Asymptotic behavior of Levin-Nohel nonlinear difference system with several delays

1.   Introduction

In the last century, difference equations have been applied to solve many problems in statistics, science and engineering. Difference equations are used to approximate ordinary and partial differential equations due to the development of computing machinery. In addition to approximating these equations, they provide a powerful method for analyzing mechanical, electrical, and other systems with repeated identical sections. Using difference equations greatly facilitates the study of insulator strings, electric-wave filters, magnetic amplifiers, multistage amplifiers, continuous beams of equal span, and acoustical filters (see ^[2,7,14,15]). Among the important equations that the researchers highlighted and which we will also study in our paper is the Levin-Nohel equation (see ^{[1,4,5,6,8,11,13,16]}).

Many researchers resort to using the fixed point theorems and the Lyapunov function to study the qualitative properties of difference systems with and without delays because these methods give impressive results and support the conformity of the conditions of the studied phenomenon with the reality (see ^{[3,4,5,6,8,9,10,11,12]}).

Let $\mathbb{Z}$ the set of integers. In this paper we denote $\mathbb{Z} _{a} = \left\{ a, a+1, ...\right\}$ and $\mathbb{Z} _{a}^{b} = \left\{ a, a+1, ..., b-1, b\right\}$ for $a, b\in \mathbb{Z}$.

Khelil in ^[1] obtained results for asymptotic stability of the following difference equation

with initial condition given by

such that

In ^[11], we considered the linear Levin-Nohel integro-differential system

with initial condition given by

and we studied the asymptotic stability.

Using the above works as motivation, we present in this paper the nonlinear Levin-Nohel difference system

with initial condition given by

where $\Delta u\left(m\right) = u\left(m+1\right) -u\left(m\right)$ is the forward difference operator for any sequence $\left\{ u\left(m\right), \ u\left(m_{0}\right) = u_{0}, \ m\in \mathbb{N} \right\}$ and $p\left(m\right), q\left(m\right) : \mathbb{Z} _{m_{0}}\rightarrow \mathbb{N},$ such that $m-p\left(m\right), m-q\left(m\right) \rightarrow \infty$ when the $m\rightarrow \infty$, $g: \mathbb{R} ^{N}\rightarrow \mathbb{R} ^{N}$ real sequence satisfies $g\left(0\right) = 0$. The $N\times N$ matrices $C: \mathbb{Z} _{m_{0}}\times \mathbb{Z} _{\theta \left(m_{0}\right) }\rightarrow \mathbb{R} ^{N^{2}}$ and $A: \mathbb{Z} _{m_{0}}\rightarrow \mathbb{R} ^{N^{2}}$ are bounded with real sequences as its elements.

The asymptotic behavior of the above system has never been investigated by applying Banach's fixed point theorem. There is known literature on the Levin-Nohel integro-differential systems, but the specific system (1.3) may not be examined yet, So we use the fixed point theorem of Banach to show stability, asymptotic stability and the exponential stability of solutions for the system (1.3).

2.   Preliminaries

Let $A$ be an $N\times N$ matrix valued sequence and consider the homogeneous linear system

Definition 1. The state transition matrix $t\rightarrow \mathcal{Q}\left(t, q\right)$ for the homogeneous linear system (2.1) on the open interval $J$ is the family of fundamental matrix solutions parameterized by $q\in J$ satisfying $\mathcal{Q}\left(q, q\right) = I$, where $I$ is the $N\times N$ identity matrix.

Throughout this manuscript, we assume that the matrix $I+A\left(m\right)$ is nonsingular and we define the forward operator $E$ by $Eu\left(m\right) = u\left(m+1\right)$. Furthermore, the fundamental matrix solution $\mathcal{Q}\left(m\right)$ of the unperturbed linear system (2.1) satisfies:

$({\rm{a}})$ $\det \mathcal{Q}\left(m\right) \neq 0.$

$({\rm{b}})$ $\mathcal{Q}\left(m+1\right) = \left(I+A\left(m\right) \right) \mathcal{Q}\left(m\right)$ and $\mathcal{Q}^{-1}\left(m+1\right) = \mathcal{Q}^{-1}\left(m\right) \left(I+A\left(m\right) \right) ^{-1}$.

In Lemma 1, we convert system (1.3) to a new convenient system to facilitate the application of the fixed point techniques.

Lemma 1. If $u\left(m\right) : \mathbb{Z} _{m_{0}}\rightarrow \mathbb{R} ^{m}$ is the solution of (1.3), then system (1.3) is equivalent to

where

Proof. First, we can write system (1.3) as the form

Let $u$ be a solution of (1.3) and $\mathcal{Q}\left(m, m_{0}\right)$ be a fundamental matrix of (2.1). Since

it follows that

This implies

On the other hand,

then

A summation of the above equation from $m_{0}$ to $m-1$ gives (2.2). It is easy to obtain the converse implication, and the proof is complete. □

3.   Main results

Let $\left(\mathcal{S}, \left\Vert \cdot \right\Vert \right)$ be the Banach space of bounded sequences $u:m\in \mathbb{Z} _{m_{0}}\rightarrow \mathbb{R} ^{N}$ with the maximum norm.

where $\left\vert \cdot \right\vert$ is the infinity norm for $u\in \mathbb{R} ^{N}$. We define the norm of $A\left(m\right) : = \left[a_{ij}\left(m\right) \right]$ by

where

In this paper, we assume that there exists a constant $L_{g} > 0$ such that for $u, v\in \mathbb{R} ^{N}$

Definition 2. We say that the zero solution of (1.3) is Lyapunov stable if for any $\epsilon > 0$ and $m_{0}\in \mathbb{Z}$ there exists $\delta > 0$ such that $\left\vert \omega \left(m\right) \right\vert \leq \delta$ for $m\in \mathbb{Z} _{\theta \left(m_{0}\right) }^{m_{0}}$, which implies $\left\vert u\left(m, m_{0}, u_{0}\right) \right\vert \leq \epsilon$ for $m\in \mathbb{Z} _{m_{0}}$.

Theorem 1. Assume there exists $M > 0$ and $\gamma \in \left(0, 1\right)$ such that for $m\in \mathbb{Z} _{m_{0}}$,

then the zero solution of (1.3) is stable.

Proof. Let $\epsilon > 0$ and choose $\delta > 0$ such that for $\left\vert u\left(m\right) \right\vert \leq \delta, \forall m\in \mathbb{Z} _{\theta \left(m_{0}\right) }^{m_{0}}$, we have

Define

then $\left(\Gamma _{\epsilon }, \left\Vert \cdot \right\Vert \right)$ is a complete metric space with the maximum norm.

We define the operator $\digamma :\Gamma _{\epsilon }\rightarrow \mathcal{S}$ due to Lemma 1 by

for $m\in \mathbb{Z} _{m_{0}}$.

We first prove that $\digamma$ maps $\Gamma _{\epsilon }$ into $\Gamma _{\epsilon }$. So, by (3.1)–(3.3)

We next prove that $\digamma$ is a contraction.

Let $u, v\in \Gamma _{\epsilon }$, then

Hence,

since $\gamma \in \left(0, 1\right)$, then $\digamma$ is a contraction.

Thus, by the fixed point of Banach, $\digamma$ has a unique fixed point $u$ in $\Gamma _{\epsilon }$, which is a solution of (1.3) with $u\left(m_{0}\right) = u_{0}$ and $\left\vert u\left(m\right) \right\vert = \left\vert u\left(m, m_{0}, u_{0}\right) \right\vert \leq \epsilon$ for $m\in \mathbb{Z} _{m_{0}}$. This proves that the zero solution of (1.3) is stable. □

Definition 3. We say that the zero solution of (1.3) is asymptotically stable if it is stable and if for any integer $m_{0}\geq 0$ there exists $\delta > 0$, such that $\left\vert \omega \left(m\right) \right\vert \leq \delta$ for $m\in \left[\theta \left(m_{0}\right), m_{0}\right]$, which implies $\left\vert u\left(m, m_{0}, u_{0}\right) \right\vert \rightarrow 0$ as $m\rightarrow \infty$.

Theorem 2. If (3.1)–(3.3) and

hold, then the zero solution of (1.3) is asymptotically stable.

Proof. We have shown by our last theorem that the zero solution of (1.3) is stable. For a given $\epsilon > 0$ define

Define $\digamma :\Gamma _{0}\rightarrow \Gamma _{\epsilon }$ by (3.4). We must prove that for $u\in \Gamma _{0}, \ \left(\digamma u\right) \left(m\right) \rightarrow 0$ when $m\rightarrow \infty$. By definition of $\Gamma _{0}$, $u\left(m\right) \rightarrow 0$, $\ $as $m\rightarrow \infty$. Thus, we get

By (3.5),

Moreover, let $u\in \Gamma _{0}$ so that for any $\epsilon _{1}\in \left(0, \epsilon \right),$ there exists $T\geq m_{0}$ large enough such that $s\geq T$ implies $\left\vert u\left(s-p\left(s\right) \right) \right\vert, \left\vert u\left(s-q\left(s\right) \right) \right\vert < \epsilon _{1}$. Hence, we get

Thus, $\Lambda \rightarrow 0$ as $m\rightarrow \infty$.

Hence, $\digamma$ maps $\Gamma _{0}$ into itself. By the fixed point of Banach, $\digamma$ has a unique fixed point $u\in \Gamma _{0}$, which solves (1.3). □

Definition 4. We say that the zero solution of (1.3) is exponentially stable if there exist $\delta, \sigma > 0$ and $\lambda \in \left(0, 1\right)$ such that

whenever $\left\vert u_{0}\right\vert < \delta$.

Theorem 3. The zero solution of (1.3) is exponentially stable if

1) The conditions (3.1) and (3.3) hold;

2) There exists $\lambda \in \left(0, 1\right)$ such that

and

Proof. Since the condition (3.7) holds, we define $\Gamma _{e}$ as the closed subspace of $\mathcal{S}$ as

We will show that $\digamma \left(\Gamma _{e}\right) \subset \Gamma _{e}$, then by (3.7), we have

since (3.8) holds. Thus, we have

then $\digamma \left(\Gamma _{e}\right) \subset \Gamma _{e}$.

Hence, there exists a unique fixed point $u\in \Gamma _{e}$ that solves (1.3), then the zero solution of (1.3) is stable exponentially. □

4.   Conclusions

In this research paper, the theoretical study of stability, asymptotic stability and exponential stability was addressed by using the fixed point theorem of Banach. Some new criteria was imposed on the fundamental matrix solution and system components to obtain the stability, asymptotic stability and exponential stability. The considered system contained two functional delays. However, the obtained results for Equation (1.3) can be extended to more than two delays.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research has been funded by Scientific Research Deanship at University of Ha'il-Saudi Arabia through project number $< <$ RG-23 045 $> >$.

Conflict of interest

The authors declare no competing interests.