Some New Results on the Uniform Asymptotic Stability for Volterra Integro-differential Equations with Delays

In this work, we establish sufficient conditions of the uniform asymptotic stability (UAS) of solutions to second-order and third-order of Volterra integro-differential equations (VIDE) with delay. Here, we prove two new theorems on the UAS of the solutions of the considered VIDEs. Our approach is based on Lyapunov’s second method. Our results improve and form a complement to some known recent results in the literature. Two illustrative examples are considered to support the results and two graphs are drawn to illustrate the asymptotic stability of the zero solution for the considered numerical equations. The obtained results are new and original.


Introduction
The integro-differential equations (IDEs), which combine differential and integral equations, have attracted more attention in recent years. Applications in mathematics, physics, biology, and engineering all heavily rely on IDEs.
The equations known as the Volterra equations were studied in the early years of the 20th century by Italian mathematician Vito Volterra. In the 1930 s, Volterra showed that mathematical models for some seasonal diseases, e.g., influenza, are formulated as integral and differential equations. The use of VIDEs is widespread in the fields of biology, ecology, medicine, physics, and other sciences. To the best of our knowledge, it has been observed in a variety of physical applications, including the glass-forming process, heat transfer, the diffusion process generally, neutron diffusion, the coexistence of biological species with varying generation rates, and wind ripple in the desert.
One of the most crucial methods for researching the qualitative characteristics of solutions to ordinary, functional, and IDEs is Lyapunov's second where A ia a constant and C : R + → R is a continuous function.
In 2015, Tunç [14] studied the stability and the boundedness of the zero solution of the non-linear VIDE with delay of the forṁ

B(t, s)g(x(s))ds + p(t).
Recently, in 2022, Appleby and Reynold [1] studied the asymptotic stability of the scalar linear VIDĖ Our goal for this paper is to create the sufficient conditions for the UAS of second and third-order VIDEs with delay for the following equations and . ..
1. Whenever,ẍ replaced byẋ, f 1 (x)ẋ replaced by Ax(t), and let v 1 (x) = x(t), in the integral term then (1.1) reduces to the equation that is considered in [25]. Thus, the stability and results obtained in (1.1) include and extend the previous results. 2. In [1], If we replaced the termẍ byẋ, f 1 (x)ẋ by ax(t), and let v 1 (x) = x(t) in the integral term, then (1.1) reduces to the equation that considered in [1]. Then the stability results of this paper include and improve the stability result obtained in [1]. Then (1.1) and (1.2) generalize and improve the results obtained in [1,25]. 3. As an application in physics, many models can be modeled by IDEs. For example, first, by the Kirchhoffs second law, the net voltage drop across a closed loop equals the voltage impressed E(t). Thus, the standard closed electric RLC circuit can be governed IDE [5], second, an Abeltype Volterra integral equation describes the temperature distribution along the surface when the heat transfer to it is balanced by radiation from it. Finally, also, Abel-type Volterra integral equation determines the temperature in a semi-infinite solid, whose surface can dissipate heat by nonlinear radiation [23].
Here, F is a continuous function of t for t 0 ≤ t ≤ ∞, whenever x t ∈ X H (t) for t 0 ≤ t ≤ ∞, and takes closed bounded sets of R × X(t) into bounded sets of R n . Theorem 2.1. [7] Let V (t, x t ) be continuous functional and locally Lipschitz for Then, the zero solution of (2.1) is uniformly asymptotically stable (UAS).
The following two theorems will be our main results for (1.1) and (1.2).
Then, the zero solution of (1.1) is UAS, provided that Together with the fundamental conditions given on the functions f 2 , H 2 and v 2 for (1.2), we assume that there exist the positive constants α 1 , α 2 , α 3 , α 4 , d 1 , d 2 , L, θ 1 , θ 2 and θ 3 , so that the following assumptions are true

Proof of Theorem 2.2.
Rewrite (1.1) as the followinġ It can be written as −2 Using the Schwarz inequality [8], we get By using the inequality |mn| ≤ 1 2 (m 2 + n 2 ), and the previous inequality, we can write (3.3) as the following form By the assumptions of Theorem 2.2, we have where W is a wedge function. Therefore, we have Therefore, one can conclude that Since ∞ 0 |H 1 (u)|du = L < 1 and by the assumption (i) of Theorem 2.2, we conclude (3.5) Thus, from (3.4) and (3.5), we conclude that the condition (i) of Theorem 2.1 is satisfied. Now, by differentiating Eq. (3.2), we obtain From Leibnitz rule [23] Pg. 17 and the identity [23] Pg. 17 and [6] Pg. 41, we have then, we get From the condition (i) and the inequality |mn| ≤ 1 2 (m 2 +n 2 ), we obtain Therefore, we conclude Consider the conditions (i)-(iv) and |H 1 (0)| ≥ β 1 , we have Therefore, we conclude for D 1 > 0, that

Proof of Theorem 2.3.
We can rewrite (1.2) as the following equivalent systeṁ (4.1) Define the LKF V 2 (t, x t , y t , z t ) as Applying the condition (i) and the inequality |mn| ≤ 1 2 (m 2 + n 2 ), we obtain Since ∞ 0 H 2 (u)du = L and from condition (i), then we get By the Schwarz inequality [8], we have MJOM Some New Results on the Uniform Asymptotic Page 9 of 17 280 Applying the conditions of Theorem 2.3, we obtain It follows that If we let then, we get Since 1 + α + α 2 + L > 0, then we have a positive constant γ 2 , such that Now, (4.2) becomes By (ii), we have ∞ 0 |H 2 (u)|du = L < 1 and by the assumption (i) of Theorem 2.3, we conclude that (4.5) |H 2 (t − s 2 )|v 2 2 (y(s 2 ))ds 2 .
Thus, the proof of Theorem 2.3 is now complete.
We can see that the behaviour of the solutions (x(t), y(t)) with the initial values (x 0 = 0, y 0 = 1) for (5.1) by Fig. 3.
Thus, all the hypotheses of Theorem 2.2 are satisfied. Then, the zero solution of (5.1) is UAS.
Thus, all the hypotheses of Theorem 2.3 are verified. Then, the zero solution of (5.2) is UAS.

Conclusion
This work emphasizes the stability of solutions to certain nonlinear secondorder and third-order VIDE with delay.
By employing Lyapunov's second method, a suitable LKF was constructed and used to establish the sufficient conditions of Theorems 2.2 and 2.3.
Two numerical examples were given and all functions were drawn to prove the sufficient conditions of Theorems 2.2 and 2.3, and also orbits of the numerical solutions were drawn with assigned initial conditions to demonstrate the effectiveness of the obtained results.
The results obtained in this paper extend many existing and exciting results on nonlinear VIDE.
Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).

Conflicts of Interest
The authors declare that they have no conflict of interest.
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