On the qualitative behaviors of stochastic delay integro-differential equations of second order

In this paper, we investigate the suﬃcient conditions that guarantee the stability, continuity, and boundedness of solutions for a type of second-order stochastic delay integro-diﬀerential equation (SDIDE). To demonstrate the main results, we employ a Lyapunov functional. An example is provided to demonstrate the applicability of the obtained result, which includes the results of this paper and obtains better results than those available in the literature.


Introduction
An integral equation is a mathematical expression that includes a required function under an integration sign.Such equations often describe an elementary or a complex physical process wherein the characteristics at a given point depend on values in the whole domain and cannot be defined only on the bases of the values near the given point.
A differential equation is said to be an integro-differential equation (IDE) if it contains the integrals of the unknown function.Most frequently, integral equations as well as IDEs are found in such problems of heat and mass transfer as diffusion, potential theory, and radiation heat transfer.Integral equations have a lot of applications such as actuarial science (ruin theory), computational electromagnetics, inverse problems, for example, Marchenko equation (inverse scattering transform), options pricing under jumpdiffusion, radiative transfer, and viscoelasticity (see, for example, [12,13,17,37,39,47] and the references cited in therein).
In biological applications, the population dynamics and genetics are modeled by a system of IDEs (see Kheybari et al. [19]).Next, initial value problems for a nonlinear system of IDEs are used to model the competition between tumor cells and the immune system (see Nicola et al. [9]).
Besides, in engineering, two systems of specific inhomogeneous IDEs are studied to examine the noise term phenomenon (see Wazwaz [46]).In addition, the scattered elec-tromagnetic fields from resistive strips and RLC circuits are governed by IDEs (see Hatamzadeh et al. [16]).
An IDE is said to have a delay when the rate of variation in the equation state depends on past states.In this case such an IDE is called delay integro-differential equation (DIDE).
Numerous sectors of science and technology, including biology, medicine, engineering, information systems, control theory, and finance mathematics, have utilized the stability and boundedness qualities of solutions for IDEs with and without delays.The Lyapunov's direct method, which includes an energy-like function, has proven to be an effective tool in the qualitative study of ordinary differential equations (ODEs).Many researchers have used this technique to solve delay differential equations (DDEs) and IDEs over the last five decades.In contrast to Lyapunov functionals, which are frequently employed in the study of DDEs and IDEs (see, for instance, Burton [11,40]).
To the best of our information, we observe that only a few excellent and interesting works on the stochastic stability and boundedness of solutions for second-, third-, and fourthorder SDDEs have been developed in [1-6, 22-24, 26, 27, 38, 45] (see also the references of these sources).
There are a number of results on the qualitative characteristics of first-, second-, and third-order IDEs with and without delays, but none on the qualitative characteristics of solutions for a particular class of second-order SDIDE.
As a result, the goal of this paper is to investigate the stability, continuity, and boundedness of solutions for a type of second-order SDIDE as follows: where τ (t) is a variable delay with 0 ≤ τ (t) ≤ γ , γ is a positive constant that will be determined later, τ (t) ≤ β, β ∈ (0, 1).
The functions Q and R are continuous differentiable functions such that Equation (1.1) can be expressed in the following system form: In addition, it is supposed that the derivatives Q x (x, y) = ∂Q ∂x (x, y) and R (x) = dR dx (x) exist and are continuous.
Let us consider the n-dimensional SDDE (see [25,43]): with the initial condition To formulate the stability and boundedness criteria, we suppose that C 1,2 (R + × R n ; R + ) denotes the family of all nonnegative Lyapunov functionals W (t, x t ) defined on R + × R n , which are twice continuously differentiable in x and one in t.By Itô's formula, we have where

Stochastic qualitative results
We introduce the following hypotheses before proving our main results.Assume that there are positive constants f 0 , g 0 , p 0 , c 0 , α 0 , α, K * , c, d, and N that satisfy the following conditions: (i) |f (t, y)| ≤ f 0 |y| for all t ∈ R + and y ∈ R; (ii) P(t, x, y) ≥ p 0 > 0 and g(t, x) ≤ g 0 x for all t ∈ R + and x, y ∈ R; (vi) There are γ > 0 and β ∈ (0, 1) such that 0 ≤ τ (t) ≤ γ and τ (t) ≤ β.The following theorem is the first result of this paper.Theorem 2.1 Let conditions (i)-(vi) hold.Then all the solutions of system (1.2) are continuous and bounded provided that Proof The proof of this theorem rests on the differentiable scalar Lyapunov functional V (t) := V (t, x t , y t ) defined as follows: where λ is a positive constant that will be determined later.
In view of assumptions (iii) and (iv), we obtain It follows that Then we obtain Hence, it is clear that there exists a sufficiently small positive constant δ 1 such that where As a result, the Lyapunov functional V (t) is positive definite at all (x, y) points and zero only at x = y = 0. Itô's formula (1.4) gives the derivative of the Lyapunov functional V (t) in (2.1) along any solution (x(t), y(t)) of system (1.2) as follows: It follows that ( By assumption (i), we get the following inequality: From the inequality 2|mn| ≤ m 2 + n 2 , we get the following relations: (2.5) In the same way, we obtain The following estimations can be confirmed using assumptions (ii)-(iv) and the inequality 2|mn| ≤ m 2 + n 2 : (2.7) Hence, in view of assumptions (iii), (iv) and by using the inequality 2|mn| ≤ m 2 + n 2 , we can conclude that x 2 (t) + y 2 (s) ds Similar to the preceding, we have By adding the above two inequalities and since 0 ≤ τ (t) ≤ γ , we get the following: Furthermore, from condition (vi), it follows that y 2 (s) ds. (2.9) By considering the preceding inequalities (2.4)-(2.8) in the derivative (2.3), we can arrive at With some rearrangement of terms, we can get Then, from condition (v), we obtain y 2 (s) ds.
If we now choose then we can observe If we take then there exists a positive constant δ 2 such that This implies that LV (t) ≤ 0. Because of all functions appearing in (1.1), it is obvious that there exists at least one solution of (1.1) defined on [t 0 , t 0 + ρ) for some ρ > 0.
It is necessary to show that the solution can be extended onto the entire interval [t 0 , ∞).We suppose on the contrary that there is a first time T < ∞ such that the solution exists on [t 0 , T) and Suppose that (x(t), y(t)) is a solution of system (1.2) with the initial condition (x 0 , y 0 ).Since the Lyapunov functional V (t) is a positive definite and decreasing functional on the trajectories of system (1.2), also we have Then we can say that V (t) is bounded on [t 0 , T).Now, integrating the above inequality from t 0 to T, we have Hence, it follows from (2.2) that This inequality implies that |x(t)| and |y(t)| are bounded on t → T -.Thus, we conclude that T < ∞ is not possible, we must have T = ∞.This completes the proof of Theorem 2.1.

Theorem 2.2
If assumptions (i)-(vi) of Sect. 2 hold, then the null solution of system (1.2) is uniformly stochastically asymptotically stable.
Proof From (2.1), using assumptions (iii) and (iv) and the inequality 2|mn| ≤ m 2 + n 2 , we have where Then, from conditions (i) and (v), we obtain (2.11) Therefore, by combining the two inequalities (2.2) and (2.11), we get (2.12) It follows from (2.10) and (2.12) that the Lyapunov functional V (t) satisfies the following inequalities: Thus, by taking note of how the discussion above developed, the stability theorems 1 and 2 in [8,30,45] were established.This completes the proof of Theorem 2.2.

Example
In this section, we consider an example of how to illustrate the results for second-order SDIDE.
Then we can express (3.1) as the equivalent system: When we compare systems (3.2) and (1.2), we see the following relationships: The functions R(x)  x and R (x) with their bounds are shown in Fig. 1.In Fig. 2, the behaviours of the functions Q(x,y) x , (x = 0), were plotted in [-20, 20] by MAT-LAB software.
The shape and path of τ (t) and τ (t) are shown in Fig. 3.  Finally, if γ = 5 64 < min{0.214,0.1482} ∼ = 0.1482, then the null solution of (3.1) is uniformly stochastically asymptotically stable.Thus, all the conditions of Theorems 2.1 and 2.2 are fulfilled.Therefore, their results hold.

Then we obtain
In Fig. 4, the nonlinear SDIDE (3.1) of second order was solved by MATLAB software.In Fig. 5, the nonlinear SDIDE (3.1) of second order without stochastic term was solved by MATLAB software.
In Fig. 6, the nonlinear SDIDE (3.1) of second order with stochastic term that equals 30 was solved by MATLAB software.
As a result, we may say that all the solutions of equation (3.1) are stable, continuous, and bounded.

Figure 1 Figure 2
Figure 1 Behaviors of the functions R(x) x and R (x) for x ∈ [-40, 40]

Figure 5 Figure 6
Figure 5 Trajectory of the solution for (3.1) without stochastic term