Abstract
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.
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1 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 method because this method is widely recognized as an excellent tool in the study of differential equations. Theoretically, this method is quite significant, and it is used in many different applications, see [24]. Lyapunov’s second method is a sufficient condition to show the stability of systems, which means the system may still be stable even if one cannot find a Lyapunov-Krasovskii functional (LKF) candidate to conclude the system stability property.
There are many interesting results have been obtained in the literature to study the behaviour of solutions for DDE by Lyapunov’s theory, see for example [4, 10, 15, 16, 22, 25].
Besides, it is worth mentioning, that according to our observation from the literature, recently we found many exciting papers on the kind of VIDEs, for example [2, 3, 9,10,11,12,13, 15,16,17,18,19,20,21,22].
In 2000, Zhang [25] investigated the uniform asymptotic stability for the linear scaler VIDE
where A ia a constant and \(C:\mathbb {R}^{+} \rightarrow \mathbb {R}\) 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 form
Recently, in 2022, Appleby and Reynold [1] studied the asymptotic stability of the scalar linear VIDE
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
where \( h_1,h_2:[0,\infty )\rightarrow (-\infty ,\infty )\) are continuous functions depend on the differences \( t-s_1, t-s_2\), respectively, and \(L^{1}(0,\infty )\), \(L^{1}\) is the space of integrable Lebesgue functions, \(s_1, s_2\) are time delays with \(s_1, s_2 \le t\), also there exist two functions \(H_1,H_2: [0,\infty )\rightarrow [0,\infty )\) such that \({\dot{H}}_1(t-s_1)=\frac{\textrm{d}}{\textrm{d}t}(H_1(t-s_1))=-h_1(t-s_1)\), \({\dot{H}}_2(t-s_2)=\frac{\textrm{d}}{\textrm{d}t}(H_2(t-s_2))=-h_2(t-s_2)\) with \(\int _{0}^{\infty }{|h_1 (u)|\textrm{d}u }, \int _{0}^{\infty }{|h_2 (u)|du} \in L^{1}[0,\infty ) \) and \( \int _{t}^{\infty }{|H_2 (u)|\textrm{d}u}, \int _{t}^{\infty }{|H_2 (u)|\textrm{d}u} \in L^{1}[0,\infty ) \). The functions \(f_1(x),f_2(y), v_1(x)\) and \( v_2(y)\) are continuous scalar functions defined on \(\mathbb {R}\) with \(f_1(0)=f_2(0)=v_1(0)= v_2(0)=0\).
Remark 1.1
We will give the following remarks:
- 1.:
-
Whenever, \(\ddot{x}\) replaced by \({\dot{x}}\), \(f_1(x) {\dot{x}}\) 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 term \(\ddot{x}\) by \({\dot{x}}\), \(f_1(x) {\dot{x}}\) 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 Abel-type 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].
2 Main Results
Consider the general functional differential system
where, \(x_t\) represents a function from \([\alpha ,t]\rightarrow \mathbb {R}^{n}, \, \, -\infty \le \alpha \le t_0\). For any \(t\ge t_0\), by \((X(t),\Vert .\Vert ),\) we shall mean the space of continuous functions \(\phi : [\alpha ,t]\rightarrow \mathbb {R}^{n}, \alpha >0, \, \text {with} \Vert \phi \Vert =\sup _{\alpha \le s \le t}|\phi (s)|, ~s \in R\) and |.| is any norm on \( \mathbb {R}^n\). The symbol \(X_H(t)\) denotes those \(\phi \in X(t) \) with \(\Vert \phi \Vert \le H\) for some \(H>0\).
Here, F is a continuous function of t for \(t_0\le t\le \infty , \) whenever \(x_t\in X_H(t) \) for \(t_0\le t\le \infty , \) and takes closed bounded sets of \(\mathbb {R}\times X(t)\) into bounded sets of \(\mathbb {R}^n\).
Theorem 2.1
[7] Let \(V(t,x_t)\) be continuous functional and locally Lipschitz for
\( t_{0} \le t < \infty \text {and} \,\, x_t\in X_H(t) \). Suppose there is a continuous function \( \Phi : [0,\infty ) \rightarrow [0,\infty )\) which is \( L^{1}[0,\infty )\) and satisfies
- (i):
-
\(W_{1}(|x|)\le V(t,x_{t})\le W_{2}(|x|)+ W_{3}\bigg {(}\int _{\alpha }^{t}{\Phi (t-s)W_{4}(|x(s)|)\textrm{d}s} \bigg {)},\) where \(W_{i}; \big {(}i=1,2,3,4\big {)}\) are wedges;
- (ii):
-
\( {\dot{V}}_{(2.1)}(t,x_{t})\le -W_{5}(|x|)\).
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).
Theorem 2.2
In addition to the basic assumptions given on the functions \(f_1\), \(H_1\) and \(v_1\) for (1.1), we suppose that there are the non-negative constants \( a_{1},\; a_{2},\;b_{1},\;b_{2}, \) \(L_{1}, \, L, \, c_{1}\), \(\beta _1,\,\beta _2\) and \(c_{2}, \) such that
- (i):
-
\( a_{2}\le f_1(x)\le a_{1}, \,\, |f_1'(x)|\le c_{1}\,\,\) and \(\, \,b_{2}\le v_1(x)\le b_{1},\,\,\) \(|v_1'(x)|\le c_{2}.\)
- (ii):
-
\(\int _{0}^{\infty }{|H_1(u)|\textrm{d}u}=L<1\) and \(\int _{t}^{\infty }{|H_1(u)|\textrm{d}u} \in L^{1}[0,\infty )\).
- (iii):
-
\(0<\beta _{1}\le |H_1(0)| \le \beta _{2}.\)
- (iv):
-
\(\int _{0}^{t}{|H_1(t-s_1)|\textrm{d}s_1}+ \int _{t}^{\infty }{|H_1(u-t)|\textrm{d}u}=L_{1}\).
Then, the zero solution of (1.1) is UAS, provided that
Theorem 2.3
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 \( \alpha _1, \alpha _2, \alpha _3, \alpha _4,\) \( d_1, d_2, L, \theta _1,\theta _2\) and \(\theta _3\), so that the following assumptions are true
- (i):
-
\(\alpha _{1} \le f_2(y)\le \alpha _2 \), \( |f'_2(y)|\le d_1\) and \(\alpha _3 \le v_2(y)\le \alpha _4\), \( |v'_2(y)|\le d_2\).
- (ii):
-
\(\int _{0}^{\infty }{|H_2(u)|\textrm{d}u}=L<1\) and \(\int _{t}^{\infty }{|H_2(u)|\textrm{d}u} \in L^{1}[0,\infty )\).
- (iii):
-
\( 0<\theta _1\le |H_2(0)|\le \theta _2. \)
- (iv):
-
\( \int _{0}^{t}{|H_2(t-s_2)|\textrm{d}s_2}+\int _{t}^{\infty }{|H_2(u-t)|\textrm{d}u}\le \frac{\theta _1\theta _3}{\theta _2}\).
Then, the zero solution of (1.2) is UAS, provided that
3 Proof of Theorem 2.2.
Rewrite (1.1) as the following
Define the LKF \( V_1(t,x_t,y_t)\) as
It can be written as
Using the Schwarz inequality [8], we get
By using the inequality \(|mn|\le \frac{1}{2}(m^{2}+n^{2})\), 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
On the other hand
Since \(\int _{0}^{\infty }{|H_1(u)|\textrm{d}u}=L<1\) and by the assumption (i) of Theorem 2.2, we conclude
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 \( |mn|\le \frac{1}{2}(m^{2}+n^{2})\), we obtain
Therefore, we conclude
Consider the conditions (i)–(iv) and \(|H_1(0)|\ge \beta _1\), we have
Therefore, we conclude for \(D_1>0\), that
where, \(D_1=\min {\{\beta _{1}b_{2}c_{1}+2\beta _{1}c_{2}-c_{1}b_{1}L, 2a_{2}-c_{1}b_{1}}\}.\)
Thus, from (3.4), (3.5) and (3.6) all the assumptions of Theorem 2.1 are satisfied. Therefore the zero solution of (1.1) is UAS. Hence, the proof of Theorem 2.2 is now complete.
4 Proof of Theorem 2.3.
We can rewrite (1.2) as the following equivalent system
Define the LKF \(V_2(t,x_t,y_t,z_t)\) as
From Eq. (4.2), we get
Applying the condition (i) and the inequality \(|mn|\le \frac{1}{2}(m^2+n^2)\), we obtain
Since \(\int _0^\infty {H_2(u)}\textrm{d}u=L\) and from condition (i), then we get
By the Schwarz inequality [8], we have
Applying the conditions of Theorem 2.3, we obtain
It follows that
If we let
then, we get
Since \(1+\alpha +\alpha _2+L>0\), then we have a positive constant \(\gamma _2\), such that
Now, (4.2) becomes
By (ii), we have \(\int _{0}^{\infty }{|H_2(u)|\textrm{d}u}=L<1\) and by the assumption (i) of Theorem 2.3, we conclude that
Differentiating the LKF \(V_2(t,x_t,y_t,z_t)\) with respect to t
From Leibnitz rule [23] Pg. 17 and the identity [6] Pg. 41, we get
By using the equivalent system (4.1), we obtain
From condition (i), we get
It follows from condition (iv) and the inequality \(|mn|\le \frac{1}{2}(m^2+n^2)\) that
Thus, one can conclude for a positive constant \(D_2>0\) that
where \(D_2=\theta _1 \min {\{ d_2+ \alpha d_2 +2\alpha _1 d_2 -\theta _3d_2^2,\alpha d_2,d_2\}}.\) From the results (4.4), (4.5) and (4.6), we note that all assumptions of Theorem 2.1 are satisfied, then the zero solution of (1.2) is UAS.
Thus, the proof of Theorem 2.3 is now complete.
5 Illustrative Examples
Example 5.1
Consider the following VIDE with delay
Note that
So, we find
and
Figure 1, shows the behaviour of \(f_1(x)\) and \(f_1'(x)\) on the interval \(t\in [2,20] \) and \(t\in [0,90]\), respectively.
Moreover, we have
and
Figure 2, illustrates the behaviour of \(v_1(x)\) and \(v_1'(x)\) through the interval \(t\in [0,90]\).
Also, we have
and
Then, we get
and
So, it is clear that
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.
Example 5.2
Consider the following VIDE with delay
It follows that
So, we get
and
Figure 4, shows the behaviour of \(f_2(x)\) and \(f_2'(x)\) on the interval \(t\in [0,50]\).
Moreover
So, we get
and
Figure 5, illustrates the path of \(v_2(x)\) and \(v_2'(x)\) on the interval \(t \in [0,50]\).
Also, we have
and
Also, we have
and
So, it is clear that
Figure 6, shows the behaviour of the solutions (x(t), y(t), z(t)) with the initial values \((x_0=0, y_0=1, z_0=1)\) for (5.2).
Thus, all the hypotheses of Theorem 2.3 are verified.
Then, the zero solution of (5.2) is UAS.
6 Conclusion
This work emphasizes the stability of solutions to certain nonlinear second-order 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.
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Taie, R.O.A., Bakhit, D.A.M. Some New Results on the Uniform Asymptotic Stability for Volterra Integro-differential Equations with Delays. Mediterr. J. Math. 20, 280 (2023). https://doi.org/10.1007/s00009-023-02489-w
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DOI: https://doi.org/10.1007/s00009-023-02489-w