1 Introduction

In artificial intelligence, neural networks play an important role. We are witnessing an increasing interest in these systems because of their great potential for solving complex problems and their numerous applications in engineering and other fields. All started by solving problems in optimization theory and quickly their use was spread to pattern recognition, system identification, parallel computing and many other fields [1,2,3, 6,7,8,9,10,11, 15, 17].

In this paper we are interested in neural networks of Cohen–Grossberg type [2, 6]. That is, we study the problem

$$\begin{aligned} \left\{ \begin{array}{l} \varkappa _{i}^{\prime }\left( \zeta \right) =-h_{i}\left( \varkappa _{i}\left( \zeta \right) \right) \left[ g_{i}\left( \varkappa _{i}\left( \zeta \right) \right) -\sum \limits _{j=1}^{n}a_{ij}f_{j}\left( \varkappa _{j}\left( \zeta \right) \right) -\sum \limits _{j=1}^{n}b_{ij}f_{j}\left( \varkappa _{j}\left( \zeta -\tau \right) \right) \right. \\ { \ \ \ \ \ \ \ \ \ \ \ \ }\left. -\sum \limits _{j=1}^{n}d_{ij}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) f_{j}\left( \varkappa _{j}\left( \zeta -s\right) \right) ds-I_{i}\right] , { \ }\zeta >0, \\ \varkappa _{i}\left( \zeta \right) =\phi _{i}\left( \zeta \right) ,{ \ \ \ }\zeta \le 0,\text { }i=1,2,\ldots ,n. \end{array} \right. \end{aligned}$$
(1)

The system is composed of n neurons, the states are denoted \(\varkappa _{i.}\left( \zeta \right) \); \(h_{i}\) represent the amplification functions; \( g_{i}\) are suitable functions; \(I_{i}\) are the external inputs, \(f_{j}\) denote the activation functions, \(a_{ij},\) \(b_{ij},\) \(d_{ij}\) are the connection strengths of the neuron j on the neuron i, respectively; \( \phi _{i}\) is the pre-history of the ith state, \(\tau \) is the transmission delay whilst \(k_{j}\) denote the retardation kernel functions. Observe that this is a general CGNNs system which includes continuously distributed delays and discrete delays. Naturally, we may also investigate the case of several different discrete delays or even variable discrete delays. We abstained to go to most general systems to avert distracting the reader from the main contribution here.

Problem (1) is a generalization of the system [12, 13, 22]

$$\begin{aligned} \varkappa _{i}^{\prime }\left( \zeta \right) =-\varkappa _{i}\left( \zeta \right) +\sum \limits _{j=1}^{n}a_{ij}f_{j}\left( \varkappa _{j}\left( \zeta \right) \right) +\sum \limits _{j=1}^{n}b_{ij}f_{j}\left( \varkappa _{j}\left( \zeta -\tau \right) \right) +\sum \limits _{j=1}^{n}d_{ij}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) f_{j}\left( \varkappa _{j}\left( \zeta -s\right) \right) ds+I_{i} \end{aligned}$$

(as well as others such as the Lotka–Volterra system in population dynamics and other systems in neurobiology) which is in turn a generalization of the simple and original Hopfield Neural Network (HNN) model

$$\begin{aligned} \varkappa _{i}^{\prime }\left( \zeta \right) =-\varkappa _{i}\left( \zeta \right) +\sum \limits _{j=1}^{n}a_{ij}f_{j}\left( \varkappa _{j}\left( \zeta \right) \right) ,{ \ }\zeta >0,\text { }i=1,2,\ldots ,n \end{aligned}$$

taking into account distributed delays and discrete delays [7, 8]. In fact, considering delta Dirac distributions as the kernels \(k_{j}\left( \zeta \right) \) in the distributed delays makes them become discrete delays.

The most important issue in this field is the stability of the equilibrium. The first results have been shown for simple HNNs with some specific activation functions like the Sigmoid function \(f(u)=\frac{1}{1+e^{-u/T}}\), Hyperbolic tangent function \(f(u)=\tanh (u/T)\), Inverse tangent function \( f(u)=\frac{2}{\pi }\tan ^{-1}(u/T)\), Threshold function \(f(u)=\left\{ \begin{array}{c} -1,\;u<0 \\ 1,\;u>0 \end{array} \right. \), Gaussian radial basis function \(f(u)=\exp \{-\left\| u-m\right\| ^{2}/\sigma ^{2}\}\) and the Linear function \(f(u)=au+b.\). Several results have appeared in the literature (especially for HNN systems). Because of the need in applications, these activation functions have been extended to bounded, monotone and differentiable functions. In turn, these conditions have been weakened to a mere Lipschitz continuity condition. These conditions on the monotonicity, boundedness and differentiability of the activation functions have been improved thereafter to simply a global (or local) Lipschitz continuity condition. There are also a fairly large number of papers dealing with different conditions on the different coefficients involved in the system. Indeed, for the parameters, the LMI method, M-Matrix and other techniques are very efficient. They have been used and improved in an impressive number of references that cannot fit in this limited size paper. Unfortunately, in spite of the many appearing cases in the applications (as mentioned in the book of Kosko), this issue has not received much attention. In this work, we want to fill this gap by establishing reasonable conditions on the kernels ensuring exponential stability and other types of stability.

The existence and uniqueness of the equilibrium has been discussed under different conditions and using different methods such as fixed point theorems. Therefore, the well-posedness is guaranteed in our case as we are assuming standard conditions for the (dominance conditions of the) parameters and also on the activation functions (Lipschitz continuity), we obtain easily the existence and uniqueness of a solution as our kernels remain of (dissipative) fading memory type.

In [18], the author considered a similar problem but with only distributed delays (discrete delays may be considered as special cases of distributed delays) and with \(f_{j}\left( \int \nolimits _{0}^{\infty }k_{j}\left( s\right) \varkappa _{j}\left( \zeta -s\right) ds\right) \) instead of \(\int \nolimits _{0}^{\infty }k_{j}\left( s\right) f_{j}\left( \varkappa _{j}\left( \zeta -s\right) \right) ds\) (and \(k_{ij}\) instead of \( k_{j}\)), namely

$$\begin{aligned} \varkappa _{i}^{\prime }\left( \zeta \right) =-h_{i}\left( \varkappa _{i}\left( \zeta \right) \right) \left[ g_{i}\left( \varkappa _{i}\left( \zeta \right) \right) -\sum \limits _{j=1}^{n}a_{ij}f_{j}\left( \int \nolimits _{0}^{\infty }k_{j}\left( s\right) \varkappa _{j}\left( \zeta -s\right) ds\right) \right] ,{ \ }\zeta >0,\text { }i=1,2,\ldots ,n \end{aligned}$$

The assumptions were: \(k_{j}\left( \zeta \right) \ge 0,\) \( \int \nolimits _{0}^{\infty }k_{j}\left( \zeta \right) d\zeta =1,\) \( \int \nolimits _{0}^{\infty }\zeta k_{j}\left( \zeta \right) d\zeta <\infty ,\) \(j=1,\ldots ,n,\) \(\phi _{i}\left( \zeta \right) \in C\left( (-\infty ,0],\mathbb { R}\right) \) are bounded, \(h_{i}\) are bounded (\(0<\underline{\alpha }_{i}\le h_{i}(\zeta )\le \overline{\alpha }_{i}\)), positive and continuous, \(g_{i}\) are continuous increasing. The activation functions \(f_{j}\) satisfy: \( f_{j}\in C^{2}({\mathbb {R}}),\) \(f_{j}^{\prime }(\varkappa )>0,\) \( sup_{\varkappa \in {\mathbb {R}}}\) \(f_{j}^{\prime }(\varkappa )=f_{j}^{\prime }(0),\) \(f_{j}(0)=0,\) \(\lim _{\varkappa \rightarrow \pm \infty }f_{j}(\varkappa )=\pm 1,\) \(f_{j}\) are globally Lipschitz continuous with \( L_{i}\) as Lipschitz constants and \(\left| f_{j}(\varkappa )\right| \le M_{j}\) for some \(M_{j}>0,\) \(j=1,\ldots ,n.\) He proved global asymptotic stability of the equilibrium under the additional assumption \(\frac{ g_{i}(u)-g_{i}(v)}{u-v}\ge \gamma _{i},\) \(u\in {\mathbb {R}},\) \(i=1,\ldots ,n\) and some conditions of the form

$$\begin{aligned} \mu :=\min _{1\le i\le n}\left\{ \underline{\alpha }_{i}\gamma _{i}q_{i}-L_{i}\sum \limits _{j=1}^{n}\overline{\alpha }_{j}\left| a_{ji}\right| q_{j}\right\} >0 \end{aligned}$$

for some positive real constants \(q_{i}.\) Assuming further that

$$\begin{aligned} k_{j}\left( \zeta \right) \le e^{-\beta \zeta },\;\zeta >\zeta _{0},\;j=1,\ldots ,n \end{aligned}$$
(2)

for some positive numbers \(\beta \) and \(\zeta _{0},\) it is proved that the stability is of exponential type. That is

$$\begin{aligned} \sum \limits _{i=1}^{n}\left| \varkappa _{i}(\zeta )-\varkappa _{i}^{*}\right| \le Ce^{-\sigma \zeta }\sum \limits _{j=1}^{n}\sup _{u\in (-\infty ,0]}\left( \left| \phi _{j}\left( u\right) -\varkappa _{j}^{*}\right| \right) ,\;\zeta >0 \end{aligned}$$
(3)

for some positive constants C and \(\sigma .\)

The condition (2) on the kernels has been extended to cover functions satisfying

$$\begin{aligned} \int \nolimits _{0}^{\infty }k_{j}\left( s\right) e^{\beta s}ds<\infty ,\;j=1,\ldots ,n \end{aligned}$$
(4)

for some positive constant \(\beta \) (just as in the HNN case with distributed delays) in [3, 14, 16, 19,20,21, 23, 24]. There are many works on stability for similar problems, but without explicit decay.

We will also cite [5] (see [4] as well) where the authors studied a Cohen–Grossberg neural network model with both time-varying delays and distributed delays, namely

$$\begin{aligned} \varkappa _{i}^{\prime }\left( t\right)= & {} -a_{i}\left( \varkappa _{i}\left( t\right) \right) \left[ b_{i}\left( \varkappa _{i}\left( t\right) \right) +\sum \limits _{j=1}^{n}\sum \limits _{p=1}^{P}\left( h_{ij}^{\left( p\right) }\left( x_{j}\left( t-\tau _{ij}^{\left( p\right) }\left( t\right) \right) \right) \right. \right. \\{} & {} \quad \left. \left. +f_{ij}^{\left( p\right) }\left( \int \nolimits _{-\infty }^{0}g_{ij}^{\left( p\right) }\left( x_{j}\left( t+s\right) \right) d\eta _{ij}^{\left( p\right) }\left( s\right) \right) \right) \right] ,{ \ } 0\le t\ne t_{k},\text { }\\{} & {} \quad \bigtriangleup \left( x_{i}\left( t_{k}\right) \right) =I_{ik}\left( x_{i}\left( t_{k}^{-}\right) \right) ,i=1,2,\ldots ,n,k\in {\mathbb {N}} \end{aligned}$$

where \(t_{k}\nearrow \infty \) as \(k\rightarrow \infty ,\) \(a_{i}: {\mathbb {R}} \rightarrow \left( 0,\infty \right) ,\) \(b_{i},h_{ij}^{\left( p\right) },f_{ij}^{\left( p\right) },g_{ij}^{\left( p\right) },I_{ik}: {\mathbb {R}} \rightarrow {\mathbb {R}},\) \(\tau _{ij}^{\left( p\right) }:[0,\infty )\rightarrow [0,\infty )\) are continuous functions with \(\tau _{ij}^{\left( p\right) }\left( t\right) \le \tau _{ij}^{\left( p\right) }\le \tau \) for some \(\tau >0,\) and \(\eta _{ij}^{\left( p\right) }:(-\infty ,0]\rightarrow {\mathbb {R}} \) are non-decreasing bounded functions, normalized so that \(\eta _{ij}^{\left( p\right) }\left( 0\right) -\eta _{ij}^{\left( p\right) }\left( -\infty \right) =1,\) for all \(i,j\in \left\{ 1,2,\ldots ,n\right\} ,\) \(p\in \left\{ 1,2,\ldots ,P\right\} .\) Furthermore, they assumed that

  • there exist constants \(\beta _{i}>0\) such that \(\frac{b_{i}\left( u\right) -b_{i}\left( v\right) }{u-v}>\beta _{i}>0,\) for all \(u,v\in {\mathbb {R}},\) \(\ u\ne v,\) \(i=1,2,\ldots ,n.\)

  • \(h_{ij}^{\left( p\right) },f_{ij}^{\left( p\right) },g_{ij}^{\left( p\right) }\) are Lipschitzian with Lipschitz constants \(\zeta _{ij}^{\left( p\right) },\) \(\mu _{ij}^{\left( p\right) },\) \(\sigma _{ij}^{\left( p\right) },\) respectively.

  • \({\widehat{I}}_{ik}\) are Lipschitz continuous, with \(\left| {\widehat{I}}_{ik}\left( u\right) -{\widehat{I}}_{ik}\left( v\right) \right| \le {\widehat{\gamma }}_{ik}\left| u-v\right| ,\) for all \(u,v\in {\mathbb {R}},\) \(i=1,2,\ldots ,n,k\in {\mathbb {N}},\) where \({\widehat{I}}_{ik}\left( u\right) =I_{ik}\left( u\right) +u,\) \( u\in {\mathbb {R}} \)

  • \(a_{i}\left( u\right) \ge \underline{a_{i}}>0\)

  • there is \(k^{*}\in {\mathbb {N}} \) such that the conditions

    $$\begin{aligned} \int \nolimits _{-\infty }^{0}e^{\gamma s}d\eta _{ij}^{\left( p\right) }\left( s\right) <\infty ,{ \ }i,j=1,2,\ldots ,n,\text { }p=1,2,\ldots ,P \end{aligned}$$

    hold for some \(\gamma >\eta :=\sup _{k\ge k^{*}}\left( \frac{\log \left( \max \left\{ 1,{\widehat{\gamma }}_{k}\right\} \right) }{t_{k}-t_{k-1}}\right) ,\) where \({\widehat{\gamma }}_{k}:=\max _{1\le i\le n}{\widehat{\gamma }}_{ik}\)

  • the matrix

    $$\begin{aligned} M=\text {diag}\left( \beta _{1}-\frac{\eta }{\underline{a_{1}}},\ldots ,\beta _{n}-\frac{\eta }{\underline{a_{n}}}\right) -\left[ n_{ij}\right] \end{aligned}$$

    where \(\left[ n_{ij}\right] =\sum \nolimits _{p=1}^{P}\left( \zeta _{ij}^{\left( p\right) }e^{\eta \tau _{ij}^{\left( p\right) }}+\mu _{ij}^{\left( p\right) }\sigma _{ij}^{\left( p\right) }\int \nolimits _{-\infty }^{0}e^{-\eta s}d\eta _{ij}^{\left( p\right) }\left( s\right) \right) ,\) is a non-singular M-matrix.

They proved that there is a unique equilibrium point \(x^{*}\) of the above system which is globally exponentially stable. Observe that the class of kernels discussed here if of subexponential type.

Here the focus will be on the kernels \(k_{j}\left( \zeta \right) \) in problem (1). We extend the class of kernels satisfying (4) to a much wider class for which we have exponential stability as well as stability with general decay. Our result is proved under rather standard conditions on the other parameters and functions in the system but is ready to be adopted for more general situations. Indeed, in the existing papers, it is either kernels of exponential type or of subexponential type which are considered. In the present work, we do not use these assumptions. Instead, we assume the following condition: let \(\eta _{j}\left( \zeta \right) \) be non-negative continuous functions satisfying

$$\begin{aligned} \lim _{\zeta \rightarrow \infty }\eta \left( \zeta \right) :=\lim _{\zeta \rightarrow \infty }\min _{1\le j\le n}\eta _{j}\left( \zeta \right) =\bar{ \eta }, \end{aligned}$$

and

$$\begin{aligned} k_{j}\left( \zeta -s\right) \ge \eta _{j}\left( \zeta \right) \int \nolimits _{\zeta }^{\infty }k_{j}\left( \sigma -s\right) d\sigma ,\;j=1,2,\ldots ,n,\;0\le s\le \zeta . \end{aligned}$$

This new class of kernels is much more wider than the existing one in the market. It contains properly the exponentially decaying functions. Moreover, it contains polynomially decaying functions and many more functions. Therefore, this improves earlier results and allows the treatment of more problems by allowing a larger class of admissible kernels. As consequence, the rates of stability are general and not necessarily exponential.

In the section below we prepare some material needed later. The equilibrium is shifted to the zero equilibrium by a change of functions in Sect. 3. We also prove a uniform boundedness of the energy result there. The last section is devoted to the statement and proof of our main result on the arbitrary stability for a large class of kernels.

2 Preliminaries

This part is dedicated to some assumptions, definitions, and necessary lemmas for our main result. Throughout this paper we assume that:

(B1):

The functions \(f_{i}\) are Lipschitz continuous on \( {\mathbb {R}},\) \(i=1,2,\ldots ,n,\) that is

$$\begin{aligned} \left| f_{i}\left( \varkappa \right) -f_{i}\left( y\right) \right| \le L_{i}\left| \varkappa -y\right| ,\text { }\forall \varkappa ,y\in {\mathbb {R}},\text { }i=1,2,\ldots ,n. \end{aligned}$$

where \(L_{i}\) are Lipschitz constants,  \(i=1,2,\ldots ,n.\)

(B2):

The functions \(g_{i}\) are monotone increasing continuous functions and there exist constants \(\beta _{i}>0\), such that

$$\begin{aligned} \frac{g_{i}\left( \varkappa \right) -g_{i}\left( y\right) }{\varkappa -y} \ge \beta _{i},\text { }i=1,2,\ldots ,n\text {, for all }\varkappa ,\text { }y\in {\mathbb {R}} \text { with }\varkappa \ne y\text {.} \end{aligned}$$
(B3):

The delay kernel functions \(k_{j}\) are nonnegative and piecewise continuous functions such that \(c_{j}=\int \nolimits _{0}^{\infty }k_{j}\left( s\right) ds<\infty ,\) \(j=1,2,\ldots ,n.\)

(B4):

\(h_{i}\) are positive, continuous functions and in addition there are constants \(\underline{\alpha }_{i}\) and \(\overline{\alpha }_{i}\), such that

$$\begin{aligned} 0<\underline{\alpha }_{i}\le h_{i}(\varkappa )\le \overline{\alpha }_{i}, \text { }i=1,2,\ldots ,n\text {, for all }\varkappa \in {\mathbb {R}}. \end{aligned}$$
(B5):

The initial data \(\phi _{i}\left( \zeta \right) ,\) \( \zeta \le 0\), are continuous and square summable functions on \((-\infty ,0)\).

In fact, we need \(\int \nolimits _{0}^{\infty }k_{j}\left( s\right) \int \nolimits _{-s}^{0}\phi _{j}^{2}\left( \sigma \right) d\sigma ds<\infty ,\) \(j=1,2,\ldots ,n.\)

Definition 2.1

The point \(\varkappa ^{*}=\left( \varkappa _{1}^{*},\varkappa _{2}^{*},\ldots ,\varkappa _{n}^{*}\right) ^{T}\) is called an equilibrium of problem (1) if \(\varkappa ^{*}\) is a solution of

$$\begin{aligned} g_{i}\left( \varkappa _{i}^{*}\right)= & {} \sum \limits _{j=1}^{n}a_{ij}f_{j}\left( \varkappa _{j}^{*}\right) +\sum \limits _{j=1}^{n}b_{ij}f_{j}\left( \varkappa _{j}^{*}\right) +\sum \limits _{j=1}^{n}d_{ij}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) f_{j}\left( \varkappa _{j}^{*}\right) ds+I_{i}\nonumber \\= & {} \sum \limits _{j=1}^{n}\left[ a_{ij}+b_{ij}+d_{ij}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) ds\right] f_{j}\left( \varkappa _{j}^{*}\right) +I_{i},\text { }i=1,2,\ldots ,n,\text { }\zeta >0. \end{aligned}$$
(5)

The fact that the equilibrium exists and is unique has been proved in [3, 14, 16, 18,19,20,21, 23, 24].

Theorem 2.2

Under the above assumptions, Problem (1) admits a unique equilibrium.

3 Uniform boundedness of the energy

In this section we shift the equilibrium to zero and prove a uniform boundedness result of the energy. Plugging \(\varkappa \left( \zeta \right) =u\left( \zeta \right) +\varkappa ^{*}\) into the equation in (1) gives

$$\begin{aligned} \left\{ \begin{array}{l} u_{i}^{\prime }\left( \zeta \right) =-h_{i}\left( u_{i}\left( \zeta \right) +\varkappa ^{*}\right) \left[ g_{i}\left( u_{i}\left( \zeta \right) +\varkappa ^{*}\right) -\sum \limits _{j=1}^{n}a_{ij}f_{j}\left( u_{i}\left( \zeta \right) +\varkappa ^{*}\right) -\sum \limits _{j=1}^{n}b_{ij}f_{j}\left( u_{j}\left( \zeta -\tau \right) +\varkappa ^{*}\right) \right. \\ { \ \ \ \ \ \ \ \ \ \ \ \ }\left. -\sum \limits _{j=1}^{n}d_{ij}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) f_{j}\left( u_{j}\left( \zeta -s\right) +\varkappa ^{*}\right) ds-I_{i} \right] ,{ \ }\zeta >0,\text { }i=1,2,\ldots ,n, \\ u_{i}\left( \zeta \right) =\psi _{i}\left( \zeta \right) :=\phi _{i}\left( \zeta \right) -\varkappa _{i}^{*},{ \ \ \ }\zeta \le 0,\text { } i=1,2,\ldots ,n, \end{array} \right. \end{aligned}$$

or

$$\begin{aligned} \left\{ \begin{array}{l} u_{i}^{\prime }\left( \zeta \right) =-H_{i}\left( u_{i}\left( \zeta \right) \right) \left[ G_{i}\left( u_{i}\left( \zeta \right) \right) -\sum \limits _{j=1}^{n}a_{ij}F_{j}\left( u_{i}\left( \zeta \right) \right) -\sum \limits _{j=1}^{n}b_{ij}F_{j}\left( u_{j}\left( \zeta -\tau \right) \right) \right. \\ { \ \ \ \ \ \ \ \ \ \ \ \ }\left. -\sum \limits _{j=1}^{n}d_{ij}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) F_{j}\left( u_{j}\left( \zeta -s\right) \right) ds\right] ,{ \ }\zeta >0,\text { }i=1,2,\ldots ,n, \\ u_{i}\left( \zeta \right) =\psi _{i}\left( \zeta \right) :=\phi _{i}\left( \zeta \right) -\varkappa _{i}^{*},{ \ \ \ }\zeta \le 0,\text { } i=1,2,\ldots ,n \end{array} \right. \end{aligned}$$
(6)

where

$$\begin{aligned} \begin{array}{l} F_{i}\left( u_{i}\left( \zeta \right) \right) =f_{i}\left( u_{i}\left( \zeta \right) +\varkappa ^{*}\right) -f_{i}\left( \varkappa ^{*}\right) , { \ }\zeta >0,\text { }i=1,2,\ldots ,n, \\ H_{i}\left( u_{i}\left( \zeta \right) \right) =h_{i}\left( u_{i}\left( \zeta \right) +\varkappa ^{*}\right) ,\text { }G_{i}\left( u_{i}\left( \zeta \right) \right) =g_{i}\left( u_{i}\left( \zeta \right) +\varkappa ^{*}\right) -g_{i}\left( \varkappa ^{*}\right) . \end{array} \end{aligned}$$

The stability of the equilibrium \(\varkappa ^{*}\) is therefore shifted to the stability of the zero solution of system (6). Consider the ’energy’ functional

$$\begin{aligned} E\left( \zeta \right) :=\sum \limits _{i=1}^{n}u_{i}^{2}\left( \zeta \right) , \text { }\zeta \ge 0. \end{aligned}$$

Lemma 3.1

Let assumption (B1) and (B2) hold. Then

$$\begin{aligned} \begin{array}{l} 2\left| u_{i}\left( \zeta \right) F_{j}\left( u_{j}\left( \zeta \right) \right) \right| \le u_{i}^{2}\left( \zeta \right) +L_{j}^{2}u_{j}^{2}\left( \zeta \right) ,\text { }\zeta \ge 0,\text { }i,j=1,2,\ldots ,n, \\ 2\left| u_{i}\left( \zeta \right) F_{j}\left( u_{j}\left( \zeta -\tau \right) \right) \right| \le u_{i}^{2}\left( \zeta \right) +L_{j}^{2}u_{j}^{2}\left( \zeta -\tau \right) ,\text { }\zeta \ge 0,\text { } i,j=1,2,\ldots ,n, \\ 2u_{i}\left( \zeta \right) G_{i}\left( u_{i}\left( \zeta \right) \right) \ge 2\beta _{i}u_{i}^{2}\left( \zeta \right) ,\text { }\zeta \ge 0,\text { } i=1,2,\ldots ,n. \end{array} \end{aligned}$$

Proof

The first two relations are straightforward. To prove the last relation, we use (B2) to obtain

$$\begin{aligned} u_{i}\left( \zeta \right) G_{i}\left( u_{i}\left( \zeta \right) \right)= & {} u_{i}\left( \zeta \right) \left[ g_{i}\left( u_{i}\left( \zeta \right) +\varkappa ^{*}\right) -g_{i}\left( \varkappa ^{*}\right) \right] =u_{i}^{2}\left( \zeta \right) \frac{\left[ g_{i}\left( u_{i}\left( \zeta \right) +\varkappa ^{*}\right) -g_{i}\left( \varkappa ^{*}\right) \right] }{u_{i}\left( \zeta \right) } \\= & {} u_{i}^{2}\left( \zeta \right) \frac{\left[ g_{i}\left( u_{i}\left( \zeta \right) +\varkappa ^{*}\right) -g_{i}\left( \varkappa ^{*}\right) \right] }{u_{i}\left( \zeta \right) +\varkappa ^{*}-\varkappa ^{*}} \ge \beta _{i}u_{i}^{2}\left( \zeta \right) ,\text { }\zeta \ge 0,\text { } i=1,2,\ldots ,n. \end{aligned}$$

\(\square \)

Lemma 3.2

Let presuppositions (B1)–(B5) hold. Then

$$\begin{aligned} E^{\prime }\left( \zeta \right)\le & {} \sum \limits _{i=1}^{n}\left[ -2 \underline{\alpha }_{i}\beta _{i}+\overline{\alpha }_{i}\sum \limits _{j=1}^{n}\left( a_{ij}+\frac{\overline{\alpha }_{j}}{\overline{ \alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij}\right) \right] u_{i}^{2}\left( \zeta \right) \\{} & {} +\sum \limits _{j=1}^{n}\lambda _{1j}u_{j}^{2}\left( \zeta -\tau \right) +\sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds,\text { }\zeta \ge 0 \end{aligned}$$

where

$$\begin{aligned} \lambda _{1j}=\left( \sum \limits _{i=1}^{n}\overline{\alpha } _{i}b_{ij}\right) L_{j}^{2},\text { }\lambda _{2j}=\left( \sum \limits _{i=1}^{n}\overline{\alpha }_{i}d_{ij}\right) L_{j}^{2}c_{j}, \text { }j=1,2,\ldots ,n. \end{aligned}$$
(7)

Proof

The differentiation of \(E\left( \zeta \right) ,\) along solutions of (6), yields

$$\begin{aligned} E^{\prime }\left( \zeta \right)= & {} 2\sum \limits _{i=1}^{n}u_{i}\left( \zeta \right) u_{i}^{\prime }\left( \zeta \right) =\sum \limits _{i=1}^{n}H_{i}\left( u_{i}\left( \zeta \right) \right) \left[ -2u_{i}\left( \zeta \right) G_{i}\left( u_{i}\left( \zeta \right) \right) +2\sum \limits _{j=1}^{n}a_{ij}u_{i}\left( \zeta \right) F_{j}\left( u_{i}\left( \zeta \right) \right) \right. \\{} & {} \left. +2\sum \limits _{j=1}^{n}b_{ij}u_{i}\left( \zeta \right) F_{j}\left( u_{j}\left( \zeta -\tau \right) \right) +2\sum \limits _{j=1}^{n}d_{ij}u_{i}\left( \zeta \right) \int \nolimits _{0}^{\infty }k_{j}\left( s\right) F_{j}\left( u_{j}\left( \zeta -s\right) \right) ds\right] ,\text { }\zeta \ge 0. \end{aligned}$$

By Lemma 3.1 and owing to the hypothesis (B4) we find the bound

$$\begin{aligned} E^{\prime }\left( \zeta \right)\le & {} -2\sum \limits _{i=1}^{n}\underline{ \alpha }_{i}\beta _{i}u_{i}^{2}\left( \zeta \right) +\sum \limits _{i,j=1}^{n} \overline{\alpha }_{i}a_{ij}\left( u_{i}^{2}\left( \zeta \right) +L_{j}^{2}u_{j}^{2}\left( \zeta \right) \right) +\sum \limits _{i,j=1}^{n} \overline{\alpha }_{i}b_{ij}\left( u_{i}^{2}\left( \zeta \right) +L_{j}^{2}u_{j}^{2}\left( \zeta -\tau \right) \right) \\{} & {} +\sum \limits _{i,j=1}^{n}\overline{\alpha }_{i}d_{ij}\left( u_{i}^{2}\left( \zeta \right) +\left[ \int \nolimits _{0}^{\infty }k_{j}\left( s\right) L_{j}\left| u_{j}\left( \zeta -s\right) \right| ds\right] ^{2}\right) ,\text { }\zeta \ge 0 \end{aligned}$$

and clearly Cauchy–Schwarz inequality implies that

$$\begin{aligned} \begin{array}{l} \left[ \int \nolimits _{0}^{\infty }k_{j}\left( s\right) L_{j}\left| u_{j}\left( \zeta -s\right) \right| ds\right] ^{2}\le \left( \int \nolimits _{0}^{\infty }k_{j}\left( s\right) ds\right) \left( \int \nolimits _{0}^{\infty }k_{j}\left( s\right) L_{j}^{2}u_{j}^{2}\left( \zeta -s\right) ds\right) \\ \quad \le c_{j}L_{j}^{2}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds,\text { }\zeta \ge 0,\text { }j=1,2,\ldots ,n. \end{array} \end{aligned}$$

Therefore,

$$\begin{aligned} E^{\prime }\left( \zeta \right)\le & {} -2\sum \limits _{i=1}^{n}\underline{ \alpha }_{i}\beta _{i}u_{i}^{2}\left( \zeta \right) +\sum \limits _{i,j=1}^{n} \overline{\alpha }_{i}a_{ij}u_{i}^{2}\left( \zeta \right) +\sum \limits _{i,j=1}^{n}\overline{\alpha }_{i}a_{ij}L_{j}^{2}u_{j}^{2}\left( \zeta \right) +\sum \limits _{i,j=1}^{n}\overline{\alpha }_{i}b_{ij}u_{i}^{2} \left( \zeta \right) \\{} & {} \qquad +\sum \limits _{i,j=1}^{n}\overline{\alpha }_{i}b_{ij}L_{j}^{2}u_{j}^{2}\left( \zeta -\tau \right) +\sum \limits _{i,j=1}^{n}\overline{\alpha } _{i}d_{ij}u_{i}^{2}\left( \zeta \right) +\sum \limits _{i,j=1}^{n}\overline{ \alpha }_{i}d_{ij}c_{j}L_{j}^{2}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds \\{} & {} \quad =-2\sum \limits _{i=1}^{n}\underline{\alpha }_{i}\beta _{i}u_{i}^{2}\left( \zeta \right) +\sum \limits _{i,j=1}^{n}\overline{\alpha }_{i}a_{ij}u_{i}^{2} \left( \zeta \right) +\sum \limits _{i,j=1}^{n}\overline{\alpha } _{j}a_{ji}L_{i}^{2}u_{i}^{2}\left( \zeta \right) +\sum \limits _{i,j=1}^{n} \overline{\alpha }_{i}b_{ij}u_{i}^{2}\left( \zeta \right) \\{} & {} \qquad +\sum \limits _{i,j=1}^{n}\overline{\alpha }_{i}b_{ij}L_{j}^{2}u_{j}^{2}\left( \zeta -\tau \right) +\sum \limits _{i,j=1}^{n}\overline{\alpha } _{i}d_{ij}u_{i}^{2}\left( \zeta \right) +\sum \limits _{i,j=1}^{n}\overline{ \alpha }_{i}d_{ij}c_{j}L_{j}^{2}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds \end{aligned}$$

or

$$\begin{aligned} E^{\prime }\left( \zeta \right)\le & {} \sum \limits _{i=1}^{n}\left[ -2 \underline{\alpha }_{i}\beta _{i}+\overline{\alpha }_{i}\sum \limits _{j=1}^{n}\left( a_{ij}+\frac{\overline{\alpha }_{j}}{\overline{ \alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij}\right) \right] u_{i}^{2}\left( \zeta \right) \\{} & {} \qquad +\sum \limits _{i,j=1}^{n}\overline{\alpha }_{i}b_{ij}L_{j}^{2}u_{j}^{2}\left( \zeta -\tau \right) +\sum \limits _{i,j=1}^{n}\overline{\alpha } _{i}d_{ij}L_{j}^{2}c_{j}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds \\{} & {} \quad =\sum \limits _{i=1}^{n}\left[ -2\underline{\alpha }_{i}\beta _{i}+\overline{ \alpha }_{i}\sum \limits _{j=1}^{n}\left( a_{ij}+\frac{\overline{\alpha }_{j}}{ \overline{\alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij}\right) \right] u_{i}^{2}\left( \zeta \right) \\{} & {} \qquad +\sum \limits _{j=1}^{n}\left( \sum \limits _{i=1}^{n}\overline{\alpha } _{i}b_{ij}\right) L_{j}^{2}u_{j}^{2}\left( \zeta -\tau \right) +\sum \limits _{j=1}^{n}\left( \sum \limits _{i=1}^{n}\overline{\alpha } _{i}d_{ij}\right) L_{j}^{2}c_{j}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds,\text { }\zeta \ge 0. \end{aligned}$$

In short, we have

$$\begin{aligned} E^{\prime }\left( \zeta \right)\le & {} \sum \limits _{i=1}^{n}\left[ -2 \underline{\alpha }_{i}\beta _{i}+\overline{\alpha }_{i}\sum \limits _{j=1}^{n}\left( a_{ij}+\frac{\overline{\alpha }_{j}}{\overline{ \alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij}\right) \right] u_{i}^{2}\left( \zeta \right) \nonumber \\{} & {} \quad +\sum \limits _{j=1}^{n}\lambda _{1j}u_{j}^{2}\left( \zeta -\tau \right) +\sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds,\text { }\zeta \ge 0. \end{aligned}$$
(8)

\(\square \)

Theorem 3.3

Let presuppositions (B1)–(B5) hold. If

$$\begin{aligned} \sum \limits _{j=1}^{n}\left[ \overline{\alpha }_{i}\left( a_{ij}+\frac{ \overline{\alpha }_{j}}{\overline{\alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij} \right) +L_{i}^{2}\overline{\alpha }_{j}\left( b_{ji}+d_{ji}c_{i}^{2}\right) \right] <2\underline{\alpha }_{i}\beta _{i},\text { }i=1,2,\ldots ,n, \end{aligned}$$

then \(E\left( \zeta \right) \) is uniformly bounded.

Proof

For \(\zeta \ge 0,\) we introduce the functionals

$$\begin{aligned} V_{1}\left( \zeta \right) :=\sum \limits _{j=1}^{n}\lambda _{1j}\int \nolimits _{\zeta -\tau }^{\zeta }u_{j}^{2}\left( s\right) ds, \end{aligned}$$
(9)

and

$$\begin{aligned} V_{2}\left( \zeta \right) :=\sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{-\infty }^{\zeta }\left( \int \nolimits _{\zeta }^{\infty }k_{j}\left( \sigma -s\right) d\sigma \right) u_{j}^{2}\left( s\right) ds=\sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) \int \nolimits _{\zeta -s}^{\zeta }u_{j}^{2}\left( \sigma \right) d\sigma ds. \end{aligned}$$
(10)

Notice that, by assumption (B5), we have

$$\begin{aligned} V_{1}\left( 0\right)= & {} \sum \limits _{j=1}^{n}\lambda _{1j}\int \nolimits _{-\tau }^{0}u_{j}^{2}\left( s\right) ds=\sum \limits _{j=1}^{n}\lambda _{1j}\int \nolimits _{-\tau }^{0}\psi _{i}^{2}\left( s\right) ds<\infty , \\ V_{2}\left( 0\right)= & {} \sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) \int \nolimits _{-s}^{0}\psi _{i}^{2}\left( \sigma \right) d\sigma ds<\infty . \end{aligned}$$

Moreover, by direct differentiation of (9) and (10), we get

$$\begin{aligned} V_{1}^{\prime }\left( \zeta \right) =\sum \limits _{j=1}^{n}\lambda _{1j}\left[ u_{j}^{2}\left( \zeta \right) -u_{j}^{2}\left( \zeta -\tau \right) \right] , { \ }\zeta \ge 0, \end{aligned}$$

and

$$\begin{aligned} V_{2}^{\prime }\left( \zeta \right)= & {} \sum \limits _{j=1}^{n}\lambda _{2j}\left( \int \nolimits _{\zeta }^{\infty }k_{j}\left( \sigma -\zeta \right) d\sigma \right) u_{j}^{2}\left( \zeta \right) -\sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{-\infty }^{\zeta }k_{j}\left( \zeta -s\right) u_{j}^{2}\left( s\right) ds \\= & {} \sum \limits _{j=1}^{n}\lambda _{2j}\left( \int \nolimits _{0}^{\infty }k_{j}\left( s\right) ds\right) u_{j}^{2}\left( \zeta \right) -\sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds \\= & {} \sum \limits _{j=1}^{n}\lambda _{2j}c_{j}u_{j}^{2}\left( \zeta \right) -\sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds,{ \ }\zeta \ge 0, \end{aligned}$$

respectively. Let

$$\begin{aligned} Y\left( \zeta \right) =E\left( \zeta \right) +V_{1}\left( \zeta \right) +V_{2}\left( \zeta \right) ,{ \ }\zeta \ge 0. \end{aligned}$$

Then \(Y\left( 0\right) <\infty \) and for \(\zeta \ge 0\)

$$\begin{aligned} \begin{array}{l} Y^{\prime }\left( \zeta \right) =E^{\prime }\left( \zeta \right) +V_{1}^{\prime }\left( \zeta \right) +V_{2}^{\prime }\left( \zeta \right) \\ \quad \le \sum \limits _{i=1}^{n}\left[ -2\underline{\alpha }_{i}\beta _{i}+ \overline{\alpha }_{i}\sum \limits _{j=1}^{n}\left( a_{ij}+\frac{\overline{ \alpha }_{j}}{\overline{\alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij}\right) \right] u_{i}^{2}\left( \zeta \right) \\ \qquad +\sum \limits _{j=1}^{n}\lambda _{1j}u_{j}^{2}\left( \zeta -\tau \right) +\sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds \\ \qquad +\sum \limits _{j=1}^{n}\lambda _{1j}\left[ u_{j}^{2}\left( \zeta \right) -u_{j}^{2}\left( \zeta -\tau \right) \right] +\sum \limits _{j=1}^{n}\lambda _{2j}c_{j}u_{j}^{2}\left( \zeta \right) -\sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds \\ \quad =\sum \limits _{i=1}^{n}\left[ -2\underline{\alpha }_{i}\beta _{i}+\overline{ \alpha }_{i}\sum \limits _{j=1}^{n}\left( a_{ij}+\frac{\overline{\alpha }_{j}}{ \overline{\alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij}\right) \right] u_{i}^{2}\left( \zeta \right) +\sum \limits _{j=1}^{n}\left[ \lambda _{1j}+\lambda _{2j}c_{j}\right] u_{j}^{2}\left( \zeta \right) . \end{array} \end{aligned}$$
(11)

This expression (11) may be rewritten simply as

$$\begin{aligned} Y^{\prime }\left( \zeta \right)\le & {} \sum \limits _{i=1}^{n}\left[ -2 \underline{\alpha }_{i}\beta _{i}+\overline{\alpha }_{i}\sum \limits _{j=1}^{n}\left( a_{ij}+\frac{\overline{\alpha }_{j}}{\overline{ \alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij}\right) +\lambda _{1i}+\lambda _{2i}c_{i}\right] u_{i}^{2}\left( \zeta \right) \nonumber \\= & {} \sum \limits _{i=1}^{n}\left[ -2\underline{\alpha }_{i}\beta _{i}+\overline{ \alpha }_{i}\sum \limits _{j=1}^{n}\left( a_{ij}+\frac{\overline{\alpha }_{j}}{ \overline{\alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij}\right) +L_{i}^{2}\left( \sum \limits _{j=1}^{n}\overline{\alpha }_{j}b_{ji}+c_{i}^{2}\sum \limits _{j=1}^{n}\overline{\alpha }_{j}d_{ji}\right) \right] u_{i}^{2}\left( \zeta \right) \nonumber \\= & {} \sum \limits _{i=1}^{n}\left\{ -2\underline{\alpha }_{i}\beta _{i}+\sum \limits _{j=1}^{n}\left[ \overline{\alpha }_{i}\left( a_{ij}+\frac{ \overline{\alpha }_{j}}{\overline{\alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij} \right) +L_{i}^{2}\overline{\alpha }_{j}\left( b_{ji}+d_{ji}c_{i}^{2}\right) \right] \right\} u_{i}^{2}\left( \zeta \right) ,{ \ }\zeta \ge 0.\nonumber \\ \end{aligned}$$
(12)

It follows, from (12) and the condition stated in the theorem, that \( Y^{\prime }\left( \zeta \right) \le 0,\) \(\zeta \ge 0\). Therefore

$$\begin{aligned} E\left( \zeta \right) \le Y\left( \zeta \right) \le Y\left( 0\right) , { \ }\zeta \ge 0, \end{aligned}$$

and thus \(E\left( \zeta \right) ,\) \(\zeta \ge 0\), is uniformly bounded as professed. \(\square \)

4 General exponential stability

To prove our second result the following assumption is needed

(B6):

Let \(\eta _{j}\left( \zeta \right) \) be nonnegative continuous functions satisfying

$$\begin{aligned} \lim _{\zeta \rightarrow \infty }\eta \left( \zeta \right) :=\lim _{\zeta \rightarrow \infty }\min _{1\le j\le n}\eta _{j}\left( \zeta \right) =\bar{ \eta }, \end{aligned}$$

and

$$\begin{aligned} k_{j}\left( \zeta -s\right) \ge \eta _{j}\left( \zeta \right) \int \nolimits _{\zeta }^{\infty }k_{j}\left( \sigma -s\right) d\sigma ,\;j=1,2,\ldots ,n,\;0\le s\le \zeta . \end{aligned}$$

Observe that this assumption (B6) is fulfilled by a vast class of functions including exponentially and polynomially decaying functions.

Theorem 4.1

Let presuppositions (B1)–(B6) hold. If, for \(i=1,2,\ldots ,n \) and some \(\varepsilon >0,\)

$$\begin{aligned} \sum \limits _{j=1}^{n}\left\{ \overline{\alpha }_{i}\left( a_{ij}+\frac{ \overline{\alpha }_{j}}{\overline{\alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij} \right) +L_{i}^{2}\overline{\alpha }_{j}\left[ \left( 1+\varepsilon \right) b_{ji}+2c_{i}^{2}d_{ji}\right] \right\} <2\underline{\alpha }_{i}\beta _{i} \end{aligned}$$

then

  1. (i)

    if  \(\lim \nolimits _{\zeta \rightarrow \infty }\eta \left( \zeta \right) ={\bar{\eta }}=0,\) we have

    $$\begin{aligned} E\left( \zeta \right) \le Me^{-\delta \int \nolimits _{0}^{\zeta }\eta \left( s\right) ds},\text { }\zeta \ge 0 \end{aligned}$$

    for some \(\delta ,M>0.\)

  2. (ii)

    In case \(0<{\bar{\eta }}\le \infty ,\) we obtain

    $$\begin{aligned} E\left( \zeta \right) \le Ne^{-\gamma \zeta },\text { }\zeta \ge 0 \end{aligned}$$

    for some \(\gamma ,N>\) 0.

Proof

For \(0<\delta <1/2,\) consider the functional

$$\begin{aligned} {\tilde{Y}}\left( \zeta \right) :=E\left( \zeta \right) +V_{3}\left( \zeta \right) +\frac{1}{1-\delta }V_{2}\left( \zeta \right) ,{ \ }\zeta \ge 0, \end{aligned}$$

where

$$\begin{aligned} V_{3}\left( \zeta \right) :=e^{-\beta \zeta }\sum \limits _{j=1}^{n}\lambda _{1j}\int \nolimits _{\zeta -\tau }^{\zeta }e^{\beta \left( s+\tau \right) }u_{j}^{2}\left( s\right) ds,{ \ }\zeta \ge 0,\text { }\beta >0, \end{aligned}$$

\(\lambda _{1j}\) as in (7), \(V_{2}\) as in (10) and \(\beta >0\) such that \(e^{\beta \tau }\le 1+\varepsilon .\)

By direct differentiation, we find

$$\begin{aligned} V_{3}^{\prime }\left( \zeta \right) =-\beta V_{3}\left( \zeta \right) +e^{\beta \tau }\sum \limits _{j=1}^{n}\lambda _{1j}u_{j}^{2}\left( \zeta \right) -\sum \limits _{j=1}^{n}\lambda _{1j}u_{j}^{2}\left( \zeta -\tau \right) ,{ \ }\zeta \ge 0 \end{aligned}$$
(13)

and

$$\begin{aligned} V_{2}^{\prime }\left( \zeta \right)= & {} \sum \limits _{j=1}^{n}\lambda _{2j}c_{j}u_{j}^{2}\left( \zeta \right) -\delta \sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{-\infty }^{\zeta }k_{j}\left( \zeta -s\right) u_{j}^{2}\left( s\right) ds \\{} & {} -\left( 1-\delta \right) \sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{-\infty }^{\zeta }k_{j}\left( \zeta -s\right) u_{j}^{2}\left( s\right) ds,{ \ }\zeta \ge 0. \end{aligned}$$

Using our new assumption on the kernels, we get

$$\begin{aligned} V_{2}^{\prime }\left( \zeta \right)\le & {} \sum \limits _{j=1}^{n}\lambda _{2j}c_{j}u_{j}^{2}\left( \zeta \right) -\delta \sum \limits _{j=1}^{n}\lambda _{2j}\eta _{j}\left( \zeta \right) \int \nolimits _{-\infty }^{\zeta }\left( \int \nolimits _{\zeta }^{\infty }k_{j}\left( \sigma -s\right) d\sigma \right) u_{j}^{2}\left( s\right) ds\nonumber \\{} & {} \qquad -\left( 1-\delta \right) \sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{-\infty }^{\zeta }k_{j}\left( \zeta -s\right) u_{j}^{2}\left( s\right) ds \nonumber \\\le & {} \sum \limits _{j=1}^{n}\lambda _{2j}c_{j}u_{j}^{2}\left( \zeta \right) -\delta \eta \left( \zeta \right) V_{2}\left( \zeta \right) -\left( 1-\delta \right) \sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{-\infty }^{\zeta }k_{j}\left( \zeta -s\right) u_{j}^{2}\left( s\right) ds,\text { }\zeta \ge 0. \end{aligned}$$
(14)

Gathering these estimations (13) and (14), we obtain

$$\begin{aligned} \begin{array}{l} {\tilde{Y}}^{\prime }\left( \zeta \right) =E^{\prime }\left( \zeta \right) +V_{3}^{\prime }\left( \zeta \right) +\frac{1}{1-\delta }V_{2}^{\prime }\left( \zeta \right) \\ \quad \le \sum \limits _{i=1}^{n}\left[ -2\underline{\alpha }_{i}\beta _{i}+\sum \limits _{j=1}^{n}\overline{\alpha }_{i}\left( a_{ij}+\frac{ \overline{\alpha }_{j}}{\overline{\alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij} \right) \right] u_{i}^{2}\left( \zeta \right) +\sum \limits _{j=1}^{n}\lambda _{1j}u_{j}^{2}\left( \zeta -\tau \right) \\ \qquad +\sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{0}^{\infty }k_{j}\left( s\right) u_{j}^{2}\left( \zeta -s\right) ds-\beta V_{3}\left( \zeta \right) +e^{\beta \tau }\sum \limits _{j=1}^{n}\lambda _{1j}u_{j}^{2}\left( \zeta \right) -\sum \limits _{j=1}^{n}\lambda _{1j}u_{j}^{2}\left( \zeta -\tau \right) \\ \qquad +\frac{1}{1-\delta }\left[ \sum \limits _{j=1}^{n}\lambda _{2j}c_{j}u_{j}^{2}\left( \zeta \right) -\delta \eta \left( \zeta \right) V_{2}\left( \zeta \right) -\left( 1-\delta \right) \sum \limits _{j=1}^{n}\lambda _{2j}\int \nolimits _{-\infty }^{\zeta }k_{j}\left( \zeta -s\right) u_{j}^{2}\left( s\right) ds\right] \end{array} \end{aligned}$$

and, simplifying terms, we end up with

$$\begin{aligned} {\tilde{Y}}^{\prime }\left( \zeta \right)\le & {} \sum \limits _{i=1}^{n}\left[ -2 \underline{\alpha }_{i}\beta _{i}+\sum \limits _{j=1}^{n}\overline{\alpha } _{i}\left( a_{ij}+\frac{\overline{\alpha }_{j}}{\overline{\alpha }_{i}} L_{i}^{2}a_{ji}+b_{ij}+d_{ij}\right) \right] u_{i}^{2}\left( \zeta \right) -\beta V_{3}\left( \zeta \right) \nonumber \\{} & {} \quad -\frac{\delta }{1-\delta }\eta \left( \zeta \right) V_{2}\left( \zeta \right) +\sum \limits _{j=1}^{n}\left[ e^{\beta \tau }\lambda _{1j}+\frac{ \lambda _{2j}c_{j}}{1-\delta }\right] u_{j}^{2}\left( \zeta \right) ,\text { } \zeta \ge 0. \end{aligned}$$
(15)

In view of (7),

$$\begin{aligned} {\tilde{Y}}^{\prime }\left( \zeta \right)\le & {} \sum \limits _{i=1}^{n}\left[ -2 \underline{\alpha }_{i}\beta _{i}+\sum \limits _{j=1}^{n}\overline{\alpha } _{i}\left( a_{ij}+\frac{\overline{\alpha }_{j}}{\overline{\alpha }_{i}} L_{i}^{2}a_{ji}+b_{ij}+d_{ij}\right) \right] u_{i}^{2}\left( \zeta \right) -\beta V_{3}\left( \zeta \right) -\frac{\delta }{1-\delta }\eta \left( \zeta \right) V_{2}\left( \zeta \right) \nonumber \\{} & {} \qquad +\sum \limits _{j=1}^{n}\left[ e^{\beta \tau }\left( \sum \limits _{i=1}^{n} \overline{\alpha }_{i}b_{ij}\right) L_{j}^{2}+\frac{c_{j}}{1-\delta }\left( \sum \limits _{i=1}^{n}\overline{\alpha }_{i}d_{ij}\right) L_{j}^{2}c_{j} \right] u_{j}^{2}\left( \zeta \right) \nonumber \\\le & {} \sum \limits _{i=1}^{n}\left\{ -2\underline{\alpha }_{i}\beta _{i}+\sum \limits _{j=1}^{n}\left[ \overline{\alpha }_{i}\left( a_{ij}+\frac{ \overline{\alpha }_{j}}{\overline{\alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij} \right) +L_{i}^{2}\overline{\alpha }_{j}\left( e^{\beta \tau }b_{ji}+\frac{ c_{i}^{2}}{1-\delta }d_{ji}\right) \right] \right\} u_{i}^{2}\left( \zeta \right) \nonumber \\{} & {} \qquad -\beta V_{3}\left( \zeta \right) -\frac{\delta }{1-\delta }\eta \left( \zeta \right) V_{2}\left( \zeta \right) \end{aligned}$$
(16)

and by the condition on the coefficients in the theorem

$$\begin{aligned} {\tilde{Y}}^{\prime }\left( \zeta \right) \le -\alpha E\left( \zeta \right) -\beta V_{3}\left( \zeta \right) -\frac{\delta }{1-\delta }\eta \left( \zeta \right) V_{2}\left( \zeta \right) ,\text { }\zeta \ge 0, \end{aligned}$$
(17)

where

$$\begin{aligned} \alpha =\min _{1\le i\le n}\left\{ 2\underline{\alpha }_{i}\beta _{i}-\sum \limits _{j=1}^{n}\left[ \overline{\alpha }_{i}\left( a_{ij}+\frac{ \overline{\alpha }_{j}}{\overline{\alpha }_{i}}L_{i}^{2}a_{ji}+b_{ij}+d_{ij} \right) +L_{i}^{2}\overline{\alpha }_{j}\left( e^{\beta \tau }b_{ji}+\frac{ c_{i}^{2}}{1-\delta }d_{ji}\right) \right] \right\} >0. \end{aligned}$$

We are lead to two cases

Case 1: \(\lim \nolimits _{\zeta \rightarrow \infty }\eta \left( \zeta \right) =0.\)

Let \(T>0\) be large enough so that

$$\begin{aligned} \eta \left( \zeta \right) \le \frac{1}{\delta }\min \left\{ \alpha ,\beta \right\} ,\text { }\zeta \ge T. \end{aligned}$$

Then, from (17), we find

$$\begin{aligned} {\tilde{Y}}^{\prime }\left( \zeta \right) \le -\delta \eta \left( \zeta \right) E\left( \zeta \right) -\delta \eta \left( \zeta \right) V_{3}\left( \zeta \right) -\frac{\delta }{1-\delta }\eta \left( \zeta \right) V_{2}\left( \zeta \right) \le -\delta \eta \left( \zeta \right) {\tilde{Y}} \left( \zeta \right) ,\text { }\zeta \ge T, \end{aligned}$$

which implies that

$$\begin{aligned} {\tilde{Y}}\left( \zeta \right) \le {\tilde{Y}}\left( T\right) e^{-\delta \int \nolimits _{T}^{\zeta }\eta \left( s\right) ds},\text { }\zeta \ge T. \end{aligned}$$
(18)

Case 2: \(0<{\bar{\eta }}\le \infty .\)

Notice that here

$$\begin{aligned} \exists \text { }T>0\text { such that }\eta \left( \zeta \right) \ge \frac{ {\bar{\eta }}}{2},\text { }\forall \text { }\zeta \ge T. \end{aligned}$$

In case \({\bar{\eta }}=+\infty \), we consider any positive constant \(\xi \), \( \eta \left( \zeta \right) \ge \xi .\)

In view of (17), we see that

$$\begin{aligned} {\tilde{Y}}^{\prime }\left( \zeta \right) \le -\alpha E\left( \zeta \right) -\beta V_{3}\left( \zeta \right) -\frac{\delta }{1-\delta }\frac{{\bar{\eta }}}{ 2}V_{2}\left( \zeta \right) \le -\gamma {\tilde{Y}}\left( \zeta \right) , \text { }\zeta \ge T, \end{aligned}$$

where

$$\begin{aligned} \gamma \le \min \left\{ \alpha ,\beta ,\frac{\delta {\bar{\eta }}}{2}\right\} >0. \end{aligned}$$

Therefore

$$\begin{aligned} {\tilde{Y}}\left( \zeta \right) \le {\tilde{Y}}\left( T\right) e^{-\gamma \left( \zeta -T\right) },\text { }\zeta \ge T. \end{aligned}$$
(19)

By continuity and decreasingness of \({\tilde{Y}}\left( T\right) \) we get similar estimates (to (18) and (19)) on [0, T]. \(\square \)

Remark 4.2

Whilst the established exponential stability in case (ii) is of exponential type, in the case (i) we have an arbitrary stability. This will depend on \( \eta \left( \zeta \right) .\) For instance, if \(\eta \left( \zeta \right) =\rho ^{\prime }(\zeta )/\rho (\zeta )\) for some differentiable positive function \(\rho (\zeta )\) with \(\lim _{\zeta \rightarrow \infty }\rho (\zeta )=+\infty ,\) then \(E\left( \zeta \right) \le C/\rho (\zeta ),\) \(\zeta \ge 0.\)

5 Numerical illustration

In this section, we shall present numerical examples validating the efficiency of the above theoretical results.

Example 5.1

Consider the following Cohen–Grossberg neural network system, composed of three neurons

$$\begin{aligned} \left\{ \begin{array}{l} \varkappa '_{i}( \zeta ) =-h_{i}( \varkappa _{i}( \zeta ) ) [ g_{i}( \varkappa _{i}( \zeta ) ) -\sum \limits _{j=1}^{3}a_{ij}f_{j}( \varkappa _{j}( \zeta ) ) -\sum \limits _{j=1}^{3}b_{ij}f_{j}( \varkappa _{j}( \zeta -\tau ) ) \\ { \ \ \ \ \ \ \ \ \ \ \ \ } -\sum \limits _{j=1}^{3}d_{ij}\int \nolimits _{0}^{\infty }k_{j}( s) f_{j}( \varkappa _{j}( \zeta -s) ) ds-I_{i}], { \ }\zeta >0, \text { } i=1,2,3, \end{array} \right. \end{aligned}$$
(20)

where the associated functions and parameters are selected as follows

$$\begin{aligned} \begin{array}{ll} h_{1}(x)= h_{3}(x)=1+ \frac{1}{1+x^{2}}, \; h_{2}(x)= 2- \frac{1}{1+x^{2}}, \; g_{1}(x)=2x, \; g_{2}(x)=2.5x, \\ g_{3}(x)=3x, \; f_{1}(x)=\frac{1}{8}(|x+1|-|x-1|), \; f_{2}(x)=\frac{1}{4} tanh(x), \; f_{3}(x)=\frac{1}{4} tanh(0.5x), \\ k_{i}(t)= \frac{1}{16} e^{-\sqrt{1+t}},\; I_{i}=0, \; i=1, 2, 3, \; \phi _{1}( x) =0.5, \; \phi _{2}( x) =-1, \; \phi _{3}( x) =1, \; x\in [-1, 0], \\ a_{11}= 0.15, \; a_{12}= 0.12, \; a_{13}= 0.17, \; a_{21}= 0.16, \; a_{22}= 0.18, \; a_{23}= 0.2, \; a_{31}= 0.14, \\ a_{32}= 0.16, \; a_{33}= 0.12, \; b_{11}= 0.17, \; b_{12}= 0.15, \; b_{13}= 0.13, \; b_{21}= 0.18, \; b_{22}= 0.12, \\ b_{23}= 0.11, \; b_{31}= 0.13, \; b_{32}= 0.19, \; b_{33}= 0.16, \; d_{11}= 0.14, \; d_{12}= 0.2, \; d_{13}= 0.18, \\ d_{21}= 0.16, \; d_{22}= 0.17, \; d_{23}= 0.14, \; d_{31}= 0.15, \; d_{32}= 0.14, \; d_{33}= 0.2, \; \tau =1. \end{array} \end{aligned}$$

Through some simple calculations, we get \(L_{1}=L_{2}=L_{3}=\frac{1}{4}, \; \beta _{1}= 2, \; \beta _{2}= 2.5, \; \beta _{3}= 3, \; c_{i}= \frac{1}{4e},\; \underline{\alpha }_{i}=1, \; {\bar{\alpha }}_{i}=2, \; i=1, 2, 3\), and we choose \(\eta _{i}(t)= \frac{1}{4(1+\sqrt{1+t})}, \; i=1, 2, 3. \)

Hence, the assumptions (B1)–(B6) are met. By virtue of Theorem 4,

then the solutions of the system (20) decay to the stationary states. These can be depicted in Figs. 1 and 2. We also observe that the related energy functional decays to zero, as shown in Fig. 3.

Fig. 1
figure 1

State trajectories of \(x_{1}(t), x_{2}(t)\) and \(x_{3}(t)\)

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Fig. 2
figure 2

State trajectories of \(x_{1}(t), x_{2}(t)\) and \(x_{3}(t)\) with random initial data

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Fig. 3
figure 3

The trajectory of the energy functional E(t)

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Example 5.2

Consider the system (20) in which the related functions and parameters are chosen as

$$\begin{aligned} \begin{array}{ll} h_{1}(x)= 1+0.2cos(x), \; h_{2}(x)= h_{3}(x)= 1+0.2sin(x), \\ g_{1}(x)=10x, \; g_{2}(x)=9x, \; g_{3}(x)=9.5x, \; f_{i}(x)=\frac{1}{16} tanh(x), \; k_{i}(t)= \frac{1}{(1+x)^{2}},\\ I_{i}=0, \; i=1, 2, 3, \; \phi _{1}( x) =0.25, \; \phi _{2}( x) =0.75, \; \phi _{3}( x) =-0.5, \; x\in [-2, 0], \\ a_{11}= 1, \; a_{12}= 0.25, \; a_{13}= 0.75, \; a_{21}= 0.5, \; a_{22}= 1, \; a_{23}= 0.4, \; a_{31}= 0.6, \\ a_{32}= 0.3, \; a_{33}= 1, \; b_{11}= 0.8, \; b_{12}= 0.25, \; b_{13}= 1, \; b_{21}= 0.75, \; b_{22}= 0.5, \\ b_{23}= 0.3, \; b_{31}= 0.6, \; b_{32}= 1, \; b_{33}= 0.25, \; d_{11}= 0.75, \; d_{12}= 1, \; d_{13}= 0.45, \\ d_{21}= 0.5, \; d_{22}= 0.8, \; d_{23}= 1, \; d_{31}= 0.5, \; d_{32}= 0.75, \; d_{33}= 0.25, \; \tau =2. \end{array} \end{aligned}$$

Via a simple calculation, we obtain \(L_{1}=L_{2}=L_{3}=\frac{1}{16}, \; \beta _{1}= 10, \; \beta _{2}= 9, \; \beta _{3}= 9.5, \; c_{i}= 1,\; \underline{\alpha }_{i}=0.8, \; {\bar{\alpha }}_{i}=1.2, \; \eta _{i}(t)= \frac{1}{2(1+t)}, \; i=1, 2, 3. \)

Therefore, as the hypotheses (B1)–(B6) of Theorem 4 are fulfilled, the solutions of the system (20) decay to steady points. We can see these in Figs. 4 and 5. Furthermore, Fig. 6 illustrates the decay of the associated energy functional to zero.

Fig. 4
figure 4

State curves of \(x_{1}(t), x_{2}(t)\) and \(x_{3}(t)\)

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Fig. 5
figure 5

State curves of \(x_{1}(t), x_{2}(t)\) and \(x_{3}(t)\) with random initial data

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Fig. 6
figure 6

The curve of the energy functional E(t)

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