Asymptotic behaviour and boundedness of solutions for third-order stochastic differential equation with multi-delay

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

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.
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 ≤ r i (t) ≤ γ i , (i = 1, 2, . . ., n), γ i > 0 are constants, ψ 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, ∞) to [0, l 1 ].w(t) ∈ R n is a standard Brownian motion.
Consider the following notations Therefore, equivalent system of (1.1) can be written as . (1.2)

Stability results
Let B(t) = (B 1 (t), . . ., B m (t)) be an m-dimensional Brownian motion defined on the probability space.Consider an n-dimensional SDDE R + × R n → R n×m satisfy the local Lipschitz and the linear growth conditions.Hence, for any given initial value x(0) = x 0 ∈ R n , it is known that equation (2.1) has a unique continuous solution on t ≥ 0, which is known as x(t; x 0 ) in this section.Suppose that N 1 (t, 0) = 0 and N 2 (t, 0) = 0, for all t ≥ 0. Hence, the SDDE admits the zero solution x(t; 0) ≡ 0. Consider a functional W (t, ϕ) that can be represented in the form and suppose that the function W ϕ (t, x) has a continuous derivative with respect to t and two continuous derivatives with respect to x.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 ε ∈ (0, 1) and r > 0, there exists a δ = δ(ε, r) > 0 such that P x(t; x 0 ) < r for all t ≥ 0 ≥ 1ε, 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 ε ∈ (0, 1), there exists a δ 0 = δ 0 (ε) > 0, such that 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 : R + × R + → 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, Theorem 2.1 Assuming that the hypotheses (h 1 )-(h 8 ) hold true provided that where ) 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 ε ≡ 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 ≤ 0, can be written as follows Since the integrals t t+s y 2 (θ ) dθ ds and 0 -r i (t) t t+s z 2 (θ ) dθ ds are positive, from the conditions (h 1 )-(h 3 ), we conclude Therefore, we get and Then, we get which tends to the following (2.5) Hence, there exists a positive constant E 1 , such that In view of the hypotheses (h 1 )-(h 4 ) and the following inequalities Therefore, we can write (2.4) as Since r i (t) ≤ γ i and l(t) ≤ l 1 , with applying the estimate 2pq ≤ (p 2 + q 2 ), we find (2.7) Then, there exists a positive constant E 2 , such that (2.8) Now, using the equivalent system (1.2) with ε = 0 and the Itô formula (2.2), the derivative of the Lyapunov functional U 1 is given by z 2 (θ ) dθ ∂Q i (x(s), y(s)) ∂x y(s) Therefore, using the definition of 2 (t) and considering the conditions (h 1 )-(h 4 ) of Theorem 2.1, we have (2.9) Suppose that Using the Schwarz inequality |pq| ≤ 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 (2.10) Thus, by (2.9), (2.10) and the fact that 2pq ≤ (p 2 + q 2 ), we obtain the following estimate z 2 (θ ) dθ. (2.11) If we let and We also have μb i -c i 2 = ab i -c i 4 > 0 and H 1 (a -1) ≥ 2μ; 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.

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 Provided that the positive constant γ i satisfies the following Then, all solutions of (1.1) are USB.
Proof Here, consider ε = 0 and define the Lyapounov functional as follows where U 1 is defined in (2.4), and we define U 2 as x 2 (s) ds.
(3.3) Since t t-l(t) x 2 (s) ds is nonnegative, recall the hypotheses (h 1 )-(h 4 ), and then U 2 becomes so the above inequality leads to the following (3.4) Therefore, from (h 2 ), we find We can find a positive constant ϕ 1 such that the last inequality gives Thus, from (2.5) and (3.4), we conclude Hence, for the positive constant ϕ 2 , we get Applying the inequality 2pq ≤ (p 2 + q 2 ) and using the condition 0 < l(t) ≤ l 1 , it tends to (3.7) Then, with ϕ 3 > 0, we have Combining the inequality (2.7) with (3.7), we conclude Hence, for the positive constant ϕ 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 ∂Q i (x(s), y(s)) ∂x y(s) Using the fact that 2pq ≤ (p 2 + q 2 ), we get If we let then from (3.5), we conclude where Considering the conditions l(t) ≤ 1 2 , y ∂ψ 1 (x,y) ∂x ≤ 0 and using equation (3.10), we find ∂Q i (x(s), y(s)) ∂x y(s) Now, from the hypotheses (h 2 ) and (h 4 ), we obtain By compiling the above inequality with ( , we conclude We take Therefore, from (2.3) and since B i = (1β 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 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 ν(t) = ζ /2, ρ 4 (t) = (3ζ /2)κ 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 ρ 3 = 0.Then, all conditions of Lemma 2.4 in [8,14]  for all t ≥ t 0 ≥ 0. Thus, condition (2.4) [8] holds.Now, since g T = 00αx tl(t) , 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.

Examples and discussion
Example 4.1 In a particular case n = 1, consider the following third-order SDDE Therefore, we get The derivative of h(y) is h (y) = -e -4y .
Then, we find We can see that Fig. 1 illustrates the behavior of h(y) in the interval x ∈ [0, 50].

Figure 1 Trajectory of h(y)
We also have the function then, we get a = 19 and a 0 = 19 + π 2 .We also obtain and  For the behavior of the functions ∂Q(x,y) ∂y , ∂Q(x,y) ∂x , and Q(x,y) y , see Fig. 3. Now, the function Therefore, we find Figure 4 gives the path of f (x) x , f (x).Finally, we obtain Figure 5 shows the behavior of the stochastic term 1 4 sin(x(t -1 2 e t )), and it also shows that |l (t)| < 1  2 on the interval [0, 30].Now, we have μ = ab + c 4b = 6.38, then abc = 100 > 0, Suppose that β = 1 2 , then we conclude  Then, since q 1 = q 2 = n = 2, we get all assumptions of Theorem 2.2 [28]  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 α = 0.25 and α = 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 -1 2 e -t )) = 1, then we get Figs.8 and 9, with α = 0.25 and α = 1, respectively.

Figure 6 25 Figure 7 1 Figure 8 25 Figure 9
Figure 6The behavior of the solutions with α = 0.25 denote the family of nonnegative functionals W (t, x t ) defined on R + × R n , which are once continuously differentiable in t and twice continuously differentiable in x.
are satisfied.It follows from the above estimates, the following inequality holds ds du ≤ 3κ 2 = 60.2, for all t ≥ t 0 ≥ 0.