Lett. 143, 56–66 (2018)
Ren, Y., Jia, X., Hu, L.: Exponential stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion. Discrete Contin. Dyn. Syst., Ser. B 20, 2157–2169 (2015)
Ren, Y., Jia, X., Sakthivel, R.: The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion. Appl. Anal. 96, 988–1003 (2017)
Ren, Y., Wang, K., Yang, H.: Stability analysis of stochastic pantograph multi-group models with dispersal driven by G-Brownian motion. Appl. Math. Comput. 355, 356–365 (2019)
Ren, Y., Wangas, J., Hu, L.: Multi-valued stochastic differential equations driven by G-Brownian motion and related stochastic control problems. Int. J. Control 90, 1132–1154 (2017)
Ren, Y., Yin, W., Lu, W.: Asymptotical boundedness for stochastic coupled systems o n networks driven by GBrownian motion. J. Math. Anal. Appl. 466, 338–350 (2018)
Ren, Y., Yin, W., Sakthivel, R.: Stabilization of stochastic differential equations driven by G-Brownian motion with feedback control based on discrete-time state observation. Automatica 95, 146–151 (2018)
Ren, Y., Yin, W., Zhu, D.: Exponential stability of SDEs driven by G-Brownian motion with delayed impulsive effects: average impulsive interval approach. Discrete Contin. Dyn. Syst., Ser. B 23, 3347–3360 (2018)
Ren, Y., Yin, W., Zhu, D.: Stabilization of SDEs and applications to synchronization of stochastic neural network driven by G-Brownian motion with state feedback control. Int. J. Syst. Sci. 50, 273–282 (2019)
Shen, Y., Liao, X.: Razumikhin theorems for exponential stability of neutral stochastic functional differential equations. Chin. Sci. Bull. 43, 2225–2228 (1998)
Ullah, R., Faizullah, F.: On existence and approximate solutions for stochastic differential equations in the framework of G-Brownian motion. Eur. Phys. J. Plus 132, 435–443 (2017)
Wu, F., Yin, G., Mei, H.: Stochastic functional differential equations with infinite delay: existence and uniqueness of solutions, solutionmaps, Markov properties, and ergodicity. J. Differ. Equ. 262, 1226–1252 (2017)
Zhou, S., Wang, Z., Feng, D.: Stochastic functional differential equations with infinite delay. J. Math. Anal. Appl. 357, 416–426 (2009)
Zhu, M., Li, J., Zhu, Y.: Exponential stability of neutral stochastic functional differential equations driven by G-Brownian motion. J. Nonlinear Sci. Appl. 4(10), 1830–1841 (2017)
Acknowledgements
We are very thankful to the reviewers for their several useful suggestions and comments which have improved the quality of this paper. We admire the financial support of NUST Pakistan.
Funding
This research work is sponsored by the Commonwealth Scholarship Commission in the United Kingdom with project ID: PKRF-2017-429.
Author information
Authors and Affiliations
Contributions
Under some useful conditions, we have proved the boundedness of solutions to neutral stochastic functional differential equations driven by G-Brownian motion in the phase space \(C_{q}((-\infty ,0]; \mathbb{R}^{n})\). The convergence of any two solutions with different initial data has been explored. We have also obtained the boundedness and convergent of solution maps in the mentioned space. Moreover, we have determined the \(L^{2}_{G}\) and exponential estimates for solutions to G-NSFDEs. Furthermore, our work can be further generalized to neutral stochastic functional differential equations with Markovian switching and NSFDES with Levy process, which could be a new direction of our further work. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Faizullah, F. On boundedness and convergence of solutions for neutral stochastic functional differential equations driven by G-Brownian motion. Adv Differ Equ 2019, 289 (2019). https://doi.org/10.1186/s13662-019-2218-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-019-2218-x