1 Introduction

In relevant literature, during the investigations of qualitative theory of ODEs, functional DEs, IDEs and impulsive DEs, three methods come forward; the direct Lyapunov method, the Lyapunov–Krasovskiĭ method and the Lyapunov–Razumikhin method. To the best of knowledge, the direct Lyapunov method and Lyapunov–Krasovskiĭ method are the most effective methods in the investigations of qualitative theory of ODEs and functional DEs, respectively, (see [3, 5, 6, 21, 22, 28,29,30,31, 43,44,45,46,47, 51, 60,61,62,63,64, 68, 69, 71, 74]). Both of these methods are much related to each the other. However, in generally, the Lyapunov–Razumikhin method is very less used than that the direct Lyapunov method and the Lyapunov–Krasovskiĭ method during the qualitative investigations of solutions. When the Lyapunov–Razumikhin method is used, it is very effective and useful to arrive the possible qualitative results (see [1, 11, 46,47,48, 60, 64]). Here, we would not like to aim details of the related information.

Next, in physics, many real situations in circuit analysis and some other topics can be modeled by IDEs. For example, by the Kirchhoffs second law, the net voltage drop across a closed loop equals to the voltage impressed \(E(t).\) Hence, the standard closed electric an RLC circuit can be governed by the following IDE:

$$ L\frac{d}{dt}I(t) + RI(t) + \frac{1}{C}\int\limits_{0}^{t} {I(s)ds = E(t),} $$

where \(I(t)\) is the electric current as a function of the time \(t,\)\(R\) is the resistance, \(L\) is the inductance and \(C\) is the capacitance [24].

In 1973, by the means of the Lyapunov–Razumikhin method, the stability of the VIDE

$$ x^{\prime}(t) = - Ax(t) + \int\limits_{0}^{t} {K(t,s,x(s))ds} $$
(1)

was discussed as an application by Seifert [49]. In [49], the researcher obtained a very interesting result on the stability of trivial solution of VIDE (1).

In a recent and very interesting paper, Sedova [48] consider the following VIDE:

$$ x^{\prime}(t) = G(t,x(t)) + \int\limits_{0}^{t} {H(t,s,x(s))ds} . $$

In [48], the specific applications of Razumikhin technique to the stability analysis of this VIDE are considered and new sufficient conditions for uniform asymptotic stability of the zero solution are given using the phase space of a special construction and constraints on the right side of the equation. In [48], at the presented constraints, it can be analyzed stability, relying not only on the behavior of the auxiliary function along the solutions, but also on the properties of the so called limiting equations.

In [4], Burton first consider a class of scalar VIDEs given as the following:

$$ x^{\prime}(t) = A(t)f(x(t)) + \int\limits_{0}^{t} {B(t,s)g(x(s))ds} , $$
$$ x^{\prime}(t) = - \int\limits_{0}^{t} {C(t,s)h(x(s))ds} $$

and

$$ x^{\prime}(t) = A(t)f(x(t)) + \int\limits_{0}^{t} {\left[ {B(t,s) - C(t,s) - D(t,s)} \right]f(x(s)ds} . $$

In [4], Burton discussed the stability of zero solution, boundedness and convergence of the bounded solutions to the zero of the first equation by a Lyapunov functional [4, Theorem 1], the stability of the zero solution, boundedness of solutions, the square integrability of \(x^{\prime}(t),\) convergence of the solution \(x(t)\) to the zero as \(t \to \infty\) of the second equation via a Lyapunov functional [4, Theorem 2] and the stability of the zero solution, boundedness of solutions and the satisfaction of the result \(\int\limits_{0}^{\infty } {f^{2} (x(t))dt < \infty }\) by a Lyapunov functional [4, Theorem 4], respectively.

Later, Burton [4] consider the following system of VIDEs and its some modified versions:

$$ x^{\prime}(t) = A(t)f(x(t)) + \int\limits_{0}^{t} {B(t,s)E(x(s)x(s)ds} . $$

In Burton [4], some sufficient conditions are established for this equation and its modified versions such that under which the zero solution is stable, uniform stable, solutions are bounded, all bounded solutions tend to zero and so on for the considered equations [4, Theorems 5–8].

In Vanualailai and Nakagiri [65], the authors use Lyapunov functionals to prove several results on the stability of the zero solution of the system of VIDEs:

$$ x^{\prime}(t) = A(t)f(x(t)) + \int\limits_{0}^{t} {B(t,s)g(x(s))ds} . $$

In [65], for the particular cases of this system of VIDEs, some discussion are done and examples are provided to show that the assumptions of the given results hold.

The main reason to do this work is that the qualitative concepts in this paper such as asymptotic stability, uniform stability, global asymptotic stability and boundedness at infinity of VIDEs have many applications in applied sciences [6, 24, 31, 32, 38, 39, 52, 68]. Next, with respect to our observations from the literature, the second Lyapunov method and the Lyapunov–Krasovskiĭ method have been used intensively to discuss the qualitative behaviors of solutions of ODEs, functional differential equations, integro-differential equations of integer order with and without delay, until now. However, as we could see the Lyapunov–Razumikhin method is less in use to discuss the mentioned concepts for the linear and non-linear VIDEs of integer order. In the papers or books given with the references numbers [2, 4,5,6,7,8,9,10, 12,13,14,15,16,17, 19,20,21,22,23, 25,26,27, 31,32,33,34,35,36,37,38,39,40,41,42, 46, 47, 49, 50, 52,53,54,55,56,57,58,59, 65, 66,67,68, 70, 72,73,74] and those in the references of them, to study the qualitative behaviors of solutions of integro-differential equations of integer order with and without delay, in generally, the second Lyapunov method and the Lyapunov–Krasovskiĭ method have been used as basic tool to prove the results therein.

In this paper, the Lyapunov–Razumikhin method is applied to VIDEs of integer order as in Seifert [49] and Sedova [48]. In spite of this connection, the results of this paper, the established conditions, the given examples and so on are different from those in Burton [4], Seifert [49], Sedova [48], Vanualailai and Nakagiri [65] and those are available in the references of this paper. Indeed, in addition to these information, the VIDEs to be considered here, the results of this paper, the given conditions, methodology, examples, etc., are completely different from those in [2, 4,5,6,7,8,9,10, 12,13,14,15,16,17, 19,20,21,22,23, 25,26,27, 31,32,33,34,35,36,37,38,39,40,41,42, 49, 50, 52,53,54,55,56,57,58,59, 65, 66,67,68, 70, 72,73,74] and the references of them.

The results of this paper are new, original and they have scientific novelty. At the above, we give some brief comparisons with respect to the related literature, the references and the results of this paper. For the sake of the brevity, we would not like to give more details on the subject.

Finally, through this paper, we would like to do some contributions to the results of [1, 2, 4,5,6,7,8,9,10, 12,13,14,15,16,17, 19, 20, 23, 25,26,27, 31,32,33,34,35,36,37,38,39,40,41,42, 49, 50, 52,53,54,55,56,57,58,59, 65,66,67,68, 70, 72, 73]. For some proper contributions of this paper, see also the discussions of the paper at the end.

Firstly, we would present the assumptions and the stability result of Seifert [49].

A. Assumptions

We have the below hypotheses:

\((A1)\) \(A \in {\mathbb{R}}^{n \times n} ,\) all eigenvalues of \(A\) have negative real parts, \(A^{T} = A,\) \(A^{T}\) represents the transpose of \(A,\) the kernel \(K\) is continuous in \((s,t,x)\) with \(0 \le s \le t < \infty\) and \(x\) in \({\mathbb{R}}^{n} .\) Further, there are positive constants \(\mu\) and \(\rho\) for which the kernel \(K\) satisfies

$$ \left| {\int\limits_{0}^{t} {K(t,s,x(s))ds} } \right| \le \mu \mathop {\sup }\limits_{0 \le s \le t} \left| {x(s)} \right| $$

for any continuous function \(x(s)\) in \(0 \le s \le t\) such that \(\left| {x(s)} \right| \le \rho\) on this interval.

\((A2)\) \(B \in {\mathbb{R}}^{n \times n} ,\) \(B\) is positive definite with \(B = (b_{ij} ),\) \(\left| B \right| = (\sum\limits_{i,j} {(b_{ij}^{2} )} )^{\frac{1}{2}} ,\) such that

$$ BA + A^{T} B = - I, $$

where \(I \in {\mathbb{R}}^{n \times n} ,\) that is, \(I\) represents the identity matrix. The eigenvalues of \(B\) satisfies

$$ \Lambda^{2} \left\| x \right\|^{2} \ge \left\langle {Bx,x} \right\rangle \ge \lambda^{2} \left\| x \right\|^{2} , $$

where \(x \in {\mathbb{R}}^{n} ,\) \(\Lambda , \, \lambda > 0,\)\(\Lambda , \, \lambda \in {\mathbb{R}}\) and \(\Lambda ,\)\(\lambda\) are the greatest and least eigenvalues of the matrix \(B,\) respectively.

Seifert [49] proved the following Theorem 1.

Theorem 1

If assumptions \(\frac{2\left| B \right|\mu \Lambda }{\lambda } < 1\) and \((A1),\)\((A2)\) hold, then the trivial solution is stable for VIDE (1).

We note that the Lyapunov function given as follows:

$$ V(t,x) = V(x) = \left\langle {x,Bx} \right\rangle . $$

This function was used as a basic tool in the proof of Theorem 1 by Seifert [49].

2 Preliminaries

We consider the following system of delay differential equations (DDEs) of the form:

$$ \frac{dx}{{dt}} = f(t,x_{t} ),x_{t} (\theta ) = x(t + \theta ), - \tau \le \theta \le 0. $$
(2)

We suppose \(f:( - \infty ,\infty ) \times C \to \Re^{n} ,\) where \(C\) is a set of continuous functions \(\phi :[ - \tau , \, 0] \to {\mathbb{R}}^{n} ,\) \(\tau > 0.\) Here, \(f\) is continuous and takes closed bounded sets into bounded sets, and \(f(t,0) = 0.\) Since \(f(t,0) = 0,\) the DDE (2) includes the solution \(x(t) \equiv 0,\) with zero initial function \(\phi \equiv 0.\)

For any \(\phi \in C([ - \tau ,0],{\mathbb{R}}^{n} ),\) we refer the usual Euclidean norm \(\left\| . \right\|,\) which is defined by

$$ \left\| \phi \right\| = \mathop {\sup }\limits_{ - \tau \le s \le 0} \left| {\phi (s)} \right|. $$

It should be noted that when we calculate the time derivative of a Lyapunov–Krasovskiĭ functional along the solutions of (2), the upper right-hand derivative of the functional will be calculated here.

Lemma 1

(Hale [21], Theorem 4.2, pp. 127] and Hale and Verduyn Lunel [22], Theorem 4.2, pp. 152]). The DDE (2) is globally uniformly asymptotically stable if there exists a continuous function \(V(t,x)\) and positive definite functions \(u,\)\(v,\)\(\omega\) and a continuous non-decreasing function \(q(s) > s\) for \(s > 0\) such that the following conditions hold:

$$ u(\left| x \right|) \le V(t,x) \le v(\left| x \right|)\;\forall t \in J,\;\forall x \in {\mathbb{R}}^{n} , $$
$$ \frac{d}{dt}V(t,x(t)) \le - \omega (\left| x \right|) $$

if

$$ V(t + s,x(t + s)) < q(V(t,x(t))),\forall s \in [ - \tau ,0]. $$

3 Asymptotically stability and uniformly stability

Firstly, we have the non-linear VIDE as follows:

$$ \frac{dx}{{dt}} = - F(t,x)x + \int\limits_{0}^{t} {K(t,s,x(s))ds,} \quad \forall t > t_{0} \ge 0, $$
(3)

where \(t \in {\mathbb{R}}^{ + } ,\) \(x \in {\mathbb{R}}^{n} ,\) \(F(t,x) \in C({\mathbb{R}}^{ + } \times {\mathbb{R}}^{n} ,{\mathbb{R}}^{n} \times {\mathbb{R}}^{n} ),\) \(D = \{ (u,v) \in {\mathbb{R}}^{2} :0 \le v \le u < \infty \} ,\) and \(K(u,v,x) \in C(D \times {\mathbb{R}}^{n} ,{\mathbb{R}}^{n} )\) and \(K(u,v,x) = 0 \Leftrightarrow x = 0.\)

Firstly, we investigate here the asymptotic and uniform stability of trivial solution of VIDE (3) by the Lyapunov–Krasovskiĭ method.

A. Assumptions

For our results, we need the conditions as follows:

\((C1)\) \(F(t,x) \in C({\mathbb{R}}^{ + } \times {\mathbb{R}}^{n} ,{\mathbb{R}}^{n} \times {\mathbb{R}}^{n} )\) and it is positive definite such that

$$ \mathop {\sup }\limits_{{(t,x) \in \Re^{ + } \times \Re^{n} }} \left\| {F(t,x)} \right\| < \infty , $$

and the eigenvalues of \(F(t,x)\) satisfy

$$ f_{1i} \ge \lambda_{i} (F(t,x)) \ge f_{0i} ,\;f_{0i} > 0,\forall t \in {\mathbb{R}}^{ + } ,{\mathbb{R}}^{ + } = [0,\infty ),\forall x \in {\mathbb{R}}^{n} ,(i = 1,..,n). $$

Let \(\beta > 0\) be a positive constant such that

$$ \begin{gathered} \left\| {K(t,s,x(s))} \right\| \le \left\| {D(t,s)} \right\| \, \left\| {f(x(s))} \right\|, \hfill \\ \left\| {f(x(s))} \right\| \le \beta \left\| {x(s)} \right\|. \hfill \\ \end{gathered} $$

\((C2) \int\limits_{0}^{t} {\left\| {D(t,s)} \right\|ds} \le \alpha_{1} (t),\;\int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\|du} \le \alpha_{2} (t), \)

where \(\alpha_{1} ,\) \(\alpha_{2} \in C({\mathbb{R}}^{ + } ,{\mathbb{R}}^{ + } )\) and are bounded functions for \(\forall t \in {\mathbb{R}},\) and

$$ \alpha (t) = f_{0} - \frac{1}{2}\int\limits_{0}^{t} {\left\| {D(t,s)} \right\| \, ds} - \frac{1}{2}\beta^{2} \int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\|du} \ge c > 0,\forall t \in {\mathbb{R}}, $$

where

$$ c \in {\mathbb{R}},\;f_{0} = \min \{ f_{01} ,f_{02} ,...,f_{0n} ). $$

Theorem 2

The zero solution of VIDE (3) is asymptotically stable if assumptions \((C1)\) and \((C2)\) hold.

Proof

To do the proof, we use the Lyapunov–Krasovskiĭ method. For the coming steps, define the functional:

$$ W = W(t,x(t)) = \frac{1}{2}\left\langle {x(t),x(t)} \right\rangle + \sigma \int\limits_{0}^{t} {\int\limits_{t}^{\infty } {\left\| {D(u,s)} \right\|} } \, \left\| {f(x(s))} \right\|^{2} duds, $$

where \(\sigma > 0,\) \(\sigma \in {\mathbb{R}},\) the constant \(\sigma\) is chosen later in the proof.

For the first step, from the given functional, we derive

$$ W(t,0) = 0 $$

and

$$ W(t,x(t)) \ge \frac{1}{2}\left\| {x(t)} \right\|^{2} . $$

Thus, clearly, we see that \(W\) is positive definite and has lower bound.

For the next step, differentiating \(W\) gives:

$$ \begin{aligned} \frac{d}{dt}W(t,x(t)) & = \frac{1}{2}\left\langle {x^{\prime}(t),x(t)} \right\rangle + \frac{1}{2}\left\langle {x(t),x^{\prime}(t)} \right\rangle \\ & \quad + \sigma \int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\| \, \left\| {f(x(t))} \right\|^{2} du} - \sigma \int\limits_{0}^{t} {\left\| {D(t,s)} \right\|} \, \left\| {f(x(s))} \right\|^{2} ds \\ & = - \frac{1}{2}\left\langle {F(t,x(t))x(t),x(t)} \right\rangle - \frac{1}{2}\left\langle {x(t),F(t,x(t))x(t)} \right\rangle \\ & = - \frac{1}{2}\left\langle {F(t,x(t))x(t),x(t)} \right\rangle - \frac{1}{2}\left\langle {x(t),F(t,x(t))x(t)} \right\rangle \\ & \quad + \frac{1}{2}\left\langle {x(t),\int\limits_{0}^{t} {K(t,s,x(s))ds} } \right\rangle + \frac{1}{2}\left\langle {\int\limits_{0}^{t} {K(t,s,x(s))ds,x(t)} } \right\rangle \\ & \quad + \sigma \int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\| \, \left\| {f(x(t))} \right\|^{2} du} - \sigma \int\limits_{0}^{t} {\left\| {D(t,s)} \right\|} \, \left\| {f(x(s))} \right\|^{2} ds. \\ \end{aligned} $$

Since

$$ f_{1i} \ge \lambda_{i} (F(t,x(t))) \ge f_{0i} ,\;f_{0i} > 0, $$

we can assume that

$$ f_{0} = \min \{ f_{01} ,f_{02} ,...,f_{0n} ). $$

Then, by assumptions \((C1),\) \((C2)\) and an elementary inequality, we have

$$ \begin{aligned} \frac{d}{dt}W(t,x(t)) & \le - f_{0} \left\| {x(t)} \right\|^{2} + \int\limits_{0}^{t} {\left\| {K(t,s,x(s))} \right\|ds} \left\| {x(t)} \right\| \\ & \quad + \sigma \int\limits_{t}^{\infty } {\left\| {f(x(t))} \right\|^{2} \left\| {D(u,t)} \right\| \, du} - \sigma \int\limits_{0}^{t} {\left\| {f(x(s))} \right\|^{2} \left\| {D(t,s)} \right\|} \, ds \\ & \le - f_{0} \left\| {x(t)} \right\|^{2} + \int\limits_{0}^{t} {\left\| {f(x(s))} \right\|\left\| {D(t,s)} \right\| \, \left\| {x(t)} \right\|ds} \\ & \quad + \sigma \int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\| \, \left\| {f(x(t))} \right\|^{2} du} - \sigma \int\limits_{0}^{t} {\left\| {D(t,s)} \right\|} \, \left\| {f(x(s))} \right\|^{2} ds \\ & \le - f_{0} \left\| {x(t)} \right\|^{2} + \frac{1}{2}\int\limits_{0}^{t} {\left\| {D(t,s)} \right\|{ [}\left\| {f(x(s))} \right\|^{2} + \left\| {x(t)} \right\|^{2} ]ds} \\ & \quad + \sigma \int\limits_{t}^{\infty } { \, \left\| {f(x(t))} \right\|^{2} \left\| {D(u,t)} \right\|du} - \sigma \int\limits_{0}^{t} {\left\| {f(x(s))} \right\|^{2} \left\| {D(t,s)} \right\|} \, ds \\ & = - f_{0} \left\| {x(t)} \right\|^{2} + \frac{1}{2}\int\limits_{0}^{t} {\left\| {f(x(s))} \right\|^{2} \left\| {D(t,s)} \right\|} \, ds + \frac{1}{2}\int\limits_{0}^{t} {\left\| {D(t,s)} \right\| \, \left\| {x(t)} \right\|^{2} ds} \\ & \quad + \sigma \int\limits_{t}^{\infty } {\left\| {f(x(t))} \right\|^{2} \left\| {D(u,t)} \right\|du} - \sigma \int\limits_{0}^{t} {\left\| {f(x(s))} \right\|^{2} \left\| {D(t,s)} \right\|} ds \\ & \le - f_{0} \left\| {x(t)} \right\|^{2} + \frac{1}{2}\int\limits_{0}^{t} {\left\| {f(x(s))} \right\|^{2} \left\| {D(t,s)} \right\|} ds + \frac{1}{2}\int\limits_{0}^{t} {\left\| {D(t,s)} \right\| \, \left\| {x(t)} \right\|^{2} ds} \\ & \quad + \sigma \beta^{2} \int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\| \, \left\| {x(t)} \right\|^{2} du} - \sigma \int\limits_{0}^{t} {\left\| {D(t,s)} \right\|} \, \left\| {f(x(s))} \right\|^{2} ds. \\ \end{aligned} $$

Let \(\sigma = \frac{1}{2}.\) Then, we rearrange this inequality as following:

$$ \begin{aligned} \frac{d}{dt}W(t,x(t)) &\le - f_{0} \left\| {x(t)} \right\|^{2} + \frac{1}{2}\int\limits_{0}^{t} {\left\| {D(t,s)} \right\| \, \left\| {x(t)} \right\|^{2} ds} + \frac{1}{2}\beta^{2} \int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\| \, \left\| {x(t)} \right\|^{2} du} \hfill \\ &= - \, [f_{0} - \frac{1}{2}\int\limits_{0}^{t} {\left\| {D(t,s)} \right\| \, ds} - \frac{1}{2}\beta^{2} \int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\|du} ]\left\| {x(t)} \right\|^{2} \hfill \\ &\le - c\left\| {x(t)} \right\|^{2} \le 0 \hfill \\ \end{aligned} $$

by \((C2).\)

The last inequality together with the discussion above imply that the trivial solution of VIDE (3) is asymptotically stable (see Xu et al. [69, Theorem 3.2] and Sinha [51, Lemma 1]).

Our next result is to discuss the uniform stability of trivial solution of VIDE (3). Firstly, it is needed to introduce an additional assumption given below.

B. Assumption

We have the following conditions:

\((C3)\) Let \(\beta > 0\) and \(\Delta > 0\) with \(\beta ,\) \(\Delta \in {\mathbb{R}}\) such that

$$ \int\limits_{0}^{t} {\left\| {D(t,s)} \right\|ds} \le \alpha_{1} (t),\forall t \in {\mathbb{R}}, $$
$$ \int\limits_{0}^{{t_{0} }} {\int\limits_{{t_{0} }}^{\infty } {\left\| {D(u,s)} \right\|} } duds = \Delta < \infty ,t_{0} > 0, $$

where \(\alpha_{1} \in C({\mathbb{R}}^{ + } ,{\mathbb{R}}^{ + } )\), \(\alpha_{1}\) is also bounded for \(\forall t \in {\mathbb{R}},\) and

$$ \alpha (t) = f_{0} - \frac{1}{2}\int\limits_{0}^{t} {\left\| {D(t,s)} \right\| \, ds} - \frac{1}{2}\beta^{2} \int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\|du} \ge 0,\forall t \in {\mathbb{R}}. $$

Theorem 3

The zero solution of VIDE (3) is uniformly stable if assumptions \((C1)\) and \((C3)\) hold.

Proof

Let \(x \in {\mathbb{R}}^{n}\) and \(\left| x \right|\) be any norm, \(C\) denote the Banach space of continuous functions \(\phi :[t_{0} - \tau ,t_{0} ] \to {\mathbb{R}}^{n} ,\) \(\tau > 0,\) with

$$ \left\| \phi \right\|_{{t_{0} }} = \mathop {\sup }\limits_{{t_{0} - \tau \le s \le t_{0} }} \left| {\phi (s)} \right|. $$

In the proof of this theorem, the main tool is the Lyapunov–Krasovskiĭ \(W = W(t,x(t)),\) which is used in the proof of Theorem 2. By the time derivative of the functional \(W\) and the conditions \((C1)\) and \((C3),\) it is easily derived that

$$ \frac{d}{dt}W(t,x(t)) \le 0. $$

This inequality can complete the proof of the uniformly stability of the zero solution of VIDE (3) (see Xu et al. [69], Theorem 3.1].

Remark 1

Additionally, as for the definition of uniform stability, we complete the proof of Theorem 3 by the following elementary calculations.

From Theorem 3, it is clear that the functional \(W(t,x(t))\) is decreasing. In view of this information, as for the next step, we can write that

$$ \frac{1}{2}\left\| {x(t)} \right\|^{2} \le W(t,x(t)) \le W(t_{0} ,\phi (t_{0} )), \, t \ge t_{0} . $$

From this point, we have

$$ \begin{aligned} W(t_{0} ,\phi (t_{0} )) & = \frac{1}{2}\left\| {\phi (t_{0} )} \right\|^{2} + \sigma \int\limits_{0}^{{t_{0} }} {\int\limits_{{t_{0} }}^{\infty } {\left\| {D(u,s)} \right\|} } \, \left\| {f(\phi (s))} \right\|^{2} duds \\ & \le \frac{1}{2}\left\| {\phi (t_{0} )} \right\|^{2} + \sigma \beta^{2} \int\limits_{0}^{{t_{0} }} {\int\limits_{{t_{0} }}^{\infty } {\left\| {D(u,s)} \right\|} } \left\| {\phi (s)} \right\|^{2} duds \\ & \le \frac{1}{2}[1 + 2\sigma \beta^{2} ]\left\| \phi \right\|_{{t_{0} }}^{2} \int\limits_{0}^{{t_{0} }} {\int\limits_{{t_{0} }}^{\infty } {\left\| {D(u,s)} \right\|} } duds \\ & = \frac{1}{2}[1 + 2\sigma \beta^{2} \Delta ]\left\| \phi \right\|_{{t_{0} }}^{2} . \\ \end{aligned} $$

Hence, we derive that

$$ \left\| {x(t)} \right\|^{2} \le [1 + 2\sigma \beta^{2} \Delta ]\left\| \phi \right\|_{{t_{0} }}^{2} . $$

Now, by the definition of the stability, for each \(\varepsilon > 0,\) we choose a constant \(\delta = \left( {\frac{1}{{\sqrt {1 + 2\sigma \beta^{2} \Delta } }}} \right)\frac{\varepsilon }{2}\) such that if \(\left\| {\phi (t)} \right\| < \delta ,\) \(\forall t \in [ - \tau ,t_{0} ],\) then

$$ \left\| {x(t,t_{0} ,\phi )} \right\| \le \left( {\sqrt {1 + 2\sigma \beta^{2} \Delta } } \right)\delta < \varepsilon ,\forall t \ge t_{0} . $$

Next, since the constant \(\delta\) does not depend on the constant \(t_{0} ,\) then the solution \(x(t) \equiv 0\) of VIDE (3) is uniformly stable. This inequality competes the proof.

4 Globally uniformly asymptotically stability

In this section, we consider the VIDE:

$$ \frac{dx}{{dt}} = - F(t,x)x + \int\limits_{t - \tau }^{t} {K(t,s,x(s))ds,} $$
(4)

with the initial condition

$$ x(t) = \phi (t)\;{\text{for}}\;t \in [ - \tau ,0],\;\phi \in C([ - \tau ,0],{\mathbb{R}}^{n} ), $$

where \(t \in [ - \tau ,\infty ),\) \(\tau\) is a given positive constant, i.e., constant delay, \(x \in {\mathbb{R}}^{n} ,\) \(F\) is defined as in VIDE (3) and \(K(t,s,x) \in C({\mathbb{R}} \times {\mathbb{R}} \times {\mathbb{R}}^{n} ,{\mathbb{R}}^{n} )\) with \(- \tau \le s \le t < \infty\) and \(K(t,s,x) = 0 \Leftrightarrow x = 0.\)

As for the last result of this paper, we need an additional assumption given by \((C4).\)

C. Assumption

We assume the following condition holds:

\((C4)\) There exist positive constants \(\beta ,\) \(f_{0}\) and \(\sigma_{1}\) such that

$$ \int\limits_{t - \tau }^{t} {\left\| {D(t,s)} \right\|} \, ds < + \infty ,\int\limits_{0}^{t} {\left\| {D(t,s)} \right\|} \, ds < + \infty , $$

and

$$ \vartheta (t) = f_{0} - \beta \int\limits_{t - \tau }^{t} {\left\| {D(t,s)} \right\|} \, ds \ge \sigma_{1} . $$

Theorem 4

The trivial solution of VIDE (4) is globally uniformly asymptotically stable if assumptions \((C1)\) and \((C4)\) are satisfied.

Proof

The proof will be done by means of the Lyapunov–Razumikhin method (see, Hale [21, 22] and Zhou and Egorov [74]). For the first step, we choose a Lyapunov function \(W_{1} = W_{1} (t,x(t))\) as follows:

$$ W_{1} (t,x(t)) = \frac{1}{2}\left\langle {x(t),x(t)} \right\rangle = \frac{1}{2}\left\| {x(t)} \right\|^{2} . $$

Next, it is clear from the Lyapunov function \(W_{1} (t,x(t))\) that

$$ W_{1} (t,0) = 0. $$

Then, it is clear that the Lyapunov function \(W_{1}\) satisfies the inequality:

$$ k_{1} \left\| {x(t)} \right\|^{2} \le W_{1} (t,x(t)) \le k_{2} \left\| {x(t)} \right\|^{2} ,\quad k_{1} = k_{2} = \frac{1}{2}. $$

We now consider an arbitrary initial data \((t_{0} ,\phi ) \in {\mathbb{R}}^{ + } \times C([ - \tau ,0],{\mathbb{R}})\) and a point \(t > t_{0}\) such that the Razumikhin condition \(W_{1} (t + \theta ,x(t + s)) < W_{1} (t,x(t)),\) \(s \in [ - \tau ,0],\) holds, i.e., \(\frac{1}{2}\left\| {x(t + s)} \right\|^{2} < \frac{1}{2}\left\| {x(t)} \right\|^{2}\) holds for \(s \in [ - \tau ,0].\) Let \(x(t) = x(t,t_{0} ,\phi )\) denote the solution of IVP for VIDE (4) such that \(x(t_{0}^{ + } + \theta ) = \phi (\theta )\) for \(\theta \in [ - \tau ,0].\)

Differentiating \(W_{1} (t,x(t)),\) we find:

$$ \begin{aligned} \frac{d}{dt}W_{1} (t,x(t)) & = - \frac{1}{2}\left\langle {F(t,x(t))x(t),x(t)} \right\rangle - \frac{1}{2}\left\langle {x(t),F(t,x(t))x(t)} \right\rangle \\ & \quad + \frac{1}{2}\left\langle {x(t),\int\limits_{t - \tau }^{t} {K(t,s,x(s))ds} } \right\rangle + \frac{1}{2}\left\langle {x(t),\int\limits_{t - \tau }^{t} {K(t,s,x(s))ds} } \right\rangle . \\ \end{aligned} $$

Next, assumptions \((C1),\) \((C4)\) and an elementary inequality give that

$$ \begin{aligned} \frac{d}{dt}W_{1} (t,x(t)) & \le - f_{0} \left\| {x(t)} \right\|^{2} + \left\| {x(t)} \right\|\int\limits_{t - \tau }^{t} {\left\| {K(t,s,x(s))} \right\|ds} \\ & \le - f_{0} \left\| {x(t)} \right\|^{2} + \left\| {x(t)} \right\|\int\limits_{t - \tau }^{t} {\left\| {f(x(s))} \right\| \, \left\| {D(t,s)} \right\|ds} \\ & \le - f_{0} \left\| {x(t)} \right\|^{2} + \beta \left\| {x(t)} \right\|\int\limits_{t - \tau }^{t} {\left\| {x(s)} \right\| \, \left\| {D(t,s)} \right\|ds} \\ & \le - f_{0} \left\| {x(t)} \right\|^{2} + \frac{\beta }{2}\int\limits_{t - \tau }^{t} {\left\| {D(t,s)} \right\|{ [}\left\| {x(s)} \right\|^{2} + \left\| {x(t)} \right\|^{2} ]ds} \\ & = - f_{0} \left\| {x(t)} \right\|^{2} + \frac{\beta }{2}\int\limits_{t - \tau }^{t} {\left\| {x(s)} \right\|^{2} \left\| {D(t,s)} \right\|} ds \\ & \quad + \frac{\beta }{2}\int\limits_{t - \tau }^{t} {\left\| {x(t)} \right\|^{2} \left\| {D(t,s)} \right\|ds} \\ & \le - f_{0} \left\| {x(t)} \right\|^{2} + \frac{\beta }{2}\int\limits_{t - \tau }^{t} {\left\| {x(s)} \right\|^{2} \left\| {D(t,s)} \right\|} ds \\ & \quad + \frac{\beta }{2}\left\| {x(t)} \right\|^{2} \int\limits_{t - \tau }^{t} {\left\| {D(t,s)} \right\|ds} . \\ \end{aligned} $$
(5)

We note that the following term:

$$ \frac{\beta }{2}\int\limits_{t - \tau }^{t} {\left\| {x(s)} \right\|^{2} \left\| {D(t,s)} \right\|} ds, $$

which is included in (5).

We now apply this integration the transformation \(s - t = \xi .\) Then, it follows that \(ds = d\xi .\) Hence, if \(s = t - \tau ,\) then \(\xi = - \tau .\) Similarly, if \(s = t,\) then \(\xi = 0.\) From this point, we have

$$ \begin{aligned} \frac{\beta }{2}\int\limits_{t - \tau }^{t} {\left\| {x(s)} \right\|^{2} \left\| {D(t,s)} \right\|} ds & = \frac{\beta }{2}\int\limits_{ - \tau }^{0} {\left\| {x(t + \xi )} \right\|^{2} \left\| {D(t,t + \xi )} \right\|} d\xi \\ & \quad \le \frac{\beta }{2}\int\limits_{ - \tau }^{0} {\left\| {x(t)} \right\|^{2} \left\| {D(t,t + \xi )} \right\|} d\xi \\ & = \frac{\beta }{2}\left\| {x(t)} \right\|^{2} \int\limits_{ - \tau }^{0} {\left\| {D(t,t + \xi )} \right\|} \, d\xi \\ & = \frac{\beta }{2}\left\| {x(t)} \right\|^{2} \int\limits_{t - \tau }^{t} {\left\| {D(t,s)} \right\|} \, ds. \\ \end{aligned} $$
(6)

Substituting the inequality (6) into the inequality (5) and using condition \((C4),\) we get

$$ \begin{aligned} \frac{d}{dt}W_{1} (t,x(t)) & \le - f_{0} \left\| {x(t)} \right\|^{2} + \frac{\beta }{2}\left\| {x(t)} \right\|^{2} \int\limits_{t - \tau }^{t} {\left\| {D(t,s)} \right\|ds} + \frac{\beta }{2}\left\| {x(t)} \right\|^{2} \int\limits_{t - \tau }^{t} {\left\| {D(t,s)} \right\|} \, ds. \\ & = - \, [f_{0} - \beta \int\limits_{t - \tau }^{t} {\left\| {D(t,s)} \right\|} \, ds]\left\| {x(t)} \right\|^{2} = - \vartheta (t)\left\| {x(t)} \right\|^{2} \\ & \le - \sigma_{1} \left\| {x(t)} \right\|^{2} . \\ \end{aligned} $$

Thus, in view of the discussion in this theorem, it follows that the conditions of Lemma 2 (Hale [21], Theorem 4.2, pp. 127] and Hale and Verduyn Lunel [22], Theorem 4.2, pp. 152]) are satisfied. Hence, we can conclude that the zero solution of VIDE (4) is globally uniformly asymptotically stable.

For the coming examples, we need the following remark.

Remark 2

Let \(x \in {\mathbb{R}}^{n}\) and \(A \in {\mathbb{R}}^{n \times n} .\) We define the vector and matrix norm by \(\left\| x \right\| = \left( {\sum\limits_{i = 1}^{n} {\left| {x_{i} } \right|} } \right)\) and \(\left\| A \right\| = \mathop {\max }\limits_{1 \le j \le n} \left( {\sum\limits_{i = 1}^{n} {\left| {a_{ij} } \right|} } \right),\) respectively. We will use the definition of both of these norms in the following examples, when it is needed.

5 Numerical applications

Example 1

We now take into account the VIDE:

$$ \begin{aligned} \left( \begin{gathered} x^{\prime}_{1} \hfill \\ x^{\prime}_{2} \hfill \\ \end{gathered} \right) & = - \, \left( {\begin{array}{*{20}c} {9 + \frac{1}{{1 + \exp (t) + x_{1}^{2} + x_{2}^{2} }}} & 1 \\ 1 & {9 + \frac{1}{{1 + \exp (t) + x_{1}^{2} + x_{2}^{2} }}} \\ \end{array} } \right)\left( \begin{gathered} x_{1} \hfill \\ x_{2} \hfill \\ \end{gathered} \right) \\ & \quad + \int\limits_{0}^{t} {\left( \begin{gathered} \exp ( - 2t + s)\frac{{\sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} }}{{1 + x_{1}^{2} (s)}} \hfill \\ 2\exp ( - 2t + s)\frac{{\sin \sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} }}{{1 + x_{2}^{2} (s)}} \hfill \\ \end{gathered} \right)} ds, \\ \end{aligned} $$
(7)

where \(t \ge 0.\)

Let compare VIDE (7) with VIDE (3). Then, we have

$$ F(t,x_{1} {,}x_{2} {) = }\left( {\begin{array}{*{20}c} {9 + \frac{1}{{1 + \exp (t) + x_{1}^{2} + x_{2}^{2} }}} & 1 \\ 1 & {9 + \frac{1}{{1 + \exp (t) + x_{1}^{2} + x_{2}^{2} }}} \\ \end{array} } \right) $$

and

$$ K(t,s,x_{1} (s),x_{2} (s)) = \left( \begin{gathered} \exp ( - 2t + s)\frac{{\sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} }}{{1 + x_{1}^{2} (s)}} \hfill \\ 2\exp ( - 2t + s)\frac{{\sin \sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} }}{{1 + x_{2}^{2} (s)}} \hfill \\ \end{gathered} \right). $$

Hence, some elementary calculations give the following relations:

$$ \begin{aligned} & \mathop {\sup }\limits_{{(t,x) \in {\mathbb{R}}^{ + } \times {\mathbb{R}}^{2} }} \left\| {F(t,x_{1} ,x_{2} )} \right\| \le 11 < \infty , \\ & \quad \lambda_{1} (F(t,x_{1} ,x_{2} )) = 8 + \frac{1}{{1 + \exp (t) + x_{1}^{2} + x_{2}^{2} }}, \\ & \quad \lambda_{2} (F(t,x_{1} ,x_{2} )) = 10 + \frac{1}{{1 + \exp (t) + x_{1}^{2} + x_{2}^{2} }}, \\ & \quad f_{0i} = 8 \le 8 + \frac{1}{{1 + \exp (t) + x_{1}^{2} + x_{2}^{2} }} \le \lambda_{i} (F(t,x_{1} ,x_{2} )), \\ & \quad \lambda_{i} (F(t,x_{1} ,x_{2} )) = 10 + \frac{1}{{1 + \exp (t) + x_{1}^{2} + x_{2}^{2} }} \le 11 = f_{1i} ,f_{0} = 8, \\ & \quad 8 \le f_{0i} \le \lambda_{i} (F(t,x)) \le f_{1i} \le 11,(i = 1,2). \\ \end{aligned} $$
$$ \begin{aligned} \left\| {K(t,s,x(s))} \right\| & \le \exp ( - 2t + s)\sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} + 2\exp ( - 2t + s)\left| {\sin \sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} } \right| \\ & \le \exp ( - 2t + s)\sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} + 2\exp ( - 2t + s)\sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} \\ & \le 3\exp ( - 2t + s)\sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} \\ & = \left\| {D(t,s)} \right\| \, \left\| {f(x(s))} \right\|, \\ \end{aligned} $$

where

$$ \left\| {D(t,s)} \right\| = 3\exp ( - 2t + s),{ 0} \le s \le t, $$
$$ \left\| {D(u,t)} \right\| = 3\exp ( - 2u + t),{ 0} \le t \le u, $$
$$ \left\| {f(x_{1} (s),x_{2} (s))} \right\| = \sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} = \left\| {x(s)} \right\|, \, \beta = 1. $$

Then, we have

$$ \int\limits_{0}^{t} {\left\| {D(t,s)} \right\|ds} = 3\int\limits_{0}^{t} {\exp ( - 2t + s)ds} = 3 \, [\exp ( - t) - \exp ( - 2t)] = \alpha_{1} (t), $$
$$ \int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\|du} = 3\int\limits_{t}^{\infty } {\exp ( - 2u + t)du} = \frac{3}{2}\exp ( - t) = \alpha_{2} (t). $$

Following these estimates, we can write that

$$ \begin{aligned} \alpha (t) & = f_{0} - \frac{1}{2}\int\limits_{0}^{t} {\left\| {D(t,s)} \right\| \, ds} - \frac{1}{2}\beta^{2} \int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\|du} \\ & = 8 - \frac{3}{2}\exp ( - t) + \frac{3}{2}\exp ( - 2t) - \frac{3}{4}\exp ( - t) \\ & \ge 8 - \frac{3}{2} - \frac{3}{4} = \frac{23}{4} = \rho > 0. \\ \end{aligned} $$

Thus, assumptions \((C1)\) and \((C2)\) hold, which implies that trivial solution of VIDE (7) is asymptotic stable.

For the next step, we derive the following relations:

$$ \begin{gathered} \left\| {D(u,s)} \right\| = 3\exp ( - 2u + s),{ 0} \le s \le u, \hfill \\ \int\limits_{0}^{{t_{0} }} {\int\limits_{{t_{0} }}^{\infty } {\left\| {D(u,s)} \right\|} } duds = 3\int\limits_{0}^{{t_{0} }} {\int\limits_{{t_{0} }}^{\infty } {\exp ( - 2u + s)} } duds = {3}/{2}[\exp (-t_{0} ) - \exp (-2t_{0} )] = \Delta < \infty ,t_{0} \ge 0, \hfill \\ \end{gathered} $$
$$ \begin{aligned} \omega (t) & = f_{0} - \beta^{2} \int\limits_{0}^{t} {\left\| {D(t,s)} \right\|} \, ds \\ & = 8 - 3 \, \exp ( - t) + 3\exp ( - 2t) \ge 8 - 3 \, \exp ( - t) \ge 5 = \sigma . \\ \end{aligned} $$

Thus, assumption \((C3)\) of Theorem 3 is satisfied. As consequence of this result, the trivial solution of VIDE (7) is uniformly stable.

Here, Example 1 was solved using MATLAB software, i.e., it was solved using the 4th order Runge–Kutta method in MATLAB. The graphs of Figs. 1, 2 show the behaviors of paths of the solutions \(x_{1} (t), \, x_{2} (t)\) of Example 1, respectively, for different initial values.

Fig. 1
figure 1

Motions of the orbits of \({x}_{1}(t)\) of VIDE (7)

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Fig. 2
figure 2

Motions of the orbits of solution \({x}_{2}(t)\) of VIDE (7)

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Example 2

We now consider the VIDE:

$$ \begin{aligned} \left( \begin{gathered} x^{\prime}_{1} \hfill \\ x^{\prime}_{2} \hfill \\ \end{gathered} \right) & = - \, \left( {\begin{array}{*{20}c} {9 + \frac{1}{{1 + \exp (t) + x_{1}^{2} + x_{2}^{2} }}} & 1 \\ 1 & {9 + \frac{1}{{1 + \exp (t) + x_{1}^{2} + x_{2}^{2} }}} \\ \end{array} } \right)\left( \begin{gathered} x_{1} \hfill \\ x_{2} \hfill \\ \end{gathered} \right) \\ & \quad + \int\limits_{t - 1}^{t} {\left( \begin{gathered} \exp ( - 2t + s)\frac{{\sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} }}{{1 + x_{1}^{2} (s)}} \hfill \\ 2\exp ( - 2t + s)\frac{{\sin \sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} }}{{1 + x_{2}^{2} (s)}} \hfill \\ \end{gathered} \right)} ds, \\ \end{aligned} $$
(8)

where \(t \ge 1\) and \(\tau = 1\) is the fixed constant.

We now compare VIDE (8) and VIDE (4). \(F(t,x)\) is the same as in Example 1 and satisfies assumption \((C1)\) of Theorem 4. Next, we have

$$ \int\limits_{t - \tau }^{t} {K(t,s,x(s))ds = } \int\limits_{t - 1}^{t} {\left( \begin{gathered} \exp ( - 2t + s)\frac{{\sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} }}{{1 + x_{1}^{2} (s)}} \hfill \\ 2\exp ( - 2t + s)\frac{{\sin \sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} }}{{1 + x_{2}^{2} (s)}} \hfill \\ \end{gathered} \right)} ds. $$

Then, it follows that

$$ K(t,s,x(s)) = \left( \begin{gathered} \exp ( - 2t + s)\frac{{\sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} }}{{1 + x_{1}^{2} (s)}} \hfill \\ 2\exp ( - 2t + s)\frac{{\sin \sqrt {x_{1}^{2} (s) + x_{2}^{2} (s)} }}{{1 + x_{2}^{2} (s)}} \hfill \\ \end{gathered} \right). $$

Similarly, as in Example 1, it is derived the following relations:

$$ \left\| {D(t,s)} \right\| = 3\exp ( - 2t + s),{ 0} \le s \le t, $$
$$ \int\limits_{0}^{t} {\left\| {D(t,s)} \right\|ds} = 3\int\limits_{0}^{t} {\exp ( - 2t + s)ds} = 3 \, [\exp ( - t) - \exp ( - 2t)] < + \infty , $$
$$ \int\limits_{t - \tau }^{t} {\left\| {D(t,s)} \right\|} \, ds = 3\int\limits_{t - 1}^{t} {\exp ( - 2t + s)} \, ds = 3[\exp ( - t) - \exp ( - t - 1)] < + \infty , $$
$$ \left\| {f(x_{1} (s),x_{2} (s))} \right\| = \left\| {x(s)} \right\|, \, \beta = 1. $$
$$ \begin{aligned} \vartheta (t) & = f_{0} - \beta \int\limits_{t - \tau }^{t} {\left\| {D(t,s)} \right\|} \, ds \\ & = 8 - 3[\exp ( - t) - \exp ( - t - 1)] \\ & \ge 8 - 3\exp ( - t) \ge 5 = \sigma_{1} . \\ \end{aligned} $$

Subject to the discussion done, assumption \((C4)\) of Theorem 4 is satisfied. Then, we conclude that the trivial solution of VIDE (8) is globally uniformly asymptotic stable.

Here, Example 2 was solved using MATLAB software using the 4th order Runge–Kutta method in MATLAB. The graphs of Figs. 3, 4 show the behaviors of paths of the solutions \(x_{1} (t), \, x_{2} (t)\) of Examples 2, respectively, for \(\tau = 1\) and different initial values.

Fig. 3
figure 3

Motions of the orbits of the solution of \({x}_{1}(t)\) of VIDE (8)

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Fig. 4
figure 4

Motions of the orbits of the solution \({x}_{2}(t)\) of VIDE (8)

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6 Discussions

We will now compare the results of this paper with the application result of Seifert [49] and some results in the references of this paper (see [1, 2, 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,17, 19, 20, 23, 25,26,27, 31,32,33,34,35,36,37,38,39,40,41,42, 48,49,50, 52,53,54,55,56,57,58,59, 65, 66,67,68, 70, 72, 73]). Indeed, some comparisons are given in the introduction of the paper.

\(1^{0} )\) VIDE (3) and VIDE (4) generalize and improve VIDE (1). In fact, this is a clear idea if \(F(t,x) = A\) in VIDE (3) and if we take \(F(t,x) = A\) and put “0” instead of \(t - \tau\) in VIDE (4). To the best of our information, the concepts in the results of this paper were not discussed for VIDE (3) and VIDE (4) in the literature until now. This is the first contribution of this paper.

\(2^{0} )\) Seifert [49] studied the stability of the trivial solution of VIDE (1) via a quadratic Lyapunov function. In this paper, together with the stability, we also investigate asymptotic stability, uniform stability and global uniform asymptotic stability of zero solution for the more general VIDE (3) and VIDE (4). These are the second contributions of this paper.

\(3^{0} )\) Seifert [49] used a quadratic Lyapunov function to prove his application result, Theorem 1, as follows:

$$ V(x) = \left\langle {x,Bx} \right\rangle . $$

However, in this paper, we defined and used the following functional and function in the proofs of Theorems 2, 3 and Theorem 4, respectively:

$$ W(t,x(t)) = \frac{1}{2}\left\| {x(t)} \right\|^{2} + \sigma \int\limits_{0}^{t} {\int\limits_{t}^{\infty } {\left\| {D(u,s)} \right\|} } \, \left\| {f(x(s))} \right\|^{2} duds $$

and

$$ W_{1} (t,x(t)) = \frac{1}{2}\left\| {x(t)} \right\|^{2} . $$

Actually, the stability result of Theorem 1 in Seifert [49] and our asymptotic stability result, Theorem 2, have connections. However, the conditions both of Theorems 1, 2 are completely different. Next, the asymptotic stability implies to the stability. In general, contrary, the stability does not imply the asymptotic stability. This means that our result is stronger and more suitable than the stability result of [49]. Furthermore, in spite of the use of Lyapunov function \(V\) in the proof of Theorem 1 (see Seifert [49]), we use the Lyapunov–Krasovskiĭ functional \(W\) in the proof of Theorem 2.

Next, Vanualailai and Nakagiri [65] used the Lyapunov functional

$$ \begin{aligned} V_{2} (t,x(.)) & = \frac{1}{2}x^{2} + \sqrt \alpha \int\limits_{0}^{x} {\sqrt {uf(u)} } du + \frac{1}{2}\alpha \int\limits_{0}^{x} {f(u)} du \\ & \quad + k\int\limits_{0}^{t} {\int\limits_{t}^{\infty } {\left| {B(u,s)} \right|} } du \, f^{2} (x(s))ds \\ \end{aligned} $$

and its a modified version to prove the stability results therein, [65], Theorems 3.2 and 4.2]. As for the asymptotic stability conditions of Theorem 2 and the others, the Lyapunov–Krasovskiĭ functional \(W\) given here is completely different from those used Vanualailai and Nakagiri [65], Theorems 3.2 and 4.2].

As we gave and summarized above, Burton [4], Theorems, 1, 2, Theorems 4–8] investigated qualitative behaviors of a class of the scalar VIDEs and systems of VIDEs. Indeed, when we compare our work with that of Burton [4], it can be followed that the considered scalar VIDEs and systems of VIDEs, constructed conditions and obtained results in [4] are completely different from those of this paper, except some minor similarity with our condition such as

$$ \alpha (t) = f_{0} - \frac{1}{2}\int\limits_{0}^{t} {\left\| {D(t,s)} \right\| \, ds} - \frac{1}{2}\beta^{2} \int\limits_{t}^{\infty } {\left\| {D(u,t)} \right\|du} > 0. $$

We would like to mean that the results of this paper are different, new, original, more general and suitable than those of [4]. These are the next contributions and originality of this work.

\(4^{0} )\) In the proof of our last result, Theorem 4, we benefit from the Lyapunov–Razumikhin method as a main tool to complete the proof of this theorem. Indeed, Theorem 4 is a new result on the global uniform asymptotic stability of the trivial solution of VIDE (4). This is the new contribution and originality of this work.

\(5^{0} )\) In Theorem 1 of Seifert [49], the kernel \(K(.)\) of VIDE (1) has to satisfy the inequality

$$ \left| {\int\limits_{0}^{t} {K(t,s,x(s))ds} } \right| \le \mu \mathop {\sup }\limits_{0 \le s \le t} \left| {x(s)} \right| $$

for any continuous function \(x(s)\) in \(0 \le s \le t\) such that \(\left| {x(s)} \right| \le \rho\) on this interval. That is, the function \(x(s)\) has to be bounded on the interval \(0 \le s \le t\) and, in addition, \(\frac{2\left| B \right|\mu \Lambda }{\lambda } < 1.\) These conditions are very restrictive than those of Theorems 2–4. We would not like to present the details. Hence, we obtain the results of Seifert [49] under less restrictive conditions here, i.e., under weaker and less conservative conditions. Thus, the results of this paper improve Theorem 1 of Seifert [49].

\(6^{0} )\) In Seifert [49], it was not given any example to applications of the results and concepts introduced therein, [49]. However, in this paper, we gave two new examples, Examples 1, 2 to explore the movements of the orbits of the solutions of the given VIDE (7) and VIDE (8). These are desirable applications for any paper on the concepts as they were introduced here.

\(7^{0} )\) To the best of our knowledge, so far the Lyapunov–Razumikhin method was not used to study the qualitative behaviors of VIDEs of integer order, except those of Sedova [48] and Seifert [49], which are related to the integro- differential equations without delay. Here, by this paper, it can be followed the effectiveness and usefulness of the Lyapunov–Razumikhin method during the investigations of that kind of results. The theorems of this paper and the Lyapunov–Razumikhin method used here can be considered as complements of other related results and methods already available in the literature [1, 2, 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,17, 19, 20, 23, 25,26,27, 31,32,33,34,35,36,37,38,39,40,41,42, 48,49,50, 52,53,54,55,56,57,58,59, 65,66,67,68, 70, 72, 73]. In conclusion, the contributions in details are ignored here for the sake of the brevity.