Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 6, 1-21; http://www.math.u-szeged.hu/ejqtde/

Oscillation and global asymptotic stability of a neuronic equation with two delays

Hassan A. El-Morshedy ^{∗}

Department of Mathematics, Abha Teachers’ College, Abha P. O. Box 249, Saudi Arabia

                       (elmorshedy@yahoo.com)

B. M. El-Matary

Department of Mathematics, Damietta Faculty of Science, New Damietta 34517, Egypt

                      (bassantmarof@yahoo.com)

Abstract

In this paper we study the oscillatory and global asymptotic stability of a single neuron model with two delays and a general activation function. New sufficient conditions for the oscillation and nonoscillation of the model are given. We obtain both delay-dependent and delay-independent global asymptotic stability criteria. Some of our results are new even for models with one delay.

Keywords. neuronic equation; oscillation; stability; single neuron. AMS Subject Classifications. 34K11; 34K20; 34K60; 92B20.

1 Introduction

Delay differential equations have been used to describe the dynamics of a single neuron to take into account the processing time. Pakdaman et al [15] considered a neuron that has a delayed self-connection with weight a > 0 and delay τ. Implementing a decay rate λ in the model, they found that the neuron activation at time t; say x(t), follows the delay differential equation

\frac{dx(t)}{dt} = −λx(t) + K + af(x(t − τ)),  (1.1)

EJQTDE, 2008 No. 6, p. 1 where K is the constant input received by the neuron and the neuron transfer function f is defined by f(x) = \frac{1}{1+e^{−x}}.

∗Corresponding author. On leave from Department of Mathematics, Damietta Faculty of Science, New Damietta 34517, Egypt

In [4] the delay differential equation

\frac{dx(t)}{dt} = −x(t) + a tanh(x(t) − bx(t − τ) + C),  (1.2)

has been proposed to describe the behavior of the activation level x(t) of a single neuron which is capable of self-activation modulated by a dynamic threshold C with a single delay τ. In the absence of the threshold effect, equation (1.2) has the form

\frac{dx(t)}{dt} = −x(t) + a tanh(x(t) − bx(t − τ)),  (1.3)

Letting y(t) = x(t) − bx(t − τ) in (1.3). Then y satisfies the equation

\frac{dy(t)}{dt} = −y(t) + a tanh(y(t)) − ab tanh(y(t − τ)),  (1.4)

the stability and/or bifurcation analysis of (1.4) have been studied in [4, 16] and [13] but with more general activation function. It is also proved by [13, 14] that (1.4) is not capable of producing chaos violating the existence of chaos conjectured by [16]. Gopalsamy and Leung [4] proved that the unique equilibrium of (1.4) and hence of (1.3) is globally asymptotically stable if

a(1 − b) < 1 and a(1 + b) < 1

when a > 0 and b ≥ 0, which agrees with the findings of [13, 16]. El-Morshedy and Gopalsamy [2] improves the above condition by allowing the equality signs to be nonstrict. In fact, Theorem 3.1 in [2] is the best known absolute (delay-independent) global asymptotic stability criteria for (1.2).

Liao et al [10] considered a single neuron model with general activation function; namely,

\frac{dx(t)}{dt} = −x(t) + af(x(t) − bx(t − τ) + C),

They discussed the local stability as well as the existence of Hopf bifurcation under the assumption that f has a continuous third derivative.

Based on [8]; Györi and Hartung [5] investigate the stability character of the single neuron model

\frac{dx(t)}{dt} = −λx(t) + Af(x(t)) + Bf(x(t − τ)) + C,

where f(x) = 0.5(|x + 1| − |x − 1|).

As one may observe; all the above models contain only one delay. It has been demonstrated by [3, 7] that models of single neuron can contain many delays. In this work we investigate the oscillatory and asymptotic stability characters of a single neuron (neuronic) equation with two delays; namely,

\frac{dx(t)}{dt} = −λx(t) + af(x(t) − bx(t − τ) + cx(t − σ)), τ, σ > 0  (1.5)

where a, b, c ∈ R, λ > 0 and with each solution of (1.5) an initial function ϕ ∈ C[−l, 0] is associated where l = max{τ, σ}. For generality reasons we will not assume that f is a tanh-like function only. Instead, we assume that f is continuous on R, f(0) = 0 and satisfies some or all of the following conditions

(H1) 0 < \frac{f(x)}{x} < 1 for all x ≠ 0.

(H2) f is differentiable near zero with f′(0) = 1.

(H3) x\frac{d}{dx} \frac{f(x)}{x} < 0 for all x ≠ 0.

(H4) |f(x)| < 1 for all x ∈ R.

It can be seen that the substitution y(t) = x(t) − bx(t − τ) + cx(t − σ) transforms (1.5) into the equation,

\frac{dy(t)}{dt} = −λy(t) + af(y(t)) − abf(y(t − τ)) + acf(y(t − σ)), t ≥ l  (1.6)

In Section 2, we investigate the oscillatory character of (1.5). We say that a solution x(t) of (1.5) is nonoscillatory if it is eventually positive or eventually negative, otherwise x(t) is called oscillatory. Equation (1.5) is called oscillatory if all its solutions are oscillatory. If equation (1.5) has at least one nonoscillatory solution, then it is called nonoscillatory. The oscillation theory of the delay differential equations can be found in [3, 6]. In contrast with the stability of these equations, there are no absolute (delayindependent) oscillation criteria for first order delay differential equations. Although the oscillatory properties of models arising from many fields as mathematical biology is now completely characterized (see [3, 6] for more details), the oscillation of equations of the form (1.5) has not yet received the deserved attention. It seems that [2] is the only work on this type of equations.

The asymptotic behavior of the trivial solution of (1.5) will be considered in Section 3. Theorem 3.1 in [2] will be extended to (1.5) and interesting delay dependent global asymptotic stability criteria are obtained which are new even for the special case (1.2).

2 The Oscillatory Behavior

Suppose that x(t) is a solution of (1.5). Define a function M as follows:

M(t) = \{ \begin{matrix}\frac{f(x(t)−bx(t−τ)+cx(t−σ))}{x(t)−bx(t−τ)+cx(t−σ)}, & if x(t) − bx(t − τ) + cx(t − σ) ≠ 0, \\ 1, & if x(t) − bx(t − τ) + cx(t − σ) = 0, \end{matrix}

for all t ≥ 0 . If (H1), (H2) hold, then M is continuous on [0, ∞), 0 < M(t) ≤ 1 for all t ≥ 0 and

f(x(t) − bx(t − τ) + cx(t − σ)) = (x(t) − bx(t − τ) + cx(t − σ))M(t).

Thus equation (1.5) can be rewritten in the form

\frac{dx(t)}{dt} = (aM(t) − λ)x(t) − abM(t)x(t − τ) + acM(t)x(t − σ),  t ≥ 0.  (2.1)

It is not difficult to see that if x(t) is a solution of (2.1), then z(t) = x(t)e^{∫_{0}^{t}(λ−aM(s))ds} is a solution of the equation

\frac{dz(t)}{dt} + abM(t)e^{∫_{t−τ}^{t} (λ−aM(s))ds}z(t − τ) − acM(t)e^{∫_{t−σ}^{t}(λ−aM(s))ds}z(t − σ) = 0. (2.2)

We will see that (2.1) and (2.2) play a key rule in the proofs of most of our results.

Theorem 2.1 Assume that (H1), (H2) hold and

bc ≤ 0.  (2.3)

Equation (1.5) is nonoscillatory if one of the following conditions is satisfied: (i) ab ≤ 0. (ii) a(1 − b + c) > λ and (H4) holds.

Proof. Assume that x(t) is a solution of (1.5) with ϕ(s) > 0 for all s ∈ [−τ, 0]. Let t_{0} > 0 be such that x(t) > 0 for all t ∈ [0, t_{0}) and x(t_{0}) = 0. From (2.2) we obtain

\frac{d}{dt} (x(t)e^{∫_{0}^{t}(λ−aM(s))ds}) = −abM(t)e^{∫_{0}^{t}(λ−aM(s))ds}x(t − τ) + acM(t)e^{∫_{0}^{t}(λ−aM(s))ds}x(t − σ).

So if (i) holds, we get

\frac{d}{dt} (x(t)e^{∫_{0}^{t}(λ−aM(s))ds}) ≥ 0 for all t ∈ [0, t_{0}).

Integrating from 0 to t_{0},

x(t_{0})e^{∫_{0}^{t_{0}}(λ−aM(s))ds} ≥ x(0) > 0,

which is impossible since x(t_{0}) = 0. Therefore x(t) > 0 for all t ≥ 0; i.e., (1.5) is nonoscillatory.

Suppose that (ii) holds. Define a function F by

F (y) = −λy + af((1 − b + c)y), y ∈ R.

Then

\frac{dF (y)}{dy}|_{y=0} = −λ + a(1 − b + c)f′(0) = a(1 − b + c) − λ > 0.

Consequently, F (y) will be positive for all sufficiently small positive values of y. Since lim_{y→∞} F (y) = −∞, then there exists a positive value c such that F (c) = 0. Set x(t) = c, it follows that x(t) satisfies (1.5); i.e., (1.5) is nonoscillatory.

In view of the idea used in the proof of Theorem 2.1(ii), a more general form from it can be obtained by replacing (ii) by the following phrase:

The equation −λy + af(1 − b + c)y) = 0 has at least one nontrivial root.

In the next oscillation results we will make use of the following theorem which is adapted from [6, Corollary 3.4.1] concerning the oscillation of the equation

\frac{dx(t)}{dt} + p(t)x(t − τ) + q(t)x(t − σ) = 0,  t ≥ 0.  (2.4)

Theorem 2.2 Assume that τ, σ > 0 and p, q ∈ C ([0, ∞), R^{+}) such that lim inf_{t→∞} (p(t)+ q(t)) > 0. Then each of the following two conditions is sufficient for the oscillation of equation (2.4):

(a) lim inf_{t→∞} (τp(t) + σq(t)) > \frac{1}{e};

(b) lim inf_{t→∞} (p(t)q(t))^{\frac{1}{2}} (τ + σ) > \frac{1}{e}.

We refer here to the fact that Theorem 2.2 is a consequence of [6, Theorem 3.4.2] but we use it here since (a) and (b) are practically easier to apply than the original condition provided in [6, Theorem 3.4.2].

Theorem 2.3 Assume that (H1), (H2), (2.3) hold,

ab > 0, a ≤ λ,  (2.5)

and either one of the conditions

abτe^{(λ−a)τ} − acσe^{(λ−a)σ} > \frac{1}{e},

(2.6)

\sqrt{|bc|a^{2}}(τ + σ)e^{\frac{1}{2} (λ−a)(τ+σ)} > \frac{1}{e}

is satisfied. Then (1.5) is oscillatory.

Proof. To the contrary let us assume that (1.5) is nonoscillatory. Without loss of generality one can assume that (2.1) has a solution x(t) such that x(t) > 0, t ≥ t_{0} for some t_{0} ≥ 0. Recalling that M(t) ≤ 1, we find from (2.1), (2.5) that

\frac{dx(t)}{dt} < 0 for all t ≥ t_{0} + l.

Thus, there exists a real number μ ≥ 0 such that lim_{t→∞} x(t) = μ and

x(t) > μ for all t ≥ t_{0} + l.  (2.7)

It follows from (2.1) and (2.7) that

\frac{dx(t)}{dt} < (aM(t) − λ)μ − abM(t)μ + acM(t)μ
= ((aM(t) − λ) − abM(t) + acM(t))μ
= (a(1 − b + c)M(t) − λ)μ, t ≥ t_{0} + l.  (2.8)

Suppose that μ ≠ 0. Then

lim_{t→∞} M(t) = \{ \begin{matrix}\frac{f((1−b+c)μ)}{(1−b+c)μ}, & if b − c ≠ 1, \\ 1, & if b − c = 1. \end{matrix}

Taking into account that

a(1 − b + c) < \{ \begin{matrix}λ, & if a, b > 0, \\ 0, & if a, b < 0, \end{matrix}

we obtain

lim_{t→∞}(a(1 − b + c)M(t) − λ)μ = −β,

where

0 < β = \{ \begin{matrix}λμ − af((1 − b + c)μ), & if b − c ≠ 1, \\ λμ, & if b − c = 1. \end{matrix}

Thus (2.8) yields

lim sup_{t→∞} \frac{dx(t)}{dt} ≤ −β.

The last inequality leads to the existence of a constant ν > 0 such that \frac{dx(t)}{dt} < −ν for all sufficiently large t. Integrating the last inequality from a suitable large t (say t_{1} > t_{0} +l) to ∞, we get

lim_{t→∞} x(t) = −∞,

which is a contradiction. Therefore μ = 0; that is,

lim_{t→∞} x(t) = 0.

Consequently

lim_{t→∞} M(t) = lim_{t→∞} \frac{f(x(t) − bx(t − τ) + cx(t − σ))}{x(t) − bx(t − τ) + cx(t − σ)} = 1.  (2.9)

Set

z(t) = x(t)e^{∫_{t_{0}+τ}^{t} (λ−aM(s))ds}, t ≥ t_{0} + l.

Then it is easy to verify that z(t) is a positive solution of equation (2.2) or equivalently equation (2.4), where

0 < p(t) = abM(t)e^{∫_{t−τ}^{t} (λ−aM(s))ds},

and

0 < q(t) = −acM(t)e^{∫_{t−σ}^{t} (λ−aM(s))ds}.

Thus (2.2) is nonoscillatory. On the other hand (2.9) yields

lim_{t→∞} p(t) = abe^{(λ−a)τ} and lim_{t→∞} q(t) = −ace^{(λ−a)σ}.  (2.10)

From (2.6) and (2.10) we see that all conditions of Theorem 2.2 are satisfied and hence equation (2.2) is oscillatory which is a contradiction.

The following oscillation results deal with the case a > λ and hence complete, partially, Theorem 2.3.

Theorem 2.4 Assume that (2.3), (H1) and (H2) are satisfied. If either one of the inequalities of (2.6) holds, a > λ and

e^{(a−λ)τ} ≤ b,  (2.11)

then (1.5) is oscillatory.

Proof. Let equation (1.5) be nonoscillatory. One can assume that (1.5) has a solution x(t) > 0, t ≥ t_{0} ≥ 0. It follows from (2.2) that

\frac{d}{dt} (x(t)e^{∫_{t_{0}+l}^{t}(λ−aM(s))ds}) < 0, t ≥ t_{0} + l.

Integrating the above inequality from t − τ to t we obtain,

x(t)e^{∫_{t_{0}+l}^{t}(λ−aM(s))ds} < x(t − τ)e^{∫_{t_{0}+l}^{t−τ}(λ−aM(s))ds}, t ≥ t_{0} + l.

Then

x(t) < x(t − τ)e^{− ∫_{t−τ}^{t} (λ−aM(s))ds} < e^{(a−λ)τ}x(t − τ) ≤ bx(t − τ), t ≥ t_{0} + l.

That is

x(t) − bx(t − τ) ≤ 0, t ≥ t_{0} + l.

Also,

cx(t − σ) ≤ 0.

Hence

x(t) − bx(t − τ) + cx(t − σ) ≤ 0, t ≥ t_{0} + l.

It follows from (1.5) that

\frac{dx(t)}{dt} < −λx(t), t ≥ t_{0} + l.

Therefore,

\frac{d}{dt}(x(t)e^{λt}) < 0, t ≥ t_{0} + l,

which implies that lim_{t→∞} x(t) = 0. Now the proof can be completed as in the proof of Theorem 2.3.

Notice that (2.3) and (2.11) imply that c ≤ 0. This restriction is very important. In fact when c > 0, the conditions a > λ and (2.3) imply that ab ≤ 0 which leads to the nonoscillation of (1.5) according to Theorem 2.1.

Lemma 2.1 Assume that a ≠ 0 and (H4) holds. If x is any solution of (1.5), then there exists t_{0} ≥ 0 such that

\frac{−|a|}{λ} < x(t) < \frac{|a|}{λ},  t ≥ t_{0}.  (2.12)

Proof. From (1.5), we have

\frac{d}{dt}(x(t)e^{λt}) = ae^{λt}f(x(t) − bx(t − τ) + cx(t − σ)).

Since |f(x)| < 1 for all x ∈ R,

−|a|e^{λt} ≤ \frac{d}{dt}(x(t)e^{λt}) ≤ |a|e^{λt}, t ≥ 0.

Integrating the above inequality from s(≥ 0) to t, we obtain

−\frac{|a|}{λ}(e^{λt} − e^{λs}) ≤ x(t)e^{λt} − x(s)e^{λs} ≤ \frac{|a|}{λ}(e^{λt} − e^{λs}), for all t ≥ s ≥ 0.

The above inequality yields

e^{λt}(x(t) − \frac{|a|}{λ}) ≤ e^{λs}(x(s) − \frac{|a|}{λ}),  for all t ≥ s ≥ 0.  (2.13)

Therefore the function N, where N(t) = e^{λt}(x(t) − \frac{|a|}{λ}), is nonincreasing on [0, ∞). This implies that N(t) must be eventually of one sign. We claim that N(t) is eventually negative. Suppose not. Then there exists T ≥ 0 such that N(t) > 0 for all t ≥ T. But N(t) is nonincreasing, then it has a nonnegative finite limit as t → ∞. Also we have e^{λt} → ∞ as t → ∞. It follows from (2.13) that

lim_{t→∞} x(t) − \frac{|a|}{λ} = 0, i.e., lim_{t→∞} x(t) = \frac{|a|}{λ}.  (2.14)

By making use of (2.14), equation (1.5) yields

lim_{t→∞} \frac{dx(t)}{dt} = −|a| + af(\frac{|a|}{λ}((1 − b + c))) < −|a| + |a| = 0.  (2.15)

As in the proof of Theorem 2.3 we conclude from (2.15) that lim_{t→∞} x(t) = −∞ which contradicts (2.14). Thus N(t) must be eventually negative as claimed; i.e., there exists t_{0} ≥ 0 such that e^{λt}(x(t) − \frac{|a|}{λ}) < 0, t ≥ t_{0}. This inequality holds only if x(t) < \frac{|a|}{λ} for all t ≥ t_{0}. The left inequality of (2.12) can be proved similarly.

Theorem 2.5 Assume that b ≥ 0, −c ≥ 0 and (H1)-(H4) hold. If either one of the conditions

(abτe^{(λ−a)τ} − acσe^{(λ−a)σ})r > \frac{1}{e},

(2.16)

\sqrt{|bc|a^{2}}(τ + σ)e^{\frac{1}{2} (λ−a)(τ+σ)}r > \frac{1}{e}

is satisfied where r = min\{\frac{f( \frac{a}{λ})}{\frac{a}{λ}}, \frac{f( \frac{−a(b−c)}{λ})}{\frac{−a(b−c)}{λ}}\}, then (1.5) is oscillatory.

Proof. To the contrary let us assume that (1.5) has a solution x(t) such that x(t) > 0, for all t ≥ t_{0} ≥ 0. Then as in the previous proofs,

z(t) = x(t)e^{∫_{t_{0}+l}^{t} (λ−aM(s))ds}

is a positive solution of equation (2.2). Set u(t) = x(t) − bx(t − τ) + cx(t − σ), t ≥ t_{1} ≥ t_{0} + l, where t_{1} is so large that (2.12) is satisfied. Then

\frac{−a}{λ}(b − c) < u(t) < \frac{a}{λ}, t ≥ t_{1}.

This inequality and (H3) yield

M(t) > \{ \begin{matrix}r & if u(t) ≠ 0,  t ≥ t_{1}, \\ 1 & if u(t) = 0,  t ≥ t_{1}. \end{matrix}

But (H1) implies

1 > r.

Then

M(t) > r,  for all t ≥ t_{1}.  (2.17)

Defining the functions p, q as in the proof of Theorem 2.3 and using similar arguments, then (2.16) and (2.17) imply that all conditions of Theorem 2.2 are satisfied and hence equation (2.2) is oscillatory which is a contradiction.

3 The Asymptotic Behavior

Theorem 3.1 Assume that f is a nondecreasing function on R, (H1), (H4) holds and either

bc ≤ 0 and a(1 + b − c) ≤ λ,  a(1 − b + c) ≤ λ,  (3.1)

or

bc ≥ 0 and a(1 + b + c) ≤ λ,  a(1 − b − c) ≤ λ  (3.2)

is satisfied. Then all solutions of (1.5) satisfy that

lim_{t→∞} x(t) = 0.  (3.3)

Proof. We will prove the theorem when (3.1) holds. The proof when (3.2) holds is similar and will be omitted to avoid repetition. First we assume that x is a solution of (1.5). It follows from Lemma 2.1 that x is bounded. Therefore there exist L, S ∈ R such that

L = lim inf_{t→∞} x(t) ≤ S = lim sup_{t→∞} x(t).

Thus for any ε > 0 there exists t_{0} ≥ 0 such that

L − ε ≤ x(t) ≤ S + ε,  t ≥ t_{0}.  (3.4)

In view of the continuity of x one can choose two sequences {t_{n}}, {t̃_{n}} such that t_{n}, t̃_{n} → ∞ as n → ∞,

\frac{d}{dt}x(t_{n}) = 0,  lim_{t→∞} x(t_{n}) = S,  (3.5)

and

\frac{d}{dt}x(t̃_{n}) = 0,  lim_{t→∞} x(t̃_{n}) = L.  (3.6)

Assume, for the sake of contradiction, that x does not satisfy (3.3); i.e., L < S. First consider the case when a > 0, b > 0 and c < 0. Using (3.4), (3.5) and (1.5) we obtain

λx(t_{n}) ≤ af(x(t_{n}) − b(L − ε) + c(L − ε)),  t ≥ t_{0} + l.  (3.7)

Also (3.4), (3.6) and (1.5) imply

λx(t̃_{n}) ≥ af(x(t̃_{n}) − b(S + ε) + c(S + ε)),  t ≥ t_{0} + l.  (3.8)

Let n → ∞ in (3.7), (3.8). Then we obtain respectively that

λS ≤ af(S − b(L − ε) + c(L − ε)),

and

λL ≥ af(L − b(S + ε) + c(S + ε)).

Since ε is arbitrary small, we get

λS ≤ af(S − (b − c)L),  (3.9)

and

λL ≥ af(L − (b − c)S).  (3.10)

Assume that S ≤ 0. Then L < 0 which, by (3.10), yields

f(L − (b − c)S) < 0.

From (H1) we conclude that

f(L − (b − c)S) > (L − (b − c)S).

Thus (3.10) yields

λL > a(L − (b − c)S).  (3.11)

Hence,

L(λ − a) > −a(b − c)S.  (3.12)

But (3.1) leads to λ − a ≥ 0, then L > 0 which is a contradiction. If S > 0, from (3.9) we get

0 < λS ≤ af(S − (b − c)L).

It follows that

f(S − (b − c)L) > 0.

From (H1), we conclude that

f(S − (b − c)L) < S − (b − c)L.

Thus

(λ − a)S < (−ab + ac)L.  (3.13)

Which implies that L < 0. Therefore, we obtain (3.11) and (3.12). Combining (3.12) and (3.13), we get

S((λ − a) + a(c − b)) < L((λ − a) + a(c − b)),

or equivalently

(S − L)(λ − a(1 + b − c)) < 0,

which is impossible in view of (3.1) and the fact that S > L. This contradiction implies that L = S.

When a < 0, b < 0 and c > 0, similar arguments as in the above case, imply easily (3.7), (3.8) as well as their consequences (3.9), (3.10). If S ≤ 0, then L < 0 and hence (3.10) yields

f(L − (b − c)S) > 0.

From (H1) and (3.10), we conclude that

λL > a(L − (b − c)S).  (3.14)

Hence,

(λ − a)L > −a(b − c)S ≥ 0.

Thus L > 0, which is contradiction. If S > 0, using similar arguments we obtain

0 < (λ − a)S < (−ab + ac)L,

which implies that L < 0. Therefore, (3.14) leads to

(λ − a)L > (−ab + ac)S.

From the previous two inequalities we get

(S − L)(λ − a(1 + b − c)) < 0

which is impossible in view of (3.1) and the fact that S > L. This contradiction implies that L = S.

When a < 0, c < 0 and b > 0, using similar arguments as before, we find

λx(t̃_{n}) ≥ af(x(t̃_{n}) − b(L − ε) + c(L − ε)).  (3.15)

Also

λx(t_{n}) ≤ af(x(t_{n}) − b(S + ε) + c(S + ε)).  (3.16)

Let n → ∞ in (3.15), (3.16). Then we obtain respectively that

λL ≥ af(L − b(L − ε) + c(L − ε)),  (3.17)

and

λS ≤ af(S − b(S − ε) + c(S − ε)).  (3.18)

Since ε is arbitrary small, we get

λL ≥ af(L(1 − b + c)),  (3.19)

and

λS ≤ af(S(1 − b + c)).  (3.20)

Suppose that S > 0 then (3.20) yields

f(S(1 − b + c)) < 0.

So (H1) and (3.20) imply that

λS ≤ af(S(1 − b + c)) < Sa(1 − b + c) ≤ λS.

which is impossible. Therefore S ≤ 0 and hence L < 0. From (3.19) we get

λL ≥ af(L(1 − b + c)) > La(1 − b + c) ≥ λL.

which is also impossible. Then L = S.

Consider now the last possible case; that is a > 0, c > 0 and b < 0,. The above reasoning implies similarly the inequalities (3.15)-(3.20). Suppose that S > 0. Then (3.20) yields

f(S(1 − b + c)) > 0.

Thus

λS ≤ af(S(1 − b + c)) < aS(1 − b + c) ≤ λS.

It follows that

λS < aS(1 − b + c) ≤ λS,

which is impossible. Consequently we have L < 0. So, in view of (3.19), we obtain

λL < aL(1 − b + c) ≤ λL,

which is a contradiction. Hence L = S.

Since the trivial solution is the unique equilibrium, due to the second inequality of (3.1), we get L = S = 0.

Lemma 3.1 If all solutions of (2.2) are bounded and λ > a > 0 or a ≤ 0, then the zero solution of (1.5) is globally exponentially stable.

Proof. We know that

z(t) = x(t)e^{∫_{0}^{t}(λ−aM(s))ds}, t ≥ 0.  (3.21)

is a solution of (2.2), for t ≥ l, if x(t) is a solution of equation (1.5). Since

e^{∫_{0}^{t}(λ−aM(s))ds} ≥ \{ \begin{matrix}e^{λt} & a ≤ 0, \\ e^{(λ−a)t} & a > 0. \end{matrix}

Then,

|x(t)| = |z(t)|e^{− ∫_{0}^{t}(λ−aM(s))ds} ≤ |z(t)| \{ \begin{matrix}e^{−λt}, & a ≤ 0 \\ e^{−(λ−a)t}, & λ > a > 0. \end{matrix}

Since all solutions of (2.2) are bounded, then there exists a constant A > 0 such that |z(t)| < A for all t ≥ 0. It follows that

|x(t)| ≤ A \{ \begin{matrix}e^{−λt}, & a ≤ 0 \\ e^{−(λ−a)t}, & λ > a > 0. \end{matrix}

for all t > 0. Then x(t) is exponentially stable.

Next we give some delay-dependent global asymptotic stability results. The first result is extracted from [11, Theorem 2.2] and [12, Theorem 2.2] for the equation

\frac{dy(t)}{dt} = −a(t)y(t) − ∑_{i=0}^{m} a_{i}(t)g_{i}(y(t − τ_{i})),  (3.22)

where as usual y(t) = ϕ(t) for all t ∈ [−l, 0], ϕ ∈ C([−l, 0], R) and l = max{τ_{i} : i = 0, 1, …, m}. Also, a, a_{i} are continuous bounded functions on [0, ∞) such that a(t) ≥ 0, a_{i}(t) ≥ 0 for all 0 ≤ i ≤ m, t ≥ 0 and ∑_{i=0}^{m} a_{i}(t) > 0, ∫_{0}^{∞} ∑_{i=0}^{m} a_{i}(t)dt = ∞. The functions g_{i} are continuous on R with the following property;

g_{i}(0) = 0 and 0 < \frac{g_{i}(x)}{x} ≤ 1, for x ≠ 0, and all 0 ≤ i ≤ m.

We use the following notations:

μ = e^{(inf_{t≥l} ∫_{t−l}^{t} a(s)ds)},

and

λ̄ = \{ \begin{matrix}sup_{t≥l} ∫_{t−l}^{t} ∑_{i=0}^{m} a_{i}(t) dt, & a(t) ≡ 0 \\ ^{\frac{exp (sup_{t≥l} ∫_{t−l}^{t} a(s)ds)−1}{inf_{t≥0} a(t)}}(sup_{t≥0} ∑_{i=0}^{m} a_{i}(t)), & inf_{t≥0} a(t) > 0. \end{matrix}

Now Muroyas’ results [11, Theorem 2.2] and [12, Theorem 2.2] can be joint into the following result.

Theorem 3.2 All Solutions of equation (3.22) are bounded from above and below if λ̄ ≤ \frac{1}{2} + μ, and the zero solution of (3.22) is globally asymptotically stable if λ̄ < \frac{1}{2} + μ.

The second result from literatures is the following one:

Theorem 3.3 [9] Consider the delay differential equation (2.4) where τ and σ are nonnegative constants and p, q are continuous functions satisfying the conditions

p(t) + q(t − τ + σ) ≠ 0,

for τ sufficiently large and

2 lim sup_{t→∞} | ∫_{t−τ}^{t−σ}|q(s + σ)|ds| + lim sup_{t→∞} ∫_{t−τ}^{t} |p(s) + q(s − τ + σ)|ds < 1.

Then every oscillatory solution of equation (2.4) tends to zero as t → ∞.

To be able to apply Theorem 3.2, we need the following lemma.

Lemma 3.2 Assume that x(t) is a solution of equation (1.5) and y(t) = x(t) − bx(t − τ) + cx(t − σ). If lim_{t→∞} y(t) = 0, then lim_{t→∞} x(t) = 0. Furthermore, if 0 < \frac{f(x)}{x} ≤ 1 and the trivial solution of (1.6) is stable, then the trivial solution of (1.5) is also stable.

Proof. From (1.5), we obtain

\frac{d(e^{λt}x(t))}{dt} = ae^{λt}f(y(t)).  (3.23)

Integrating from t − l to t and rearranging, we obtain

x(t) = e^{−λl}x(t − l) + ae^{−λt} ∫_{t−l}^{t} e^{λs}f(y(s))ds,

which yields

|x(t)| ≤ e^{−λl}|x(t − l)| + \frac{|a|}{λ}(1 − e^{−λl}) max_{t−l≤s≤t}\{|f(y(s))|\}, t ≥ l.  (3.24)

Since lim_{t→∞} max_{t−l≤s≤t}{|f(y(s))|} = 0, then for any ϵ > 0 there exists t_{ϵ} > l such that

|x(t)| ≤ (e^{−λl} + ϵ)|x(t − l)| + ϵ, for t ≥ t_{ϵ}.

Therefore lim_{t→∞} |x(t)| = 0 according to [6, Lemma 1.5.3].

Now if 0 < \frac{f(x)}{x} ≤ 1 and the trivial solution of (1.6) is stable, then for any ϵ_{1} > 0, there exists δ_{1} > 0 such that for the initial function ϕ_{y} associated with a solution y of (1.6) we have |y(t)| < ϵ_{1} for all t ≥ l when ||ϕ_{y}|| < δ_{1} where ||ϕ_{y}|| = max{ϕ_{y}(t) : 0 ≤ t ≤ l}. Let ϵ be an arbitrary positive number. Choose ϵ_{1} = \frac{λ}{|a|}ϵ and δ < δ_{1} < ϵ_{1}. Assume that ϕ_{x} is the initial function associated with a solution x of (1.5) such that y(t) = x(t) − bx(t − τ) + cx(t − σ). We claim that if ||ϕ_{x}|| = max{ϕ_{x}(t) : −l ≤ t ≤ 0} < δ < ϵ, then |x(t)| < ϵ for all t > 0. Suppose not, then there exists t_{0} > 0 such that |x(t)| < ϵ for all t < t_{0} and |x(t_{0})| = ϵ. Therefore, for t_{0} ≥ l, (3.24) yields

ϵ ≤ \frac{|a|}{λ} max_{t_{0}−l≤s≤t_{0}}\{|f(y(s)|\} ≤ \frac{|a|}{λ} max_{t_{0}−l≤s≤t_{0}}\{|y(s)|\} < ϵ,

which is impossible. When t_{0} < l, we see from (3.23) that

x(t_{0}) = e^{−λt_{0}}x(0) + ae^{−λt_{0}} ∫_{0}^{t_{0}} e^{λs}f(y(s))ds,

Then

ϵ ≤ \frac{|a|}{λ} max_{0≤s≤t_{0}}\{|f(y(s)|\} ≤ \frac{|a|}{λ} max_{0≤s≤t_{0}}\{|y(s)|\} < \frac{|a|}{λ}δ_{1} < ϵ,

which is also impossible. Thus we get our claim which means that the trivial solution of (1.5) is stable.

Now assume that m = 2, a(t) = λ, g_{i} = f for all i = 0, 1, 2, a_{0}(t) = −a, a_{1}(t) = ab and a_{2}(t) = −ac. Then, applying Theorem 3.2 on (1.6) and using Lemma 3.2, we obtain the following result.

Theorem 3.4 Assume that (H1), a < 0, b ≤ 0 and c ≥ 0 are satisfied. If

−a(1 − b + c)\frac{e^{lλ} − 1}{λ} < \frac{1}{2} + e^{lλ},

then the zero solution of (1.5) is globally asymptotically stable.

The next result can also be obtained using Theorem 3.2 and Lemma 3.2. In this case we assume that m = 3, a(t) = 0, g_{0}(x) = x, g_{i} = f for all i = 1, 2, 3, a_{0}(t) = λ, a_{1}(t) = −a, a_{2}(t) = ab, and a_{3}(t) = −ac.

Theorem 3.5 Assume that (H1), a < 0, b ≤ 0 and c ≥ 0 are satisfied. If

(λ − a(1 − b + c))l < \frac{3}{2},

then the zero solution of (1.5) is globally asymptotically stable.

As in the proofs of Theorem 2.3 and Lemma 3.1; any solution x(t) of (1.5) can be related to a solution y(t) of equation (2.4) where

p(t) = abM(t)e^{∫_{t−τ}^{t} (λ−aM(s))ds},

and

q(t) = −acM(t)e^{∫_{t−σ}^{t} (λ−aM(s))ds}.

If (H4) holds then the continuity of M and the boundedness of all solutions of (1.5) (according to Lemma 2.1) lead to the existence of a constant B > 0 such that M(t) > B for all t ≥ 0. If ab > 0, we have

p(t) > B \{ \begin{matrix}e^{(λ−a)τ}, & if λ ≥ a > 0, \\ e^{λτ}, & if a ≤ 0. \end{matrix}

Now when ab, −ac > 0, we have p(t) + q(t) > 0 for all t ≥ 0, ∫_{0}^{∞}(p(t) + q(t))dt = ∞ and

λ̄ ≤ \{ \begin{matrix}able^{(λ−a)τ} − acle^{(λ−a)σ}, & if a < 0, \\ able^{λτ} − acle^{λσ}, & if a ≥ 0. \end{matrix}

Applying Theorem 3.2 on equation (2.4) and using Lemma 3.1 we obtain the following result.

Theorem 3.6 Assume that (2.3), (H1), (H2), (H4) hold and ab > 0. Then the zero solution of (1.5) is globally exponentially stable if either one of the following conditions is satisfied:

\begin{matrix}able^{(λ−a)τ} − acle^{(λ−a)σ} ≤ \frac{3}{2}, & if a ≤ 0, \\ able^{λτ} − acle^{λσ} ≤ \frac{3}{2}, & if λ > a > 0. \end{matrix}

Now by making use of Theorem 3.3, we obtain the following result. Theorem 3.7 Assume that (H1), (H2), (2.3) and

ab > 0  (3.25)

are satisfied. If x(t) is any solution of (1.5), then lim_{t→∞} x(t) = 0 provided that either

abτe^{(λ−a)τ} − ace^{(λ−a)σ}(2|τ − σ| + τ) < 1, when a ≤ 0,

or

abτe^{λτ} − ace^{λσ}(2|τ − σ| + τ) < 1, when λ ≥ a > 0.

Proof. We know that (2.2) has the form (2.4) with

p(t) = abM(t)e^{∫_{t−τ}^{t} (λ−aM(s))ds}, q(t) = −acM(t)e^{∫_{t−σ}^{t} (λaM(s))ds}.

Thus the functions p and q are continuous and

p(t) + q(t − τ + σ) = abM(t)e^{∫_{t−τ}^{t} (λ−aM(s))ds} − acM(t − τ + σ)e^{∫_{t−τ}^{t−τ+σ}(λaM(s))ds} ≠ 0.

Taking into account that

λ − aM(s) < \{ \begin{matrix}λ − a, & if a ≤ 0, \\ λ, & if a > 0. \end{matrix}  (3.26)

From (3.26) we have,

2|q(s + σ)| ≤ 2 \{ \begin{matrix}−ace^{∫_{s}^{s+σ}(λ−a)ds}, & a ≤ 0, \\ −ace^{∫_{s}^{s+σ}λds}, & a > 0. \end{matrix}

= 2 \{ \begin{matrix}−ace^{(λ−a)σ}, & a ≤ 0, \\ −ace^{λσ}, & a > 0. \end{matrix}

Also,

|p(s) + q(s − τ + σ)| = |abM(s)e^{∫_{s−τ}^{s} (λ−aM(s))ds} − acM(s − τ + σ)e^{∫_{s−τ}^{s−τ+σ}(λ−aM(s))ds} |
≤ abe^{∫_{s−τ}^{s} (λ−aM(s))ds} − ace^{∫_{s−τ}^{s−τ+σ}(λ−a M(s))ds}.

Then

|p(s) + q(s − τ + σ)| ≤ \{ \begin{matrix}abe^{∫_{s−τ}^{s} (λ−a)ds} − ace^{∫_{s−τ}^{s−τ+σ}(λ−a)ds} & if a ≤ 0 \\ abe ^{∫_{s−τ}^{s} λds} − ace ^{∫_{s−τ}^{s−τ+σ} λds} & if a > 0. \end{matrix}

= \{ \begin{matrix}abe^{(λ−a)τ} − ace^{(λ−a)σ}, & if a ≤ 0 \\ abe^{λτ} − ace^{λσ}, & if a > 0. \end{matrix}

Thus

Λ < \{ \begin{matrix}abτe^{(λ−a)τ} − ace^{(λ−a)σ}(2|τ − σ| + τ) < 1, & if a ≤ 0 \\ abτe^{λτ} − ace^{λσ}(2|τ − σ| + τ) < 1, & if a > 0. \end{matrix}

where

Λ = 2 lim sup_{t→∞} | ∫_{t−τ}^{t−σ}| q(s + σ)|ds| + lim sup_{t→∞} | ∫_{t−τ}^{t} | p(s) + q(s − τ + σ)|ds|.

Then every oscillatory solution of equation (2.4) tends to zero as t → ∞ according to Theorem 3.3. Therefore every oscillatory solution of equation (1.5) tends to zero as t → ∞ according to Lemma 3.1. To complete the proof. We consider the case when equation (1.5) has a nonoscillatory solution, say x(t). As usual, we assume that x(t) > 0, t ≥ t_{0} for some t_{0} ≥ 0. Therefore

y(t) = x(t)e^{∫_{t_{0}+l}^{t}(λ−aM(s))ds} > 0.

It follows from equation (2.4) that

\frac{dy(t)}{dt} = \frac{d}{dt}(x(t)e^{∫_{t_{0}+l}^{t}(λ−aM(s))ds}) < 0, t ≥ t_{0} + l.

Then there is exists L ≥ 0 such that lim_{t→∞} y(t) = L. Thus x(t) is bounded and hence, as before, a number B > 0 exists such that M(t) > B for t ≥ t_{0} + l. Assume that σ ≥ τ. Then t − σ ≤ t − τ and

x(t − σ) > x(t − τ)e^{∫_{t−σ}^{t−τ}(λ−aM(s))ds}

If L > 0, there exists t′ ≥ t_{0} such that y(t) > L for all t ≥ t′ + l and hence equation (2.4) yields

\frac{dy(t)}{dt} = −abM(t)x(t − τ)e^{∫_{t_{0}+l}^{t}(λ−aM(s))ds} + acM(t)x(t − σ)e^{∫_{t_{0}+l}^{t}(λ−aM(s))ds},

≤ B(−abe^{∫_{t_{0}+l}^{t}(λ−aM(s))ds} + ace^{∫_{t_{0}+l}^{t}(λ−aM(s))ds}e^{∫_{t−σ}^{t−τ}(λ−aM(s))ds})x(t − τ),

= Be^{∫_{t−τ}^{t} (λ−aM(s))ds}(−ab + ace^{∫_{t−σ}^{t−τ}(λ−aM(s))ds})y(t − τ),

< BLe^{∫_{t−τ}^{t} (λ−aM(s))ds}(−ab + ace^{∫_{t−τ}^{t} (λ−aM(s))ds}).

Thus

\frac{dy(t)}{dt} < −K, for all t ≥ t^{‵} + l

where K = BL(ab − ac) > 0. Integrating the last inequality from t^{‵} + l to ∞ we get

lim_{t→∞} y(t) = −∞.

Which is a contradiction. Therefore L = 0.

When σ < τ, similar arguments leads to the above conclusion (L = 0). Thus lim_{t→∞} y(t) = 0 and lim_{t→∞} x(t) = 0; i.e., every nonoscillatory solution of equation (1.5) tends to zero as t → ∞.

Remark 3.1 It should be noted that there are many interesting linear stability criteria that can be applied here (see, e.g., [1, 17] and the references cited therein) but of course it is not possible to apply all these results due to space limitation.

We conclude our results with the following consequences of Theorems 3.4-3.6 (with c = 0, l = τ) on the single delay model (1.3). As far as the authors know these results are new.

Corollary 3.1 Assume that a < 0, b ≤ 0. Then the zero solution of (1.3) is globally asymptotically stable if either one of the following conditions is satisfied:

−a(1 − b)(e^{τ} − 1) < \frac{1}{2} + e^{τ},

(1 − a(1 − b))τ < \frac{3}{2}.

Corollary 3.2 Assume that ab > 0. Then the zero solution of (1.3) is globally exponentially stable if either one of the following conditions is satisfied:

\begin{matrix}abτe^{(1−a)τ} ≤ \frac{3}{2}, & if a ≤ 0, \\ abτe^{τ} ≤ \frac{3}{2}, & if 1 > a > 0. \end{matrix}

Acknowledgments. The authors would like to thank the referee for his/her careful reading and valuable comments.

References

[1] L. Berezansky and E. Braverman, On stability of some linear and nonlinear delay differential equations, J. Math. Anal. Appl. 314(2006), 391-411.

[2] H. A. El-Morshedy and K. Gopalsamy, On the Oscillation and Asymptotic Behaviour of Solutions of a Neuronic Equation, Funkcialaj Ekvacioj, 44(2001) 83-98.

[3] K. Gopalsamy, Stability and oscillation in delay differential equations of population dynamics, Kluwer Academic, Netherlands, 1992.

[4] K. Gopalsamy and I. K. C. Leung, Convergence under dynamical thresholds with delays, IEEE Trans. Neural Networks, 8(1997), 341-348.

[5] I. Györi and F. Hartung, Stability of a single neuron model with delay, J. Comput. Appl. Math. 157(2003), 73-92.

[6] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.

[7] Y. Hamaya and A. Redkina, On global asymptotic stability of nonlinear stochastic difference equations with delays, International Journal of Difference Equations, 1(2006), 101-118.

[8] U. an der Heiden, M. C. Mackey, H. O. Walther, Complex oscillations in a simple deterministic neuronal network, Lectures in Appl. Math. 19(1981), 355-360.

[9] G. Ladas and Y. G. Sficas, Asymptotic behaviour of oscillatory solutions, Hiroshima Math. J. 18(1988), 351-359.

[10] X. Liao, K.-W. Wong, C.-S. Leung and Z. Wu, Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function, Chaos, Solitons & Fractals, 12(2001), 1535-1547.

[11] Y. Muroya, Global stability for separable nonlinear delay differential equations, Comput. Math. Appl. 49(2005), 1913-1927.

[12] Y. Muroya, A global stability criterion in nonautonomous delay differential equations, J. Math. Anal. Appl. 326(2007), 209-227.

[13] K. Pakdaman and C. P. Malta, A note on convergence under dynamical threshold with delays, IEEE Trans. Neural Networks, 9(1998), 231-233.

[14] K. Pakdaman and C. P. Malta, Comment on ”Dynamics of some neural network models with delays”, Phys. Rev. E 66, 043901 (2002).

[15] K. Pakdaman, C. P. Malta, C. Grotta-Ragazzo and J.-F. Vibert, Effect of delay on the boundary of the basin of attraction in a self-excited single graded-response neuron, Neural Computa. 9(1997), 319-336.

[16] J. Ruan, L. Li and W. Lin, Dynamics of some neural network models with delay, Phys. Rev. E 63, 051906 (2001).

[17] B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal. 63(2005), e233-e242.

(Received May 9, 2007)