ON THE SIGN DEFINITENESS OF LIAPUNOV FUNCTIONALS AND STABILITY

OF A LINEAR DELAY EQUATION

L.B.Knyazhishche and V.A.Shcheglov

Institute of Mathematics National Academy of Sciences of Belarus Surganov str. 11, Minsk, 220072, Belarus e-mail: imanb@imanb.belpak.minsk.by

Abstract. In this paper we give a new definition of the positive-definiteness of the Liapunov functional involved in the stability and asymptotic stability investigation. Using this notion we prove Liapunov type theorems and apply these results to the scalar equation ẋ(t) = b(t)x(t − r(t)), where b(t) and r(t) may be unbounded.

AMS (MOS) subject classification 34A37, 34K20 Key words: stability, functional differential equations, Liapunov functionals

1. Introduction

This paper deals with stability of delay equations in terms of Liapunov functionals. We consider the system

ẋ(t) = f(t, x_{t}), f(t, 0) ≡ 0,  (1.1)

where x ∈ R^{n}, t ∈ R, f : R_{+} × C([−r(t), 0], R^{n}) → R^{n}, x_{t}(τ) = x(t + τ), τ ∈ [−r(t), 0], C([−r(t), 0], R^{n}) is the space of continuous functions on the interval [−r(t), 0] at any point t (see, for example, [6,17])

It is supposed that there exists a solution x(t, t_{0}, φ) of (1.1) with initial data (t_{0}, φ), t_{0} ∈ R, φ ∈ B_{t}(0, H) and it depends continuously on initial data. (B_{t}(0, H) =

{φ ∈ C([−r(t), 0], R^{n}), ||φ|| < H}, ||φ|| = max_{τ∈[−r(t),0]} |φ(τ)|).

Definition 1.1: The zero solution of (1.1) is called stable if for each ε > 0 and each t_{0} ≥ 0 there exists δ = δ(ε, t_{0}) > 0 such that ||φ|| < δ implies |x(t, t_{0}, φ)| < ε for all t ≥ t_{0}.

The zero solution is called uniformly stable if δ in Definition 1.1 does not depend on t_{0}.

Definition 1.2: The zero solution of (1.1) is called attractive if for each t_{0} ≥ 0 there exists ε_{1} = ε_{1}(t_{0}) such that ||φ|| < ε_{1} implies x(t, t_{0}, φ) → 0 as t → ∞.

The zero solution is called asymptotically stable if it is stable and attractive.

Let V : R_{+} × C → R (V (t, 0) ≡ 0) be a continuous, Lipschitzian with respect to the second argument functional. V̇ (t, φ) denotes the derivative of V (t, φ) with respect to (1.1).

Stability and asymptotic stability conditions have been studied in numerous papers [1]-[20]. It is well-known [6,17] that conditions

u(|φ(0)|) ≤ V (t, φ),  (1.2)
V̇ (t, φ) ≤ 0,  (1.3)

guarantee stability of the zero solution of (1.1), where u : R_{+} → R_{+} is continuous and u(α) > 0 for all α > 0, u(0) = 0 (functions with these properties are called Hahn functions).

To study asymptotic stability the following requirements were proposed in [8]

u(|φ(0)|) ≤ V (t, φ) ≤ v(||φ||),  (1.4)
V̇ (t, φ) ≤ −p(t)w(|φ(0)|),  (1.5)

where v, w are Hahn functions, p(t) is an integrally positive function, i.e.,

∑_{i=1}^{∞} ∫_{c_{i}}^{d_{i}} p(t)dt = ∞  (1.6)

holds for any set of intervals (c_{i}, d_{i}), i ∈ Z_{+} if d_{i} − c_{i} > σ > 0, c_{i+1} > d_{i}.

In fact, there are numerous papers and books (see,[1-4],[6],[8-10],[13-17]) dealing with improvements of requirements for V and V̇. The purpose of this paper is to weaken condition (1.2) and to study asymptotic stability for (1.1) with unbounded right hand side allowing that p(t) in (1.5) is vanishing at infinity and the second inequality in (1.4) is not fulfilled.

2. Stability

To investigate stability we need the following condition: for any t_{0} ≥ 0 there exists T > 0 such that t_{0} − r(t_{0}) ≤ t − r(t) for t > t_{0} + T.

Theorem 2.1: Suppose that for equation (1.1) there exist a Liapunov functional V (t, φ), a number H > 0 and Hahn functions u, v such that

(i) u(|φ(0)|) ≤ V (t, φ), provided that t ∈ R_{+} and φ satisfies |φ(0)| =

max_{τ∈[−r(t),0]} |φ(τ)|, ||φ|| < H;

(ii) V̇ (t, φ) ≤ 0, provided that t ∈ R_{+} and V (t, φ) > 0, ||φ|| < H; Then the zero solution of (1.1) is stable. If, in addition, r′(t) ≤ 1 and

(iii) V (t, φ) ≤ v(||φ||) for t ∈ R_{+}, ||φ|| < H; then the zero solution of (1.1) is uniformly stable.

Proof: Let us suppose that the equilibrium is not stable at some initial time t_{0}. Without loss of generality we can assume t_{0} = 0. Then there is ε > 0 such that there exists sequences {t_{n}}_{n=1}^{∞}, {φ_{n}}_{n=1}^{∞}, ||φ_{n}|| → 0 (n → ∞) such that |x(t_{n}, 0, φ_{n})| = ε. It is easy to see that the sequences {t_{n}}_{n=1}^{∞}, {φ_{n}}_{n=1}^{∞} can be chosen so that |x(t_{n}, 0, φ_{n})| < ε for all t ∈ [−r(0), t_{n}). Note that continuous dependence of the solutions on the initial data guarantees that t_{n} → ∞ if n → ∞. So we can choose n_{0} such that −r(0) < t − r(t) for t ≥ t_{n_{0}}. It is obvious that |x(t_{n}, 0, φ_{n})| ≤ max_{θ∈[−r(t_{n}),0]} |x(t_{n} + θ, 0, φ_{n})| = ε for n > n_{0}. Hence, condition (i) guarantees that V (t_{n}, x_{t_{n}}(φ_{n})) ≥ u(ε) > 0. Since ||φ_{n}|| → 0 if n → ∞ and the functional V is continuous we can indicate a sequence {α_{n}}_{n=1}^{∞} such that α_{n} > 0 V (0, φ_{n}) ≤ α_{n} and α_{n} → 0 if n → ∞. On the other hand, according to (ii), the functional V does not increase along the solution x(t, 0, φ_{n}) at any point t if V (t, x_{t}(φ_{n})) > 0. If there exist a point 0 < t′_{n} < t_{n} such that V (t′_{n}, x_{t′_{n}}(φ_{n})) ≤ 0 then V (t, x_{t}(φ_{n})) ≤ 0 for all t ∈ [t′_{n}, t_{n}] by (ii). So we can conclude that V (t_{n}, x_{t_{n}}(φ_{n})) ≤ α_{n} → 0 if n → ∞. Obtained contradiction proves the stability.

Uniform stability can be proved in a similar way.

Remark 2.1: Condition (i) extends the class of functionals which are admissible for stability investigation because functionals with sign changes can be used (see also [5]).

3. Asymptotic stability

Let q : R_{+} → R_{+} be a continuous function and let I be a set of the form

I = ⋃_{i=1}^{∞}(c_{i}, d_{i}), c_{i} > d_{i}, d_{i} → ∞, if i → ∞

and there is σ > 0 such that ∫_{c_{i}}^{d_{i}} q(t)dt > σ for any i ∈ Z_{+}

Definition 3.1: A function p : R_{+} → R_{+} is said to be I — integrally positive with respect to q(t) if for any σ′ > 0, k whenever I′ = ⋃_{i=k}^{∞}(c′, d′) with

c_{i} < c′_{i}, d′_{i} < d_{i}, ∫_{c′_{i}}^{d′_{i}} q(t)dt > σ′, then ∫_{I′} p(t)dt = ∞.

A function p(t) is called I-integrally positive if q(t) in Definition 3.1 is a constant function (see [12] for ordinary differential equations).

Definition 3.2: A delay function r(t) is said to be embedded into I if there are points t_{i} ∈ (c_{i}, d_{i}) such that t_{i} − r(t_{i}) ∈ (c_{i}, d_{i}) for all i ∈ Z_{+}.

Note that the class of the functions I-integrally positive with respect to q(t) is wider then that of integrally positive functions, and includes, for example, functions which decrease like 1/t on some appropriate sequence of intervals and have arbitrary behavior outside of these intervals.

In what follows the condition r′(t) ≤ δ < 1 is assumed. Let β be an increasing Hahn function and β(s) > s.

Theorem 3.1: If for equation (1.1) there are a Liapunov functional V (t, φ), a number H > 0, Hahn functions u, v, w and a set I such that:

(i) ||f(t, φ)|| ≤ F (t) for t ∈ I, ||φ|| ≤ H;

(ii) V (t, φ) ≤ v(||φ||) for t ∈ I, ||φ|| ≤ H;

(iii) u(|φ(0)|) ≤ V (t, φ), provided that t ∈ R_{+} and φ satisfies max_{τ∈[−r(t),0]} |φ(τ)| ≤

β|φ(0)|), ||φ|| ≤ H; (iv) V̇ (t, φ) ≤ −p(t)w(|φ(0)|) provided that t ∈ R_{+}, V (t, φ) > 0, ||φ|| ≤ H;

(v) p(t) is I—integrally positive with respect to F (t);

(vi) r(t) is embedded into I; then the zero solution of (1.1) is asymptotically stable.

Proof: It remains to prove the attractivity. Without loss of generality we can consider the initial time t_{0} = 0. Let us take ε ∈ (0, H) and σ > 0 such that for every φ, ||φ|| < σ the inequality |x(t, φ)| ≤ ε is satisfied. Let S(t) = {φ : V (t, φ) ≤ 0, ||φ|| < ε}. First we investigate asymptotic behavior of the solutions x(t, φ), ||φ|| < σ which enter into S(t′) at some t′ and remain (by condition (iv)) in S(t) if t > t′. Assume that there is a solution x(t, φ) such that V (t′, x_{t′}) ≤ 0 for some t′, but x(t, φ) does not tend to zero when t → ∞. Then there are a sequence {t_{n}}, t_{n} → ∞ and 0 < ε_{1} < ε, for which |x(t_{n}, φ)| > ε_{1}. Take Δ > 0 such that β(s) − s > Δ for s > ε_{1}. Since (iii) holds and V (t_{n}, x_{t_{n}}) ≤ 0 we have that for every point t_{n} there is t_{n}^{1} such that t_{n} − r(t_{n}) ≤ t_{n}^{1} ≤ t_{n} and, besides, |x(t_{n}^{1}, φ)| > |x(t_{n}, φ)| + Δ. Also, for every t under sufficiently large n (namely, such that t_{n} − r(t_{n}) is sufficiently large) there is t_{n}^{2} such that t_{n}^{1} − r(t_{n}^{1}) ≤ t_{n}^{2} ≤ t_{n}^{1} and |x(t_{n}^{2}, φ)| > |x(t_{n}, φ)| + 2Δ. Thus, we have |x(t_{n}^{i}, φ)| > iΔ > ε for sufficiently large n and i, which is a contradiction showing that if |φ| < σ and x(t, φ) enters into S(t′) at some t′ then |x(t, φ)| → 0 if t → ∞.

Consider now the solutions of (1.1) which remain in the set S_{+}(t) = {φ : V (t, φ) > 0, ||φ|| < ε}. It follows from condition (iv) of the theorem that the functional V (t, x_{t}) is not increasing for any such solution x(t, φ) because V (t, x_{t}) > 0 if t ≥ 0.

We prove that V (t, x_{t}) → 0 if t → ∞. Choose \overline{t}_{i} ∈ (c_{i}, d_{i}) such that \overline{t}_{i} − r(\overline{t}_{i}) ∈ (c_{i}, d_{i}). First we show that there exist intervals (\overline{t}_{i_{k}} − r(\overline{t}_{i_{k}}), \overline{t}_{i_{k}}) such that |x(t, φ)| → 0 if t ∈ ⋃_{k=1}^{∞}(\overline{t}_{i_{k}} − r(\overline{t}_{i_{k}}), \overline{t}_{i_{k}}). Suppose the contrary. Then there

exists \overline{ε} > 0 such that for every n ∈ Z_{+} there is a point t̂_{n} such that |x(t̂_{n}, φ)| > \overline{ε} and t̂_{n} ∈ (\overline{t}_{n} − r(\overline{t}_{n}), \overline{t}_{n}). Let (c′_{n}, d′_{n}) be the maximal connected interval such that |x(t, φ)| > \overline{ε}/2 for t ∈ (c′_{n}, d′_{n}) and c_{n} ≤ c′_{n}, d′_{n} ≤ d_{n}. From (i) and requirements for I it follows that ∫_{c′_{n}}^{d′_{n}} F (t)dt > min\{\overline{ε}/2, σ\}, where σ is taken from the description of I.

Since (iv) holds and p(t) is I-integrally positive with respect to F (t) we have

∑_{n=1}^{∞} ∫_{c′_{n}}^{d′_{n}} V̇ (t, x_{t})dt = −∞.

This contradicts the assumption that V (t, x_{t}) > 0 if t > 0.

Thus it is proved that ||x_{\overline{t}_{i_{k}}} || → 0 when k → ∞ at least for some subsequence \{\overline{t}_{i_{k}}\} of the sequence \{\overline{t}_{i}\}. From condition (ii) of the theorem it follows that V (\overline{t}_{i_{k}}, x_{\overline{t}_{i_{k}}}) → 0 if k → ∞. Hence, V (t, x_{t}) → 0 when t → ∞.

Suppose that |x(t, φ)| does not tend to zero if t → ∞. Then, taking into account that ||x_{\overline{t}_{i_{k}}} || → 0 when k → ∞, it is easy to see that there are ε_{1} > 0 and points t_{n} → ∞, n → ∞ such that |x(t_{n}, φ)| = ε_{1} and |x(t, φ)| < ε_{1} for t ∈ [t_{n} − r(t_{n}), t_{n}]. By (iii), V (t_{n}, x_{t_{n}}) does not tend to zero.

This contradiction shows that x(t, φ) → 0 if t → ∞. So asymptotic stability is proved.

In the sequel we construct Liapunov functionals meeting conditions in Theorem 3.1.

4. The equation ẋ(t) = ax(t) + bx(t − r).

We shall consider the scalar linear equation

ẋ(t) = a(t)x(t) + b(t)x(t − r(t)), t > 0.  (4.1)

For the sake of simplicity, at first we are dealing with the case when a(t) ≡ a, b(t) ≡ b, r(t) ≡ r. So (4.1) looks like

ẋ(t) = ax(t) + bx(t − r), t > 0.  (4.2)

The region of asymptotic stability for (4.2) is well-known [6]. This result was obtained by the method of the characteristic equation. On the other hand, the way for obtaining a near result with a Liapunov functional was given in [6]. However, in the process of constructing one must find a solution of a similar delay equation. This does not allow to generate the scheme for more complicated equations. We show that the use of condition (i) of Theorem 2.1 overcomes this difficulty.

Usually, V has the form

V (t, x_{t}) = x^{2}(t) + ∫_{t−r}^{t} K(u)x^{2}(u)du, K(u) ≥ 0,

which immediately provides (1.2) and condition (i) of Theorem 2.1 but the stability conditions are far from the best. There is also considered (see, for example, [1]-[3]) functional V (t, x_{t}) in the form

V (t, x_{t}) = V_{1}(t, x_{t}) + V_{2}(t, x_{t}),

where

V_{1}(t, x_{t}) = \frac{1}{2}(x(t) + ∫_{t−r}^{t} K(u)x(u)du)^{2}

Since

(∫_{t−r}^{t} K(u)x(u)du)^{2} ≤ ∫_{t−r}^{t} |K(u)|du ∫_{t−r}^{t} |K(u)|x^{2}(u)du

where the function K(u) is not necessarily of a constant sign, V_{1}(t, x_{t}) does not satisfy (1.2), in general. So the additional term

V_{2}(t, x_{t}) = ∫_{−r}^{0} ∫_{t+s}^{t} D(s, u, x_{u})duds, D(s, u, x_{u}) ≥ 0,

is used to satisfy (1.2). To this end strict relations between D and K have to be required.

Now we suggest the condition

sup_{t≥0} ∫_{t−r}^{t} |K(t, u)|du < 1,  (4.3)

guaranteeing that V_{1}(t, x_{t}) ≥ u(|x(t)|) for any x(t) such that |x(t)| = max_{t−r≤u≤t} |x(u)|,

i.e. condition (i) of Theorem 2.1, is fulfilled without any relation between K and D.

We have

V̇_{1}(t, x_{t}) = (x(t) + ∫_{t−r}^{t} K(t, u)x(u)du)

(∫_{t−r}^{t} \frac{∂K(t, u)}{∂t}x(u)du + (a + K(t, t))x(t) + (b − K(t, t − r))x(t − r))

Let K(t, t − r) = b. To have V̇ (t, x_{t}) as a quadratic of x(t) and ∫_{t−r}^{t} K(t, u)x(u)du we require that the function K(t, u) is a solution of the equation

\frac{∂K(t, u)}{∂t} = ϕ(t)K(t, u),

where ϕ(t) is still undefined function.

For the case of constant coefficients a, b, consider a constant function ϕ(t). Then

K(t, u) = be^{α(u−t+r)}, α − const.

Now (4.3) (i.e. condition (i) of Theorem 2.1) is guaranteed by the inequality |b|α^{−1}(e^{αr} − 1) < 1. This is satisfied if α < 0 and

|b| < \frac{|α|}{1 − e^{−|α|r}}  (4.4)

Now we try to choose the best α. Denote

x_{1}(t) = x_{1}(t, x_{t}) = b ∫_{t−r}^{t} e^{α(u−t+r)}x(u)du, a_{1} = a + be^{αr}.

Then

V̇_{1}(t, x_{t}) = a_{1}x^{2}(t) + (a_{1} − α)x(t)x_{1}(t) − αx_{1}^{2}(t).

To ensure the condition V̇ (t, x_{t}) < 0 let us take

D(s, u, x_{u}) = pe^{αs}x^{2}(u),

where p > 0 is constant. Then

V̇_{2}(t, x_{t}) = px^{2}(t) ∫_{−r}^{0} e^{αs}ds − pe^{αt} ∫_{t−r}^{t} e^{αu}x^{2}(u)du

By the Cauchy - Schwarz integral inequality we have

∫_{t−r}^{t} e^{αu}x^{2}(u)du ≥ \frac{(∫_{t−r}^{t} e^{αu}x(u)du)^{2}}{∫_{t−r}^{t} e^{αu}du} = \frac{e^{αt−2αr}}{b^{2} ∫_{−r}^{0} e^{αs}ds}x_{1}^{2}(t),

whence

V̇_{2}(t, x_{t}) ≤ \frac{p(1 − e^{−αr})}{α}x^{2}(t) − \frac{pαe^{−2αr}}{b^{2}(1 − e^{−αr})}x_{1}^{2}(t),

V̇ (t, x_{t}) ≤ (a_{1} + \frac{p(1 − e^{−αr})}{α})x^{2}(t) + (a_{1} − α)x(t)x_{1}(t) − α(1 + \frac{pe^{−2αr}}{b^{2}(1 − e^{−αr})})x_{1}^{2}(t)

Clearly, V̇ (t, x_{t}) ≤ 0 if there exists p such that

b^{2}(e^{αr} − e^{2αr}) ≤ p ≤ \frac{αa_{1}}{e^{−αr} − 1},  (4.5)

and

b^{2}(e^{αr} − 1) + \frac{αa_{1}}{e^{αr} − 1} − |b|(a_{1} + α) ≤ 0,  (4.6)

By virtue of (4.5) and (4.6), such a p exists if

a_{1} ≤ 0, b^{2}(e^{αr} − e^{2αr}) ≤ \frac{αa_{1}}{e^{−αr} − 1},  (4.7)

and

b^{2}(e^{αr} − 1) + \frac{αa_{1}}{e^{αr} − 1} − |b|(a_{1} + α) ≤ 0,  (4.8)

Inequality (4.8) may be reduced to

(b + a)(−b + \frac{α}{e^{αr} − 1}) ≤ 0 for b > 0

and

(b(2e^{αr} − 1) + a)(b + \frac{α}{e^{αr} − 1}) ≤ 0 for b ≤ 0.

Analyzing above inequalities one can see that conditions (4.4),(4.7), (4.8) hold everywhere on the domain −a ≥ |b| if |α| is taken sufficiently large. In case when −b ≥ |a|, requirements (4.7),(4.8) are satisfied if α = \frac{1}{2}(a + b) and (4.4) holds for the pair (a, b).

Remark 4.1: It should be noted that the functional V (t, x_{t}) satisfies (1.2). In fact, we have in the above example

V (t, x_{t}) = (x(t) + b ∫_{t−r}^{t} e^{α(u−t+r)}x(u)du)^{2} + ∫_{−r}^{0} ∫_{t+s}^{t} ke^{αs}x^{2}(u)duds

We will show that V (t, x_{t}) → 0 implies x(t) → 0. If

∫_{−r}^{0} ∫_{t+s}^{t} D(s, u, x_{u})dsdu = ∫_{−r}^{0} ∫_{t+s}^{t} ke^{αs}x^{2}(u)duds → 0  (4.9)

then

x_{1}(t) = b ∫_{t+s}^{t} e^{α(u−t+r)}x(u)du → 0  (4.10)

In fact, since k, e^{α(s)}, x^{n}(u) are bounded, (4.9) implies

∫_{t−r}^{t} ke^{−αr}(x^{n}(u))^{2}du → 0  (4.11)

Using (4.11) and the Cauchy-Schwarz inequality it is easy to see that (4.10) is valid.

Note that the main step here is that (4.9) implies (4.11). If coefficients of the delay equation are not constant (unbounded, in general) we have to choose k as an unbounded, in general, function. In such a case (4.9) does not imply (4.11) and (1.2) is not true. This situation is realized, for example, if a ≡ 0, b depends on t and

lim sup_{t→∞} b(t) = ∞.

5. The equation ẋ(t) = b(t)x(t − r)

Let us consider the equation

ẋ(t) = b(t)x(t − r)  (5.1)

and modify the form of V_{1} and V_{2}. We set V_{1}(t, x_{t}) = \frac{1}{2}(x(t) + x_{1}(t))^{2}, where

x_{1}(t) = e^{−α(t−r)} ∫_{t−r}^{t} b(u + r)e^{α(u)}x(u)du,

and the function α(t) is defined by

α(t) = c ∫_{0}^{t+r} b(u + r)du, b(t) ≤ 0, c > 0  (5.2)

Denote B(t) = ∫_{t}^{t+r} b(u + r)du. We have

V̇_{1}(t, x_{t}) = b(t + r)(e^{cB(t)}x^{2}(t) + (e^{cB(t)} − c)x(t)x_{1}(t) − cx^{2}(t)).

Let

V_{2}(t, x_{t}) = p ∫_{−r}^{0} ∫_{t+s}^{t} b(|s − u| + r)e^{−α(|s−u|−r)}b(u + r)e^{2α(u)−α(u−r)}x^{2}(u)duds

Then

V̇_{2}(t, x_{t}) = p(x^{2}(t)b(t + r)e^{α(t)+cB(t)} ∫_{−r}^{0} b(|s − t| + r)e^{−α(|s−t|−r)}−

b(t + r)e^{−α(t−r)} ∫_{t−r}^{t} b(u + r)e^{α(u)+cB(u)}x^{2}(u)du) ≤

pb(t + r)(x^{2}(t)e^{α(t)+cB(t)} ∫_{t}^{t+r} b(u + r)e^{−α(u−r)}du+

\frac{e^{α(t−r)}}{∫_{t−r}^{t} |b(u + r)|e^{α(u−r)}du}(∫_{t−r}^{t} b(u + r)e^{α(u)}x(u)du)^{2}) =

pb(t + r)(\frac{1}{c}e^{cB(t)}(e^{cB(t)} − 1)x^{2}(t) + \frac{c}{e^{−cB(t−r)} − 1}x_{1}^{2}(t)).

Suppose that function B(t) satisfies the inequality

b_{s} = sup_{t≥−r} |B(t)| = sup_{t≥−r} ∫_{t}^{t+r} |b(u + r)|du < ∞  (5.3)

Then

V̇ (t, x_{t}) ≤ b(t + r)((e^{−cb_{s}} + \frac{p}{c}e^{−cb_{s}}(e^{−cb_{s}} − 1))x^{2}(t)+

(e^{−cb_{s}} − c)x(t)x_{1}(t) + c(\frac{pe^{−cb_{s}}}{1 − e^{−cb_{s}}} − 1)x_{1}^{2}(t)).  (5.4)

Evidently, V̇ (t, x_{t}) ≤ 0 if there exists p such that

e^{cb_{s}} − 1 ≤ p ≤ \frac{c}{1 − e^{−cb_{s}}}  (5.5)

and

4e^{−2cb_{s}}p^{2} − 4pe^{−cb_{s}}(\frac{c}{e^{cb_{s}} − 1} + 1 − e^{−cb_{s}}) + (c + e^{−cb_{s}})^{2} ≤ 0  (5.6)

Inequalities (5.5), (5.6) hold with some p if the estimate b_{s} ≤ ln 4 is true and c = \frac{1}{2}. Everywhere below (see (5.2)) we set c = \frac{1}{2}.

Theorem 5.1: Suppose that b(t) ≤ 0,

sup_{t≥0} ∫_{t−r}^{t} |b(u + r)|du ≤ ln 4.

and

sup_{t≥0} ∫_{t−r}^{t} |b(u + r)|e^{α(u)−α(t−r)}du < 1,  (5.7)

Then the zero solution of (5.1) is uniformly stable.

Remark 5.1: Condition (5.7) is sufficient for condition (i) of Theorem 2.1 (see (4.3)).

Let us find sufficient conditions for the asymptotic stability using Theorem 3.1. Show, at first, that the functional V satisfies condition (ii) of Theorem 3.1. In fact,

V_{1}(t, x_{t}) ≤ \frac{1}{2}||x_{t}||^{2}(1 + ∫_{t−r}^{t} |b(u + r)|e^{α(u)−α(t−r)}du)^{2} ≤

\frac{1}{2}||x_{t}||^{2}(1 + ∫_{t−r}^{t} |b(u + r)|du)^{2} ≤ \frac{(1 + b_{s})^{2}}{2} ||x_{t}||^{2},

V_{2}(t, x_{t}) ≤ pb_{s}^{2}e^{\frac{bs}{2}} ||x_{t}||^{2}.

Furthermore, it is easy to see that (5.7) implies condition (iii) with β(s) = s(1 + δ_{1})/(2δ_{1}), where δ_{1} is chosen from the inequality

sup_{t≥0} ∫_{t−r}^{t} |b(u + r)|e^{α(u)−α(t−r)}du < δ_{1} < 1

Besides, from (5.3)-(5.6) we can conclude that b_{s} < ln 4 implies the existence of a constant c_{1} > 0 such that

V̇ (t, x_{t}) ≤ c_{1}b(t + r)x^{2}(t).

Thus we have proved the following

Theorem 5.2: Suppose that all requirements of Theorem 5.1 are satisfied and there is a set I such that r is embedded into I and |b(t + r)| is I-integrally positive with respect to |b(t)|. Then the zero solution of (5.1) is asymptotically stable.

Note here that the estimate b_{s} ≤ ln 4 is not the best one [19] and requirement (5.7) is not needed [2] if b(t) is bounded.

6. The equation ẋ(t) = b(t)x(t − r(t)).

The results can be extended to the case r = r(t). Consider the equation

ẋ(t) = b(t)x(t − r(t)),  (6.1)

where r′(t) ≤ δ < 1. Take again V_{1}(t, x_{t}) = (x(t) + x_{1}(t))^{2}, where

x_{1}(t) = e^{−α(t−r(t))} ∫_{t−r(t)}^{t} β(u)e^{α(u)}x(u)du,

β(t − r(t)) = \frac{b(t)}{1 − r′(t)}, α(t) = \frac{1}{2} ∫_{0}^{t} ξ(s)ds, ξ(t − r(t)) = \frac{β(t)}{1 − r′(t)}.

By the assumption on r′(t), the functions β, ξ are well defined and negative, provided b(t) < 0.

We have

V̇_{1}(t, x_{t}) = β(t)(e^{\frac{1}{2} ∫_{t−r(t)}^{t} ξ(τ)dτ}x^{2}(t) + (e^{\frac{1}{2} ∫_{t−r(t)}^{t} ξ(τ)dτ} − \frac{1}{2})x(t)x_{1}(t) − \frac{1}{2}x_{1}^{2}(t)),

∫_{−r(0)}^{t−r(t)} ξ(τ)dτ = ∫_{0}^{t} β(τ)dτ  (6.2)

Let

V_{2}(t, x_{t}) = p ∫_{t−r(t)}^{t} F_{1}(s) ∫_{s}^{t} F_{2}(u)x^{2}(u)duds,

where F_{1}(s) = ξ(s)e^{−α(s)}, F_{2}(u) = β(u)e^{2α(u)−α(u−r(u))}. Using (6.2) and previous integral estimates we have

V̇_{2}(t, x_{t}) ≤ pβ(t)(2x^{2}(t)e^{\frac{1}{2} ∫_{t−r(t)}^{t} ξ(τ)dτ}(e^{\frac{1}{2} ∫_{t−r(t)}^{t} ξ(τ)dτ} − 1) + \frac{x_{1}^{2}(t)}{2(e^{− \frac{1}{2} ∫_{t−r(t)}^{t} β(τ)dτ} − 1)}).

Since

sup_{t≥0} ∫_{t−r(t)}^{t} |β(s)|ds = sup_{t≥0} ∫_{t−r(t)}^{t} |ξ(s − r(s))|(1 − r′(s))ds =

sup_{t≥0} ∫_{t−r(t)−r(t−r(t))}^{t−r(t)} |ξ(s)|ds ≥ sup_{t≥0} ∫_{t−r(t)}^{t} |ξ(s)|ds

and

sup_{t≥t_{1}} ∫_{t−r(t)}^{t} |b(s)|ds = sup_{t≥t_{1}} ∫_{t−r(t)−r(t−r(t))}^{t−r(t)} |β(s)|ds ≥ sup_{t≥0} ∫_{t−r(t)}^{t} |β(s)|ds,

we have

b_{s} = sup_{t≥t_{1}} ∫_{t−r(t)}^{t} |b(s)|ds

where the point t_{1} is defined from the equation t_{1} − r(t_{1}) = 0.

Now we want to apply Theorem 3.1. According to condition (v), function |β(t)| is to be I-integrally positive with respect to |b(t)| for some I. But β in not known explicitly, so consider the set

I_{r} = ⋃_{i=1}^{∞}(c_{i} − r(c_{i}), d_{i} − r(d_{i})).

Since ∫_{c_{i}}^{d_{i}} |b(s)|ds = ∫_{c_{i}−r(c_{i})}^{d_{i}−r(d_{i})} |β(s)|ds, one can state that, for given I = ⋃_{i=1}^{∞}(c_{i}, d_{i}), the

function |β(t)| is I_{r}-integrally positive with respect to |b(t)| if and only if |b(t)| is I-integrally positive with respect to |b(t − r(t))|(1 − r′(t)).

Thus, if we demand that r(t) is embedded into I_{r} and |b(t)| is I-integrally positive with respect to |b(t − r(t))|(1 − r′(t)) then conditions (v) and (vi) of Theorem 3.1 will be satisfied.

A possible way to satisfy condition (i) of Theorem 2.1 is to require

sup_{t≥t_{2}} ∫_{t−r(t)}^{t} |b(u − r(u))|(1 − r′(u))e^{α_{1}(u,t−r(t))}du < 1,  (6.3)

where α_{1}(u, t−r(t)) = \frac{1}{2} ∫_{t−r(t)}^{u} b(s)ds and the point t_{2} satisfies t_{2}−r(t_{2})−r(t_{2}−r(t_{2})) = 0.

If we follow the scheme of section 5 we can obtain the next conclusion:

Theorem 6.1: If b(t) ≤ 0, b_{s} ≤ ln 4 and (6.3) holds, then x ≡ 0 is uniformly stable. If, in addition, b_{s} ≠ ln 4, and there is I such that r(t) is embedded into I_{r} and |b(t)| is I-integrally positive with respect to |b(t − r(t))|(1 − r′(t)); then x ≡ 0 is asymptotically stable.

Remark 6.1: The best estimate for b_{s} by using Liapunov-Krasovskii functionals is b_{s} < 1 (see [1]-[3],[16]). On the other hand, in [7] it was proved by Liapunov-Razumikhin method that for bounded r(t) the zero solution of (6.1) is stable if b_{s} ≤ 3/2. Here, we guarantee stability in a more general situation but only under b_{s} ≤ ln 4.

Let us give an example which illustrates the conditions of Theorem 6.1.

Example 6.1: Consider equation (6.1) with r(t) = λ_{1}t, 0 ≤ λ_{1} < 1, b(t) = −(\frac{1}{t+1} + c(t)), where

c(t) = \{ \begin{matrix}γt sin t^{2}, & t ∈ T, \\ 0, & t ∉ T, \end{matrix}

γ ≥ 0, T = ⋃_{n≥n_{1}}(\sqrt{2πk}, \sqrt{π(2k + 1)}), k = [\frac{λ^{1−4n}}{2π}], λ = 1 − λ_{1}, n_{1} is such that

\frac{λ^{2−4n}}{2π} < [\frac{λ^{1−4n}}{2π}] < \frac{λ^{−4n}}{2π} − \frac{1}{2}, n ≥ n_{1}

Clearly, c(t) is a continuous, unbounded, nonnegative function with one ”peak” on every interval (λ^{1−2n}, λ^{−2n}), n ≥ n_{1} and the area of each ”peak” is equal to γ.

Let us check conditions of theorem 6.1. We have t_{1} = t_{2} = 0 and

b_{s} ≤ γ + sup_{t≥0} ∫_{λt}^{t} \frac{1}{u + 1}du = γ + sup_{t≥0} ln \frac{t + 1}{λt + 1} = γ − ln λ.

Hence b_{s} ≤ ln 4 if γ − ln λ ≤ ln 4. Condition (6.3) is also fulfilled if

sup_{t≥t_{2}} ∫_{t−r(t)}^{t} |b(u − r(u))|(1 − r′(u))e^{∫_{t−r(t)}^{u} b(s)ds}du ≤ γ+

sup_{t≥0} ∫_{λt}^{t} \frac{1}{u + 1/λ}e^{− ∫_{λt}^{u} \frac{1}{s+1} ds}du < γ + 1 − λ < 1.

Finally, |b(t)| is I-integrally positive with respect to |b(t − r(t))|(1 − r′(t)) for

I = ⋃_{n≥n_{1}}(λ^{1−2n}, λ^{−2n}).

Thus the zero solution is asymptotically stable if 1 > λ > 1/4 and γ < min{λ, ln(4λ)}.

Remark 6.2: Note that Example 6.1 was considered in [7],[18] under γ = 0, where the asymptotic stability was proved for 1 > λ > 1/e.

References

1 L.C.Becker, T.A.Burton and S.Zhang, Functional Differential Equations and Jensen’s Inequality, J. Math. Anal. Appl., 138, (1989), 137-156.

2 T.A.Burton and L.Hatvani, Stability theorems for non-autonomous functional differential equations by Liapunov functionals, Tohoku Math. J., 41, (1989), 65-104.

3 T.A.Burton, A.Casal and A.Somolinos, Upper and lower bounds for Liapunov functionals, Funcial. Ekvac., 32, (1989), 23-55.

4 S.V.Busenberg and K.L.Cooke, Stability conditions for linear non-autonomous delay differential equations, Quart. Appl. Math., 42, (1984), 295-306.

5 I.V.Gaishun and L.B.Knyazhishche, Nonmonotone Lyapunov functionals. Stability conditions for delay differential equations, Differentsial’nye Uravneniya, 30, (1994), 1291-1298.

6 J.K.Hale, Theory of Functional Differential Equations, Springer Verlag, New York, 1977.

7 T.Hara, T.Yoneyama and R.Miyazaki, Some refinements of Razumikhin’s method and their applications, Funkcial. Ekvac., 35, (1992), 279-305.

8 L.Hatvani, On the asymptotic stability of the solutions of functional differential equations, Colloquia Math. Soc. J. Bolyai, 53 ”Qualitative Theory of Differential Equations”, North-Holland, Amsterdam, 1990; 227-238.

9 L.Hatvani, Annulus arguments in the stability theory for functional differential equations, Differential and Integral Equation, 10, (1997), 975-1002.

10 L.Hatvani and T.Krisztin, Asymptotic stability for a differential- difference equation containing terms with and without a delay, Acta. Sci. Math.(Szeged), 60, (1995), 371-384.

11 J.Kato, On Liapunov-Razumikhin type theorems for functional differential equations, Funkcial. Ekvac., 16, (1973), 225-239.

12 L.B.Knyazhishche and Raptunovich N.A., Partial localization of limit sets and asymptotic stability of nonautonomous equations. II, Differentsial’nye Uravneniya, 25, (1989), 388-400.

13 Y.Ko, An asymptotic stability and a uniform asymptotic stability for functional differential equations, Proc. Amer. Math. Soc., 119, (1993), 535-545.

14 K.Kobayashi, Stability and boundedness in functional differential equations with finit delay, Math. Japonica, 40, (1994), 423-432

15 V.B.Kolmanovskii and A.D.Myshkis, Applied theory of functional differential equations, Dordrecht: Kluwer Academic Publishers, 1992.

16 V.B.Kolmanovskii, L.Torelli and R.Vermiglio, Stability of some test equations with delay, SIAM J. Math. Anal., 25, (1994), 948-961.

17 N.N.Krasovskii, Stability of Motion, Stanford Univ. Press., Stanford, 1963. 18 M.Parrott, The limiting behavior of solutions of infinite delay differential

equations, J. Math. Anal. Appl., 87, (1982), 603-627. 19 T.Yoneyama, On the 3/2 stability theorem for one-dimensional delay-differential

equations, J. Math. Anal. Appl., 125, (1987), 161-173. 20 J.A.Yorke, Asymptotic stability for one-dimensional differential-delay equations,

J.Differential Equations, 7, (1970), 189-202.