Lyapunov and Comparison Methods for Nonlinear Distributed-Delay Equations with Variable Delay and Perturbations

1. Introduction

Functional and delay differential equations arise naturally in mathematical models in which the present evolution of a system depends not only on its current state but also on its past history. Such equations occur in population dynamics, control theory, viscoelasticity, neural networks, epidemiology, engineering, and other areas in which aftereffects and hereditary mechanisms cannot be neglected. The qualitative theory of functional differential equations, including existence, boundedness, stability, and asymptotic behavior, has therefore developed into a substantial area of research; see, for example, [5,11,12,13] and the references therein. Closely related ideas occur in the theory of Volterra integral and integro-differential equations, where memory is represented through integral terms; see [4,14].

An important class of hereditary equations consists of equations with distributed delay. In contrast with a discrete-delay equation, in which the dynamics depend on one or several particular past states, a distributed-delay equation incorporates an aggregate of the past over an interval. A representative scalar model, studied in particular by Burton [1], is

$$x^{\prime }\left(t\right)=-{\int }_{t-r}^{t}a(t,s)g\left(x\left(s\right)\right)ds,$$

(1)

where $r>0$ is constant. Of particular relevance to the present paper is Burton’s fixed-point treatment of (1) in the classical paper [1]. Burton considered (1) with a nonconvolution kernel and developed a fixed-point approach to the stability and asymptotic stability of its solutions. The kernel $a(t,s)$ describes the relative influence of the past state at time s on the present dynamics, while the nonlinear function g represents the response of the system. Depending on the assumptions imposed on the kernel and the nonlinearity, one may seek boundedness of solutions, stability or asymptotic stability of the zero solution, decay estimates, or more general asymptotic properties.

A variety of methods have been developed for the stability analysis of delay and functional differential equations. Among the classical tools are Lyapunov functions, Lyapunov–Krasovskii functionals, Razumikhin techniques, comparison principles, limiting-equation methods, and differential inequalities; see [5,11,12,13]. Lyapunov functionals are particularly attractive because they permit the geometry of the memory term to be incorporated directly into an energy-like quantity. Nevertheless, the construction of a suitable functional can be technically demanding, particularly when the kernel is nonconvolution, the delay depends on time, or the equation contains nonlinear perturbations.

A closely related and especially useful comparison device is the Halanay inequality. In its classical form, a nonnegative function satisfying an inequality of the type

$$D^{+}V\left(t\right)\le -\alpha V\left(t\right)+\beta \underset{t-r\le s\le t}{\sup }V\left(s\right),\alpha >\beta >0,$$

decays exponentially. Since its introduction by Halanay [12], this principle has become an important tool in the stability theory of equations with delay. Numerous extensions have been developed for nonlinear, nonautonomous, neutral, and time-varying delay equations; see, for example, [15,16,17] and the references therein. The importance of such inequalities for the present work is that they naturally separate an instantaneous dissipative contribution from the destabilizing effect of the delayed terms. This separation is precisely the mechanism that we exploit below.

Another approach to stability, particularly influential in the work of Burton and collaborators, is based on fixed-point theory. Fixed-point methods have proved useful for ordinary, functional, integral, and integro-differential equations because they can sometimes establish stability without constructing a classical Lyapunov functional; see [1,2,3] and the references therein. The underlying idea is to rewrite the differential equation as an equivalent integral equation and then study the associated solution operator on a suitable complete metric space. Banach’s contraction principle, Krasnosel’skii-type fixed-point theorems, and related compactness arguments have consequently provided useful alternatives to direct Lyapunov methods.

Of particular relevance to the present paper is Burton’s treatment of the nonconvolution equation (1) in [1]. In that approach an integrated kernel $A(t,s)$ is introduced and the differential equation is transformed into an integral equation. Stability is then obtained by requiring the resulting operator to be contractive. A central smallness condition can be expressed in the form

$$2\underset{t\ge 0}{\sup }{\int }_{t-r}^t\left|A(t,u)\right|du<1.$$

(2)

An important advantage of the fixed-point formulation is that it avoids some of the sign and differentiability requirements appearing in certain classical Lyapunov-functional arguments. On the other hand, condition (2) requires the memory contribution to be uniformly small enough for the complete solution operator to be a contraction. Thus, an equation may possess substantial intrinsic dissipation and nevertheless fail the contraction criterion simply because the memory operator is too large when considered independently.

This observation motivates the present work. Rather than asking that the complete memory contribution be small in an absolute sense, we ask whether its size can be measured relative to the instantaneous dissipation available in the equation. This point of view is closer in spirit to Lyapunov and Halanay comparison methods, where stability results from a balance between a negative instantaneous contribution and a delayed contribution. It also suggests that one may treat equations for which the associated fixed-point operator is not a contraction.

We consider the nonlinear perturbed distributed-delay equation

$$x^{\prime }\left(t\right)=-{\int }_{t-r\left(t\right)}^{t}a(t,s)g\left(x\left(s\right)\right)ds+p(t,x\left(t\right))+f\left(t\right),$$

(3)

where $r\left(t\right)$ is a bounded time-dependent delay, $a(t,s)$ is a possibly nonconvolution kernel, g is a nonlinear restoring term, $p(t,x)$ represents a state-dependent perturbation, and $f\left(t\right)$ is an external forcing term. Equation (3) therefore extends the classical equation (1) in several directions simultaneously. Our principal objectives are to obtain boundedness, stability of the zero solution, asymptotic and exponential stability, and robustness estimates in the presence of perturbations and forcing.

The starting point of our analysis is a simple decomposition which reveals the dissipative structure hidden in the distributed-delay term. Define

$$B\left(t\right)={\int }_{t-r\left(t\right)}^{t}a(t,s)ds.$$

(4)

Adding and subtracting $B\left(t\right)g\left(x\right(t\left)\right)$ in (3) gives

$$x^{\prime }\left(t\right)=-B\left(t\right)g\left(x\left(t\right)\right)+R\left(t\right)+p(t,x\left(t\right))+f\left(t\right),$$

(5)

where

$$R\left(t\right)={\int }_{t-r\left(t\right)}^{t}a(t,s)\left[g\left(x\right(t\left)\right)-g\left(x\right(s\left)\right)\right]ds.$$

(6)

The significance of (5) is that the original distributed memory term has been separated into two qualitatively different parts. The term

$$-B\left(t\right)g\left(x\right(t\left)\right)$$

acts instantaneously and is dissipative when $B\left(t\right)>0$ and $xg\left(x\right)>0$, whereas $R\left(t\right)$ measures the error generated by replacing the past values $g\left(x\right(s\left)\right)$ by the present value $g\left(x\right(t\left)\right)$.

When g is Lipschitz, the remainder $R\left(t\right)$ can be estimated through the difference $x\left(t\right)-x\left(s\right)$. Since

$$|x\left(t\right)-x\left(s\right)|\le {\int }_s^t\left|x^{\prime }\left(u\right)\right|du,$$

the resulting estimate involves the first moment of the kernel,

$${\int }_{t-r\left(t\right)}^t(t-s)\left|a(t,s)\right|ds.$$

This quantity measures not merely the total magnitude of the memory kernel but also the temporal distance over which the past acts on the present state. The resulting differential inequality has the general form

$$D^{+}V\left(t\right)\le -\alpha \left(t\right)V\left(t\right)+\beta \left(t\right)\underset{t-\tau \le s\le t}{\sup }V\left(s\right)+q\left(t\right),$$

and is therefore naturally suited to Lyapunov–Halanay comparison. The essential stability requirement becomes a dissipation-versus-memory condition: the effective instantaneous dissipation must dominate the contribution generated by the memory remainder. This differs fundamentally from requiring the entire solution operator to be contractive.

A further advantage of the decomposition is particularly important for time-dependent delays. We do not differentiate an integral having the moving lower endpoint $t-r\left(t\right)$. Consequently, no derivative $r^{\prime }\left(t\right)$ is produced in the estimates. The method can therefore be formulated for bounded continuous delays without imposing the frequently used differentiability restriction on r. This permits delays whose derivatives are unavailable or poorly behaved and separates the stability mechanism from unnecessary regularity of the delay itself. The treatment of time-varying delays by comparison and generalized Halanay inequalities has received considerable attention in the literature; see, for example, [16,17]. The present approach combines this comparison philosophy with the special structure of distributed memory.

The results developed below may be viewed as providing a bridge between two classical philosophies. The fixed-point approach controls the memory term by requiring a suitable integral operator to be a contraction, whereas the Lyapunov–Halanay approach permits a larger memory contribution provided that it is compensated by sufficiently strong dissipation. Our aim is not to replace the fixed-point method, which remains useful under assumptions where Lyapunov constructions may be difficult, but rather to identify a complementary regime in which dissipation can be exploited explicitly.

The distinction can already be seen in a simple linear example. Consider

$$x^{\prime }\left(t\right)=-c{\int }_{t-r}^{t}x\left(s\right)ds,c>0.$$

(7)

For this equation, Burton’s contraction hypothesis (2) reduces to

$$c{r}^2<1.$$

In contrast, the dissipation-versus-memory criterion developed in this paper yields

$$c{r}^2<2.$$

Consequently, the interval

$$1\le c{r}^2<2$$

provides an explicit class of equations for which the stability theorem proved here applies while Burton’s contraction criterion does not, see [1]. Thus the gain is not merely methodological: separating dissipation from memory produces a strictly larger admissible parameter region.

The paper is organized around several broad stages. We first establish the fundamental estimates associated with decomposition (5) and identify the kernel quantities governing the memory remainder. We then obtain boundedness results under integrable growth conditions and develop stability criteria based on the competition between dissipation and memory. Halanay-type comparison arguments yield exponential asymptotic stability under a uniform dissipative gap, while more general time-dependent conditions lead to asymptotic results under nonuniform dissipation. The theory is then extended to state-dependent perturbations and external forcing, including ultimate boundedness and convergence under vanishing forcing. Finally, explicit examples are presented to compare the new conditions with the contraction criterion and to illustrate the role played by variable delay, nonlinear damping, and perturbations.

The principal contribution of the paper is therefore a Lyapunov–comparison framework for nonlinear distributed-delay equations with bounded variable delay in which stability is governed by the balance between instantaneous dissipation and effective memory. The approach avoids differentiation of the delay, accommodates perturbations and forcing, and, even in the elementary linear case, produces stability conditions beyond the range obtained from the corresponding contraction argument.

2. Preliminaries and the Fundamental Decomposition

Let $r:[0,\infty )\to [0,r_0]$ be continuous, where $r_0>0$. For simplicity of notation we assume that an initial function

$$\varphi \in C([-r_0,0],\mathbb{R})$$

is prescribed and that solutions of (3) satisfy $x\left(t\right)=\varphi \left(t\right)$ for $-r_0\le t\le 0$. When necessary one may begin at an arbitrary initial time $t_0$ with the usual translation of the history interval. We assume throughout that the hypotheses imposed on a, g, p, and f guarantee local existence of solutions. The estimates below then provide continuation whenever they give an a priori bound.

We impose the following basic assumptions, locally or globally according to the theorem under consideration.

(H1)

$a(t,s)$ is continuous on the region $t\ge 0$, $t-r\left(t\right)\le s\le t$, and

$$K\left(t\right):={\int }_{t-r\left(t\right)}^t\left|a(t,s)\right|ds<\infty .$$

(H2)

The averaged coefficient

$$B\left(t\right):={\int }_{t-r\left(t\right)}^{t}a(t,s)ds$$

is nonnegative. For stability results we shall usually assume $B\left(t\right)\ge B_0>0$.

(H3)

$g\left(0\right)=0$ and, on the region under consideration, there are constants $m>0$ and $L>0$ such that

$$xg\left(x\right)\ge m{x}^2,\left|g\left(x\right)-g\left(y\right)\right|\le L\left|x-y\right|.$$

In particular, $\left|g\left(x\right)\right|\le L\left|x\right|$.

(H4)

The perturbation satisfies $p(t,0)=0$ and

$$\left|p(t,x)\right|\le \rho \left(t\right)\left|x\right|,$$

where $\rho$ is nonnegative and continuous.

The quantity that naturally measures the memory remainder is the first absolute moment

$$J\left(t\right):={\int }_{t-r\left(t\right)}^t(t-s)\left|a(t,s)\right|ds.$$

(8)

Notice that

$$0\le J\left(t\right)\le r_{0}K\left(t\right).$$

(9)

The first moment $J\left(t\right)$ contains more information than the mass $K\left(t\right)$: memory located very near the present time is less costly than memory concentrated near the far end of the delay interval.

For $t\ge 0$ define

$$M_2\left(t\right):=\underset{\max \{-r_0,t-2r_0\}\le u\le t}{\sup }\left|x\left(u\right)\right|.$$

(10)

The appearance of $2r_0$ in the definition of $M_2\left(t\right)$ is essential. Indeed, if $s\in [t-r(t),t]$, then $s\ge t-r_0$, and in estimating $x\left(t\right)-x\left(s\right)$ we integrate $x^{\prime }\left(u\right)$ over $u\in [s,t]$. Since $x^{\prime }\left(u\right)$ contains delayed states $x\left(v\right)$ with

$$u-r\left(u\right)\le v\le u,$$

and since $u\ge t-r_0$ and $r\left(u\right)\le r_0$, it follows that

$$v\ge u-r\left(u\right)\ge t-2r_0.$$

Taking into account the prescribed initial interval $[-r_0,0]$, the appropriate envelope is therefore $M_2\left(t\right)$ as defined above. Thus the subscript 2 reflects the effective history length $2r_0$ generated by estimating the memory remainder.

Lemma 1.

Assume (H1)–(H4) Let

$$\widehat{K}\left(t\right)=\underset{\max \{0,t-r_0\}\le u\le t}{\sup }K\left(u\right),\widehat{\rho }\left(t\right)=\underset{\max \{0,t-r_0\}\le u\le t}{\sup }\rho \left(u\right),$$

and

$$F_1\left(t\right)=\underset{\max \{0,t-r_0\}\le u\le t}{\sup }\left|f\left(u\right)\right|.$$

Then the remainder (6) satisfies

$$\left|R\left(t\right)\right|\le LJ\left(t\right)\left[L\widehat{K}\left(t\right)+\widehat{\rho }\left(t\right)\right]M_2\left(t\right)+LJ\left(t\right)F_1\left(t\right).$$

(11)

Proof. 

By the Lipschitz property of g,

$$\begin{array}{cc}\hfill \left|R\left(t\right)\right|& \le L{\int }_{t-r\left(t\right)}^t\left|a(t,s)\right|\left|x\left(t\right)-x\left(s\right)\right|ds\hfill \\ & \le L{\int }_{t-r\left(t\right)}^t\left|a(t,s)\right|{\int }_s^t\left|x^{\prime }\left(u\right)\right|duds.\hfill \end{array}$$

For $u\in [t-r_0,t]$, equation (3) gives

$$\begin{array}{cc}\hfill |x^{\prime }\left(u\right)|& \le {\int }_{u-r\left(u\right)}^u\left|a(u,v)\right|\left|g\left(x\right(v\left)\right)\right|dv+\left|p(u,x(u\left)\right)\right|+\left|f\left(u\right)\right|\hfill \\ & \le LK\left(u\right)M_2\left(t\right)+\rho \left(u\right)M_2\left(t\right)+F_1\left(t\right)\hfill \\ & \le \left[L\widehat{K}\left(t\right)+\widehat{\rho }\left(t\right)\right]M_2\left(t\right)+F_1\left(t\right).\hfill \end{array}$$

Substitution into the preceding estimate yields

$$\begin{array}{cc}\hfill \left|R\left(t\right)\right|& \le L\left([L\widehat{K}\left(t\right)+\widehat{\rho }\left(t\right)]M_2\left(t\right)+F_1\left(t\right)\right){\int }_{t-r\left(t\right)}^t(t-s)\left|a(t,s)\right|ds,\hfill \end{array}$$

which is (11). □

It is convenient to introduce

$$\Gamma \left(t\right):=LJ\left(t\right)\left[L\widehat{K}\left(t\right)+\widehat{\rho }\left(t\right)\right].$$

(12)

Then (5) and Lemma 1 imply, whenever $x\left(t\right)\ne 0$,

$$D^{+}\left|x\left(t\right)\right|\le -mB\left(t\right)\left|x\left(t\right)\right|+\rho \left(t\right)\left|x\left(t\right)\right|+\Gamma \left(t\right)M_2\left(t\right)+\left[1+LJ\left(t\right)\right]F_1\left(t\right).$$

(13)

The same inequality holds in the upper Dini sense at $x\left(t\right)=0$. Thus, with

$$\alpha \left(t\right):=mB\left(t\right)-\rho \left(t\right),q\left(t\right):=\left[1+LJ\left(t\right)\right]F_1\left(t\right),$$

(14)

we obtain the fundamental comparison inequality

$$D^{+}\left|x\left(t\right)\right|\le -\alpha \left(t\right)\left|x\left(t\right)\right|+\Gamma \left(t\right)M_2\left(t\right)+q\left(t\right).$$

(15)

This inequality is the basis of all subsequent results.

We shall use the following standard form of Halanay’s inequality.

Lemma 2

(Halanay comparison). Let $h>0$ and let $u:[-h,\infty )\to [0,\infty )$ be continuous. Suppose

$$D^{+}u\left(t\right)\le -\alpha u\left(t\right)+\beta \underset{t-h\le s\le t}{\sup }u\left(s\right),t\ge 0,$$

(16)

where $\alpha >\beta \ge 0$. Then there exists a unique $\eta >0$ satisfying

$$\eta =\alpha -\beta e^{\eta h},$$

(17)

and

$$u\left(t\right)\le \left(\underset{-h\le s\le 0}{\sup }u\left(s\right)\right)e^{-\eta t},t\ge 0.$$

(18)

We also use an envelope observation for estimates without a negative instantaneous term.

Lemma 3

(Envelope estimate). Let $h>0$ and let

$$u:[-h,\infty )⟶[0,\infty )$$

be continuous. Suppose that

$$D^{+}u\left(t\right)\le \nu \left(t\right)\underset{t-h\le s\le t}{\sup }u\left(s\right),t\ge 0,$$

where $\nu \ge 0$ is locally integrable. Then

$$\underset{-h\le s\le t}{\sup }u\left(s\right)\le \left(\underset{-h\le s\le 0}{\sup }u\left(s\right)\right)\exp \left({\int }_0^t\nu \left(s\right)ds\right),t\ge 0.$$

(19)

In particular, if $\nu \in L^1(0,\infty )$, then u is uniformly bounded on $[-h,\infty )$.

Proof. 

Define the running envelope

$$U\left(t\right):=\underset{-h\le s\le t}{\sup }u\left(s\right),t\ge 0,$$

and put

$$U\left(0\right)=\underset{-h\le s\le 0}{\sup }u\left(s\right).$$

Since u is continuous, U is finite, continuous, and nondecreasing. Moreover,

$$u\left(t\right)\le U\left(t\right)$$

and

$$\underset{t-h\le s\le t}{\sup }u\left(s\right)\le U\left(t\right).$$

Consequently, the assumed differential inequality implies

$$D^{+}u\left(t\right)\le \nu \left(t\right)U\left(t\right).$$

(20)

We claim that

$$D^{+}U\left(t\right)\le \nu \left(t\right)U\left(t\right),t\ge 0.$$

(21)

To establish this claim, fix $t\ge 0$. There are two cases.

First suppose that

$$u\left(t\right)<U\left(t\right).$$

By continuity of u, there exist $\delta >0$ and $\eta >0$ such that

$$u\left(s\right)\le U\left(t\right)-\eta ,t\le s\le t+\delta .$$

Indeed, since $u\left(t\right)<U\left(t\right)$, one may choose

$$0<\eta <{\displaystyle \frac{U\left(t\right)-u\left(t\right)}{2}}$$

and then use continuity at t.

For $0<q<\delta$, the values of u on $[t,t+q]$ therefore remain strictly below $U\left(t\right)$. Since the interval $[-h,t]$ is contained in $[-h,t+q]$, we have

$$U(t+q)=\max \left\{U\left(t\right),\underset{t\le s\le t+q}{\sup }u\left(s\right)\right\}=U\left(t\right).$$

Hence

$$D^{+}U\left(t\right)=0.$$

Since $\nu \left(t\right)U\left(t\right)\ge 0$, it follows immediately that

$$D^{+}U\left(t\right)\le \nu \left(t\right)U\left(t\right).$$

Now suppose that

$$u\left(t\right)=U\left(t\right).$$

For $q>0$,

$$U(t+q)=\max \left\{U\left(t\right),\underset{t\le s\le t+q}{\sup }u\left(s\right)\right\}.$$

Therefore

$$U(t+q)-U\left(t\right)\le \underset{t\le s\le t+q}{\sup }{\left(u\left(s\right)-u\left(t\right)\right)}^{+},$$

where

$$z^{+}:=\max \{z,0\}.$$

Taking the upper right Dini derivative gives

$$D^{+}U\left(t\right)\le \max \{0,D^{+}u\left(t\right)\}.$$

Using (20) and the fact that $\nu \left(t\right)U\left(t\right)\ge 0$, we obtain

$$D^{+}U\left(t\right)\le \max \{0,D^{+}u\left(t\right)\}\le \nu \left(t\right)U\left(t\right).$$

Thus (21) holds in both cases.

We may now apply the Dini-derivative form of Gronwall’s inequality. For completeness, this step can also be seen directly. Define

$$N\left(t\right):={\int }_0^t\nu \left(s\right)ds$$

and

$$W\left(t\right):=e^{-N\left(t\right)}U\left(t\right).$$

The inequality (21) implies, in the Dini sense,

$$D^{+}W\left(t\right)\le e^{-N\left(t\right)}\left[D^{+}U\left(t\right)-\nu \left(t\right)U\left(t\right)\right]\le 0.$$

Hence W is nonincreasing, and consequently

$$e^{-N\left(t\right)}U\left(t\right)\le U\left(0\right).$$

Equivalently,

$$U\left(t\right)\le U\left(0\right)e^{N\left(t\right)}=U\left(0\right)\exp \left({\int }_0^t\nu \left(s\right)ds\right).$$

Recalling the definition of U, we obtain

$$\underset{-h\le s\le t}{\sup }u\left(s\right)\le \left(\underset{-h\le s\le 0}{\sup }u\left(s\right)\right)\exp \left({\int }_0^t\nu \left(s\right)ds\right),$$

which proves (19).

Finally, if $\nu \in L^1(0,\infty )$, then

$${\int }_0^t\nu \left(s\right)ds\le {\int }_0^{\infty }\nu \left(s\right)ds=:N_{\infty }<\infty .$$

Therefore

$$U\left(t\right)\le U\left(0\right)e^{N_{\infty }},t\ge 0.$$

Since $u\left(t\right)\le U\left(t\right)$, we conclude that

$$u\left(t\right)\le \left(\underset{-h\le s\le 0}{\sup }u\left(s\right)\right)\exp \left({\int }_0^{\infty }\nu \left(s\right)ds\right),t\ge 0,$$

and hence u is uniformly bounded on $[-h,\infty )$. □

3. Boundedness Under an Integrable Growth Majorant

We begin with a result whose purpose is different from asymptotic stability. The negative term in (15) may be discarded, and the remaining growth can still be controlled if its coefficient is integrable. This is useful when the available dissipation is not uniformly strong enough to dominate the memory at every time.

Theorem 1

(Boundedness). Assume (H1)–(H4), let $f\equiv 0$, and suppose

$$\alpha \left(t\right)\ge 0,\Gamma \in L^1(0,\infty ).$$

(22)

Then every solution that remains in the region where the hypotheses hold satisfies

$$\left|x\left(t\right)\right|\le {∥\varphi ∥}_{\infty }\exp \left({\int }_0^t\Gamma \left(s\right)ds\right)\le e^{{∥\Gamma ∥}_1}{∥\varphi ∥}_{\infty }.$$

(23)

If the hypotheses are global, every local solution is bounded on its interval of existence; under the usual continuation property it is therefore global.

Proof. 

With $f\equiv 0$, (15) and $\alpha \left(t\right)\ge 0$ give

$$D^{+}\left|x\left(t\right)\right|\le \Gamma \left(t\right)M_2\left(t\right).$$

Apply Lemma 3 with $h=2r_0$ and $u\left(t\right)=\left|x\right(t\left)\right|$. This gives

$$\underset{-2r_0\le s\le t}{\sup }\left|x\left(s\right)\right|\le {∥\varphi ∥}_{\infty }\exp \left({\int }_0^t\Gamma \left(s\right)ds\right),$$

where the history may be extended harmlessly on $[-2r_0,-r_0]$ by its endpoint bound if only $[-r_0,0]$ is prescribed. The stated estimate follows. □

Remark 1.

Theorem 1 illustrates the difference between an integrable-growth condition and a contraction condition. No number is required to be uniformly less than one. A coefficient may be large on finite intervals, or on a sparse collection of intervals, provided that its total destabilizing contribution measured by Γ is integrable.

A useful local version follows immediately.

Corollary 1.

Suppose (H1)–(H4) hold for $\left|x\right|\le R$, $f\equiv 0$, (22) holds, and

$$e^{{∥\Gamma ∥}_1}{∥\varphi ∥}_{\infty }<R.$$

Then the solution cannot leave the ball $\left|x\right|<R$. Consequently, the estimate (23) is valid for all $t\ge 0$ for which the standard continuation theorem applies.

4. Stability and Dissipation Versus Memory

We now retain the negative term in (15). The resulting criterion compares the instantaneous damping $mB\left(t\right)-\rho \left(t\right)$ with the memory coefficient $\Gamma \left(t\right)$. This is the central structural condition of the paper.

Theorem 2

(Uniform exponential stability). Assume (H1)–(H4), let $f\equiv 0$, and suppose there are constants ${\alpha }_0>{\beta }_0\ge 0$ such that

$$\alpha \left(t\right)=mB\left(t\right)-\rho \left(t\right)\ge {\alpha }_0,\Gamma \left(t\right)\le {\beta }_0,t\ge 0.$$

(24)

Then the zero solution of (3) is uniformly exponentially stable. More precisely, if $\eta >0$ is the solution of

$$\eta ={\alpha }_0-{\beta }_0{e}^{2\eta r_0},$$

(25)

then

$$\left|x\left(t\right)\right|\le {∥\varphi ∥}_{\infty }{e}^{-\eta t},t\ge 0.$$

(26)

for every solution remaining in the region on which the hypotheses hold.

Proof. 

From (15), with $f=0$,

$$D^{+}\left|x\left(t\right)\right|\le -{\alpha }_0\left|x\left(t\right)\right|+{\beta }_0\underset{t-2r_0\le s\le t}{\sup }\left|x\left(s\right)\right|.$$

Lemma 2, with $h=2r_0$, yields (26). □

Remark 2.

Condition (24) does not ask that the memory operator itself be a contraction. The quantity ${\beta }_0$ is allowed to be positive and even substantial; what matters is that the effective instantaneous dissipation ${\alpha }_0$ dominate it. This distinction is responsible for the improvement in the examples below.

The preceding theorem has a convenient global form when K, J, and $\rho$ have uniform bounds.

Corollary 2.

Assume $f\equiv 0$ and suppose

$$B\left(t\right)\ge B_0>0,K\left(t\right)\le K_0,J\left(t\right)\le J_0,\rho \left(t\right)\le {\rho }_0.$$

If

$$m{B}_0-{\rho }_0>L{J}_0(L{K}_0+{\rho }_0),$$

(27)

then the zero solution is uniformly exponentially stable.

Proof. 

We have

$$\alpha \left(t\right)\ge m{B}_0-{\rho }_0$$

and

$$\Gamma \left(t\right)\le L{J}_0(L{K}_0+{\rho }_0).$$

Thus (27) gives the strict gap required in Theorem 2. □

The theorem also yields local nonlinear stability under local sector conditions.

Theorem 3

(Local asymptotic stability). Suppose there is $R>0$ such that (H1)–(H4) hold whenever $\left|x\right|,\left|y\right|\le R$, and suppose (24) holds on this region. Then there exists $\delta >0$ such that

$${∥\varphi ∥}_{\infty }<\delta$$

implies that the solution exists globally, remains in $\left|x\right(t\left)\right|<R$, and satisfies $x\left(t\right)\to 0$ exponentially.

Proof. 

Choose $\delta <R$. As long as the solution remains in the ball, Theorem 2 gives

$$\left|x\left(t\right)\right|\le {∥\varphi ∥}_{\infty }{e}^{-\eta t}<R.$$

Thus a first exit from the ball is impossible. The estimate therefore continues for all time and implies convergence to zero. □

5. Asymptotic Stability with Time-Dependent Dissipation

Uniform domination is convenient but not always necessary. In applications the damping and memory coefficients may vary substantially with time. We therefore record a variable-coefficient comparison formulation. The preceding estimates reduce the asymptotic analysis to a variable-coefficient delay differential inequality. We therefore record the following Halanay-type comparison lemma, which provides an explicit decay estimate when the instantaneous dissipation dominates the delayed contribution by a fixed relative gap. An important feature of the result is that the coefficient of dissipation need not be bounded away from zero; it is sufficient that its cumulative effect diverges while its mass over each delay interval remains uniformly bounded.

Lemma 4

(A variable-coefficient Halanay inequality). Let $h>0$, let

$$u:[-h,\infty )⟶[0,\infty )$$

be continuous, and suppose that

$$D^{+}u\left(t\right)\le -\lambda \left(t\right)u\left(t\right)+\theta \lambda \left(t\right)\underset{t-h\le s\le t}{\sup }u\left(s\right),t\ge 0,$$

(28)

where $\lambda :[0,\infty )\to (0,\infty )$ is continuous and $0\le \theta <1$. Assume that

$${\Lambda }_h:=\underset{t\ge 0}{\sup }{\int }_{{(t-h)}^{+}}^t\lambda \left(s\right)ds<\infty ,$$

(29)

where ${(t-h)}^{+}=\max \{0,t-h\}$.

Let $\eta \in (0,1]$ be the unique positive solution of

$$\eta +\theta e^{\eta {\Lambda }_h}=1.$$

(30)

Then

$$u\left(t\right)\le \left(\underset{-h\le s\le 0}{\sup }u\left(s\right)\right)\exp \left(-\eta {\int }_0^t\lambda \left(s\right)ds\right),t\ge 0.$$

(31)

Consequently, if

$${\int }_0^{\infty }\lambda \left(s\right)ds=\infty ,$$

(32)

then

$$u\left(t\right)⟶0\mathrm{as}t\to \infty .$$

Proof. 

Set

$$U_0:=\underset{-h\le s\le 0}{\sup }u\left(s\right)$$

and define

$$L\left(t\right):={\int }_0^t\lambda \left(s\right)ds,t\ge 0.$$

Since $\lambda \left(t\right)>0$, the function L is continuous and strictly increasing.

We first verify that the number $\eta$ appearing in (30) exists and is unique. Define

$$F\left(\eta \right)=\eta +\theta e^{\eta {\Lambda }_h}-1,\eta \ge 0.$$

If $0\le \theta <1$, then

$$F\left(0\right)=\theta -1<0.$$

Moreover,

$$F\left(1\right)=\theta e^{{\Lambda }_h}\ge 0.$$

Thus, by continuity, there exists

$$\eta \in (0,1]$$

such that $F\left(\eta \right)=0$. Furthermore,

$$F^{\prime }\left(\eta \right)=1+\theta {\Lambda }_h{e}^{\eta {\Lambda }_h}>0,$$

so F is strictly increasing. Hence the solution is unique.

We now construct an explicit comparison function. Define

$$v\left(t\right)=\left\{\begin{array}{cc}{U}_0,\hfill & -h\le t\le 0,\hfill \\ U_0{e}^{-\eta L\left(t\right)},\hfill & t\ge 0.\hfill \end{array}\right.$$

Clearly,

$$u\left(t\right)\le v\left(t\right)=U_0,-h\le t\le 0.$$

For $t>0$,

$$v^{\prime }\left(t\right)=-\eta \lambda \left(t\right)v\left(t\right).$$

We next estimate the delayed envelope of v. Since L is nondecreasing, v is nonincreasing on $[0,\infty )$. We claim that

$$\underset{t-h\le s\le t}{\sup }v\left(s\right)\le e^{\eta {\Lambda }_h}v\left(t\right),t\ge 0.$$

(33)

To verify this, first suppose that $t\ge h$. Then $t-h\ge 0$, and since v is nonincreasing,

$$\underset{t-h\le s\le t}{\sup }v\left(s\right)=v(t-h).$$

Therefore

$$\begin{array}{cc}\hfill {\displaystyle \frac{v(t-h)}{v\left(t\right)}}& =\exp \left(\eta \left[L\right(t)-L(t-h\left)\right]\right)\hfill \\ \hfill & =\exp \left(\eta {\int }_{t-h}^t\lambda \left(s\right)ds\right)\hfill \\ \hfill & \le e^{\eta {\Lambda }_h}.\hfill \end{array}$$

Hence

$$\underset{t-h\le s\le t}{\sup }v\left(s\right)\le e^{\eta {\Lambda }_h}v\left(t\right).$$

If $0\le t<h$, part of the delay interval lies in the initial interval $[-h,0]$. On that part,

$$v\left(s\right)=U_0.$$

Thus

$$\underset{t-h\le s\le t}{\sup }v\left(s\right)=U_0.$$

Since

$$v\left(t\right)=U_0{e}^{-\eta L\left(t\right)},$$

we obtain

$${\displaystyle \frac{U_0}{v\left(t\right)}}=e^{\eta L\left(t\right)}=\exp \left(\eta {\int }_0^t\lambda \left(s\right)ds\right)\le e^{\eta {\Lambda }_h}.$$

Thus (33) also holds when $0\le t<h$.

Using (33), we have

$$\begin{array}{cc}\hfill -\lambda \left(t\right)v\left(t\right)+\theta \lambda \left(t\right)\underset{t-h\le s\le t}{\sup }v\left(s\right)& \le -\lambda \left(t\right)v\left(t\right)+\theta \lambda \left(t\right)e^{\eta {\Lambda }_h}v\left(t\right)\hfill \\ \hfill & =-\lambda \left(t\right)\left(1-\theta e^{\eta {\Lambda }_h}\right)v\left(t\right).\hfill \end{array}$$

(34)

By the defining equation (30),

$$1-\theta e^{\eta {\Lambda }_h}=\eta .$$

Consequently,

$$-\lambda \left(t\right)v\left(t\right)+\theta \lambda \left(t\right)\underset{t-h\le s\le t}{\sup }v\left(s\right)\le -\eta \lambda \left(t\right)v\left(t\right)=v^{\prime }\left(t\right).$$

(35)

Thus v is a supersolution of the Halanay inequality.

We now prove that

$$u\left(t\right)\le v\left(t\right),t\ge 0.$$

Although this follows from the standard comparison principle, we give the argument because it is central to the estimate.

Fix $\epsilon >0$ and define

$$v_{\epsilon }\left(t\right)=\left\{\begin{array}{cc}{U}_0+\epsilon ,\hfill & -h\le t\le 0,\hfill \\ (U_0+\epsilon )e^{-\eta L\left(t\right)},\hfill & t\ge 0.\hfill \end{array}\right.$$

Then

$$u\left(s\right)<v_{\epsilon }\left(s\right),-h\le s\le 0.$$

The preceding calculation applies equally to $v_{\epsilon }$.

Suppose, contrary to the desired conclusion, that u reaches $v_{\epsilon }$ at some positive time. Let $t_{*}>0$ be the first such time. Then

$$u\left(t\right)<v_{\epsilon }\left(t\right),0\le t<t_{*},$$

and

$$u\left(t_{*}\right)=v_{\epsilon }\left(t_{*}\right).$$

By the first-contact property,

$$D^{+}\left(u-v_{\epsilon }\right)\left(t_{*}\right)\ge 0.$$

Moreover, for every

$$s\in [t_{*}-h,t_{*}],$$

we have

$$u\left(s\right)\le v_{\epsilon }\left(s\right),$$

where the inequality is strict whenever $s<t_{*}$ and equality may occur at $s=t_{*}$. Hence

$$\underset{t_{*}-h\le s\le t_{*}}{\sup }u\left(s\right)\le \underset{t_{*}-h\le s\le t_{*}}{\sup }{v}_{\epsilon }\left(s\right).$$

Applying (28) at $t_{*}$ gives

$$\begin{array}{cc}\hfill D^{+}u\left(t_{*}\right)& \le -\lambda \left(t_{*}\right)u\left(t_{*}\right)+\theta \lambda \left(t_{*}\right)\underset{t_{*}-h\le s\le t_{*}}{\sup }u\left(s\right)\hfill \\ \hfill & \le -\lambda \left(t_{*}\right)v_{\epsilon }\left(t_{*}\right)+\theta \lambda \left(t_{*}\right)\underset{t_{*}-h\le s\le t_{*}}{\sup }{v}_{\epsilon }\left(s\right).\hfill \end{array}$$

By (35),

$$D^{+}u\left(t_{*}\right)\le v_{\epsilon }^{\prime }\left(t_{*}\right).$$

Therefore

$$D^{+}(u-v_{\epsilon })\left(t_{*}\right)\le 0.$$

To obtain a strict contradiction, one may replace $\eta$ temporarily by any number ${\eta }_{\epsilon }>0$ satisfying

$${\eta }_{\epsilon }+\theta e^{{\eta }_{\epsilon }{\Lambda }_h}<1$$

and then let ${\eta }_{\epsilon }\uparrow \eta$. Equivalently, the standard non-strict Dini comparison principle applied to the supersolution $v_{\epsilon }$ yields

$$u\left(t\right)\le v_{\epsilon }\left(t\right),t\ge 0.$$

Letting $\epsilon \downarrow 0$ gives

$$u\left(t\right)\le U_0{e}^{-\eta L\left(t\right)}.$$

Thus

$$u\left(t\right)\le \left(\underset{-h\le s\le 0}{\sup }u\left(s\right)\right)\exp \left(-\eta {\int }_0^t\lambda \left(s\right)ds\right),$$

which proves (31).

Finally, if

$${\int }_0^{\infty }\lambda \left(s\right)ds=\infty ,$$

then

$$L\left(t\right)⟶\infty .$$

Since $\eta >0$,

$$\exp [-\eta L(t\left)\right]⟶0.$$

Therefore

$$u\left(t\right)⟶0\mathrm{as}t\to \infty .$$

This completes the proof. □

Before stating the weighted-gap result, we emphasize the length of the delay interval appearing in the comparison inequality. Although the original equation involves the variable delay $r\left(t\right)\le r_0$, the estimate of the memory remainder requires a larger history interval. Indeed, for $s\in [t-r(t),t]$, the difference $x\left(t\right)-x\left(s\right)$ is estimated by integrating $x^{\prime }$ over $[s,t]$. For each $u\in [s,t]$, the term $x^{\prime }\left(u\right)$ depends in turn on values $x\left(v\right)$ with

$$u-r\left(u\right)\le v\le u.$$

Since $s\ge t-r_0$, $u\ge s$, and $r\left(u\right)\le r_0$, we have

$$v\ge u-r\left(u\right)\ge t-2r_0.$$

Consequently, the remainder estimate naturally involves

$$M_2\left(t\right)=\underset{\max \{-r_0,t-2r_0\}\le s\le t}{\sup }\left|x\left(s\right)\right|.$$

Thus the effective delay length in the Halanay comparison is $2r_0$, rather than $r_0$. This is the same delay length used in Theorem 2. Accordingly, in the variable-coefficient comparison below we take

$$h=2r_0$$

and define

$${\Lambda }_{2r_0}:=\underset{t\ge 0}{\sup }{\int }_{{(t-2r_0)}^{+}}^t\lambda \left(s\right)ds.$$

(36)

This choice makes the weighted Halanay comparison consistent with the fundamental estimate (15). “`latex

Theorem 4

(Asymptotic stability under a weighted gap). Let $f\equiv 0$ and assume (H1)–(H4). Suppose there exist a continuous function

$$\lambda :[0,\infty )⟶(0,\infty )$$

and a constant $\theta \in [0,1)$ such that

$$\Gamma \left(t\right)\le \theta \lambda \left(t\right),\alpha \left(t\right)\ge \lambda \left(t\right),t\ge 0.$$

(37)

Assume further that

$${\Lambda }_{2r_0}:=\underset{t\ge 0}{\sup }{\int }_{{(t-2r_0)}^{+}}^t\lambda \left(s\right)ds<\infty$$

(38)

and

$${\int }_0^{\infty }\lambda \left(s\right)ds=\infty .$$

(39)

Let $\eta \in (0,1]$ be the unique solution of

$$\eta +\theta e^{\eta {\Lambda }_{2r_0}}=1.$$

(40)

Then every solution with initial function ϕ satisfies

$$\left|x\left(t\right)\right|\le {∥\varphi ∥}_{\infty }\exp \left(-\eta {\int }_0^t\lambda \left(s\right)ds\right),t\ge 0.$$

(41)

Consequently, the zero solution is asymptotically stable.

Proof. 

Since $f\equiv 0$, the fundamental comparison inequality (15) gives

$$D^{+}\left|x\left(t\right)\right|\le -\alpha \left(t\right)\left|x\left(t\right)\right|+\Gamma \left(t\right)M_2\left(t\right),$$

where

$$M_2\left(t\right)=\underset{\max \{-r_0,t-2r_0\}\le s\le t}{\sup }\left|x\left(s\right)\right|.$$

Using (37), we obtain

$$D^{+}\left|x\left(t\right)\right|\le -\lambda \left(t\right)\left|x\left(t\right)\right|+\theta \lambda \left(t\right)M_2\left(t\right).$$

(42)

The important point is that the history appearing in (42) extends at most $2r_0$ units into the past. Hence, with $u\left(t\right)=\left|x\right(t\left)\right|$, the relevant variable-coefficient Halanay inequality has delay length

$$h=2r_0.$$

Therefore the local accumulated dissipation appearing in Lemma 4 is precisely

$${\Lambda }_h={\Lambda }_{2r_0}=\underset{t\ge 0}{\sup }{\int }_{{(t-2r_0)}^{+}}^t\lambda \left(s\right)ds.$$

Applying Lemma 4 with $h=2r_0$ yields

$$\left|x\left(t\right)\right|\le {∥\varphi ∥}_{\infty }\exp \left(-\eta {\int }_0^t\lambda \left(s\right)ds\right),$$

where $\eta >0$ is determined by

$$\eta +\theta e^{\eta {\Lambda }_{2r_0}}=1.$$

This proves (41).

Since

$${\int }_0^{\infty }\lambda \left(s\right)ds=\infty ,$$

we have

$$\exp \left(-\eta {\int }_0^t\lambda \left(s\right)ds\right)⟶0\mathrm{as}t\to \infty .$$

Thus

$$x\left(t\right)⟶0.$$

Moreover,

$$\exp \left(-\eta {\int }_0^t\lambda \left(s\right)ds\right)\le 1,$$

and therefore

$$\left|x\left(t\right)\right|\le {∥\varphi ∥}_{\infty },t\ge 0.$$

Hence, given $\epsilon >0$, choosing $\delta =\epsilon$ gives

$${∥\varphi ∥}_{\infty }<\delta ⟹\left|x\left(t\right)\right|<\epsilon \mathrm{for}\mathrm{all}t\ge 0.$$

The zero solution is therefore stable and attractive, and hence asymptotically stable. □

Remark 3.

Theorem 4 does not require the effective dissipation $\lambda \left(t\right)$ to be bounded away from zero. Thus it applies to situations not covered by the uniform exponential-stability criterion of Theorem 2. The decay rate is determined by the accumulated dissipation

$${\int }_0^t\lambda \left(s\right)ds.$$

For example, if

$$\lambda \left(t\right)={\displaystyle \frac{a}{1+t}},a>0,$$

then

$${\int }_0^t\lambda \left(s\right)ds=a\log (1+t),$$

and (41) yields the algebraic decay estimate

$$\left|x\left(t\right)\right|\le {∥\varphi ∥}_{\infty }{(1+t)}^{-a\eta }.$$

Thus the weighted-gap theorem includes nonexponential decay as a natural part of the theory.

A particularly useful alternative is to combine a uniform gap with an integrable perturbation of that gap.

Corollary 3.

Suppose

$$\alpha \left(t\right)\ge {\alpha }_0-\nu \left(t\right),\Gamma \left(t\right)\le {\beta }_0+\nu \left(t\right),$$

where ${\alpha }_0>{\beta }_0\ge 0$, $\nu \ge 0$, and $\nu \in L^1(0,\infty )$. Then the zero solution is stable under the associated integrable-growth comparison. If, in addition, $\nu \left(t\right)\to 0$, then after a sufficiently large time the coefficients satisfy a strict uniform gap, and every sufficiently small solution converges exponentially from that time onward.

Proof. 

Since $\nu \in L^1$, the integrable-growth estimate prevents amplification by more than a finite factor. Also, for every $\epsilon >0$ the set on which $\nu$ is persistently larger than $\epsilon$ cannot have infinite measure without violating integrability. If one strengthens the hypothesis to $\nu \left(t\right)\to 0$, then for sufficiently large T,

$$\alpha \left(t\right)\ge {\alpha }_0-\epsilon ,\Gamma \left(t\right)\le {\beta }_0+\epsilon ,t\ge T.$$

Choosing $\epsilon <({\alpha }_0-{\beta }_0)/2$ gives a uniform Halanay gap on $[T,\infty )$. Thus, under the additional natural condition $\nu \left(t\right)\to 0$, exponential convergence follows. Without pointwise decay of $\nu$, stability remains valid but the final exponential assertion requires an averaged Halanay theorem. □

6. Perturbations, Forcing, and Robustness

We now return to the full equation (3). The perturbation $p(t,x)$ has already been incorporated into the effective damping $\alpha \left(t\right)$ and the memory coefficient $\Gamma \left(t\right)$. The forcing enters through

$$q\left(t\right)=\left[1+LJ\left(t\right)\right]\underset{\max \{0,t-r_0\}\le s\le t}{\sup }\left|f\left(s\right)\right|.$$

Thus the same comparison structure yields robustness estimates.

Theorem 5

(Bounded forcing and ultimate boundedness). Assume the hypotheses of Theorem 2 except that f need not vanish. Suppose

$$\underset{t\ge 0}{\sup }J\left(t\right)\le J_0,\underset{t\ge 0}{\sup }\left|f\left(t\right)\right|\le F.$$

Let

$$Q=(1+L{J}_0)F.$$

Then every solution satisfies

$$\underset{t\to \infty }{\mathrm{lim\; sup}}\left|x\left(t\right)\right|\le {\displaystyle \frac{Q}{{\alpha }_0-{\beta }_0}}.$$

(43)

In particular, the size of the ultimate bound tends to zero with the forcing amplitude F.

Proof. 

From (15) and the hypotheses of the theorem,

$$D^{+}\left|x\left(t\right)\right|\le -{\alpha }_0\left|x\left(t\right)\right|+{\beta }_0{M}_2\left(t\right)+Q.$$

Set

$$R_{*}:={\displaystyle \frac{Q}{{\alpha }_0-{\beta }_0}}$$

and define

$$z\left(t\right):={\left(\left|x\left(t\right)\right|-R_{*}\right)}^{+}.$$

We claim that

$$D^{+}z\left(t\right)\le -{\alpha }_{0}z\left(t\right)+{\beta }_0\underset{\max \{-r_0,t-2r_0\}\le s\le t}{\sup }z\left(s\right).$$

Indeed, whenever $\left|x\left(t\right)\right|>R_{*}$, we have

$$z\left(t\right)=\left|x\left(t\right)\right|-R_{*},$$

and

$$M_2\left(t\right)\le R_{*}+\underset{\max \{-r_0,t-2r_0\}\le s\le t}{\sup }z\left(s\right).$$

Consequently,

$$\begin{array}{cc}\hfill D^{+}z\left(t\right)& =D^{+}\left|x\left(t\right)\right|\hfill \\ \hfill & \le -{\alpha }_0\left(z\left(t\right)+R_{*}\right)+{\beta }_0\left(R_{*}+\underset{\max \{-r_0,t-2r_0\}\le s\le t}{\sup }z\left(s\right)\right)+Q\hfill \\ \hfill & =-{\alpha }_{0}z\left(t\right)+{\beta }_0\underset{\max \{-r_0,t-2r_0\}\le s\le t}{\sup }z\left(s\right)-({\alpha }_0-{\beta }_0)R_{*}+Q\hfill \\ \hfill & =-{\alpha }_{0}z\left(t\right)+{\beta }_0\underset{\max \{-r_0,t-2r_0\}\le s\le t}{\sup }z\left(s\right),\hfill \end{array}$$

where the last equality follows from

$$({\alpha }_0-{\beta }_0)R_{*}=Q.$$

At points where $\left|x\left(t\right)\right|\le R_{*}$, the same inequality holds in the upper-Dini sense for the positive part $z\left(t\right)$.

Since ${\alpha }_0>{\beta }_0$, the Halanay comparison theorem, with effective delay length $2r_0$, implies that

$$z\left(t\right)⟶0\mathrm{as}t\to \infty .$$

Therefore

$$\underset{t\to \infty }{\mathrm{lim\; sup}}\left|x\left(t\right)\right|\le R_{*}={\displaystyle \frac{Q}{{\alpha }_0-{\beta }_0}},$$

which proves (43). □

Theorem 6

(Vanishing forcing). Under the hypotheses of Theorem 5, if

$$f\left(t\right)\to 0(t\to \infty ),$$

then

$$x\left(t\right)\to 0(t\to \infty ).$$

Proof. 

Because $f\left(t\right)\to 0$, we also have $F_1\left(t\right)\to 0$ and hence $q\left(t\right)\to 0$. Given $\epsilon >0$, choose T so large that

$$q\left(t\right)\le ({\alpha }_0-{\beta }_0)\epsilon ,t\ge T.$$

Applying the ultimate-bound argument from time T gives

$$\underset{t\to \infty }{\mathrm{lim\; sup}}\left|x\left(t\right)\right|\le \epsilon .$$

Since $\epsilon$ is arbitrary, $x\left(t\right)\to 0$. □

The preceding theorem may also be expressed in a weighted-growth form. This is useful when f is not bounded.

Theorem 7

(Growth under unbounded forcing). Assume (24) and $J\left(t\right)\le J_0$. Let $w:[-2r_0,\infty )\to [1,\infty )$ be locally absolutely continuous and nondecreasing, and suppose

$$\left|f\right(t\left)\right|\le Fw\left(t\right).$$

Then there is a constant $C>0$, depending only on ${\alpha }_0$, ${\beta }_0$, L, $J_0$, and $r_0$, such that

$$\left|x\left(t\right)\right|\le C\left({∥\varphi ∥}_{\infty }+F\right)w\left(t\right),t\ge 0.$$

(44)

provided the weighted Halanay comparison remains in the region where the structural hypotheses hold.

Proof. 

Since w is nondecreasing and

$$\left|f\right(t\left)\right|\le Fw\left(t\right),$$

we have, for $s\le t$,

$$\left|f\right(s\left)\right|\le Fw\left(s\right)\le Fw\left(t\right).$$

Hence

$$F_1\left(t\right)\le Fw\left(t\right),$$

and therefore

$$q\left(t\right)\le (1+L{J}_0)Fw\left(t\right).$$

Set

$$Q_0:=(1+L{J}_0)F$$

and define

$$y\left(t\right):={\displaystyle \frac{\left|x\right(t\left)\right|}{w\left(t\right)}}.$$

Since w is locally absolutely continuous and nondecreasing, we have $w^{\prime }\left(t\right)\ge 0$ for almost every $t\ge 0$. At every point at which w is differentiable, the quotient rule and the fundamental comparison inequality give

$$\begin{array}{cc}\hfill D^{+}y\left(t\right)& \le {\displaystyle \frac{D^{+}\left|x\left(t\right)\right|}{w\left(t\right)}}-{\displaystyle \frac{\left|x\left(t\right)\right|w^{\prime }\left(t\right)}{w^2\left(t\right)}}\hfill \\ \hfill & \le -{\alpha }_{0}y\left(t\right)+{\beta }_0{\displaystyle \frac{M_2\left(t\right)}{w\left(t\right)}}+Q_0-{\displaystyle \frac{w^{\prime }\left(t\right)}{w\left(t\right)}}y\left(t\right).\hfill \end{array}$$

Since $w^{\prime }\left(t\right)\ge 0$, the last term is nonpositive and may be discarded. Thus

$$D^{+}y\left(t\right)\le -{\alpha }_{0}y\left(t\right)+{\beta }_0{\displaystyle \frac{M_2\left(t\right)}{w\left(t\right)}}+Q_0.$$

Now, for

$$s\in [\max \{-r_0,t-2r_0\},t],$$

the monotonicity of w gives $w\left(s\right)\le w\left(t\right)$. Consequently,

$${\displaystyle \frac{\left|x\right(s\left)\right|}{w\left(t\right)}}={\displaystyle \frac{\left|x\right(s\left)\right|}{w\left(s\right)}}{\displaystyle \frac{w\left(s\right)}{w\left(t\right)}}\le y\left(s\right),$$

and hence

$${\displaystyle \frac{M_2\left(t\right)}{w\left(t\right)}}\le \underset{\max \{-r_0,t-2r_0\}\le s\le t}{\sup }y\left(s\right).$$

Therefore

$$D^{+}y\left(t\right)\le -{\alpha }_{0}y\left(t\right)+{\beta }_0\underset{\max \{-r_0,t-2r_0\}\le s\le t}{\sup }y\left(s\right)+Q_0.$$

Thus y satisfies a forced Halanay inequality with effective delay length $2r_0$ and bounded forcing $Q_0$.

Since ${\alpha }_0>{\beta }_0$, the bounded-forcing comparison of Theorem 5 yields a uniform bound

$$y\left(t\right)\le C\left({∥\varphi ∥}_{\infty }+F\right),t\ge 0,$$

for some constant $C>0$ depending only on ${\alpha }_0,{\beta }_0,L,J_0$, and $r_0$. Since

$$\left|x\right(t\left)\right|=w\left(t\right)y\left(t\right),$$

we obtain

$$\left|x\left(t\right)\right|\le C\left({∥\varphi ∥}_{\infty }+F\right)w\left(t\right),t\ge 0,$$

which is (44). □

7. Examples and Comparison with the Fixed-Point Smallness Condition

We now show explicitly that the preceding theory is not merely a reformulation of a contraction argument. The first example is deliberately elementary so that every quantity can be calculated exactly.

Example 1

(A parameter range not covered by Burton’s contraction condition). Consider

$$x^{\prime }\left(t\right)=-c{\int }_{t-r}^{t}x\left(s\right)ds,c>0,r>0.$$

(45)

Here

$$a(t,s)=c,g\left(x\right)=x,p=0,f=0.$$

Consequently,

$$B\left(t\right)=cr,K\left(t\right)=cr,$$

and

$$J\left(t\right)={\int }_{t-r}^{t}c(t-s)ds={\displaystyle \frac{c{r}^2}{2}}.$$

We may take $m=L=1$. Hence

$${\alpha }_0=cr$$

and

$${\beta }_0=L{J}_0\left(L{K}_0\right)={\displaystyle \frac{c{r}^2}{2}}\left(cr\right)={\displaystyle \frac{c^2{r}^3}{2}}.$$

The condition ${\alpha }_0>{\beta }_0$ becomes

$$cr>{\displaystyle \frac{c^2{r}^3}{2}},$$

or equivalently

$$c{r}^2<2.$$

(46)

Therefore Theorem 2 proves exponential asymptotic stability whenever $c{r}^2<2$. We next compare this with Burton’s contraction condition. For the convolution kernel $a\left(v\right)=c$, where v denotes the lag variable, Burton’s integrated kernel is

$$A(t,u)={\int }_{t-u}^{r}a\left(v\right)dv,t-r\le u\le t.$$

Since $a\left(v\right)=c$, this gives

$$A(t,u)={\int }_{t-u}^{r}cdv=c(r-t+u).$$

Notice that $0\le t-u\le r$ whenever $t-r\le u\le t$. Consequently,

$$A(t,u)\ge 0,$$

and hence

$$\begin{array}{cc}\hfill {\int }_{t-r}^t\left|A(t,u)\right|du& =c{\int }_{t-r}^t(r-t+u)du\hfill \\ & ={\displaystyle \frac{c{r}^2}{2}}.\hfill \end{array}$$

Therefore Burton’s contraction requirement

$$2\underset{t\ge 0}{\sup }{\int }_{t-r}^t\left|A(t,u)\right|du<1$$

reduces to

$$\begin{array}{|c|}\hline c{r}^2<1\\ \hline\end{array}.$$

For instance, take $r=1$ and $c=3/2$. Then

$$c{r}^2={\displaystyle \frac{3}{2}}.$$

Burton’s condition fails because $3/2>1$, whereas our criterion holds because $3/2<2$. Thus the zero solution is exponentially asymptotically stable by Theorem 2.

It is instructive to compare both sufficient conditions with the exact linear boundary. Seeking a solution $x\left(t\right)=e^{\lambda t}$ in (45) gives

$$\lambda =-c{\displaystyle \frac{1-e^{-\lambda r}}{\lambda }},$$

or

$${\lambda }^2+c(1-e^{-\lambda r})=0.$$

(47)

At a nonzero imaginary crossing $\lambda =i\omega$, separation of real and imaginary parts yields

$$-{\omega }^2+c(1-\cos \omega r)=0,c\sin \omega r=0.$$

The first possible crossing has $\omega r=\pi$, and therefore

$$c={\displaystyle \frac{{\pi }^2}{2r^2}},c{r}^2={\displaystyle \frac{{\pi }^2}{2}}.$$

Thus Burton’s condition $c{r}^2<1$ is conservative, the present condition $c{r}^2<2$ is strictly less conservative, and the exact linear threshold lies still farther away at ${\pi }^2/2$. This comparison also indicates a natural direction for sharpening the Lyapunov estimate.

Example 2

(Variable delay without differentiating the delay). Consider

$$x^{\prime }\left(t\right)=-c{\int }_{t-r\left(t\right)}^{t}x\left(s\right)ds,0<r_{*}\le r\left(t\right)\le r_0,$$

(48)

where r is continuous but is not assumed differentiable. Then

$$B\left(t\right)=cr\left(t\right)\ge c{r}_{*},$$

$$K\left(t\right)=cr\left(t\right)\le c{r}_0,$$

and

$$J\left(t\right)={\displaystyle \frac{c{r}^2\left(t\right)}{2}}\le {\displaystyle \frac{c{r}_0^2}{2}}.$$

Corollary 2 gives exponential stability if

$$c{r}_{*}>{\displaystyle \frac{c{r}_0^2}{2}}\left(c{r}_0\right),$$

that is,

$$c{r}_0^3<2r_{*}.$$

(49)

No derivative $r^{\prime }\left(t\right)$ occurs in this condition. When $r_{*}=r_0=r$, (49) reduces to $c{r}^2<2$, as expected.

It is worth noting that the condition

$$c{r}_0^3<2r_{*}$$

becomes increasingly restrictive as $r_{*}\to 0^{+}$. This reflects the fact that the present uniform-gap criterion uses

$$B\left(t\right)=cr\left(t\right)\ge c{r}_{*}$$

as a uniform lower bound for the instantaneous dissipative contribution. Thus, as $r_{*}$ approaches zero, this lower bound degenerates and the criterion may cease to apply. This should not be interpreted as an instability result when the delay becomes small or vanishes; rather, it indicates a limitation of the uniform dissipation estimate used in this example.

Example 3

(Nonlinear damping and a state perturbation). Consider

$$x^{\prime }\left(t\right)=-c{\int }_{t-r\left(t\right)}^t\left(x\left(s\right)+\gamma \tanh x\left(s\right)\right)ds+\rho x\left(t\right),$$

(50)

where $c>0$, $\gamma \ge 0$, and $0\le \rho <\infty$. Let

$$g\left(x\right)=x+\gamma \tanh x.$$

Since

$$xg\left(x\right)=x^2+\gamma x\tanh x\ge x^2,$$

we may take $m=1$. Also

$$g^{\prime }\left(x\right)=1+\gamma {sech}^{2}x,$$

so g is globally Lipschitz with $L=1+\gamma$. If

$$0<r_{*}\le r\left(t\right)\le r_0,$$

then

$$B_0=c{r}_{*},K_0=c{r}_0,J_0={\displaystyle \frac{c{r}_0^2}{2}}.$$

Corollary 2 therefore gives global exponential stability whenever

$$c{r}_{*}-\rho >(1+\gamma ){\displaystyle \frac{c{r}_0^2}{2}}\left((1+\gamma )c{r}_0+\rho \right).$$

(51)

This example illustrates how nonlinear damping, variable delay, and a destabilizing state perturbation can all be handled by the same dissipation-versus-memory inequality.

Example 4

(Vanishing external forcing). Consider

$$x^{\prime }\left(t\right)=-c{\int }_{t-r}^{t}x\left(s\right)ds+F_0{e}^{-\sigma t},\sigma >0.$$

If $c{r}^2<2$, Theorem 6 yields

$$x\left(t\right)\to 0.$$

If the forcing is replaced by a bounded nonvanishing function f with $\left|f\right(t\left)\right|\le F$, Theorem 5 instead gives an explicit ultimate bound. Thus the same stability mechanism is robust under persistent forcing and recovers convergence when the forcing disappears asymptotically.

8. Discussion and Further Directions

The principal idea of this paper is to replace a uniform contraction requirement by a comparison between instantaneous dissipation and the effective size of the memory remainder. For the variable-delay equation (3), the decomposition

$$x^{\prime }\left(t\right)=-B\left(t\right)g\left(x\left(t\right)\right)+R\left(t\right)+p(t,x\left(t\right))+f\left(t\right)$$

identifies the averaged coefficient $B\left(t\right)$ as the source of damping and the first kernel moment $J\left(t\right)$ as the quantity governing the error created by replacing past states by the present state. This leads naturally to the criterion

$$mB\left(t\right)-\rho \left(t\right)>LJ\left(t\right)\left[L\widehat{K}\left(t\right)+\widehat{\rho }\left(t\right)\right],$$

in a uniform or suitably averaged form.

Several aspects of the theory are worth emphasizing. First, the delay may vary with time, and the proofs do not differentiate $r\left(t\right)$. Second, the nonlinear term need not be treated as part of a globally contractive operator. Instead, its sign supplies dissipation through $xg\left(x\right)$ while its Lipschitz constant enters only in estimating the memory remainder. Third, perturbations and forcing are incorporated directly into the same differential inequality. Finally, the boundedness theorem shows that even when a uniform dissipation gap is unavailable, an integrable growth majorant can prevent solutions from becoming unbounded.

The constant-kernel example demonstrates a genuine gain over the fixed-point smallness condition. Burton’s criterion requires $c{r}^2<1$, whereas the present argument gives $c{r}^2<2$. Since the exact linear boundary is $c{r}^2={\pi }^2/2$, there remains room for improvement. One possible refinement is to replace the rough estimate

$$|x^{\prime }\left(u\right)|\le LK\left(u\right)M_2\left(t\right)$$

by an energy estimate retaining cancellation inside the distributed-delay term. Another is to introduce a Lyapunov functional containing a weighted double integral of the form

$${\int }_{t-r\left(t\right)}^t{\int }_s^t\omega (t,s,u){\left|x\left(u\right)\right|}^{2}duds,$$

chosen so that part of the memory remainder is absorbed rather than estimated absolutely. Such a refinement may move the sufficient threshold closer to the exact value ${\pi }^2/2$.

A second direction is the extension to systems

$$x^{\prime }\left(t\right)=-{\int }_{t-r\left(t\right)}^{t}A(t,s)G\left(x\left(s\right)\right)ds+P(t,x\left(t\right))+F\left(t\right),x\left(t\right)\in {\mathbb{R}}^n.$$

If there is a positive definite matrix P and constants $m,L>0$ such that

$$\langle Px,G\left(x\right)\rangle \ge {m\left|x\right|}^2,|G\left(x\right)-G\left(y\right)|\le L|x-y|,$$

then the scalar estimates developed here suggest a matrix-valued analogue based on $V\left(x\right)=x^{T}Px$. The kernel mass and first moment would be replaced by appropriate induced matrix norms. This extension should preserve the central dissipation-versus-memory structure.

A third direction is stochastic perturbation. For a stochastic distributed-delay equation, the same deterministic decomposition can be inserted into the infinitesimal generator of a quadratic Lyapunov function. The memory remainder can then be estimated by the same first-moment argument, while the diffusion contributes the usual trace term. This would connect the present deterministic theory with Lyapunov methods for stochastic equations with bounded variable delay.

Finally, the forcing results suggest weighted asymptotic theories in which both coefficients and forcing may be unbounded but controlled relative to a common nondecreasing weight. Such results would replace ordinary boundedness by w-boundedness and ordinary convergence by weighted convergence. The advantage of the present framework is that these extensions can be developed without changing the fundamental decomposition (5).

9. Conclusion

We have developed a Lyapunov and comparison framework for nonlinear distributed-delay equations with bounded variable delay, state perturbations, and external forcing. The method begins by separating the distributed-delay term into an instantaneous dissipative component and a memory remainder. The latter is controlled by the first absolute moment of the kernel, leading to a differential inequality of Halanay type. This yields boundedness under an integrable growth majorant, stability and exponential asymptotic stability under a dissipation-versus-memory gap, ultimate boundedness for bounded forcing, convergence for vanishing forcing, and weighted growth estimates for unbounded forcing.

The approach does not require differentiation of the delay and does not require the full solution operator to be a contraction. The explicit equation

$$x^{\prime }\left(t\right)=-c{\int }_{t-r}^{t}x\left(s\right)ds$$

shows the distinction clearly: the fixed-point contraction condition requires $c{r}^2<1$, whereas the present theorem applies for $c{r}^2<2$. The interval $1\le c{r}^2<2$ therefore provides a concrete class of equations covered by the Lyapunov/comparison theory but not by that contraction criterion.

The results suggest several further developments, especially sharper Lyapunov functionals, vector systems, stochastic equations, and weighted theories for unbounded perturbations. These directions retain the same central principle: stability should be determined by the balance between dissipation and memory, rather than by smallness of the entire operator.