Asymptotic Behavior of Solutions in Nonlinear Neutral System with Two Volterra Terms

Abstract

In this manuscript, we generalise previous results in the literature by providing sufficient conditions for the matrix measure to guarantee the stability, asymptotic stability and exponential stability of a neutral system of differential equations. This is achieved by constructing a suitable operator from our system and applying the Banach fixed point theorem.

1. Introduction

The basic applications of Volterra equation, which was introduced in 1928 by Volterra himself, can be found in [1,2,3]. After that, the Volterra integro-differential equation appeared, which has several applications in many fields, such as nanohydrodynamics, glassforming process, diffusion process in general, heat transfer, neutron diffusion, and wind ripple in the desert. More details about these equations and their applications can be found in [4,5,6,7,8].

Many researchers studied the existence of solutions and asymptotic behavior of these equations; for example, in [9,10,11,12], the authors considered the exponential stability of continuous, discrete, and stochastic differential equations. In [13,14,15,16,17,18,19], the authors dealt with the existence and stability of one or n-dimensional equations.

A wide variety of differential equations were successfully stabilized using Lyapunov’s direct method. Over the last century, Lyapunov theory has been a very fruitful field, studying a qualitative theory of differentials and systems. However, the expressions of Lyapunov functionals are very complicated and hard to construct in many problems, as is the case when the differential equation has unbounded terms or when it contains unbounded functional delays. To overcome the shortcomings of Lyapunov’s direct method, in recent years, several researchers have investigated different aspects of the qualitative theory of differential equations using the fixed-point method, which was found to be significantly advantageous compared to Lyapunov’s direct method; see instance [20,21,22].

In [23], Dung studied the following problem of mixed linear Levin–Nohel integro-differential equations.

$$\left\{\begin{array}{c}{u}^{\prime }\left(p\right)+{\int }_{p-\beta \left(p\right)}^{p}c\left(p,\eta \right)u\left(\eta \right)d\eta +b\left(p\right)u\left(p-\gamma \left(p\right)\right)=0,p\ge p_0,\\ u\left(p\right)=\psi \left(p\right)\mathrm{for}p\in \left[m\left(p_0\right),p_0\right],\end{array}\right.$$

(1)

while the following n-dimensional version of the above problem

$$\left\{\begin{array}{c}{u}^{\prime }\left(p\right)+{\int }_{p-\beta \left(p\right)}^{p}C\left(p,\eta \right)u\left(\eta \right)d\eta +B\left(p\right)u\left(p-\gamma \left(p\right)\right)=0,p\ge p_0,\hfill \\ u\left(p\right)=\psi \left(p\right)\mathrm{for}p\in \left[m\left(p_0\right),p_0\right],\hfill \end{array}\right.$$

(2)

with

$$m\left(p_0\right)=min\left(\underset{\eta \ge p_0}{inf}\left\{\eta -\beta \left(\eta \right)\right\},\underset{\eta \ge p_0}{inf}\left\{\eta -\gamma \left(\eta \right)\right\}\right),$$

has been considered in [19], as well as its asymptotic stability.

In this work, we propose generalizing these problems by considering the following nonlinear neutral differential system with two Volterra terms, in addition to the study of the exponential stability

$$\left\{\begin{array}{c}\begin{array}{c}{\left(u\left(p\right)-{\int }_{p-\alpha \left(p\right)}^p{C}_1\left(p,\eta \right)u\left(\eta \right)d\eta \right)}^{\prime }={\int }_{p-\beta \left(p\right)}^p{C}_2\left(p,\eta \right)u\left(\eta \right)d\eta \\ +Q\left(p,u\left(p\right),u\left(p-\gamma \left(p\right)\right)\right),p\ge p_0,\end{array}\hfill \\ u\left(p\right)=\psi \left(p\right)\mathrm{for}p\in \left[\tau \left(p_0\right),p_0\right],\hfill \end{array}\right.$$

(3)

with

$$\tau \left(p_0\right)=min\left(\underset{\eta \ge p_0}{inf}\left\{\eta -\alpha \left(\eta \right)\right\},\underset{\eta \ge p_0}{inf}\left\{\eta -\beta \left(\eta \right)\right\},\underset{\eta \ge p_0}{inf}\left\{\eta -\gamma \left(\eta \right)\right\}\right),$$

where

• $u:\left[\tau \left(p_0\right),\infty \right)⟶{\mathbb{R}}^n$ and $Q:\left[p_0,\infty \right)\times {\mathbb{R}}^n\times {\mathbb{R}}^n⟶{\mathbb{R}}^n$ are continuous vector functions, such that $Q\left(p,0,0\right)\equiv 0$.
• $\alpha ,\beta ,\gamma :\mathbb{R}⟶{\mathbb{R}}^{+}$ is a continuous real function such that $p-\alpha \left(p\right),p-\beta \left(p\right),p-\gamma \left(p\right)⟶\infty$ when $n⟶\infty$.
• $C_1,C_2:\left[p_0,\infty \right)\times \left[\tau \left(p_0\right),\infty \right)\to {\mathbb{R}}^{n^2}$ are bounded with a continuous real function such that $C_2\left(p,p\right)$ is nonsingular for all $p\in \left[p_0,\infty \right)$.

In this context, the main contribution of this paper is to apply the Banach fixed-point theorem to the show stability, asymptotic stability and exponential stability of solutions for the system (3), which is the generalization of the papers [19,23]. The paper is organized as follows. In Section 2, we present some previous results, and construct the integral form of the problem. After this, we prove the stability, asymptotic stability, and exponential stability of solutions for the system (3) in Section 3, Section 4 and Section 5, respectively. An example is given in Section 6 to illustrate our results, and a conclusion is provided at the end of this article.

2. Preliminaries

We start this section by transforming the system (3) to a more tractable system. Therefore, in the analysis, we use the fundamental matrix solution of

$$y^{\prime }\left(p\right)=C_2\left(p,p\right)y\left(p\right),$$

(4)

from which we obtain a fixed-point mapping.

Throughout this paper, $\mathcal{Q}\left(p,p_0\right)$ will denote a fundamental matrix solution of the homogeneous (unperturbed) linear problem (4).

Lemma 1.

We say that u is a solution of the problem (3) if and only if

$$\begin{array}{cc}\hfill u\left(p\right)& =\mathcal{Q}\left(p,p_0\right)\left(u\left(p_0\right)-{\int }_{p_0-\alpha \left(p_0\right)}^{p_0}{C}_1\left(p_0,\eta \right)u\left(\eta \right)d\eta \right)\hfill \\ \hfill & +{\int }_{p_0}^p\mathcal{Q}\left(p,\eta \right)\left[C_2\left(\eta ,\eta \right)\left({\int }_{\eta -\alpha \left(\eta \right)}^{\eta }{C}_1\left(\eta ,r\right)u\left(r\right)dr-u\left(\eta \right)\right)d\eta \right.\hfill \\ \hfill & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }{C}_2\left(\eta ,r\right)u\left(r\right)dr+Q\left(\eta ,u\left(\eta \right),u\left(\eta -\gamma \left(\eta \right)\right)\right)\right]d\eta \hfill \end{array}$$

(5)

for $p\ge p_0$ and $u\left(p\right)=\psi \left(p\right)$ for $p\in \left[\tau \left(p_0\right),p_0\right]$.

Proof. 

Let

$$y\left(p\right)=u\left(p\right)-{\int }_{p-\alpha \left(p\right)}^p{C}_1\left(p,\eta \right)u\left(\eta \right)d\eta .$$

Then, we can write the system (3) as

$$\begin{array}{cc}\hfill y\left(p\right)& =C_2\left(p,p\right)y\left(p\right)+{\int }_{p-\beta \left(p\right)}^p{C}_2\left(p,\eta \right)u\left(\eta \right)d\eta \hfill \\ \hfill & -C_2\left(p,p\right)y\left(p\right)+Q\left(p,u\left(p\right),u\left(p-\gamma \left(p\right)\right)\right).\hfill \end{array}$$

Hence

$$\begin{array}{ccc}\hfill y\left(p\right)& =& \mathcal{Q}\left(p,p_0\right)y\left(p_0\right)-{\int }_{p_0}^p\mathcal{Q}\left(p,\eta \right)\hfill \\ & & \times \left[{\int }_{p-\beta \left(p\right)}^p{C}_2\left(p,\eta \right)u\left(\eta \right)d\eta -C_2\left(\eta ,\eta \right)y\left(\eta \right)+Q\left(\eta ,u\left(\eta \right),u\left(\eta -\gamma \left(\eta \right)\right)\right)\right]d\eta ,\hfill \end{array}$$

witch equivalent to (5). □

Lemma 2.

([24]). The state transition matrix $\mathsf{\Phi }\left(p,\eta \right)$ of Equation (4) satisfies

$$∥\mathsf{\Phi }\left(p,\eta \right)∥\le e^{{\int }_{\eta }^p\mu \left(C_2\left(p,r\right)\right)dr},p\ge \eta ,$$

where $\mu \left(C_2\left(p,r\right)\right)$ is the Matrix measure of $C_2\left(p,r\right)$.

Let $\left(\mathcal{S},∥·∥\right)$ be the Banach space of vector continuous functions $u:\left[\tau \left(p_0\right),\infty \right)⟶{\mathbb{R}}^n$ with the supremum norm.

$$\parallel u\parallel =\underset{p\in \left[\tau \left(p_0\right),\infty \right)}{sup}\left|u\left(p\right)\right|,$$

where $\left|·\right|$ is the infinity norm for $u\in {\mathbb{R}}^n$. If A is an $n\times n$ matrix valued function A given by $A\left(p\right):=\left[a_{ij}\left(p\right)\right]$, then we define the norm of A by

$$∥A∥:=\underset{p\in \left[\tau \left(p_0\right),\infty \right)}{sup}\left|A\left(p\right)\right|,$$

where

$$\left|A\left(p\right)\right|=\underset{1\le i\le n}{max}\sum _{j=1}^n\left|a_{ij}\left(p\right)\right|.$$

Using Lemma 1, we define the operator $\mathcal{Ϝ}:{\mathrm{\Gamma }}_{ϵ}\to \mathcal{S}$ by

$$\begin{array}{cc}\hfill \left(\mathcal{Ϝ}u\right)\left(p\right)& =\mathcal{Q}\left(p,p_0\right)\left(\psi \left(p_0\right)-{\int }_{p_0-\alpha \left(p_0\right)}^{p_0}{C}_1\left(p_0,\eta \right)\psi \left(\eta \right)d\eta \right)\hfill \\ \hfill & +{\int }_{p_0}^p\mathcal{Q}\left(p,\eta \right)\left[{\int }_{\eta -\beta \left(\eta \right)}^{\eta }{C}_2\left(\eta ,r\right)u\left(r\right)dr\right.\hfill \\ \hfill & +C_2\left(\eta ,\eta \right)\left({\int }_{\eta -\alpha \left(\eta \right)}^{\eta }{C}_1\left(\eta ,r\right)u\left(r\right)dr-u\left(\eta \right)\right)\hfill \\ \hfill & +\left.Q\left(\eta ,u\left(\eta \right),u\left(\eta -\gamma \left(\eta \right)\right)\right)\right]d\eta \hfill \end{array}$$

(6)

for $p\in \left[p_0,\infty \right)$. Therefore, in this paper, we need the following conditions

(C1)

The function Q satisfies

$$\begin{array}{cc}\hfill & \left|Q\left(\eta ,u_1\left(\eta \right),u_1\left(\eta -\gamma \left(\eta \right)\right)\right)-Q\left(\eta ,u_2\left(\eta \right),u_2\left(\eta -\gamma \left(\eta \right)\right)\right)\right|\hfill \\ \hfill & \le q_1\left|u_1\left(\eta \right)-u_2\left(\eta \right)\right|+q_2\left|u_1\left(\eta -\gamma \left(\eta \right)\right)-u_2\left(\eta -\gamma \left(\eta \right)\right)\right|,\hfill \end{array}$$

for all $u_1,u_2\in \mathcal{S}$ and $p\in \left[p_0,\infty \right)$.

(C2)

There is a positive constant $L\in \left(0,1\right)$ such that

$$\begin{array}{cc}\hfill L& :={\displaystyle \underset{p\in \left[\tau \left(p_0\right),\infty \right)}{sup}}{\int }_{p_0}^p\left(∥C_2\left(\eta ,\eta \right)∥\left({\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥dr+1\right)\right.\hfill \\ \hfill & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥dr+q_1+q_2\right)d\eta .\hfill \end{array}$$

3. Stability

Definition 1.

The zero solution of (3) is Lyapunov stable if, for any $ϵ>0$ and any integer $p\in \left[p_0,\infty \right)$, there exists $\delta >0$ such that $\left|\psi \left(p\right)\right|\le \delta$ for $p\in \left[\tau \left(p_0\right),p_0\right]$ implies $\left|u\left(p,p_0,\psi \right)\right|\le ϵ$ for $p\in \left[p_0,\infty \right)$.

Theorem 1.

Suppose the conditions (C1), (C2) and

$${\int }_{\eta }^p\mu \left(C_2\left(r,r\right)\right)dr\le M,p\ge \eta ,$$

where M is a positive constant. Then, the zero solution of (3) is stable.

Proof. 

Let $ϵ>0$ be given. Choose $\delta >0$, such that for

$$\left|\psi \left(p_0\right)-{\int }_{p_0-\alpha \left(p_0\right)}^{p_0}{C}_1\left(p_0,\eta \right)\psi \left(\eta \right)d\eta \right|\le \delta ,\forall p\in \left[\tau \left(p_0\right),p_0\right],$$

and

$$e^M\left(\delta +Lϵ\right)\le ϵ.$$

Define complete metric space

$${\mathrm{\Gamma }}_{ϵ}=\left\{u\in \mathcal{S}:∥u∥\le \delta ,\forall p\in \left[\tau \left(p_0\right),p_0\right]\mathrm{and}∥u∥\le ϵ,\forall p\in \left[p_0,\infty \right)\right\}.$$

We first prove that $\mathcal{Ϝ}$ maps ${\mathrm{\Gamma }}_{ϵ}$ into ${\mathrm{\Gamma }}_{ϵ}$. Therefore, using the conditions (C1) and (C2), we have

$$\begin{array}{cc}\hfill \left|\left(\mathcal{Ϝ}u\right)\left(p\right)\right|& \le ∥\mathcal{Q}\left(p,p_0\right)∥\left|\psi \left(p_0\right)-{\int }_{p_0-\alpha \left(p_0\right)}^{p_0}{C}_1\left(p_0,\eta \right)\psi \left(\eta \right)d\eta \right|\hfill \\ \hfill & +{\int }_{p_0}^p∥\mathcal{Q}\left(p,\eta \right)∥\left[∥C_2\left(\eta ,\eta \right)∥\left({\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+\left|u\left(\eta \right)\right|\right)\right.\hfill \\ \hfill & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+\left|Q\left(\eta ,u\left(\eta \right),u\left(\eta -\gamma \left(\eta \right)\right)\right)\right|\right]d\eta \hfill \\ \hfill & \le e^{{\int }_{p_0}^p\mu \left(C_2\left(r,r\right)\right)dr}\left|\psi \left(p_0\right)-{\int }_{p_0-\alpha \left(p_0\right)}^{p_0}{C}_1\left(p_0,\eta \right)\psi \left(\eta \right)d\eta \right|\hfill \\ \hfill & +{\int }_{p_0}^p{e}^{{\int }_{\eta }^p\mu \left(C_2\left(r,r\right)\right)dr}\left[∥C_2\left(\eta ,\eta \right)∥\left({\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+\left|u\left(\eta \right)\right|\right)\right.\hfill \\ \hfill & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+\left|Q\left(\eta ,u\left(\eta \right),u\left(\eta -\gamma \left(\eta \right)\right)\right)\right|\right]d\eta \hfill \\ \hfill & \le e^M\left(\delta +Lϵ\right)\le ϵ,\hfill \end{array}$$

hence

$$∥\mathcal{Ϝ}u∥\le ϵ.$$

We next prove that $\mathcal{Ϝ}$ is a contraction.

Let $u,v\in {\mathrm{\Gamma }}_{ϵ}$; then,

$$\begin{array}{cc}\hfill \left|\left(\mathcal{Ϝ}u\right)\left(p\right)-\left(\mathcal{Ϝ}v\right)\left(p\right)\right|& \le {\int }_{p_0}^p{e}^{{\int }_{\eta }^p\mu \left(C_2\left(r,r\right)\right)dr}\left[∥C_2\left(\eta ,\eta \right)∥\left({\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥dr+1\right)\right.\hfill \\ \hfill & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥dr+q_1+q_2\right]d\eta ∥u-v∥\hfill \\ \hfill & \le L∥u-v∥.\hfill \end{array}$$

Hence,

$$∥\mathcal{Ϝ}u-\mathcal{Ϝ}v∥\le L∥u-v∥,$$

since $L\in \left(0,1\right)$, then $\mathcal{Ϝ}$ is a contraction.

Thus, using the fixed point of Banach, $\mathcal{Ϝ}$ has a unique fixed point u in ${\mathrm{\Gamma }}_{ϵ}$, which is a solution of (3). This proves that the zero solution of (3) is stable. □

4. Asymptotic Stability

Definition 2.

The zero solution of (3) is asymptotically stable if it is Lyapunov stable and if, for any integer $p_0\ge 0$, there exists $\delta >0$ such that $\left|\psi \left(p\right)\right|\le \delta$ for $p\in \left[\tau \left(p_0\right),p_0\right]$ implies $\left|u\left(p\right)\right|\to 0$ as $p\to \infty$.

Theorem 2.

Assume that the hypotheses (C1), (C2) and

$${\int }_{\eta }^p\mu \left(C_2\left(r,r\right)\right)dr\to -\infty ,p\ge \eta ,,\mathrm{as}p\to \infty ,$$

(7)

hold. Then, the zero solution of (3) is asymptotically stable.

Proof. 

In the last Theorem, we proved that the zero solution of (3) is stable. For a given $ϵ>0$, define

$${\mathrm{\Gamma }}_0=\left\{u\in {\mathrm{\Gamma }}_{ϵ}\mathrm{such}\mathrm{that}u\left(p\right)\to 0,\mathrm{as}p\to \infty \right\}.$$

Define $\mathcal{Ϝ}:{\mathrm{\Gamma }}_0\to {\mathrm{\Gamma }}_{ϵ}$ by (6). We must prove that, for $u\in {\mathrm{\Gamma }}_0,\left(\mathcal{Ϝ}u\right)\left(p\right)\to 0$ when $p\to \infty$. Using the definition of ${\mathrm{\Gamma }}_0$, $u\left(p\right)\to 0$,as $p\to \infty$. Thus, we obtain

$$\begin{array}{cc}\hfill \left(\mathcal{Ϝ}u\right)\left(p\right)& \le \delta e^{{\int }_{p_0}^p\mu \left(C_2\left(r,r\right)\right)dr}\hfill \\ \hfill & {\int }_{p_0}^p{e}^{{\int }_{\eta }^p\mu \left(C_2\left(r,r\right)\right)dr}\left[∥C_2\left(\eta ,\eta \right)∥\left({\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+\left|u\left(\eta \right)\right|\right)\right.\hfill \\ \hfill & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+q_1\left|u\left(\eta \right)\right|+q_2\left|u\left(\eta -\gamma \left(\eta \right)\right)\right|\right]d\eta \hfill \end{array}$$

By (7)

$$\delta e^{{\int }_{p_0}^p\mu \left(C_2\left(r,r\right)\right)dr}\to 0\mathrm{when}n\to \infty .$$

Moreover, let $u\in {\mathrm{\Gamma }}_0$ so that, for any ${ϵ}_1\in \left(0,ϵ\right),$ there exists $T\ge p_0$ large enough such that $\eta \ge T-\tau \left(p_0\right)$ implies $\left|u\left(\eta \right)\right|,\left|u\left(\eta -\gamma \left(\eta \right)\right)\right|<{ϵ}_1$. Hence, we obtain

$$\begin{array}{cc}\hfill \mathrm{\Lambda }& ={\int }_{p_0}^p{e}^{{\int }_{\eta }^p\mu \left(C_2\left(r,r\right)\right)dr}\left[∥C_2\left(\eta ,\eta \right)∥\left({\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+\left|u\left(\eta \right)\right|\right)\right.\hfill \\ \hfill & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+q_1\left|u\left(\eta \right)\right|+q_2\left|u\left(\eta -\gamma \left(\eta \right)\right)\right|\right]d\eta \hfill \\ \hfill & \le e^{{\int }_{p_0}^p\mu \left(C_2\left(r,r\right)\right)dr}{\int }_{p_0}^T{e}^{{\int }_{\eta }^{p_0}\mu \left(C_2\left(r,r\right)\right)dr}\left[∥C_2\left(\eta ,\eta \right)∥\left({\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+\left|u\left(\eta \right)\right|\right)\right.\hfill \\ \hfill & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+q_1\left|u\left(\eta \right)\right|+q_2\left|u\left(\eta -\gamma \left(\eta \right)\right)\right|\right]d\eta \hfill \\ \hfill & +{\int }_T^p{e}^{{\int }_{\eta }^p\mu \left(C_2\left(r,r\right)\right)dr}\left[∥C_2\left(\eta ,\eta \right)∥\left({\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+\left|u\left(\eta \right)\right|\right)\right.\hfill \\ \hfill & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥\left|u\left(r\right)\right|dr+q_1\left|u\left(\eta \right)\right|+q_2\left|u\left(\eta -\gamma \left(\eta \right)\right)\right|\right]d\eta \hfill \\ \hfill & \le L{e}^{{\int }_{p_0}^p\mu \left(C_2\left(r,r\right)\right)dr}∥u∥+L{ϵ}_1.\hfill \end{array}$$

Since (7) holds, then $\mathrm{\Lambda }\to 0$, as $p\to \infty$.

Hence, $\mathcal{Ϝ}$ maps ${\mathrm{\Gamma }}_0$ into itself. Using the fixed point of Banach, $\mathcal{Ϝ}$ has a unique fixed point $u\in {\mathrm{\Gamma }}_0$, which solves (3). □

5. Exponential Stability

Definition 3.

We can say that the zero solution of (3) is exponentially stable if there exists$\sigma ,\delta ,\lambda >0$, such that

$$\left|u\left(p\right)\right|<\sigma e^{-\lambda \left(p-p_0\right)},p\ge p_0,$$

(8)

whenever $\left|\psi \left(p\right)\right|<\delta$ for $p\in \left[\tau \left(p_0\right),p_0\right]$.

Theorem 3.

Assume that conditions (C1) and (C2) hold if there exists $\lambda >0$, such that

$$\mu \left(C_2\left(p,p\right)\right)\le -\lambda ,\forall p\ge p_0,$$

(9)

and

$$\begin{array}{c}{\displaystyle \underset{p\in \left[\tau \left(p_0\right),\infty \right)}{sup}}{\int }_{p_0}^p\left[e^{\lambda \eta }∥C_2\left(\eta ,\eta \right)∥{\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥e^{-\lambda r}dr+∥C_2\left(\eta ,\eta \right)∥\right.\\ +\left.e^{\lambda \eta }{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥e^{-\lambda r}dr+q_1+q_2{e}^{\lambda \gamma \left(\eta \right)}\right]d\eta \le \frac{1}{2}\end{array}$$

(10)

hold. Then, the zero solution of (3) is exponentially stable.

Proof. 

Since the condition (9) holds. We define ${\mathrm{\Gamma }}_e$, the closed subspace of $\mathcal{S}$, as

$${\mathrm{\Gamma }}_e=\left\{u\in \mathcal{S}:\mathrm{such}\mathrm{that}\left|u\left(p\right)\right|\le \sigma e^{-\lambda \left(p-p_0\right)},\forall p\ge \tau \left(p_0\right)\mathrm{and}\sigma \ge 2\delta \right\}.$$

We will show that $\mathcal{Ϝ}\left({\mathrm{\Gamma }}_e\right)\subset {\mathrm{\Gamma }}_e$. Then, using (9), we have

$$\begin{array}{cc}\hfill \left(\mathcal{Ϝ}u\right)\left(p\right)& \le e^{{\int }_{p_0}^p\mu \left(B\left(r,r\right)\right)dr}\delta +{\int }_{p_0}^p{e}^{{\int }_{\eta }^p\mu \left(B\left(r,r\right)\right)dr}\hfill \\ \hfill & \times \left[∥C_2\left(\eta ,\eta \right)∥{\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥\sigma e^{-\lambda \left(r-p_0\right)}dr+∥C_2\left(\eta ,\eta \right)∥\sigma e^{-\lambda \left(\eta -p_0\right)}\right.\hfill \\ \hfill & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥\sigma e^{-\lambda \left(r-p_0\right)}dr++q_1\sigma e^{-\lambda \left(\eta -p_0\right)}+q_2\sigma e^{-\lambda \left(\eta -\gamma \left(\eta \right)-p_0\right)}\right]d\eta \hfill \\ \hfill & \le e^{-\lambda \left(p-p_0\right)}\delta +{\int }_{p_0}^p{e}^{-\lambda \left(p-\eta \right)}\hfill \\ \hfill & \times \left[∥C_2\left(\eta ,\eta \right)∥{\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥\sigma e^{-\lambda \left(r-p_0\right)}dr+∥C_2\left(\eta ,\eta \right)∥\sigma e^{-\lambda \left(\eta -p_0\right)}\right.\hfill \\ \hfill & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥\sigma e^{-\lambda \left(r-p_0\right)}dr+q_1\sigma e^{-\lambda \left(\eta -p_0\right)}+q_2\sigma e^{-\lambda \left(\eta -\gamma \left(\eta \right)-p_0\right)}\right]d\eta \hfill \\ \hfill & =e^{-\lambda \left(p-p_0\right)}\delta +\sigma e^{-\lambda \left(p-p_0\right)}\hfill \\ \hfill & \times {\int }_{p_0}^p\left[e^{\lambda \eta }∥C_2\left(\eta ,\eta \right)∥{\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥e^{-\lambda r}dr+∥C_2\left(\eta ,\eta \right)∥\right.\hfill \\ \hfill & +\left.e^{\lambda \eta }{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥e^{-\lambda r}dr+q_1+q_2{e}^{\lambda \gamma \left(\eta \right)}\right]d\eta \hfill \end{array}$$

since (10) holds. Then, we have

$$\left|\left(\mathcal{Ϝ}u\right)\left(p\right)\right|\le \frac{1}{2}\sigma e^{-\lambda \left(p-p_0\right)}+\frac{1}{2}\sigma e^{-\lambda \left(p-p_0\right)}=\sigma e^{-\lambda \left(p-p_0\right)}.$$

Then $\mathcal{Ϝ}\left({\mathrm{\Gamma }}_e\right)\subset {\mathrm{\Gamma }}_e$.

Hence, $\mathcal{Ϝ}$ has a unique fixed point $u\in {\mathrm{\Gamma }}_e$, which solves (3). Then, the zero solution of (3) is exponentially stable. □

6. Example

Consider the two-dimensional nonlinear neutral differential system of the form

$$\left\{\begin{array}{c}\begin{array}{c}{\left(u\left(p\right)-{\int }_{p-\frac{1}{1+p^2}+1}^p{C}_1\left(p,\eta \right)u\left(\eta \right)d\eta \right)}^{\prime }={\int }_{p-\frac{1}{1+p^2}+12}^p{C}_2\left(p,\eta \right)u\left(\eta \right)d\eta \\ +Q\left(p,u\left(p\right),u\left(p-ln{2}^8\right)\right),p\ge 0,\end{array}\hfill \\ u\left(p\right)=\psi \left(p\right)\mathrm{arbitrary}\mathrm{for}p\in \left[-\frac{1}{3},0\right],\hfill \end{array}\right.$$

(11)

where

$$C_1\left(p,\eta \right)=\left(\begin{array}{cc}-2& sin\left(p-\eta \right)\\ sin\left(p-\eta \right)& -2\end{array}\right),$$

$$C_2\left(p,\eta \right)=\frac{1}{8}\left(\begin{array}{cc}-1& \frac{1}{2}cos\left(p-\eta \right)\\ \frac{1}{2}cos\left(p-\eta \right)& -1\end{array}\right),$$

and

$$Q\left(p,x,y\right)=\left(\begin{array}{c}\frac{1}{4}{x}_{2}sint\\ \frac{1}{2}{y}_{1}sint\end{array}\right)⟹q_1=\frac{1}{4},q_2=\frac{1}{2}.$$

Since for $A={\left(a_{ij}\right)}_{n\times n}\in {\mathbb{R}}^{n\times n}$, we have

$$\mu \left(A\right)=\underset{1\le j\le n}{max}\left\{a_{jj}+\sum _{i\ne j}^n\left|a_{ij}\right|\right\}.$$

Then

$$\mu \left(C_2\left(p,\eta \right)\right)=\frac{1}{8}\left(-2+cos\left(p-\eta \right)\right)\le -\frac{1}{8},$$

$$\left|C_1\left(p,\eta \right)\right|=\underset{1\le i\le n}{max}\sum _{j=1}^n\left|C_2{}_{ij}\left(p,\eta \right)\right|=\frac{1}{16}\left(-1+\frac{1}{2}sin\left(p-\eta \right)\right)\Rightarrow ∥C_1\left(p,\eta \right)∥\le 1,$$

$$\left|C_2\left(p,\eta \right)\right|=\underset{1\le i\le n}{max}\sum _{j=1}^n\left|C_2{}_{ij}\left(p,\eta \right)\right|=\frac{1}{8}\left(-1+\frac{1}{2}cos\left(p-\eta \right)\right)\Rightarrow ∥C_2\left(p,\eta \right)∥\le \frac{1}{16},$$

and

$$\begin{array}{ccc}\hfill L& :& ={\displaystyle \underset{p\in \left[\tau \left(p_0\right),\infty \right)}{sup}}{\int }_{p_0}^p\left(∥C_2\left(\eta ,\eta \right)∥\left({\int }_{\eta -\alpha \left(\eta \right)}^{\eta }∥C_1\left(\eta ,r\right)∥dr+1\right)\right.\hfill \\ & & +\left.{\int }_{\eta -\beta \left(\eta \right)}^{\eta }∥C_2\left(\eta ,r\right)∥dr+q_1+q_2\right)d\eta \hfill \\ & =& {\displaystyle \underset{p\in \left[\frac{-1}{3},\infty \right)}{sup}}{\int }_0^p\left(\frac{1}{16}\left(\left(\frac{1}{1+{\eta }^2}-1\right)+1\right)+\frac{1}{16}\left(\frac{1}{1+{\eta }^2}-12\right)+\frac{3}{4}\right)d\eta \hfill \\ & =& {\displaystyle \underset{p\in \left[\frac{-1}{3},\infty \right)}{sup}}{\int }_0^p\left(\frac{1}{16}\frac{1}{1+{\eta }^2}+\frac{1}{16}\frac{1}{1+{\eta }^2}\right)d\eta \hfill \\ & =& {\displaystyle \underset{p\in \left[\frac{-1}{3},\infty \right)}{sup}}\frac{1}{8}{tan}^{-1}\eta =\frac{\pi }{16}<1.\hfill \end{array}$$

Therefore, using Theorem 2, system (11) is asymptotically stable.

7. Conclusions

This paper investigated the stability, asymptotic stability, and exponential stability using the Banach fixed point theorem. The matrix measure, with some conditions, was the key tool used to tackle these three types of stability for the system (3). The considered system contains two Volterra terms and a nonlinear term; hence, the obtained results generalise the findings of [19,23]. Our work is another example of the advantage of using the fixed-point method instead of the Lyapunov’s method.