Asymptotic Properties of Solutions to Linear Non-Homogeneous Neutral-Delay Differential Equations with Variable Delays and Coefficients

Abstract

This article investigates the asymptotic properties of the solutions to first-order linear non-homogeneous neutral-delay differential equations that have variable coefficients and variable delays. Firstly, the aforementioned asymptotic properties were established using the solution of the characteristic equation. Subsequently, by examining specific cases regarding the variable coefficients of the neutral-delay differential equation, two distinct results were obtained. Finally, as a special case, the linear non-homogeneous neutral-delay differential equation with constant coefficients and delays was considered. The findings are presented in terms of the solution to the characteristic equation. Furthermore, nine examples are provided to elaborate on the finding that the hypotheses examined in the main theorem and in the special case hold.

1. Introduction

This paper examines the linear neutral-delay differential equation (NDDE) with variable coefficients:

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)+\alpha \left(t\right)f\left(t\right)+{\displaystyle \sum _{j=1}^m}{\gamma }_j\left(t\right)f^{\prime }\left(t-{\sigma }_j\left(t\right)\right)+{\displaystyle \sum _{i=1}^n}{\beta }_i\left(t\right)f\left(t-{\tau }_i\left(t\right)\right)=\delta \left(t\right),t\ge 0,\end{array}$$

(1)

where the coefficients $\alpha$, ${\gamma }_j$, ${\sigma }_j$, ${\beta }_i$, ${\tau }_i$, and $\delta$ are continuous and real-valued functions on the interval $[0,\infty )$, and the delay functions are ${\sigma }_j>0$ and ${\tau }_i>0$. Furthermore, assume that at least one ${\gamma }_j$ and at least one ${\beta }_i$ are not identically zero on $[0,\infty )$.

Let us presume $\sigma$ and $\tau$ are positive, such that

$$\begin{array}{c}\hfill \sigma =\sup \left\{{\sigma }_j\left(t\right):t\ge 0,j=1, 2, \dots ,m\right\}<\infty ,\\ \hfill \tau =\sup \left\{{\tau }_i\left(t\right):t\ge 0,i=1, 2, \dots , n\right\}<\infty \end{array}$$

and we denote

$$\begin{array}{c}\hfill r=\max \{\sigma ,\tau \}.\end{array}$$

By a solution f to the NDDE (1), we give a continuous function having real values on $[-r,\infty )$, which is continuously differentiable on $[-\sigma ,\infty )$ and meets the conditions of (1).

Let $C^1([-r,0],\mathbb{R})$ express the set of continuously differentiable real-valued functions on the initial interval $[-r,0].$ As is convention, we provide an initial condition of the form as follows along with the NDDE (1)

$$f\left(t\right)=\xi \left(t\right)\mathrm{for}-r\le t\le 0,$$

(2)

where $\xi \in C^1([-r,0],\mathbb{R})$ satisfying the “consistency condition”

$$\begin{array}{c}\hfill {\xi }^{\prime }\left(0\right)+\alpha \left(0\right)\xi \left(0\right)+{\displaystyle \sum _{j=1}^m}{\gamma }_j\left(0\right){\xi }^{\prime }\left(-{\sigma }_j\left(0\right)\right)+{\displaystyle \sum _{i=1}^n}{\beta }_i\left(0\right)\xi (-{\tau }_i\left(0\right))=\delta \left(0\right).\end{array}$$

More details on the consistency condition may be accessed by referring to some previous studies [1,2,3].

Equation (1) and initial function (2) denote an “initial value problem” (henceforth IVP). It is known well that there exists a unique solution f of the NDDE (1) satisfying the given initial condition (2), and this unique solution f shall be denoted the “solution” to the IVP (1) and (2) [4].

For the case where the function $\delta$ is identical to zero on $[0,\infty )$, we reduce the NDDE (1) to

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)+\alpha \left(t\right)f\left(t\right)+{\displaystyle \sum _{j=1}^m}{\gamma }_j\left(t\right)f^{\prime }\left(t-{\sigma }_j\left(t\right)\right)+{\displaystyle \sum _{i=1}^n}{\beta }_i\left(t\right)f\left(t-{\tau }_i\left(t\right)\right)=0,t\ge 0.\end{array}$$

(3)

Besides the NDDE (3), the following equation is associated

$$\begin{array}{c}\hfill \mu \left(t\right)+\alpha \left(t\right)+{\displaystyle \sum _{j=1}^m}{\gamma }_j\left(t\right)\mu \left(t-{\sigma }_j\left(t\right)\right)\exp \left[-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right]\\ \hfill +{\displaystyle \sum _{i=1}^n}{\beta }_i\left(t\right)\exp \left[-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right]=0,t\ge 0;\\ \hfill \mu \left(t\right)={\mu }_0\left(t\right),-r\le t\le 0,\end{array}$$

(4)

which shall be expressed as the generalized characteristic equation of (3). Equation (4) is obtained based on (3) by searching solutions to the form given below.

$$\begin{array}{c}\hfill f\left(t\right)=\exp \left[{\int }_0^t\mu \left(s\right)ds\right]\mathrm{for}t\ge -r,\end{array}$$

where $\mu$ is a function, which is continuous and to which real values are assigned, defined on $[-r,\infty )$.

A solution to the generalized characteristic Equation (4) is a real-valued continuous function $\mu$ defined on $[-r,\infty )$ which meets the condition in (4) for all $t\ge 0$.

The importance of NDDEs extends beyond theoretical considerations, as evidenced by their application in the modelling of a wide range of scientific and engineering problems. These include electrical power grids, high-speed computer circuits, mechanical systems, chemistry and process control systems, biological models, the spread of infectious diseases, and economic price dynamics [5,6,7,8]. The presence of delay in the derivative terms renders these equations more complex than classical delayed differential equations. Consequently, analysing the stability and asymptotic behaviour of neutral systems is a significant research topic in the literature [1,9,10,11,12,13,14,15]. In this context, various special cases of NDDEs have been utilised, particularly in the study of dynamic systems such as Hopfield-type neural networks. The qualitative properties of this class of equations including stability, exponential stability, asymptotic stability, oscillation, and non-oscillation have been extensively investigated by numerous researchers [1,8,11,16,17,18,19,20,21]. In recent years, Berezansky and Braverman have derived new solution estimates, exponential stability conditions, and practical tests of stability for linear neutral differential equations [22,23,24,25,26,27]. More recent studies on linear neutral differential equations and systems have also attracted considerable attention. Berezansky and Braverman investigated the neutral-type linear systems in terms of their asymptotic properties and established conditions concerning the behaviour and stability of their solutions [28]. Berezansky et al. derived criteria of exponential stability for linear neutral systems and applied the obtained results to neutral-type neutral network models [29]. Ağarcık et al. developed an equivalence transformation for canonical and non-canonical types of linear neutral differential equations and employed this transformation to investigate the oscillatory behaviour of their solutions [30]. Furthermore, recent contributions have addressed the global existence and unique nature of solutions to neutral differential equations having state-dependent delays [31], exponential stability findings regarding non-linear- and linear-delay differential equations [32], and criteria of oscillation in first-order neutral-delay differential equations having several positive and negative coefficients [33]. However, a significant proportion of extant studies focuses primarily on neutral differential equations with constant delay, homogeneous equations, or equations with specific structures. It is evident that, despite the efforts of Philos and colleagues [34] who have obtained asymptotic analyses for equations with constant delay and homogeneous structure, the investigation of the asymptotic behaviour of solutions and their derivatives for differential equations that are linear, non-homogeneous, and delayed with coefficients and delays that are variable has not yet been sufficiently explored. It is this study’s objective to address the asymptotic characteristics of the aforementioned equation class within the framework of the method of roots of the characteristic equation, based on the identified gap.

In this paper, we expand some of the findings obtained for homogeneous neutral differential equations with constant delays in a past study [34] to non-homogeneous neutral differential equations that have variable delays. To put it in other terms, if $\left|\delta \left(t\right)\right|\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]$ is asymptotically zero, we reach a sharper condition for the asymptotic characteristics of the solutions of Equation (1), which is an elaboration upon the studies of some other authors [34]. Expressions regarding the solutions of (1) are presented for the solutions of a generalized characteristic equation. We formulate the main findings of our study in the form of Theorem 1 in the following section and then prove it in the same section.

The present paper includes five sections. Section 1 comprises the introduction and preliminary material. In Section 2, the proof of the asymptotic characteristics of the solutions to the NDDE (1) is presented. Then, Section 3 provides two case illustrations to support and expand upon these findings. The linear non-homogeneous neutral differential equation that has constant delays and constant coefficients was considered as a special case, and three examples are offered in Section 4. Section 5 comprises discussions and conclusions.

2. Statement of the Results

In a prior study [34], the authors demonstrated that a unique solution to the characteristic Equation (4) with $\tau \left(t\right)=\tau$, $\sigma \left(t\right)=\sigma$ (both $\tau$ and $\sigma$ are positive constants) is found. A similar step-based technique is implemented for the general Equation (4) for the delay functions $\tau >0$ and $\sigma >0$. Lemma 1 shall be employed to prove the main finding of the article, i.e., Theorem 1, which is provided following the Lemma. The Lemma in question fundamentally establishes the principal output of this paper through a solution given for the generalized characteristic Equation (4). More in-depth information concerning the step-based technique may be found in the books cited in [35,36].

Lemma 1.

For every ${\mu }_0$ real-valued function on $[-r,0]$, the generalized characteristic Equation (4) possesses a unique global solution.

Proof. 

Let us apply consecutive steps on the intervals $[{\gamma }_0\left(0\right),{\gamma }_1\left(0\right)]$, $[{\gamma }_1\left(0\right),{\gamma }_2\left(0\right)]$, … to identify the solution to the characteristic equation. Here,

$$\begin{array}{c}\hfill \gamma \left(t\right)=inf\left\{\lambda \in [0,\infty ):\max \left\{(\underset{j}{\max }\left(\lambda -{\sigma }_j\left(\lambda \right)\right),\underset{i}{\max }\left(\lambda -{\tau }_i\left(\lambda \right)\right)\right\}>t\right\}\end{array}$$

and for $t=1, 2, \dots$

$$\begin{array}{c}\hfill 0={\gamma }_0\left(0\right),{\gamma }_1\left(0\right)=\gamma \left(0\right),\cdots ,{\gamma }_j\left(0\right)=\gamma \left({\gamma }_{j-1}\left(0\right)\right),\cdots .\end{array}$$

Let us explore the way in which the solution $\mu ={\mu }_1$ to problem (4) is defined on the entirety of the interval $\left[0,{\gamma }_1\left(0\right)\right].$ Assuming

$$\begin{array}{c}\hfill w\left(t\right)=\exp \left\{{\int }_0^t{\mu }_0\left(s\right)ds\right\}\mathrm{for}-r\le t\le 0,\end{array}$$

and

$$\begin{array}{c}\hfill \psi \left(t\right)=\exp \left\{{\int }_0^t\mu \left(s\right)ds\right\}\mathrm{for}t\ge 0.\end{array}$$

Then, with (4), for $0\le t\le {\gamma }_1\left(0\right)$, we reach the linear differential equation

$$\begin{array}{c}\hfill {\psi }^{\prime }\left(t\right)=\mu \left(t\right)\psi \left(t\right)=-\alpha \left(t\right)\psi \left(t\right)-{\displaystyle \sum _{j=1}^m}{\gamma }_j\left(t\right)\mu (t-{\sigma }_j\left(t\right))w(t-{\sigma }_j\left(t\right))-{\displaystyle \sum _{i=1}^n}{\beta }_i\left(t\right)w(t-{\tau }_i\left(t\right))\end{array}$$

with $\psi \left(0\right)=1$. The solution to the equation can be given as follows

$$\begin{array}{cc}\hfill \psi \left(t\right)=& \exp \left[-{\int }_0^t\alpha \left(u\right)du\right]\{1-{\int }_0^t{\displaystyle \sum _{j=1}^m}{\gamma }_j\left(s\right)\mu (t-{\sigma }_j\left(t\right))\exp [{\int }_0^{s-{\sigma }_j\left(s\right)}{\mu }_0\left(u\right)du\hfill \\ \hfill & +{\int }_0^s\alpha \left(u\right)du]ds-{\int }_0^t{\displaystyle \sum _{i=1}^n}{\beta }_i\left(s\right)\exp \left[{\int }_0^{s-{\tau }_i\left(s\right)}{\mu }_0\left(u\right)du+{\int }_0^s\alpha \left(u\right)du\right]ds\}\hfill \end{array}$$

which allows us to define ${\mu }_1\left(t\right)={\psi }^{\prime }\left(t\right)/\psi \left(t\right)$ on $[0,{\gamma }_1\left(0\right)].$ Like us, the function ${\mu }_1$ is defined on the first interval. At the next stage of the this procedure, let $w\left(t\right)=\psi \left(t\right)$ on $[-r,{\gamma }_1\left(0\right)].$ Next, using (4) for ${\gamma }_1\left(0\right)\le t\le {\gamma }_2\left(0\right)$, we acquire the linear differential equation

$$\begin{array}{c}\hfill {\psi }^{\prime }\left(t\right)=\mu \left(t\right)\psi \left(t\right)=-\alpha \left(t\right)\psi \left(t\right)-{\displaystyle \sum _{j=1}^m}{\gamma }_j\left(t\right)\mu (t-{\sigma }_j\left(t\right))w(t-{\sigma }_j\left(t\right))-{\displaystyle \sum _{i=1}^n}{\beta }_i\left(t\right)w(t-{\tau }_i\left(t\right))\end{array}$$

whose solution let us define ${\mu }_2\left(t\right)={\psi }^{\prime }\left(t\right)/\psi \left(t\right)$ on $[{\gamma }_1\left(0\right),{\gamma }_2\left(0\right)].$ Advancing in the same manner,

$$\mu \left(t\right)=\left\{\begin{array}{ccc}{\mu }_0\left(t\right)& \mathrm{if}& t\in [-r,0],\\ {\mu }_1\left(t\right)& \mathrm{if}& t\in [0,{\gamma }_1\left(0\right)],\\ {\mu }_2\left(t\right)& \mathrm{if}& t\in [{\gamma }_1\left(0\right),{\gamma }_2\left(0\right)],\\ ⋮& ⋮& ⋮\end{array}\right.$$

denotes a unique global solution for the characteristic Equation (4) and finalizes the proof. □

Theorem 1.

Let μ be a solution to the generalized characteristic Equation (4). Supposing as follows

$$\begin{array}{cc}\hfill \underset{t\to \infty }{\mathrm{lim\; sup}}\{& {\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t\right)\right|\left[1+\left|\mu \left(t-{\sigma }_j\left(t\right)\right)\right|{\sigma }_j\left(t\right)\right]\exp \left[-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t\right)\right|{\tau }_i\left(t\right)\exp \left[-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right]\}<1\hfill \end{array}$$

(5)

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|\delta \left(t\right)\right|\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]=0.\end{array}$$

(6)

The solution f for the IVP (1) and (2) then satisfies

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }f\left(t\right)\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]=L(\mu ;\xi ),\end{array}$$

(7)

where $L(\mu ;\xi )$ is a real number depending on μ and is defined according to ξ, and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }{\left\{f\left(t\right)\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]\right\}}^{\prime }=0.\end{array}$$

(8)

The proof of Theorem 1 is established upon the integral expression of $g^{\prime }$.

Proof of Theorem 1.

Let f be the solution to (1) and (2), and let $\mu$ as well be the solution to the characteristic Equation (4). Here, the following is defined

$$\begin{array}{c}\hfill g\left(t\right)=f\left(t\right)\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]\mathrm{for}t\ge -r.\end{array}$$

(9)

It can be proven that ${\lim }_{t\to \infty }g\left(t\right)$ exists, and ${\lim }_{t\to \infty }{g}^{\prime }\left(t\right)=0$. After differentiating g and utilizing (1) and (4), we reach the following

$$\begin{array}{cc}\hfill g^{\prime }\left(t\right)=& \left(f^{\prime }\left(t\right)-f\left(t\right)\mu \left(t\right)\right)\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]\hfill \\ \hfill =& \{\delta \left(t\right)-{\displaystyle \sum _{j=1}^m}{\gamma }_j\left(t\right)f^{\prime }(t-{\sigma }_j\left(t\right))-{\displaystyle \sum _{i=1}^n}{\beta }_i\left(t\right)f(t-{\tau }_i\left(t\right))\hfill \\ \hfill & + f\left(t\right){\displaystyle \sum _{j=1}^m}{\gamma }_j\left(t\right)\mu (t-{\sigma }_j\left(t\right))\exp \left[-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill & + f\left(t\right){\displaystyle \sum _{i=1}^n}{\beta }_i\left(t\right)\exp \left[-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right]\}\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]\hfill \end{array}$$

for every $t\ge 0$. Using the equation below,

$$\begin{array}{c}\hfill f(t-{\tau }_i\left(t\right))=g(t-{\tau }_i\left(t\right))\exp \left[{\int }_0^{t-{\tau }_i\left(t\right)}\mu \left(s\right)ds\right]\end{array}$$

and

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)=\left(g^{\prime }\left(t\right)+g\left(t\right)\mu \left(t\right)\right)\exp \left[{\int }_0^t\mu \left(s\right)ds\right],\end{array}$$

the equality presented above gives us

$$\begin{array}{cc}\hfill g^{\prime }\left(t\right)=& \delta \left(t\right)\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]-{\displaystyle \sum _{j=1}^m}{\gamma }_j\left(t\right)\{g^{\prime }(t-{\sigma }_j\left(t\right))\hfill \\ \hfill & -\left[g\left(t\right)-g(t-{\sigma }_j\left(t\right))\right]\mu (t-{\sigma }_j\left(t\right))\}\exp \left[-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}{\beta }_i\left(t\right)\left[g\left(t\right)-g(t-{\tau }_i\left(t\right))\right]\exp \left[-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \end{array}$$

(10)

for $t\ge 0$. Further, f meets the initial condition (2) if and only if g meets the condition given below

$$\begin{array}{c}\hfill g\left(t\right)=\xi \left(t\right)\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]\mathrm{for}-r\le t\le 0.\end{array}$$

Here, as per (10), the expression below directly follows

$$\begin{array}{cc}\hfill g^{\prime }\left(t\right)=& \delta \left(t\right)\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]\hfill \\ \hfill & -{\displaystyle \sum _{j=1}^m}{\gamma }_j\left(t\right)\left[g^{\prime }(t-{\sigma }_j\left(t\right))-\mu (t-{\sigma }_j\left(t\right)){\int }_{t-{\sigma }_j\left(t\right)}^t{g}^{\prime }\left(s\right)ds)\right]\exp \left[-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}{\beta }_i\left(t\right){\int }_{t-{\tau }_i\left(t\right)}^t{g}^{\prime }\left(s\right)ds\exp \left[-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \end{array}$$

(11)

for $t\ge r-\sigma$.

Besides the expression given above, we see the function $g^{\prime }$ on $[0,r]$ depends on f and $\mu$, and therefore, it is dependent on ${\mu }_0$ and $\xi$. It is guaranteed by the assumptions (5) and (6) that there is an integer $p\ge 1$, such that

$$\begin{array}{cc}\hfill {\theta }_1\left({\mu }_0\right)=& \underset{t\ge pr-\sigma }{\sup }\{{\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t\right)\right|\left[1+|\mu (t-{\sigma }_j\left(t\right))|{\sigma }_j\left(t\right)\right]\exp \left[-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t\right)\right|{\tau }_i\left(t\right)\exp \left[-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right]\},\hfill \\ \hfill {\theta }_2\left({\mu }_0\right)=& \underset{t\ge pr-\sigma }{\sup }\left\{\left|\delta \left(t\right)\right|\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]\right\}\hfill \end{array}$$

and

$$\begin{array}{c}\hfill {\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)<1.\end{array}$$

It shall be pointed out that because it is assumed that at least one ${\beta }_i$ and at least one ${\gamma }_j$ are not identically zero on $[0,\infty )$, it is clearly apparent that ${\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)>0$.

Additionally, noting (6), an integer $\mathsf{\ell }\ge 1$ for which the sequence of points $t_m=(m+p-1)\mathsf{\ell }r-\sigma$ can be defined inductively, where for all $m=\mathsf{\ell }, \mathsf{\ell }+1, \mathsf{\ell }+2, \dots$ can be found,

$$\begin{array}{c}\hfill \left|\delta \left(t\right)\right|\exp \left\{-{\int }_0^t\mu \left(s\right)ds\right\}\le {\theta }_2\left({\mu }_0\right){\left({\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right)}^{m-\mathsf{\ell }}\mathrm{for}\mathrm{every}t\ge t_m.\end{array}$$

(12)

Set $t_{\mathsf{\ell }-1}=(\mathsf{\ell }+p-2)\mathsf{\ell }r-\sigma$. We remark that the maximum value of $g^{\prime }$ on $[t_{\mathsf{\ell }-1},t_{\mathsf{\ell }}]$ depends on $\mu$ and f; thus, it depends on ${\mu }_0$ and $\xi$. Set

$$\begin{array}{c}\hfill K({\mu }_0;\xi )=\max \left\{1,\underset{[t_{\mathsf{\ell }-1},t_{\mathsf{\ell }}]}{\max }\left|g^{\prime }\left(t\right)\right|\right\}.\end{array}$$

(13)

Then,

$$\begin{array}{c}\hfill |g^{\prime }\left(t\right)|\le K({\mu }_0;\xi )\mathrm{for}{t}_{\mathsf{\ell }-1}\le t\le t_{\mathsf{\ell }}.\end{array}$$

(14)

where we shall express that $K({\mu }_0;\xi )$ is also a bound of $|g^{\prime }|$ on the entire interval $[t_{\mathsf{\ell }-1},\infty )$, i.e.,

$$\begin{array}{c}\hfill |g^{\prime }\left(t\right)|\le K({\mu }_0;\xi )\mathrm{for}\mathrm{all}t\ge t_{\mathsf{\ell }-1}.\end{array}$$

(15)

For this purpose, let us take an arbitrary number $ϵ>0$. It shall be claimed that

$$\begin{array}{c}\hfill \left|g^{\prime }\left(t\right)\right|<K({\mu }_0;\xi )+ϵ\mathrm{for}t\ge t_{\mathsf{\ell }-1}.\end{array}$$

(16)

We will be assuming inequality (16) does not hold true here. In the case here, since (14) and by the continuity of $g^{\prime }$, there exists a point $t^{*}>t_{\mathsf{\ell }}$, where

$$\begin{array}{c}\hfill |g^{\prime }\left(t\right)|<K({\mu }_0;\xi )+ϵ,t\in [t_{\mathsf{\ell }-1},t^{*})\mathrm{and}\left|g^{\prime }\left(t^{*}\right)\right|=K({\mu }_0;\xi )+ϵ.\end{array}$$

Subsequently, considering the definitions of ${\theta }_1\left({\mu }_0\right)$ and ${\theta }_2\left({\mu }_0\right)$, and the fact that $K({\mu }_0;\xi )\ge 1$, from (11), we obtain

$$\begin{array}{cc}\hfill K({\mu }_0;\xi )+ϵ=& |g^{\prime }\left(t^{*}\right)|\hfill \\ \hfill \le & |\delta \left(t^{*}\right)|\exp \left[-{\int }_0^{t^{*}}\mu \left(s\right)ds\right]+{\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t^{*}\right)\right|[\left|g^{\prime }(t^{*}-{\sigma }_j\left(t^{*}\right))\right|\hfill \\ \hfill & +\left|\mu \left(t^{*}-{\sigma }_j\left(t^{*}\right)\right)\right|{\int }_{t^{*}-{\sigma }_j\left(t^{*}\right)}^{t^{*}}\left|g^{\prime }\left(s\right)\right|ds]\exp \left[-{\int }_{t^{*}-{\sigma }_j\left(t^{*}\right)}^{t^{*}}\mu \left(s\right)ds\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t^{*}\right)\right|\left[{\int }_{t^{*}-{\tau }_i\left(t^{*}\right)}^{t^{*}}\left|g^{\prime }\left(s\right)\right|ds\right]\exp \left[-{\int }_{t^{*}-{\tau }_i\left(t^{*}\right)}^{t^{*}}\mu \left(s\right)ds\right]\hfill \\ \hfill \le & \left(K({\mu }_0;\xi )+ϵ\right)\left|\delta \left(t^{*}\right)\right|\exp \left[-{\int }_0^{t^{*}}\mu \left(s\right)ds\right]\hfill \\ \hfill & +\left(K({\mu }_0;\xi )+ϵ\right){\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t^{*}\right)\right|\left[1+{\sigma }_j\left(t^{*}\right)\left|\mu (t^{*}-{\sigma }_j\left(t^{*}\right))\right|\right]\hfill \\ \hfill & \times \exp \left[-{\int }_{t^{*}-{\sigma }_j\left(t^{*}\right)}^{t^{*}}\mu \left(s\right)ds\right]\hfill \\ \hfill & +\left(K({\mu }_0;\xi )+ϵ\right){\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t^{*}\right)\right|{\tau }_i\left(t^{*}\right)\exp \left[-{\int }_{t^{*}-{\tau }_i\left(t^{*}\right)}^{t^{*}}\mu \left(s\right)ds\right]\hfill \\ \hfill \le & \left(K({\mu }_0;\xi )+ϵ\right)\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]\hfill \\ \hfill <& K({\mu }_0;\xi )+ϵ,\hfill \end{array}$$

which consequently represents a contradiction. Hence, the inequality (16) must hold true, and the inequality (15) must also hold true. Moreover, by virtue of (15), from (11), we reach, for all $t\ge t_{\mathsf{\ell }}$,

$$\begin{array}{cc}\hfill |g^{\prime }\left(t\right)|\le & \left|\delta \left(t\right)\right|\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]+{\displaystyle \sum _{j=1}^m}|{\gamma }_j\left(t\right)|[|g^{\prime }(t-{\sigma }_j\left(t\right))|\hfill \\ \hfill & +\left|\mu \left(t-{\sigma }_j\left(t\right)\right)\right|{\int }_{t-{\sigma }_j\left(t\right)}^t\left|g^{\prime }\left(s\right)\right|ds]\exp \left[-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t\right)\right|\left[{\int }_{t-{\tau }_i\left(t\right)}^t\left|g^{\prime }\left(s\right)\right|ds\right]\exp \left[-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill \le & K({\mu }_0;\xi )\left|\delta \left(t\right)\right|\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]\hfill \\ \hfill & + K({\mu }_0;\xi ){\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t\right)\right|\left[1+{\sigma }_j\left(t\right)\left|\mu \left(t-{\sigma }_j\left(t\right)\right)\right|\right]\exp \left[-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill & + K({\mu }_0;\xi ){\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t\right)\right|{\tau }_i\left(t\right)\exp \left[-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill \le & K({\mu }_0;\xi )\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right].\hfill \end{array}$$

Thus, we have

$$\begin{array}{c}\hfill |g^{\prime }\left(t\right)|\le K({\mu }_0;\xi )\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]\mathrm{for}\mathrm{every}t\ge t_{\mathsf{\ell }}.\end{array}$$

(17)

Through induction and by using ${\theta }_1\left({\mu }_0\right)$, ${\theta }_2\left({\mu }_0\right)$, (15), and (17), we demonstrate $g^{\prime }$ satisfies the inequality below:

$$\begin{array}{c}\hfill |g^{\prime }\left(t\right)|\le K({\mu }_0;\xi ){\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{m-\mathsf{\ell }+1}\mathrm{for}\mathrm{every}t\ge t_m\end{array}$$

(18)

where $t_m=(m+p-1)\mathsf{\ell }r-\sigma$ for all $m=\mathsf{\ell }, \mathsf{\ell }+1, \mathsf{\ell }+2, \dots$. It is observed that (18) with $m=\mathsf{\ell }-1$ coincides with (15), while (18) with $m=\mathsf{\ell }$ is identical to (17). Assuming here that (18) is found true for $m=k$, where k is an integer that has a positive value and is larger compared to ℓ, i.e.,

$$\begin{array}{c}\hfill |g^{\prime }\left(t\right)|\le K({\mu }_0;\xi ){\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{k-\mathsf{\ell }+1},t\ge t_k.\end{array}$$

Using (12) and the fact that $K({\mu }_0;\xi )\ge 1$, from (11), it then holds that, for $t\ge t_{k+1}$,

$$\begin{array}{cc}\hfill |g^{\prime }\left(t\right)|\le & \left|\delta \left(t\right)\right|\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t\right)\right|\left[|g^{\prime }(t-{\sigma }_j\left(t\right))|+|\mu (t-{\sigma }_j\left(t\right))|{\int }_{t-{\sigma }_j\left(t\right)}^t\left|g^{\prime }\left(s\right)\right|ds\right]\hfill \\ \hfill & \times \exp \left[-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t\right)\right|\left[{\int }_{t-{\tau }_i\left(t\right)}^t\left|g^{\prime }\left(s\right)\right|ds\right]\exp \left[-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill \le & {\theta }_2\left({\mu }_0\right){\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{k-\mathsf{\ell }+1}\hfill \\ & + K({\mu }_0;\xi ){\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{k-\mathsf{\ell }+1}\{{\displaystyle \sum _{j=1}^m}|{\gamma }_j\left(t\right)\left|\right[1+{\sigma }_j\left(t\right)|\mu (t-{\sigma }_j\left(t\right))\left|\right]\hfill \\ \hfill & \times \exp \left[-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right]+{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t\right)\right|{\tau }_i\left(t\right)\exp \left[-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right]\}\hfill \\ \hfill \le & K({\mu }_0;\xi ){\theta }_2\left({\mu }_0\right){\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{k-\mathsf{\ell }+1}+K({\mu }_0;\xi ){\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{k-\mathsf{\ell }+1}{\theta }_1\left({\mu }_0\right)\hfill \\ \hfill =& K({\mu }_0;\xi ){\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{k-\mathsf{\ell }+2}.\hfill \end{array}$$

Therefore, for $m=k+1$, we get what follows

$$\begin{array}{c}\hfill |g^{\prime }\left(t\right)|\le K({\mu }_0;\xi ){\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{k-\mathsf{\ell }+2},t\ge t_{k+1}.\end{array}$$

Based on the inductive principle, we thus reach the conclusion that (18) is true for all non-negative integers m. So, since (18) is true for all $m=\mathsf{\ell }-1, \mathsf{\ell }, \mathsf{\ell }+1, \dots$. Now, one shall demonstrate that

$$\begin{array}{c}\hfill |g^{\prime }\left(t\right)|\le K({\mu }_0;\xi ){\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{{\displaystyle \frac{t+\sigma }{\mathsf{\ell }r}}-(2\mathsf{\ell }+p-2)}\end{array}$$

(19)

for every $t\ge (\mathsf{\ell }+p-2)\mathsf{\ell }r-\sigma$. For this purpose, an arbitrary point t with $t\ge (\mathsf{\ell }+p-2)\mathsf{\ell }r-\sigma$ may be taken. Let us take $m=\left[\right|(t+\sigma )/\left(\mathsf{\ell }r\right)\left|\right]-(\mathsf{\ell }+p-2)$ ($\left[\right|(t+\sigma )/\left(\mathsf{\ell }r\right)\left|\right]$ as the greatest integer which is smaller than or equal to $\mathsf{\ell }+p-2$). Taking into account $(t+\sigma )/\left(\mathsf{\ell }r\right)\ge \mathsf{\ell }+p-2$, we get $\left[\right|(t+\sigma )/\left(\mathsf{\ell }r\right)\left|\right]\ge (\mathsf{\ell }+p-2)$, and accordingly, m is an integer that has a non-negative value. Because $\left[\right|(t+\sigma )/\left(\mathsf{\ell }r\right)\left|\right]\le \left[\right(t+\sigma )/(\mathsf{\ell }r\left)\right]<\left[\right|(t+\sigma )/\left(\mathsf{\ell }r\right)\left|\right]+1$, it holds that

$$\begin{array}{c}\hfill m+\mathsf{\ell }+p-2\le (t+\sigma )/\left(\mathsf{\ell }r\right)<m+\mathsf{\ell }+p-1.\end{array}$$

Therefore, we get $t\ge (m+\mathsf{\ell }+p-2)\mathsf{\ell }r-\sigma$ and $m>(t+\sigma )/\left(\mathsf{\ell }r\right)-\left(\mathsf{\ell }+p-1\right)$. Thus, applying (18) and using $0<{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)<1$, we get

$$\begin{array}{cc}\hfill |g^{\prime }\left(t\right)|& \le K({\mu }_0;\xi ){\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{m-\mathsf{\ell }+1}\hfill \\ \hfill & \le K({\mu }_0;\xi ){\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{{\displaystyle \frac{t+\sigma }{\mathsf{\ell }r}}-\left(2\mathsf{\ell }+p-2\right)}.\hfill \end{array}$$

So, (19) has been proven. As $t\to \infty$, we have $m\to \infty$. As ${\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{m-\mathsf{\ell }+1}\to 0$, we can directly obtain

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }{g}^{\prime }\left(t\right)=0\end{array}$$

which is a proof of the second limit in Theorem 1.

For proving that ${\lim }_{t\to \infty }g\left(t\right)$ exists (as a real number), the Cauchy convergence criterion can be used. For this objective, we can utilize (19) to reach the following, for $t\ge u\ge (\mathsf{\ell }+p-2)\mathsf{\ell }r-\sigma$

$$\begin{array}{cc}\hfill \left|g\right(t)-g(u\left)\right|\le & {\int }_u^t\left|g^{\prime }\left(s\right)\right|ds\le K({\mu }_0;\xi ){\int }_u^t{\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{{\displaystyle \frac{s+\sigma }{\mathsf{\ell }r}}-\left(2\mathsf{\ell }+p-2\right)}ds\hfill \\ \hfill =& K({\mu }_0;\xi ){\displaystyle \frac{\mathsf{\ell }r}{\ln \left({\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right)}}{\left[{\left({\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right)}^{{\displaystyle \frac{s+\sigma }{\mathsf{\ell }r}}-\left(2\mathsf{\ell }+p-2\right)}\right]}_u^t\hfill \\ \hfill =& K({\mu }_0;\xi ){\displaystyle \frac{\mathsf{\ell }r}{\ln \left({\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right)}}\hfill \\ \hfill & \times \left[{\left({\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right)}^{{\displaystyle \frac{t+\sigma }{\mathsf{\ell }r}}-\left(2\mathsf{\ell }+p-2\right)}-{\left({\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right)}^{{\displaystyle \frac{u+\sigma }{\mathsf{\ell }r}}-\left(2\mathsf{\ell }+p-2\right)}\right]\hfill \\ \hfill \le & K({\mu }_0;\xi ){\displaystyle \frac{\mathsf{\ell }r}{-\ln \left({\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right)}}\left[{\left({\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right)}^{{\displaystyle \frac{u+\sigma }{\mathsf{\ell }r}}-\left(2\mathsf{\ell }+p-2\right)}\right].\hfill \end{array}$$

Thus, we get, for $t\ge u\ge (\mathsf{\ell }+p-2)\mathsf{\ell }r-\sigma ,$

$$\begin{array}{c}\hfill |g\left(t\right)-g\left(u\right)|\le {\displaystyle \frac{\mathsf{\ell }rK({\mu }_0;\xi )}{-\ln \left({\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right)}}\left[{\left({\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right)}^{{\displaystyle \frac{u+\sigma }{\mathsf{\ell }r}}-\left(2\mathsf{\ell }+p-2\right)}\right].\end{array}$$

(20)

(Note that, since $0<{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)<1$, it follows that $-\ln \left({\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right)>0$.) In view of $0<{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)<1$, we see that

$$\begin{array}{c}\hfill \underset{u\to \infty }{\lim }{\left[{\theta }_1\left({\mu }_0\right)+{\theta }_2\left({\mu }_0\right)\right]}^{{\displaystyle \frac{u+\sigma }{\mathsf{\ell }r}}-\left(2\mathsf{\ell }+p-2\right)}=0.\end{array}$$

Thus, ${\lim }_{u\to \infty }|g\left(t\right)-g\left(u\right)|=0$ which suggests the existence of ${\lim }_{t\to \infty }g\left(t\right)$ by the Cauchy criterion of convergence. Moreover, we can denote this limit as $L(\mu ;\xi )$ as it depends on g, which, in turn, depends on the initial function $\xi$ and $\mu$. The step here explains the first limit in Theorem 1 and concludes the proof. □

After the proof of Theorem 1, the significant outcomes shall be expressed as follows:

Remark 1.

According to the condition of Theorem 1, a solution to (1) cannot grow more rapidly in comparison to the exponential function defined by the characteristic equation; i.e., there exists some positive real constant $K(\mu ;\xi )$, where

$$\begin{array}{c}\hfill \left|f\left(t\right)\right|\le K(\mu ;\xi )\exp \left[{\int }_0^t\mu \left(s\right)ds\right]\mathit{for}\mathit{all}t\ge -r.\end{array}$$

In addition, for the solution f to the IVP (1) and (2), we reach the following:

(i) 

f is stated to be bounded if

$$\begin{array}{c}\hfill \underset{t\to \infty }{\mathrm{lim\; sup}}{\int }_0^t\mu \left(s\right)ds<\infty ;\end{array}$$

(ii) 

f tends toward zero at ∞ if

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }{\int }_0^t\mu \left(s\right)ds=-\infty .\end{array}$$

Remark 2.

For the case where the solution to (4) is a constant ${\mu }_0$ meeting the conditions of (5) and (6), then, the solution f to the IVP (1) and (2) meets the conditions of the following

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }f\left(t\right)e^{-t{\mu }_0}=L({\mu }_0;\xi ).\end{array}$$

Particularly, when the solution to (4) is zero, ${\lim }_{t\to \infty }f\left(t\right)=L(0;\xi )$.

Remark 3.

In the above theorem, if $\delta \equiv 0$, then the f solution of (2) and (3) automatically satisfies (7) and (8).

According to properties (5) and (6), it is clear $\mu =0$ is a solution to (4) if and only if the following conditions are true:

Corollary 1.

Assuming

$$\begin{array}{c}\hfill \alpha \left(t\right)+{\displaystyle \sum _{i=1}^n}{\beta }_i\left(t\right)=0,\end{array}$$

$$\begin{array}{c}\hfill \underset{t\to \infty }{\mathrm{lim\; sup}}\left\{{\displaystyle \sum _{j=1}^m}|{\gamma }_j\left(t\right)|+{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t\right)\right|{\tau }_i\left(t\right)\right\}<1\end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|\delta \left(t\right)\right|=0.\end{array}$$

The solution f to the IVP (1) and (2) then satisfies

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }f\left(t\right)=L(0;\xi )\end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }{f}^{\prime }\left(t\right)=0.\end{array}$$

Given an important corollary given below, it is not necessary to get a solution to the characteristic Equation (4). This corollary, therefore, will be more practical than Theorem 1.

Corollary 2.

Assuming that there exists a positive number c, where

$$\begin{array}{c}\hfill \left|\alpha \left(t\right)\right|+{\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t\right)\right|c{e}^{c{\sigma }_j\left(t\right)}+{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t\right)\right|e^{c{\tau }_i\left(t\right)}\le c\mathit{for}t\ge 0.\end{array}$$

(21)

If

$$\begin{array}{c}\hfill \underset{t\to \infty }{\mathrm{lim\; sup}}\left\{{\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t\right)\right|\left[1+c{\sigma }_j\left(t\right)\right]e^{c{\sigma }_j\left(t\right)}+{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t\right)\right|{\tau }_i\left(t\right)e^{c{\tau }_i\left(t\right)}\right\}<1\end{array}$$

(22)

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|\delta \left(t\right)\right|e^{ct}=0,\end{array}$$

(23)

the solution f to the IVP (1) and (2) then satisfies (7) and (8).

Proof. 

Let us select any positive number c satisfying the condition (21), and let

$$\begin{array}{c}\hfill \left|{\mu }_0\left(t\right)\right|\le c\mathrm{for}t\in [-r,0].\end{array}$$

(24)

We thus claim

$$\begin{array}{c}\hfill \left|\mu \left(t\right)\right|\le c\mathrm{for}\mathrm{all}t\ge -r.\end{array}$$

(25)

Or else, based on (24) and by the continuity of $\mu$, a point $t_0>0$ exists, for which

$$\begin{array}{c}\hfill \left|\mu \left(t\right)\right|\le c,t\in [-r,t_0)\mathrm{and}\left|\mu \left(t_0\right)\right|>c.\end{array}$$

Let us write the characteristic Equation (4) as the following:

$$\begin{array}{cc}\hfill -\mu \left(t\right)=& \alpha \left(t\right)+{\displaystyle \sum _{j=1}^m}{\gamma }_j\left(t\right)\mu (t-{\sigma }_j\left(t\right))\exp \left\{-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right\}\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}{\beta }_i\left(t\right)\exp \left\{-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right\}\hfill \end{array}$$

(26)

Then, using (21), from (26), we see that

$$\begin{array}{cc}\hfill c<\left|\mu \left(t_0\right)\right|\le & \left|\alpha \left(t_0\right)\right|+{\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t_0\right)\right|\left|\mu (t_0-{\sigma }_j\left(t_0\right))\right|\exp \left[-{\int }_{t_0-{\sigma }_j\left(t_0\right)}^{t_0}\mu \left(s\right)ds\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t_0\right)\right|\exp \left[-{\int }_{t_0-{\tau }_i\left(t_0\right)}^{t_0}\mu \left(s\right)ds\right]\hfill \\ \hfill \le & \left|\alpha \left(t_0\right)\right|+{\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t_0\right)\right|c{e}^{c{\sigma }_j\left(t_0\right)}+{\displaystyle \sum _{i=1}^n}{\beta }_i\left(t_0\right)e^{c{\tau }_i\left(t_0\right)}\le c,\hfill \end{array}$$

which is a contradiction. Thus, the inequality (25) must hold true. Accordingly, by utilizing (25), the expressions below follow

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t\right)\right|\left[1+\left|\mu (t-{\sigma }_j\left(t\right))\right|{\sigma }_j\left(t\right)\right]\exp \left[-{\int }_{t-{\sigma }_j\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t\right)\right|{\tau }_i\left(t\right)\exp \left[-{\int }_{t-{\tau }_i\left(t\right)}^t\mu \left(s\right)ds\right]\hfill \\ \hfill \le & {\displaystyle \sum _{j=1}^m}\left|{\gamma }_j\left(t\right)\right|\left[1+c{\sigma }_j\left(t\right)\right]e^{c{\sigma }_j\left(t\right)}+{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\left(t\right)\right|{\tau }_i\left(t\right)e^{c{\tau }_i\left(t\right)}\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \left|\delta \left(t\right)\right|\exp \left[-{\int }_0^t\mu \left(s\right)ds\right]\le \left|\delta \left(t\right)\right|e^{ct}.\end{array}$$

with (22) and (23), we reach (5) and (6), and this finalizes the proof. □

3. Examples

For demonstrating the hypothesis applicable to Theorem 1, we offer six examples of an NDDE of the form (1) that exhibits an explicit solution to the characteristic Equation (4) and satisfies (5) and (6). In the following examples, $\xi$ is an arbitrary function satisfying the consistency condition.

Example 1.

Considering the differential equation (1) with

$$\begin{array}{c}\hfill \alpha \left(t\right)=-{\displaystyle \frac{1}{(t+1)}},\beta \left(t\right)=-{\displaystyle \frac{1}{4t+3}},\sigma \left(t\right)={\displaystyle \frac{1}{6}},\tau \left(t\right)={\displaystyle \frac{1}{4}},\gamma \left(t\right)={\displaystyle \frac{1}{4}},\delta \left(t\right)=e^{-4t}\end{array}$$

for $t\ge 0$, i.e., the NDDE

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)-{\displaystyle \frac{1}{(t+1)}}f\left(t\right)+{\displaystyle \frac{1}{4}}{f}^{\prime }(t-{\displaystyle \frac{1}{6}})-{\displaystyle \frac{1}{4t+3}}f\left(t-{\displaystyle \frac{1}{4}}\right)=e^{-4t}\end{array}$$

(27)

for $t\ge 0$. For this example, note that $r=0.25$. In the case here, the generalized characteristic Equation (4) with $n=1$ turns into the following

$$\begin{array}{cc}\hfill \mu \left(t\right)& -{\displaystyle \frac{1}{(t+1)}}+{\displaystyle \frac{1}{4}}\mu \left(t-{\displaystyle \frac{1}{6}}\right)\exp \left[-{\int }_{t-{\displaystyle \frac{1}{6}}}^t\mu \left(s\right)ds\right]\hfill \\ \hfill & +\left(-{\displaystyle \frac{1}{4t+3}}\right)\exp \left[-{\int }_{t-{\displaystyle \frac{1}{4}}}^t\mu \left(s\right)ds\right]=0\mathit{for}t\ge 0.\hfill \end{array}$$

(28)

It may be seen immediately here that (28) bears the solution

$$\begin{array}{c}\hfill \mu \left(t\right)={\displaystyle \frac{1}{t+1}}\mathit{for}t\ge -0.25.\end{array}$$

We find that assumptions (5) and (6) take the form

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{\mathrm{lim\; sup}}\left\{\left|{\displaystyle \frac{1}{4}}\right|\left[1+\left|{\displaystyle \frac{1}{t-\frac{1}{6}+1}}\right|{\displaystyle \frac{1}{6}}\right]\exp \left[-{\int }_{t-{\displaystyle \frac{1}{6}}}^t{\displaystyle \frac{1}{s+1}}ds\right]+\left|-{\displaystyle \frac{1}{4t+3}}\right|{\displaystyle \frac{1}{4}}\exp \left[-{\int }_{t-{\displaystyle \frac{1}{4}}}^t{\displaystyle \frac{1}{s+1}}ds\right]\right\}\hfill \\ \hfill & =\underset{t\to \infty }{\mathrm{lim\; sup}}\left\{{\displaystyle \frac{1}{4}}\left({\displaystyle \frac{6t+6}{6t+5}}\right)\exp \left[-\ln \left({\displaystyle \frac{6t+6}{6t+5}}\right)\right]+\left({\displaystyle \frac{1}{4t+3}}\right){\displaystyle \frac{1}{4}}\exp \left[-\ln \left({\displaystyle \frac{4t+4}{4t+3}}\right)\right]\right\}\hfill \\ \hfill & =\underset{t\to \infty }{\mathrm{lim\; sup}}\left\{{\displaystyle \frac{1}{4}}\left({\displaystyle \frac{6t+6}{6t+5}}\right)\left({\displaystyle \frac{6t+5}{6t+6}}\right)+\left({\displaystyle \frac{1}{4t+3}}\right){\displaystyle \frac{1}{4}}\left({\displaystyle \frac{4t+3}{4t+4}}\right)\right\}\hfill \\ \hfill & =\underset{t\to \infty }{\mathrm{lim\; sup}}\left\{{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{16(t+1)}}\right\}<1\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left\{|e^{-4t}|\exp \left\{-{\int }_0^t{\displaystyle \frac{1}{s+1}}ds\right\}\right\}=\underset{t\to \infty }{\lim }\left\{e^{-4t}\left({\displaystyle \frac{1}{t+1}}\right)\right\}=0,\end{array}$$

in other words, assumptions (5) and (6) are satisfied all the time. At this point, using the NDDE (27), the following initial condition is associated

$$\begin{array}{c}\hfill f\left(t\right)=\xi \left(t\right)\mathit{for}-0.25\le t\le 0,\end{array}$$

(29)

where ξ is a given continuously differentiable real-valued function on $[-0.25,0]$. When we apply Theorem 1, we obtain the conclusion that the solution f to the NDDE (27) and (29) meets the following conditions

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left[{\displaystyle \frac{f\left(t\right)}{t+1}}\right]=L\left(\xi \right),\end{array}$$

where $L\left(\xi \right)$ is some real number that depends on ξ, and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }{\left[{\displaystyle \frac{f\left(t\right)}{t+1}}\right]}^{\prime }=0.\end{array}$$

In addition, taking into account Remark 1, we obtain

$$\begin{array}{c}\hfill \left|f\right(t\left)\right|\le K\left(\xi \right)(t+1)\mathit{for}\mathit{all}t\ge -0.25,\end{array}$$

where $K\left(\xi \right)$ is some real number defined by ξ.

Example 2.

Consider the NDDE

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)-{\displaystyle \frac{1}{3e^t}}f\left(t\right)+\left({\displaystyle \frac{\sin t+1}{6}}\right)f^{\prime }\left(t-{\displaystyle \frac{1}{t+1}}\right)+{\displaystyle \frac{1}{3e^t}}f\left(t-{\displaystyle \frac{1}{t+2}}\right)\\ \hfill =e^{-2t},\mathit{for}t\ge 0.\end{array}$$

(30)

In Equation (1), $\sigma \left(t\right)=1/(t+1)$, $\tau \left(t\right)=1/(t+2)$, $\alpha \left(t\right)=-1/\left(3e^t\right)$, $\beta \left(t\right)=1/\left(3e^t\right)$, $\gamma \left(t\right)=(\sin t+1)/6$ and $\delta \left(t\right)=e^{-2t}$.

It is clear that in Equation (30), $r=1$ and $\alpha \left(t\right)+\beta \left(t\right)=0$. Then, the homogeneous Equation (30) is

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)-{\displaystyle \frac{1}{3e^t}}f\left(t\right)+\left({\displaystyle \frac{\sin t+1}{6}}\right)f^{\prime }\left(t-{\displaystyle \frac{1}{t+1}}\right)+{\displaystyle \frac{1}{3e^t}}f\left(t-{\displaystyle \frac{1}{t+2}}\right)=0.\end{array}$$

(31)

The characteristic equation of (31) is

$$\begin{array}{cc}\hfill \mu \left(t\right)& -{\displaystyle \frac{1}{3e^t}}+\left({\displaystyle \frac{\sin t+1}{6}}\right)\mu \left(t-{\displaystyle \frac{1}{t+1}}\right)\exp \left[-{\int }_{t-{\displaystyle \frac{1}{t+1}}}^t\mu \left(s\right)ds\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{3e^t}}\exp \left[-{\int }_{t-{\displaystyle \frac{1}{t+2}}}^t\mu \left(s\right)ds\right]=0,\mathit{for}t\ge 0.\hfill \end{array}$$

It can be simply verified in this case that

$$\begin{array}{c}\hfill \mu \left(t\right)=0\mathit{for}t\ge -1\end{array}$$

is a solution to the characteristic equation. Therefore, this satisfies the conditions of Corollary 1. Indeed, one can simply verify that

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{\mathrm{lim\; sup}}\left\{\left|{\displaystyle \frac{\sin t+1}{6}}\right|+\left|{\displaystyle \frac{1}{3e^t}}\right|{\displaystyle \frac{1}{t+2}}\right\}<1\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }{e}^{-2t}=0.\end{array}$$

Therefore, we conclude that the solution f of the NDDE (30), together with the initial condition

$$\begin{array}{c}\hfill f\left(t\right)=\xi \left(t\right)\mathit{for}-1\le t\le 0,\end{array}$$

(32)

where ξ is a given real-valued and continuously differentiable function on the interval $[-1,0]$, satisfies the following:

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }f\left(t\right)=L(0;\xi )\end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }{f}^{\prime }\left(t\right)=0,\end{array}$$

where $L(0;\xi )$ is some real number defined by ξ.

In addition, if Remark 1 is considered, we obtain

$$\begin{array}{c}\hfill \left|f\right(t\left)\right|\le K(0;\xi )\mathit{for}\mathit{all}t\ge -1,\end{array}$$

where $K(0;\xi )$ is some real number defined by ξ.

Example 3.

Considering the NDDE

$$\begin{array}{cc}\hfill f^{\prime }\left(t\right)& +0.1\left(1+0.2\sin t\right)f\left(t\right)-0.05\cos t{f}^{\prime }\left(t\right)\left(t-0.1(1+{\sin }^{2}t)\right)\hfill \\ \hfill & +0.08\left(1+0.25\cos t\right)f\left(t-0.1(1+{\cos }^{2}t)\right)=e^{-3t},\mathit{for}t\ge 0.\hfill \end{array}$$

(33)

In Equation (1), $\sigma \left(t\right)=0.1(1+{\sin }^{2}t)$, $\tau \left(t\right)=0.1(1+{\cos }^{2}t)$, $\alpha \left(t\right)=0.1\left(1+0.2\sin t\right)$, $\beta \left(t\right)=0.08\left(1+0.25\cos t\right)$, $\gamma \left(t\right)=-0.05\cos t$ and $\delta \left(t\right)=e^{-3t}$.

In Equation (33), it is clear that $r=0.2$. In this example, let us show that condition (21) in Corollary 2 holds for $c=1$,

$$\begin{array}{cc}\hfill & \left|\left(0.1\right)\left(1+0.2\sin t\right)\right|+\left|(-0.05)\cos t\right|e^{\left(0.1\right)(1+{\sin }^{2}t)}+\left|\left(0.08\right)\left(1+0.25\cos t\right)\right|e^{\left(0.1\right)(1+{\cos }^{2}t)}\hfill \\ \hfill & \le 0.12+0.05e^{\left(0.2\right)}+0.1e^{\left(0.2\right)}\approx 0.3032<1,\mathit{for}t\ge 0.\hfill \end{array}$$

Therefore, as in the proof demonstrated for Corollary 2, it is evident that we get

$$\begin{array}{c}\hfill \left|\mu \right(t\left)\right|\le 1\mathit{for}t\ge -0.2.\end{array}$$

Also, using the NDDE (33), the following initial condition is associated

$$\begin{array}{c}\hfill f\left(t\right)=\xi \left(t\right)\mathit{for}-0.2\le t\le 0,\end{array}$$

(34)

where ξ is a given real-valued function that is continuously differentiable on the interval $[-0.2,0]$.

Now let us apply Corollary 2. Above, we calculate the condition (21) for $c=1$. Similarly, for $c=1$ conditions (22) and (23) are satisfied. Here, it easy to check that

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{\mathrm{lim\; sup}}\{\left|(-0.05)\cos t\right|[1+0.1(1+{\sin }^{2}t)]e^{\left(0.1\right)(1+{\sin }^{2}t)}\hfill \\ \hfill & +\left|\left(0.08\right)\left(1+0.25\cos t\right)\right|\left(0.1\right)(1+{\cos }^{2}t)e^{\left(0.1\right)(1+{\cos }^{2}t)}\}\hfill \\ \hfill & \le \left(0.05\right)\left(1.2\right)e^{\left(0.2\right)}+\left(0.08\right)\left(1.25\right)\left(0.2\right)e^{\left(0.2\right)}\approx 0.0977<1\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|\delta \left(t\right)\right|e^{ct}=\underset{t\to \infty }{\lim }\left\{e^{-3t}.e^t\right\}=0.\end{array}$$

Thus, conditions (22) and (23) are satisfied. According to Corollary 2, the results in Theorem 1 are, specifically, (7) and (8) for the NDDE (33) even without finding the solution $\mu \left(t\right)$ to the characteristic equation.

In addition to this, considering Remark 1, we get

$$\begin{array}{c}\hfill \left|f\left(t\right)\right|\le K(\mu ;\xi )\exp \left[{\int }_0^t\mu \left(s\right)ds\right]\mathit{for}t\ge -0.2,\end{array}$$

where $K(\mu ;\xi )$ is some real number, which is defined by ξ.

Example 4

([23], Example 2). Taking the NDDE

$$f^{\prime }\left(t\right)-0.6f^{\prime }(t-0.1)+{\displaystyle \frac{1}{4e}}\left(1+0.1\sin t\right)f(t-\tau \left(t\right))=0,\mathit{for}t\ge 0$$

(35)

into consideration, in Equation (1), $\sigma \left(t\right)=0.1$, $\alpha \left(t\right)=0$, $\beta \left(t\right)=(1/\left(4e\right))\left(1+0.1\sin t\right)$, $\gamma \left(t\right)=-0.6$, $\delta \left(t\right)=0$ and $0.9\le \tau \left(t\right)\le 1$.

It is clear that in Equation (35), $r=1$. In this example, let us show that condition (21) in Corollary 2 holds for $c=1/2$,

$$\begin{array}{cc}\hfill |-0.6|{\displaystyle \frac{1}{2}}{e}^{\left(0.1\right){\displaystyle \frac{1}{2}}}+\left|{\displaystyle \frac{1}{4e}}\left(1+0.1\sin t\right)\right|e^{{\displaystyle \frac{1}{2}}\tau \left(t\right)}& \le {\displaystyle \frac{0.6}{2}}{e}^{{\displaystyle \frac{\left(0.1\right)}{2}}}+{\displaystyle \frac{1}{4e}}(1+0.1)e^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \approx 0.4822<{\displaystyle \frac{1}{2}},\mathit{for}t\ge 0.\hfill \end{array}$$

Therefore, as demonstrated in the proof Corollary 2, we automatically get

$$\begin{array}{c}\hfill \left|\mu \right(t\left)\right|\le 1/2\mathit{for}t\ge -1.\end{array}$$

Also, using the NDDE (35), the initial condition is associated

$$\begin{array}{c}\hfill f\left(t\right)=\xi \left(t\right)\mathit{for}-1\le t\le 0,\end{array}$$

(36)

where ξ is a given continuously differentiable real-valued function on $[-1,0]$.

Now let us apply Corollary 2. Above, we calculate the condition (21) for $c=1/2$. Similarly, for $c=1/2$ conditions (22) and (23) are met, and, accordingly, it can be simply checked that

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{\mathrm{lim\; sup}}\left\{|-0.6|\left(1+{\displaystyle \frac{1}{2}}\left(0.1\right)\right)e^{\left(0.1\right){\displaystyle \frac{1}{2}}}+\left|{\displaystyle \frac{1}{4e}}\left(1+0.1\sin t\right)\right|\tau \left(t\right)e^{{\displaystyle \frac{1}{2}}\tau \left(t\right)}\right\}\hfill \\ \hfill & \le \left(0.6\right)\left(1.05\right)e^{\left(0.05\right)}+{\displaystyle \frac{1}{4e}}\left(1.1\right)e^{{\displaystyle \frac{1}{2}}}\approx 0.8291<1\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|\delta \left(t\right)\right|e^{ct}=0.\end{array}$$

Thus, conditions (22) and (23) are satisfied. According to Corollary 2, the results in Theorem 1 are, specifically, (7) and (8) for the NDDE (35) even without finding the solution $\mu \left(t\right)$ to the characteristic equation.

Additionally, taking into account Remark 1, we obtain

$$\begin{array}{c}\hfill \left|f\left(t\right)\right|\le K(\mu ;\xi )\exp \left[{\int }_0^t\mu \left(s\right)ds\right]\mathit{for}t\ge -1,\end{array}$$

where $K(\mu ;\xi )$ is some real number defined by ξ. It is worth noting that, in Example 2 of [23], the parameter denoted by r is taken as $r=1/\left(4e\right)$, and exponential stability is established since $1/\left(4e\right)<0.307$.

Example 5

([24], Example 6.1). Considering the NDDE

$$\begin{array}{cc}\hfill f^{\prime }\left(t\right)& -{\displaystyle \frac{1}{3}}{f}^{\prime }(t-\sigma \left(t\right))+{\displaystyle \frac{1}{3}}f\left(t\right)(t-{\displaystyle \frac{1}{3}})=0,\mathit{for}t\ge 0.\hfill \end{array}$$

(37)

In Equation (1), $0<\sigma \left(t\right)\le 1/3$, $\alpha \left(t\right)=0$, $\beta \left(t\right)=1/3$, $\gamma \left(t\right)=-1/3$, $\delta \left(t\right)=0$, and $\tau \left(t\right)=1/3$.

It is clear that in Equation (37), $r=1/3$. In this example, let us show that condition (21) in Corollary 2 holds for $c=1$,

$$\begin{array}{c}\hfill \left|-{\displaystyle \frac{1}{3}}\right|e^{\sigma \left(t\right)}+\left|{\displaystyle \frac{1}{3}}\right|e^{{\displaystyle \frac{1}{3}}}\le {\displaystyle \frac{1}{3}}\left(e^{{\displaystyle \frac{1}{3}}}+e^{{\displaystyle \frac{1}{3}}}\right)\approx 0.9301<1\mathit{for}t\ge 0.\end{array}$$

Therefore, as demonstrated in the proof Corollary 2, it is clearly seen that we reach

$$\begin{array}{c}\hfill \left|\mu \left(t\right)\right|\le 1\mathit{for}t\ge -{\displaystyle \frac{1}{3}}.\end{array}$$

Also, using the NDDE (37), associating the initial condition

$$\begin{array}{c}\hfill f\left(t\right)=\xi \left(t\right)\mathit{for}-1/3\le t\le 0,\end{array}$$

(38)

where ξ is a given continuously differentiable real-valued function on $[-1/3,0]$.

Now let us apply Corollary 2. Above, we calculate the condition (21) for $c=1$. Similarly, conditions (22) and (23) are met for $c=1$, and here, it simple to check that

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{\mathrm{lim\; sup}}\left\{\left|-{\displaystyle \frac{1}{3}}\right|\left(1+\sigma \left(t\right)\right)e^{\sigma \left(t\right)}+\left|{\displaystyle \frac{1}{3}}\right|{\displaystyle \frac{1}{3}}{e}^{{\displaystyle \frac{1}{3}}}\right\}\hfill \\ \hfill & \le {\displaystyle \frac{1}{3}}\left(1+{\displaystyle \frac{1}{3}}\right)e^{{\displaystyle \frac{1}{3}}}+{\displaystyle \frac{1}{9}}{e}^{{\displaystyle \frac{1}{3}}}\approx 0.7753<1\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|\delta \left(t\right)\right|e^{ct}=0.\end{array}$$

Thus, conditions (22) and (23) are satisfied. According to Corollary 2, the results in Theorem 1 are, specifically, (7) and (8) for the NDDE (37) even without finding the solution $\mu \left(t\right)$ to the characteristic equation.

In addition to the above, accounting for Remark 1, we reach

$$\begin{array}{c}\hfill \left|f\left(t\right)\right|\le K(\mu ;\xi )\exp \left[{\int }_0^t\mu \left(s\right)ds\right]\mathit{for}t\ge -1,\end{array}$$

where $K(\mu ;\xi )$ is some real number defined by ξ.

Example 6

([37], Example 4). Let us take the delay differential equation

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)+0.2(0.5-\sin t)f(t-{\displaystyle \frac{2-\sin t}{6}})+0.2(0.5+\sin t)f(t-{\displaystyle \frac{2-\sin t}{12}})=0,\\ \hfill \mathit{for}t\ge 0.\end{array}$$

(39)

In Equation (1), $\alpha \left(t\right)=0$, ${\beta }_1\left(t\right)=0.2(0.5-\sin t)$, ${\beta }_2\left(t\right)=0.2(0.5+\sin t)$, $\delta \left(t\right)=0$, ${\tau }_1\left(t\right)=(2-\sin t)/6$, and ${\tau }_2\left(t\right)=(2-\sin t)/12$.

It is clear that in Equation (39), $r=0.5$. In this example, let us show that condition (21) in Corollary 2 holds for $c=1$,

$$\begin{array}{c}\hfill \left|0.2(0.5-\sin t)\right|\exp \left\{{\displaystyle \frac{2-\sin t}{6}}\right\}+\left|0.2(0.5+\sin t)\right|\exp \left\{{\displaystyle \frac{2-\sin t}{12}}\right\}\\ \hfill \le 0.2(0.5+1)e^{3/6}+0.2(0.5+1)e^{3/12}\approx 0.8798<1,\mathit{for}t\ge 0.\end{array}$$

Therefore, as demonstrated in the proof Corollary 2, we evidently get,

$$\begin{array}{c}\hfill \left|\mu \right(t\left)\right|\le 1\mathit{for}t\ge -0.5.\end{array}$$

Also, using the NDDE (39), we associate the initial condition

$$\begin{array}{c}\hfill f\left(t\right)=\xi \left(t\right)\mathit{for}-0.5\le t\le 0,\end{array}$$

(40)

where ξ is a given continuously differentiable real-valued function on $[-0.5,0]$.

Now let us apply Corollary 2. Above, we calculate the condition (21) for $c=1$. Similarly, conditions (22) and (23) are met for $c=1$, and here, one can simply check that

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{\mathrm{lim\; sup}}\{\left|-0.2(0.5-\sin t)\right|\left({\displaystyle \frac{2-\sin t}{6}}\right)\exp \left\{{\displaystyle \frac{2-\sin t}{6}}\right\}\hfill \\ \hfill & +\left|0.2(0.5+\sin t)\right|\left({\displaystyle \frac{2-\sin t}{12}}\right)\exp \left\{{\displaystyle \frac{2-\sin t}{12}}\right\}\}\hfill \\ \hfill & \le {\displaystyle \frac{3}{10}}{\displaystyle \frac{1}{2}}{e}^{{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{3}{10}}{\displaystyle \frac{1}{4}}{e}^{{\displaystyle \frac{1}{4}}}\approx 0.3436<1\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|\delta \left(t\right)\right|e^{ct}=0.\end{array}$$

Thus, conditions (22) and (23) are satisfied. According to Corollary 2,the results in Theorem 1 are, specifically, (7) and (8) for the NDDE (39) even without finding the solution $\mu \left(t\right)$ to the characteristic equation.

In addition to the above, taking Remark 1 into account, we obtain

$$\begin{array}{c}\hfill \left|f\left(t\right)\right|\le K(\mu ;\xi )\exp \left[{\int }_0^t\mu \left(s\right)ds\right]\mathit{for}t\ge -0.5,\end{array}$$

where $K(\mu ;\xi )$ is some real number defined by ξ. Indeed, in [37], it was found that

$$\left|f\left(t\right)\right|\le 4.89e^{-0.02t}.$$

4. The Special Case of Linear Non-Homogeneus Neutral-Delay Differential Equations with Constant Coefficients

Philos et al. [38] investigated the asymptotic behaviours of solutions to first-order linear homogeneous NDDE having constant coefficients. In this paper, however, the non-homogeneous case is considered, and the asymptotic behaviour of the relevant solutions is presented.

In this section, we address the special case of the first-order linear nonhomogeneus NDDE with constant coefficients. Here, the linear NDDE that has constant coefficients signifies a specific variant for (1).

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)+\alpha f\left(t\right)+{\displaystyle \sum _{j=1}^m}{\gamma }_j{f}^{\prime }(t-{\sigma }_j)+{\displaystyle \sum _{i=1}^n}{\beta }_{i}f(t-{\tau }_i)=\delta \left(t\right)\end{array}$$

(41)

where the coefficients $\alpha$, ${\gamma }_j$, and ${\beta }_i$, $i=1, 2, \dots , n$, $j=1, 2, \dots , m$ are real constants; ${\sigma }_j$ and ${\tau }_i$ are positive real numbers. Moreover, $\delta$ is real-valued continuous function on $[0,\infty )$. Moreover, we also we assume that the delays are positive and denote

$$\begin{array}{c}\hfill \sigma =\max \{{\sigma }_j:j\in J\},\tau =\max \{{\tau }_i:i\in I\},r=\max \{\sigma ,\tau \}.\end{array}$$

Let $C^1([-r,0],\mathbb{R})$ express the set of continuously differentiable functions with real values on the initial interval $[-r,0].$ Along with the NDDE (41), as is convention, we exhibit an initial condition of the form as follows

$$f\left(t\right)=\xi \left(t\right)\mathrm{for}-r\le t\le 0,$$

(42)

where $\xi \in C^1([-r,0],\mathbb{R})$, satisfying the “consistency condition”

$$\begin{array}{c}\hfill {\xi }^{\prime }\left(0\right)+\alpha \xi \left(0\right)+{\displaystyle \sum _{j=1}^m}{\gamma }_j{\xi }^{\prime }\left(-{\sigma }_j\right)+{\displaystyle \sum _{i=1}^n}{\beta }_i\xi (-{\tau }_i)=\delta \left(0\right).\end{array}$$

Let us write the characteristic equation for the homogeneous equation of (41) as follows:

$$\begin{array}{c}\hfill \alpha +{\displaystyle \sum _{i=1}^n}{\beta }_i{e}^{-\mu {\tau }_i}+\mu \left(1+{\displaystyle \sum _{j=1}^m}{\gamma }_j{e}^{-\mu {\sigma }_j}\right)=0.\end{array}$$

(43)

Theorem 2.

Let ${\mu }_0$ be a real root of the characteristic Equation (43) with the following property:

$$\begin{array}{c}\hfill d{\mu }_0={\displaystyle \sum _{j=1}^m}|{\gamma }_j\left|\right(1+|{\mu }_0|{\sigma }_j)e^{-{\mu }_0{\sigma }_j}+{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\right|{\tau }_i{e}^{-{\mu }_0{\tau }_i}<1\end{array}$$

(44)

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|\delta \left(t\right)\right|e^{-\mu t}=0.\end{array}$$

(45)

For any $\xi \in C^1([-r,0],\mathbb{R})$, then it holds that

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left[e^{-{\mu }_{0}t}f(\xi ;t)\right]={\displaystyle \frac{L({\mu }_0;\xi )}{1+Y\left({\mu }_0\right)}}\end{array}$$

(46)

where

$$\begin{array}{cc}\hfill L({\mu }_0;\xi )=& \xi \left(0\right)+{\displaystyle \sum _{j=1}^m}{\gamma }_j\left[\xi (-{\sigma }_j)-{\mu }_0{e}^{-{\mu }_0{\sigma }_j}{\int }_{-{\sigma }_j}^0{e}^{-{\mu }_{0}s}\xi \left(s\right)ds\right]\hfill \\ \hfill & -{\displaystyle \sum _{i=1}^n}{\beta }_i{e}^{-{\mu }_0{\tau }_i}{\int }_{-{\tau }_i}^0{e}^{-{\mu }_{0}s}\xi \left(s\right)ds\hfill \end{array}$$

and

$$\begin{array}{c}\hfill Y\left({\mu }_0\right)={\displaystyle \sum _{j=1}^m}{\gamma }_j(1-{\mu }_0{\sigma }_j)e^{-{\mu }_0{\sigma }_j}-{\displaystyle \sum _{i=1}^n}{\beta }_i{\tau }_i{e}^{-{\mu }_0{\tau }_i}.\end{array}$$

Note: Property (44) guarantees that $1+Y\left({\mu }_0\right)>0$.

Remark 4.

Let ${\mu }_0$ be a real root of the characteristic Equation (43) showing property (44). The solution f to the Equation (41) then satisfies the following:

$$\begin{array}{c}\hfill \left|f\left(t\right)\right|\le K({\mu }_0;\xi )e^{{\mu }_{0}t}\mathit{for}\mathit{all}t\ge -r\end{array}$$

where $K(\mu ;\xi )$ is a positive real constant.

Clearly, $\mu =0$ is a solution to (43) based on properties (44) and (45) if and only if the conditions given below are true:

Corollary 3.

Assuming here that

$$\begin{array}{c}\hfill \alpha +{\displaystyle \sum _{i=1}^n}{\beta }_i=0,\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle \sum _{j=1}^m}|{\gamma }_j|+{\displaystyle \sum _{i=1}^n}\left|{\beta }_i\right|{\tau }_i<1\end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|\delta \left(t\right)\right|=0.\end{array}$$

The solution f for Equation (41) then satisfies

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }f\left(t\right)=L(0,\xi ).\end{array}$$

To establish the hypothesis regarding to Theorem 2, three examples of an NDDE of the form (41) that holds an explicit solution to the characteristic Equation (43) and meets the conditions of Theorem 2 shall be provided. In the following examples, $\xi$ is an arbitrary function satisfying the consistency condition.

Example 7

([39], Example 5.1). For the following NDDE

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)-{\displaystyle \frac{1}{3e}}{f}^{\prime }(t-{\displaystyle \frac{1}{2}})-{\displaystyle \frac{1}{3}}f\left(t\right)+{\displaystyle \frac{1}{e}}f(t-{\displaystyle \frac{1}{5}})=0\mathit{for}t\ge 0.\end{array}$$

(47)

Let us set $\alpha =-1/3$, $\gamma =-1/\left(3e\right)$, $\sigma =1/2$, $\beta =1/e$, $\tau =1/5$, and $\delta \left(t\right)=0$ in Equation (41). Note that we set $r=0.5$ in Equation (47). Therefore, the characteristic equation of Equation (47) is

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{e}}{e}^{-\mu /5}+\mu \left(1-{\displaystyle \frac{1}{3e}}{e}^{-\mu /2}\right)=0.\end{array}$$

(48)

In [39], the solutions of (48) are given as ${\mu }_1\approx -3.94$ and ${\mu }_2\approx -0.043$.

Let us apply the Theorem 2 for ${\mu }_0=-0.043$.

$$\begin{array}{cc}\hfill d{\mu }_0=& \left|-{\displaystyle \frac{1}{3e}}\right|\left(1+|-0.043|{\displaystyle \frac{1}{2}}\right)e^{-(-0.043)(1/2)}+\left|{\displaystyle \frac{1}{e}}\right|{\displaystyle \frac{1}{5}}{e}^{-(-0.043)(1/5)}\approx 0.2022<1\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|\delta \left(t\right)\right|e^{0.043t}=0.\end{array}$$

Therefore, we conclude the solution f of the NDDE (47), together with the initial condition

$$\begin{array}{c}\hfill f\left(t\right)=\xi \left(t\right)\mathit{for}-0.5\le t\le 0,\end{array}$$

(49)

where ξ is a given continuously differentiable function with real values defined on the interval $[-0.5,0]$. According to Theorem 2, the solution f of the NDDE (47) holds,

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left[e^{0.043t}f(\xi ;t)\right]={\displaystyle \frac{L(-0.043;\xi )}{1+Y(-0.043)}},\end{array}$$

where

$$\begin{array}{cc}\hfill L(-0.043;\xi )=& \xi \left(0\right)+\left(-{\displaystyle \frac{1}{3e}}\right)\left\{\xi (-{\displaystyle \frac{1}{2}})-(-0.043)e^{(-0.043)(1/2)}\right\}{\int }_{-{\displaystyle \frac{1}{2}}}^0{e}^{-(-0.043s)}\xi \left(s\right)ds\hfill \\ \hfill & -\left({\displaystyle \frac{1}{e}}\right)e^{(-0.043)(1/5)}{\int }_{-{\displaystyle \frac{1}{5}}}^0{e}^{-(-0.043s)}\xi \left(s\right)ds\hfill \\ \hfill =& \xi \left(0\right)-0.1226\left[\xi (-{\displaystyle \frac{1}{2}})+0.0439\right]{\int }_{-{\displaystyle \frac{1}{2}}}^0{e}^{0.043s}\xi \left(s\right)ds-0.3711{\int }_{-{\displaystyle \frac{1}{5}}}^0{e}^{0.043s}\xi \left(s\right)ds\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill Y(-0.043)=& \left(-{\displaystyle \frac{1}{3e}}\right)\left[1-(-0.043)\left({\displaystyle \frac{1}{2}}\right)\right]e^{(-0.043)(1/2)}-\left({\displaystyle \frac{1}{e}}\right)\left({\displaystyle \frac{1}{5}}\right)e^{(-0.043)(1/5)}\approx -0.2022.\hfill \end{array}$$

Additionally, if Remark 4 is considered, we obtain

$$\begin{array}{c}\hfill \left|f\left(t\right)\right|\le K(-0.043;\xi )e^{-0.043t}\mathit{for}t\ge -0.5\end{array}$$

where $K(-0.043;\xi )$ is some real number defined by ξ.

Example 8

([39], Example 5.2). Consider the NDDE as follows

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)-{\displaystyle \frac{1}{3}}{f}^{\prime }(t-\sigma )-{\displaystyle \frac{1}{2}}f\left(t\right)+{\displaystyle \frac{1}{2}}f\left(t-{\displaystyle \frac{1}{2}}\right)={\displaystyle \frac{1}{t+1}}\mathit{for}t\ge 0.\end{array}$$

(50)

The homogeneous form of Equation (50) is given in [39]. In this example, we considered the nonhomogeneous Equation (50).

Let us set $\alpha =-1/2$, $\gamma =-1/3$, $\beta =1/2$, $\tau =1/2$, and $\delta \left(t\right)=1/(t+1)$ in Equation (41), also σ is any positive real number. Note that we set $r=\max \{\sigma ,1/2\}$ in Equation (50). Therefore, the characteristic equation for Equation (50) is

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}{e}^{-\mu /2}+\mu \left(1-{\displaystyle \frac{1}{3}}{e}^{-\mu \sigma }\right)=0.\end{array}$$

(51)

In [39], the solutions of (51) are given as ${\mu }_0=0$. Therefore, we can apply the Corollary 3. Note that

$$\alpha +\beta =0.$$

$$\begin{array}{c}\hfill \left|-{\displaystyle \frac{1}{3}}\right|+\left|{\displaystyle \frac{1}{2}}\right|{\displaystyle \frac{1}{2}}={\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{4}}={\displaystyle \frac{7}{12}}<1\mathit{and}\underset{t\to \infty }{\lim }\left|{\displaystyle \frac{1}{t+1}}\right|=0.\end{array}$$

Therefore, we conclude the solution f of the NDDE (50), together with the initial condition

$$\begin{array}{c}\hfill f\left(t\right)=\xi \left(t\right)\mathit{for}-r\le t\le 0,\end{array}$$

(52)

where ξ is a given real-valued function with continuous differentiability on $[-r,0]$. According to Corollary 3, the solution f of the NDDE (50) holds as

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }f\left(t\right)=L(0;\xi )\end{array}$$

Example 9

([40], Example 3.1). Consider the following NDDE

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)-{\displaystyle \frac{1}{2e}}{f}^{\prime }(t-1)+{\displaystyle \frac{1}{2e}}f(t-1)=e^{-t}\mathit{for}t\ge 0.\end{array}$$

(53)

The homogeneous form of Equation (53) is given in [40]. In this example, we considered the nonhomogeneous Equation (53).

Let us set $\alpha =0$, $\gamma =-1/\left(2e\right)$, $\sigma =1$, $\beta =1/\left(2e\right)$, $\tau =1$, and $\delta \left(t\right)=e^{-t}$ in Equation (41). Note that we set $r=1$ in Equation (53). Therefore, the characteristic equation of Equation (53) is

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2e}}{e}^{-\mu }+\mu \left(1-{\displaystyle \frac{1}{2e}}{e}^{-\mu }\right)=0.\end{array}$$

(54)

We obtain the solutions of (54) as ${\mu }_1=-1$ and ${\mu }_2\approx -0.356$. Let us apply the Theorem 2 for ${\mu }_0=-0.356$.

$$\begin{array}{c}\hfill d{\mu }_0=\left|-{\displaystyle \frac{1}{2e}}\right|\left(1+|-0.356|\right)e^{-(-0.356)}+\left|{\displaystyle \frac{1}{2e}}\right|e^{-(-0.356)}\approx 0.6185<1\end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|e^{-t}\right|e^{0.356t}=\underset{t\to \infty }{\lim }{e}^{-0.664t}=0\end{array}$$

Therefore, we conclude the solution f of the NDDE (53), together with the initial condition

$$\begin{array}{c}\hfill f\left(t\right)=\xi \left(t\right)\mathit{for}-1\le t\le 0,\end{array}$$

(55)

where ξ is a given continuously differentiable real-valued function on the interval $[-1,0]$. According to Theorem 2,

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left[e^{0.356t}f(\xi ,t)\right]={\displaystyle \frac{L(-0.356,\xi )}{1+Y(-0.356)}}\end{array}$$

where

$$\begin{array}{cc}\hfill L(-0.356,\xi )=& \xi \left(0\right)+\left(-{\displaystyle \frac{1}{2e}}\right)\left[\xi (-1)-(-0.356)e^{-(-0.356)}\right]{\int }_{-1}^0{e}^{-(-0.356s)}\xi \left(s\right)ds\hfill \\ \hfill & -\left({\displaystyle \frac{1}{2e}}\right)e^{-(-0.356)}{\int }_{-1}^0{e}^{-(-0.356s)}\xi \left(s\right)ds\hfill \\ \hfill =& \xi \left(0\right)-[0.1839\xi (-1)+0.356]{\int }_{-1}^0{e}^{0.356s}\xi \left(s\right)ds\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill Y(-0.356)=& \left(-{\displaystyle \frac{1}{2e}}\right)\left[1-(-0.356)\right]e^{-(-0.356)}-\left({\displaystyle \frac{1}{2e}}\right)e^{-(-0.356)}\approx -0.6185.\hfill \end{array}$$

Additionally, taking into account Remark 4, we reach

$$\begin{array}{c}\hfill \left|f\left(t\right)\right|\le K(-0.356;\xi )e^{-0.356t}\mathit{for}t\ge -1\end{array}$$

where $K(-0.356;\xi )$ is some real number defined by ξ.

5. Discussion and Conclusions

The identification of the exact solution to the characteristic Equation (4) is frequently a challenging task. Nevertheless, if one of the Corollary 1 or Corollary 2 for the coefficients in Equation (1) is valid based on the conditions that are given, one can verify, the validity of Theorem 1 without having to acquire the solution to the characteristic Equation (4). As long as the conditions that are given hold, these corollaries have utility. For instance, let us reconsider the NDDE given in Example 1. We can apply Corollary 2 to Equation (27). To do this, let us set $c=3$ and check the condition (21) of Corollary 2; we see that

$$\begin{array}{cc}\hfill \left|-{\displaystyle \frac{1}{t+1}}\right|+\left|{\displaystyle \frac{1}{4}}\right|3e^{3(1/6)}+\left|-{\displaystyle \frac{1}{4t+3}}\right|e^{3/4}& ={\displaystyle \frac{1}{t+1}}+{\displaystyle \frac{3}{4}}{e}^{1/2}+{\displaystyle \frac{1}{4t+3}}{e}^{3/4}\hfill \\ \hfill & \le 1+{\displaystyle \frac{3}{4}}{e}^{1/2}+{\displaystyle \frac{1}{3}}{e}^{3/4}\approx 2.9422<3,\mathrm{for}t\ge 0,\hfill \end{array}$$

holds. From this, we have $\left|\mu \left(t\right)\right|\le 3$ for $t\ge -0.25$. Let us show that conditions (22) and (23) are also satisfied for $c=3$.

$$\begin{array}{cc}\hfill & \underset{t\to \infty }{\mathrm{lim\; sup}}\left(\left|{\displaystyle \frac{1}{4}}\right|\left(1+3.{\displaystyle \frac{1}{6}}\right)e^{3(1/6)}+\left|-{\displaystyle \frac{1}{4t+3}}\right|{\displaystyle \frac{1}{4}}{e}^{3/4}\right)=\underset{t\to \infty }{\mathrm{lim\; sup}}\left({\displaystyle \frac{3}{8}}{e}^{1/2}+{\displaystyle \frac{1}{4t+3}}{\displaystyle \frac{1}{4}}{e}^{3/4}\right)<1\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\left|\delta \left(t\right)\right|e^{ct}=\underset{t\to \infty }{\lim }\left\{e^{-4t}{e}^{3t}\right\}=0.\end{array}$$

Thus, (22) and (23) are satisfied. According to Corollary 2, the results in Theorem 1, specifically, (7) and (8), hold for the NDDE in (27) even without finding the solution $\mu \left(t\right)=1/(t+1)$ to the characteristic equation. We examined the asymptotic behaviours of solutions to non-homogeneous linear NDDE that have bounded variable delays. Notwithstanding, the general case of variable delays is a somewhat demanding issue. The asymptotic behaviours of solutions to NDDE with variable delays and variable coefficients have been explored with methods that do not employ characteristic equations in other studies [9,10,12,13,14,16,18,21,22,23,24,25,26,27]. Expanding the results reached here to neutral-delay non-homogeneous linear differential equations with variable coefficients and unbounded variable delays would be the topic of future studies.