Abstract
In the present paper, we study stochastic stability and stochastic boundedness for the stochastic differential equation (SDE) with multi-delay of third order. The derived results extend and improve some earlier results in the relevant literature, which are related to the qualitative properties of solutions to third-order delay differential equations (DDEs) and SDEs with multi-delay. Two examples are given to illustrate the results.
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1 Introduction
During the past several years, the DDEs and the differential equations (DEs) with multiple delays have received more attention because of their widely applied backgrounds, such as population ecology, heat exchanges, mechanics, and economics. Here, we can mention the books by Burton [15], Hale [17], Hale and Verduyn Lunel [18], Iannelli [19] and numerous researchers activities such that, Abdel-Razek et al. [1], Abou-El-Ela et al. [2], Ademola and Arawomo [10], Ademola et al [11, 13], Mahmoud and Bakhit [24] Omeike [35, 36] Remili and Beldjerd [37], Remili and Oudjedi [38–40], Remili [41], Tunç [43–48], and the references therein.
Moreover, another kind of the DEs is the stochastic delay differential equations (SDDEs), where relevant parameters are modeled as suitable stochastic processes; see the book by Gikhman and Skorokhod [16]. The SDDE is a DE whose coefficients are random numbers or random functions of the independent variable (or variables). It is the appropriate tool for describing systems with external noise. The models of SDDEs play an important role in a range of application areas, including biology, chemistry, epidemiology, mechanics, microelectronics, economics, and finance. For example, in biology, we see that recently, Fathalla A. Rihan [42] studied the SDDEs for the spread of Coronavirus Infection COVID-19.
Furthermore, SDDEs are crucial in ecology, epidemiology, and many other fields; see, for example, Arnold [14], Mao [29–33], Øksendal [34], and references therein.
In the last few decades, the theory of SDDEs has attracted much attention, and numerous papers have been published. Here, we can mention the works by Abou-El-Ela et al. [3–7], Ademola [8], Ademola et al. [9, 12], Liu [21], Liu and Raffoul [22], Luo [23], Mahmoud and Tunç [26–28], Tunç [49], Zhi and Liping [20], and the references therein. Recently, Mahmoud and Bakhit [25] established the properties of solutions for nonautonomous third-order stochastic differential equation with a constant delay
The main purpose of this note is to establish new criteria for the uniformly stochastic asymptotical stability (USAS) and uniformly stochastic boundedness (USB) for solutions of the following more general third-order SDE with multi-delay as the form
where \(r_{i}(t)\) is continuously differentiable functions with \(0 \leq r_{i}(t) \leq \gamma _{i}\), (\(i=1,2,\ldots,n\)), \(\gamma _{i} > 0\) are constants, \(\psi _{1}\), \(Q_{i}\), \(f_{i}\) and ε are continuous functions in their respective arguments, with \(Q_{i}(x,0)=Q(0,y)=0\) and \(f_{i}(0)=0\). In addition, \(l(t) \) is a continuous function and defined from \([0,\infty )\) to \([0,l_{1}]\). \(w(t) \in \mathbb{R}^{n}\) is a standard Brownian motion.
Consider the following notations
Therefore, equivalent system of (1.1) can be written as
Remarks
-
(1)
Whenever \(\alpha x(t-l(t))w'(t)=0\), and we consider the case that \(i=1\), then equation (1.1) reduces to a DDE of third order discussed in [39].
-
(2)
Suppose that \(\alpha =0\), \(h(x'(t))=g(x''(t))\), \(\psi _{1}(x,x') =h(x'(t))\), and with \(i=1\) if we let \(Q(x(t-r(t)),x'(t-r(t)))=(\varphi (x(t))x(t))'\), then (1.1) can be reduced to the equation studied in [41].
-
(3)
In the case \(i=1\), \(\alpha =0\) and if \(h(x'(t))=1\), \((\psi _{1}(x,x') x')'= f(x,x')x''\), then equation (1.1) specialises to that considered in [2]. Our results generalize all the previous results.
-
(4)
Whenever, \(h(x'(t))=1\), \((\psi _{1}(x,x')x')'= a(t)f(x(t),x'(t))x''(t)\), and when \(i=1\), \(Q_{i}(x(t-r_{i}(t)),x'(t-r_{i}(t)))= b(t)\phi (x(t))x'(t) \), \(f_{i}(x(t-r_{i}(t)))=c(t)\psi (x(t-r))\), and \(\alpha x(t-l(t))=g(t,x) \), then (1.1) reduces to the studied equation in [25]. Thus, equation (1.1) generalizes the results obtained in [25]. Hence, our results include and extend all the previous results.
2 Stability results
Let \(B(t) = (B_{1}(t),\ldots ,B_{m}(t))\) be an m-dimensional Brownian motion defined on the probability space. Consider an n-dimensional SDDE
with initial value \(x(0) = x_{0}\in \mathcal{C}([-r,0];\mathbb{R}^{n} )\). Suppose that \(N_{1}: \mathbb{R}^{+}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) and \(N_{2} :\mathbb{R}^{+}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n\times m}\) satisfy the local Lipschitz and the linear growth conditions. Hence, for any given initial value \(x(0) = x_{0}\in \mathbb{R}^{n}\), it is known that equation (2.1) has a unique continuous solution on \(t\geq 0\), which is known as \(x(t; x_{0})\) in this section. Suppose that \(N_{1}(t,0) = 0\text{ and }N_{2}(t,0) = 0\), for all \(t\geq 0\). Hence, the SDDE admits the zero solution \(x(t;0)\equiv 0\).
Consider a functional \(W(t, \varphi )\) that can be represented in the form \(W(t, \varphi )=W(t, \varphi (0), \varphi (s))\), \(s<0 \), for \(\varphi =x_{t}\), put
and suppose that the function \(W_{\varphi}(t, x) \) has a continuous derivative with respect to t and two continuous derivatives with respect to x.
Let \(C^{1,2}(\mathbb{R}^{+} \times \mathbb{R}^{n};\mathbb{R}^{+})\) denote the family of nonnegative functionals \(W(t,x_{t})\) defined on \(\mathbb{R}^{+}\times \mathbb{R}^{n}\), which are once continuously differentiable in t and twice continuously differentiable in x.
By the Itô formula, we have
where
such that
Furthermore,
Now, we will give some definitions
Definition 2.1
[32] The zero solution of (2.1) is said to be stochastically stable or stable in probability if for every pair of \(\varepsilon \in (0,1)\) and \(r > 0\), there exists a \(\delta = \delta (\varepsilon ,r ) > 0\) such that
whenever \(|x_{0}| <\delta \). Otherwise, it is said to be stochastically unstable.
Definition 2.2
[32] The zero solution of (2.1) is said to be stochastically asymptotically stable if it is stochastically stable, and, moreover, for every \(\varepsilon \in (0,1)\), there exists a \(\delta _{0} = \delta _{0}(\varepsilon ) > 0\), such that
whenever \(|x_{0}| <\delta _{0}\).
Definition 2.3
[22] (Stochastic boundedness) A solution \(x(t; t_{0}, x_{0})\) of (2.1) is said to be stochastically bounded, or bounded in probability, if it satisfies
where \(E^{x_{0}}\) denotes the expectation operator with respect to the probability law associated with \(x_{0} \), and \(C : \mathbb{R}^{+} \times \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) is a constant depending on \(t_{0}\) and \(x_{0}\). We say that solutions of (2.1) are uniformly stochastically bounded if C is independent of \(t_{0}\).
Hypotheses
Suppose that there exist positive constants \(a_{0}\), a, μ, D, C, \(b_{i}\), \(c_{i}\), \(d_{i}\), \(L_{i}\), \(M_{i}\), \(N_{i}\), \(A_{i}\), \(B_{i}\), \(C_{i}\), \(D_{i}\), \(\gamma _{i}\), \(H_{1}\), \(H_{2}\) and \(l_{1}\), such that
- (\(h_{1}\)):
-
\(1 < a \leq \psi _{1}(x,y)\leq a_{0}\), \(y \frac{\partial \psi _{1}}{\partial x}\leq 0 \) for all \(x,y \in \mathbb{R}\).
- (\(h_{2} \)):
-
\(\frac{Q_{i}(x,y)}{y}\geq b_{i}>0 \), \(y\neq 0\); \(f_{i}(x)\geq d_{i} x \) with \(\sup \{f'_{i}(x)\} =\frac{c_{i}}{2}\), \(f_{i}(x) \operatorname{sgn} x>0 \) for \(x\neq 0\) and \(|f'_{i}(x)| \leq L_{i} \).
- (\(h_{3} \)):
-
\(H_{1} \leq H(t) \leq H_{2} \leq 1\), \(H_{1}(a-1)\geq 2\mu \).
- (\(h_{4} \)):
-
\(|\frac{\partial Q_{i}}{\partial x} | \leq M_{i}\), \(| \frac{\partial Q_{i}}{\partial y}| \leq N_{i}\) and \(r_{i}(t) \leq \gamma _{i}\), \(r'_{i}(t)\leq \beta _{i}\), \(0<\beta _{i} \leq 1\).
- (\(h_{5} \)):
-
\(ab_{i}-c_{i}> 2 M_{i}+2b_{i}+6\).
- (\(h_{6}\)):
-
\(0< l(t) \leq l_{1}\), \(|l'(t)|\leq \frac{1}{2}\).
- (\(h_{7}\)):
-
\(2\alpha ^{2} \leq 2H_{1} d_{i}-H_{1}(a+b_{i}+2) \).
- (\(h_{8}\)):
-
\(\int _{-\infty}^{\infty}{ | \frac{\partial \psi _{1}(u,v)}{\partial u}|\,du}+\int _{-\infty}^{ \infty}{ |\frac{\partial \psi _{1}(u,v)}{\partial v}|\,dv}\leq D< \infty \), \(\int _{-\infty}^{\infty}{|h'(u)|\,du}\leq C <\infty \).
Theorem 2.1
Assuming that the hypotheses \((h_{1})\)–\((h_{8})\) hold true provided that
where
with
Then, the zero solution of (1.1) is USAS.
Proof
The main tool of the stability results is the continuously differentiable functional \(W_{1}=W_{1}(x_{t},y_{t},z_{t})\), defined as
where
Considering \(\varepsilon \equiv 0\), we can observe that the Lyapunov functional \(U_{1}=U_{1}(x_{t},y_{t},z_{t})\), where \(x_{t}= x(t+s)\), \(s \leq 0\), can be written as follows
Since the integrals \(\int _{t-l(t)}^{t}{x^{2}(s)\,ds}\), \(\int ^{0}_{-r_{i}(t)}{ \int ^{t}_{t+s}{y^{2}(\theta )\,d\theta \,ds}}\) and \(\int ^{0}_{-r_{i}(t)}{\int ^{t}_{t+s}{z^{2}(\theta )\,d\theta \,ds }}\) are positive, from the conditions \((h_{1})\)–\((h_{3})\), we conclude
Therefore, we get
Since \(\mu =\sum_{i=1}^{n}{\frac{ab_{i}+c_{i}}{4b_{i}}}\) and \(\sup \{f'(x)\}=\frac{c_{i}}{2}\), it follows that
and
Then, we get
which tends to the following
Hence, there exists a positive constant \(E_{1}\), such that
In view of the hypotheses \((h_{1})\)–\((h_{4})\) and the following inequalities
and
Therefore, we can write (2.4) as
Since \(r_{i}(t)\leq \gamma _{i}\) and \(l(t) \leq l_{1}\), with applying the estimate \(2pq\leq (p^{2}+q^{2})\), we find
Then, there exists a positive constant \(E_{2}\), such that
Now, using the equivalent system (1.2) with \(\varepsilon =0\) and the Itô formula (2.2), the derivative of the Lyapunov functional \(U_{1}\) is given by
Therefore, using the definition of \(\Phi _{2}(t)\) and considering the conditions \((h_{1})\)–\((h_{4})\) of Theorem 2.1, we have
Suppose that
Using the Schwarz inequality \(|pq|\leq \frac{1}{2}(p^{2}+q^{2})\) and \((h_{3})\), we can write the above equation as
Therefore, we get
For the positive constant \(E_{3}\), the last inequality becomes
where
It follows from (2.6) that
Thus, by (2.9), (2.10) and the fact that \(2pq\leq ( p^{2}+q^{2})\), we obtain the following estimate
If we let
and
We also have \(\mu b_{i}-\frac{c_{i}}{2}=\frac{ab_{i}-c_{i}}{4}>0 \) and \(H_{1}(a-1)\geq 2\mu \); therefore, (2.11) becomes
Now, in view of (2.3), the last inequality becomes
Hence, for the positive constant \(E_{4}>0\), we obtain
Now, if we let
then we get
From the condition \((h_{6})\), it follows that
Because of
The stochastic derivative of the above equation is
Therefore, for the positive constant \(D_{1}\), we conclude that
Hence, from the results (2.6), (2.8), and (2.13), all conditions of the Lemma of the stability in [8, 14] are satisfied. Therefore, the proof of Theorem 2.1 is now complete. □
3 Uniformly stochastically boundedness results
Theorem 3.1
Assume that the hypotheses \((h_{1})\)–\((h_{8})\) hold true and suppose that there exist positive constants \(F_{i}\), \(K_{i}\), and m such that
and
Furthermore, we assume that
and
Provided that the positive constant \(\gamma _{i} \) satisfies the following
Then, all solutions of (1.1) are USB.
Proof
Here, consider \(\varepsilon \neq 0 \) and define the Lyapounov functional as follows
where \(U_{1}\) is defined in (2.4), and we define \(U_{2}\) as
Since \(\int _{t-l(t)}^{t}{x^{2}(s)\,ds}\) is nonnegative, recall the hypotheses \((h_{1})\)–\((h_{4})\), and then \(U_{2}\) becomes
Furthermore, we have \(H_{2}\leq H(t)\leq H_{1} \leq 1\), \(\frac{1}{H_{1}}\geq 1 \), so the above inequality leads to the following
Therefore, from \((h_{2})\), we find
We can find a positive constant \(\varphi _{1}\) such that the last inequality gives
Thus, from (2.5) and (3.4), we conclude
Hence, for the positive constant \(\varphi _{2}\), we get
Since \(|\frac{\partial Q_{i}}{\partial x}| \leq M_{i}\), \(|f'_{i}(x)|\leq L_{i}\), \(a \leq \psi _{1} \leq a_{0} \) and \(H_{1}\leq H(t) \leq H_{2} \leq 1 \), we can rewrite (3.3) in the following form
Applying the inequality \(2pq\leq (p^{2}+q^{2})\) and using the condition \(0< l(t)\leq l_{1}\), it tends to
Then, with \(\varphi _{3}>0\), we have
Combining the inequality (2.7) with (3.7), we conclude
Hence, for the positive constant \(\varphi _{4}\), the last inequality gives
In view of the hypothesis of Theorem 3.1 and the Itô formula, the derivative of the Lyapunov functional (3.3) with respect to the system (1.2) becomes
Now, we choose
Since \(|f_{i}'(x)|\leq L_{i}\), we obtain
Using the fact that \(2pq\leq (p^{2}+q^{2})\), we get
If we let
then from (3.5), we conclude
where
Considering the conditions \(l(t)\leq \frac{1}{2}\), \(y\frac{\partial \psi _{1}(x,y)}{\partial x}\leq 0\) and using equation (3.10), we find
Now, from the hypotheses \((h_{2}) \) and \((h_{4})\), we obtain
By compiling the above inequality with (2.11), from (3.1) and (3.2), we conclude
We take
Therefore, from (2.3) and since \(B_{i}=(1-\beta _{i})\), we obtain
Therefore, we can write the above inequality as follows
where
From (2.8) and (3.8), we obtain the following estimate
where
According to inequality (3.6), we conclude
It follows that
Define the Lyapunov functional \(W_{2}(x_{t},y_{t},z_{t})\) as follows
where
Then, from the hypotheses \(h_{1} \) and \(h_{3}\) and (2.12), we conclude
It follows form \((h_{8})\) that
Then, the stochastically derivative of \(W_{2}\) becomes
Hence, from (3.12), we find
Thus, from inequalities (3.6) and (3.9) and by taking \(\nu (t)=\zeta /2\), \(\rho _{4}(t)=(3\zeta /2)\kappa ^{2}\) and \(n=2\), we see that the conditions (i) and (ii) of Lemma 2.4 in [8, 14] are satisfied. As well as we can test that the condition (iii) is satisfied with \(q_{1}=q_{2}=n=2\) with \(\rho _{3} =0\). Then, all conditions of Lemma 2.4 in [8, 14] are achieved.
So, with \(\nu (t)=\zeta /2\), \(\beta (t)=(3\zeta /2)\kappa ^{2}\), \(n=2\), and \(\rho _{3} =0\), we note that
for all \(t\geq t_{0}\geq 0\). Thus, condition (2.4) [8] holds. Now, since
we have
Thus, condition (2.3) in [8, 14] is satisfied. Using Lemma 2.4 in [8, 14], we find that all solutions of (1.1) are USB, and we can also conclude
Hence, the proof of Theorem 3.1 is now complete. □
4 Examples and discussion
Example 4.1
In a particular case \(n=1\), consider the following third-order SDDE
The equivalent system of (4.1) is
Comparing equation (1.2) with (4.2), we have
Therefore, we get
The derivative of \(h(y)\) is
Then, we find
We can see that Fig. 1 illustrates the behavior of \(h(y)\) in the interval \(x\in [0,50]\).
We also have the function
then, we get \(a=19\) and \(a_{0}=19+\frac{\pi}{2}\).
We also obtain
and
Therefore, we can conclude
Figure 2 shows the behavior of the function \(\psi _{1}(x,y)\) through the interval \(x\in [-4,4]\), \(y\in [-4,4]\), and also it shows that \(y \frac{\partial \psi _{1}}{\partial x}<0\), for all x, y.
The function
fulfills
The derivatives of \(Q(x,y)\) are defined as follows
For the behavior of the functions \(\frac{\partial Q(x,y)}{\partial y}\), \(\frac{\partial Q(x,y)}{\partial x}\), and \(\frac{Q(x,y)}{y}\), see Fig. 3.
Now, the function
It follows that
Therefore, we find
Figure 4 gives the path of \(\frac{f(x)}{x}\), \(f'(x)\).
Finally, we obtain
Figure 5 shows the behavior of the stochastic term \(\frac{1}{4} \sin ({x(t-\frac{1}{2}e^{t})})\), and it also shows that \(|l'(t)|<\frac{1}{2}\) on the interval \([0,30] \).
Now, we have
and
Since \(\alpha ^{2}= \frac{1}{16}\), we have
and
Suppose that \(\beta =\frac{1}{2}\), then we conclude
Therefore, we get
Hence all hypotheses of Theorem 2.1 are achieved, then the zero solution of (4.1) is USAS.
Example 4.2
Consider the following SDDE
Using the estimates in Example 4.1, we get
Since
then we get
Let \(m=0.01 \), so we obtain
provided that
If we take \(\zeta =0.2\) and \(m=0.01\), then we find
Now, we can satisfy the condition (ii) of Theorem 2.2 in [28] by taking
Then, since \(q_{1}=q_{2}=n=2\), we get all assumptions of Theorem 2.2 [28] are satisfied.
It follows from the above estimates, the following inequality holds
And we also get
Hence, Lemma 2.4 in [28] implies that the zero solution of (4.5) is USB.
Now, in view of Figs. 6 and 7, we find that the behavior for the solutions of (4.2) and (4.5) are asymptotically stable, such that the Figs. 6 and 7 illustrate the behavior of the solution, when \(\alpha =0.25\) and \(\alpha =1\), respectively. We note that, when α is increased, the stochasticity becomes more pronounced. On the other hand, if we take the function \(\sin ( x(t-\frac{1}{2}e^{-t}))=1\), then we get Figs. 8 and 9, with \(\alpha =0.25\) and \(\alpha =1\), respectively.
Data Availability
No datasets were generated or analysed during the current study.
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All authors reviewed the manuscript A. M. Mahmoud D. A. Eisa, R. O. A. Taie and D. A. M. Bakhit1*
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Mahmoud, A.M., Eisa, D.A., Taie, R.O.A. et al. Asymptotic behaviour and boundedness of solutions for third-order stochastic differential equation with multi-delay. Bound Value Probl 2024, 49 (2024). https://doi.org/10.1186/s13661-024-01849-z
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DOI: https://doi.org/10.1186/s13661-024-01849-z