Construction of Liapunov Functionals for Linear Volterra Integrodifferential Equations and Stability of Delay Systems

Bo Zhang

Department of Mathematics and Computer Science Fayetteville State University Fayetteville, NC 28301

Abstract Inspired by the construction of a new Liapunov functional for linear Volterra integrodifferential equations, we prove some general stability theorems for functional differential equations with infinite delay and weaken the usual requirement for positive definiteness of Liapunov functionals involved in stability theory.

AMS (MOS) subject classification: 34K20, 45J05 Key words: stability of delay systems, construction of Liapunov functionals, linear Volterra equations

1 Introduction.

Consider a system of functional differential equations

x′(t) = F (t, x_{t}),  (1.1)

in which F (t, ϕ), F (t, 0) = 0, is a functional defined for t ≥ 0 and ϕ ∈ C, where C is the set of bounded continuous functions ϕ : R^{−} → R^{n}, R^{−} = (−∞, 0], with the supremum norm ∥ · ∥ = sup{|ϕ(s)| : s ∈ R^{−}}, where | · | is the Euclidean norm on R^{n}. We assume that for each t_{0} ≥ 0 and each ϕ ∈ C_{H}, C_{H} = {ϕ ∈ C : ∥ϕ∥ < H} for some H > 0, there is at least one solution x = x(t, t_{0}, ϕ) of (1.1) defined on an interval [t_{0}, α) with x_{t_{0}} = ϕ. We may also denote the solution by x_{t} = x_{t}(t_{0}, ϕ). Here x_{t}(s) = x(t + s) for s ≤ 0. Moreover, if the solution remains bounded, then α = +∞. We denote by C(X, Y) the set of continuous functions ϕ : X → Y.

This paper is in final form and no version of it will be submitted for publication elsewhere

The object of this paper is to give conditions on Liapunov functionals to ensure the asymptotic stability of the zero solution of (1.1) and to provide a new method of constructing Liapunov functionals for the linear Volterra integrodifferential equation

x′ = Ax(t) + ∫_{0}^{t} C(t − s)x(s)ds  (1.2)

where A is a constant and C ∈ C(R^{+}, R), R^{+} = [0, +∞). In fact, it was equation (1.2) and its nonlinear perturbations which inspired this investigation. There are many known results and applications on stability and basic theory of systems (1.1) and (1.2) in the literature. For reference and history, we refer to the books of Burton [3,4], Driver [8], Gripenberg, Londen, and Staffans [9], Hale [11], Hino, Murakami and Naito [12], Kolmanovskii [16], Krasovskii [17], Kuang [19], Lakshmikantham, Wen, and Zhang [20], Yoshizawa [24].

Let V : R × C → R^{+} be continuous. The upper right-hand derivative along a solution of (1.1) is defined by

V′_{(1.1)}(t, ϕ) = lim sup_{δ→0^{+}}{V (t + δ, x_{t+δ}(t, ϕ)) − V (t, ϕ)}/δ

Definition 1.1. The zero solution of (1.1) is said to be stable if for each ε > 0 and t_{0} ≥ 0, there is a δ = δ(ε, t_{0}) > 0 such that [ϕ ∈ C, ∥ϕ∥ < δ, t ≥ t_{0}] imply |x(t, t_{0}, ϕ)| < ε. The zero solution of (1.1) is uniformly stable if it is stable and δ is independent of t_{0}.

Definition 1.2. The zero solution of (1.1) is said to be asymptotically stable if it is stable and if for each t_{0} ≥ 0 there exists δ_{0} > 0 such that ∥ϕ∥ < δ_{0} implies that x(t, t_{0}, ϕ) → 0 as t → +∞. The zero solution is uniformly asymptotically stable if it is uniformly stable and if there is δ_{1} > 0 and for each ε > 0 there exists T > 0 such that [t_{0} ≥ 0, ϕ ∈ C, ∥ϕ∥ < δ_{1}, t ≥ T + t_{0}] imply that |x(t, t_{0}, ϕ)| < ε.

Definition 1.3. W : R^{+} → R^{+} is called a wedge if W is continuous and strictly increasing with W (0) = 0. Throughout of this paper W and W_{j}(j = 1, 2, ⋯) will denote the wedges.

Definition 1.4. A continuous function g : R^{+} → R^{+} is convex downward if g[(t + s)/2] ≤ [g(t) + g(s)]/2 for all t, s ∈ R^{+}.

Jensen’s inequality. Let g be convex downward and let x, p : [a, b] → R^{+} be continuous with ∫_{a}^{b} p(s)ds > 0. Then

∫_{a}^{b} p(s)ds g[ ∫_{a}^{b} p(s)x(s)ds/ ∫_{a}^{b} p(s)ds] ≤ ∫_{a}^{b} p(s)g(x(s))ds.

For reference on Jensen’s inequality and its applications in stability theory, we refer to Becker, Burton, and Zhang [1] and Natanson [22].

Lemma 1.1. Let W_{1} be a wedge. For any L > 0, define W_{0}(r) = ∫_{0}^{r} W_{1}(s)ds/L on [0, L]. Then W_{0} is a convex downward wedge such that W_{0}(r) ≤ W_{1}(r) for all r ∈ [0, L].

It is well-known (see [7]) that conditions

W_{1}(|ϕ(0)|) ≤ V (t, ϕ), V (t, 0) = 0,  (1.3)
V′_{(1.1)}(t, ϕ) ≤ 0

guarantee the stability of the zero solution of (1.1) and

W_{1}(|x(t)|) ≤ V (t, x_{t}) ≤ W_{2}(|x(t)|) + W_{3}(∫_{0}^{t} Φ(t − s)W_{4}(|x(s)|)ds), (1.4)
V′_{(1.1)}(t, x_{t}) ≤ −W_{5}(|x(t)|)

yield the uniform asymptotic stability, where Φ : R^{+} → R^{+} is continuous, bounded, and Φ ∈ L^{1}(R^{+}).

The goal here is to weaken the lower bound of V (t, ϕ) to

W_{1}(|x(t) + ∫_{0}^{t} G(t − s)x(s)ds|) ≤ V (t, x_{t})  (1.5)

and drop the boundedness condition on Φ(t). This will be done in section 2. As an application of the main theorem, we construct a new Liapunov functional for (1.2) in section 3.

2 A General Stability Theorem

Liapunov functional of the form

V (t, x_{t}) = |x(t)|^{2} + ∫_{0}^{t} Φ(t − s)|x(s)|^{2}ds, Φ(s) ≥ 0

may work for (1.2), but the stability condition may not be the best. Many concrete examples and applications (see [5]) suggest that V (t, ϕ) should take the form

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

where

V_{1}(t, x_{t}) = (x(t) + ∫_{0}^{t} G(t − s)x(s)ds)^{2}

or a more general integral operator. A complete discussion on the construction of such functionals for (1.2) will be given in Section 3. The following lemmas are needed for our main theorem.

Lemma 2.1 ([9], p.14). Let G, Z : R^{+} → R^{n×n} be continuous and satisfy

Z(t) = G(t) − ∫_{0}^{t} G(t − s)Z(s)ds.  (2.1)

If y, x : R^{+} → R^{n} satisfy

y(t) = x(t) + ∫_{0}^{t} G(t − s)x(s)ds,

then

x(t) = y(t) − ∫_{0}^{t} Z(t − s)y(s)ds

for all t ∈ R^{+}.

Lemma 2.2. For all the functions given in Lemma 2.1, if ∫_{0}^{+∞} |G(u)|du < 1, then ∫_{0}^{+∞} |Z(u)|du < +∞ and if y(t) → 0 as t → +∞, then x(t) → 0 as t → +∞.

Proof. It follows from (2.1) that

|Z(t)| ≤ |G(t)| + ∫_{0}^{t} |G(t − s)||Z(s)|ds.

Integrate the above inequality from 0 to ∞ and interchange variables of the second term to obtain

∫_{0}^{+∞} |Z(t)|dt ≤ ∫_{0}^{+∞} |G(t)|dt + ∫_{0}^{+∞} |Z(s)|ds ∫_{0}^{+∞} |G(t)|dt.

This yields ∫_{0}^{+∞} |Z(t)|dt < +∞ since ∫_{0}^{+∞} |G(t)|dt < 1. The statement y(t) → 0 as t → +∞ implies x(t) → 0 as t → +∞ is a known result ([3], p.48). The proof is complete.

The following result has a general lower bound on V (t, ϕ) and removes the boundedness restriction on Φ(t) in ([7], p.144).

Theorem 2.1. Suppose that there exists a continuous functional V : R^{+} × C → R^{+}, V (t, 0) = 0, functions G, Φ ∈ C(R^{+}, R) with ∫_{0}^{+∞} |G(u)|du < 1 and Φ ∈ L^{1}(R^{+}, R^{+}), and wedges W_{j} such that the following conditions hold for t ≥ 0 and ∥x_{t}∥ < H,

(i) W_{1}(|x(t) + ∫_{0}^{t} G(t − s)x(s)ds|) ≤ V (t, x_{t}),

(ii) V (t, x_{t}) ≤ W_{2}(|x(t)|) + W_{3}(∫_{0}^{t} Φ(t − s)W_{4}(|x(s)|)ds)

(iii) V′_{(1.1)}(t, x_{t}) ≤ −W_{5}(|x(t)|).

Then the zero solution of (1.1) is uniformly asymptotically stable.

Proof. Let J = ∫_{0}^{+∞} Φ(u)du. For each 0 < B < H, we choose δ > 0 with 0 < δ < B and

W_{2}(δ) + W_{3}[JW_{4}(δ)] < W_{1}[B(1 − ∫_{0}^{+∞} |G(u)|du)].  (2.2)

Let x(t) = x(t, t_{0}, ϕ) be a solution of (1.1) with ∥ϕ∥ < δ and V (t) = V (t, x_{t}). Then for t ≥ t_{0}, we have

W_{1}(|x(t) + ∫_{0}^{t} G(t − s)x(s)ds|) ≤ V (t) ≤ V (t_{0})
≤ W_{2}(δ) + W_{3}(JW_{4}(δ)).

We claim that |x(t)| < B for all t ≥ t_{0}. Note that |x(u)| < δ < B for all 0 ≤ u ≤ t_{0}. If there exists the first t̂> t_{0} such that |x(t̂)| = B and |x(s)| < B for t_{0} ≤ s < t̂, then

W_{1}[B(1 − ∫_{0}^{+∞} |G(u)|du)] ≤ W_{1}(|x(t̂) + ∫ t̂ ; 0G(t̂− s)x(s)ds|)
≤ W_{2}(δ) + W_{3}[JW_{4}(δ)],

which contradicts (2.2). Thus, the zero solution of (1.1) is uniformly stable. By (iii), we have V (t) − V (τ) ≤ − ∫_{τ}^{t} W_{5}(|x(s)|)ds for t ≥ τ ≥ t_{0}. Since

V (τ) ≤ W_{2}(B) + W_{3}(JW_{4}(B)) =: B^{∗}

for any τ ≥ t_{0}, we have

∫_{τ}^{t} W_{5}(|x(s)|)ds ≤ V (τ) ≤ B^{∗}.

This implies for each σ > 0 there exists L > 0 such that W_{5}(|x(τ^{∗})|) < σ for some τ^{∗} ∈ [τ, τ + L]. Now let σ > 0 be fixed and find h > 0 so that

(B + W_{4}(B)) ∫_{h}^{+∞} Φ(u)du < σ.  (2.3)

By the definition of L, we can choose a sequence {t_{n}} satisfying the following conditions:

t_{n−1} + h ≤ t_{n} ≤ t_{n−1} + h + L

and

|x(t_{n})| < σ

for n = 1, 2, · · ·. Define K = sup{Φ(u) : 0 ≤ u ≤ h}. By (i), we have

V (t_{j}) ≤ W_{2}(|x(t_{j})|)
+W_{3}( ∫_{0}^{t_{j}−h} Φ(t_{j} − s)W_{4}(|x(s)|)ds + ∫_{t_{j}−h}^{t_{j}} Φ(t_{j} − s)W_{4}(|x(s)|)ds)
< W_{2}(σ) + W_{3}(W_{4}(B) ∫_{h}^{+∞} Φ(s)ds + K ∫_{t_{j}−h}^{t_{j}} W_{4}(|x(s)|)ds)
≤ W_{2}(σ) + W_{3}(σ + K ∫_{t_{j}−h}^{t_{j}} W_{4}(|x(s)|)ds).

Since

V (t_{n}) − V (t_{0}) ≤ − ∫_{t_{0}}^{t_{n}} W_{5}(|x(s)|)ds ≤ − ∑_{j=1}^{n} ∫_{t_{j}−h}^{t_{j}} W_{5}(|x(s)|)ds,

we have

∑_{j=1}^{+∞} ∫_{t_{j}−h}^{t_{j}} W_{5}(|x(s)|)ds ≤ V (t_{0}) ≤ B^{∗}.

This implies there exists a positive integer N such that

∫_{t_{N} −h}^{t_{N}} W_{5}(|x(s)|)ds < σ.

By Lemma 1.1 , there exists a convex downward wedge W_{6} with W_{6}(r) ≤ W_{5}[W_{4}^{−1}(r)] for 0 ≤ r ≤ W_{4}(B). This yields

∫_{t_{N} −h}^{t_{N}} W_{5}(|x(s)|)ds = ∫_{t_{N} −h}^{t_{N}} W_{5}(W_{4}^{−1})[W_{4}(|x(s)|)]ds ≥ ∫_{t_{N} −h}^{t_{N}} W_{6}[W_{4}(|x(s)|)ds.

Apply Jensen’s inequality to obtain

∫_{t_{N} −h}^{t_{N}} W_{6}[W_{4}(|x(s)|)]ds ≥ hW_{6}[\frac{1}{h} ∫_{t_{N} −h}^{t_{N}} W_{4}(|x(s)|)ds].

Thus,

∫_{t_{N} −h}^{t_{N}} W_{4}(|x(s)|)ds < hW_{6}^{−1}(σ/h)

and

V (t) ≤ V (t_{N}) ≤ W_{2}(σ) + W_{3}(σ + KhW_{6}^{−1}(σ/h))

for all t ≥ t_{N}. Therefore,

W_{1}(|x(t) + ∫_{0}^{t} G(t − s)x(s)ds|) ≤ W_{2}(σ) + W_{3}(σ + KhW_{6}^{−1}(σ/h))

and

|x(t) + ∫_{0}^{t} G(t − s)x(s)ds| ≤ W_{1}^{−1}(W_{2}(σ) + W_{3}[σ + KhW_{6}^{−1}(σ/h)]) =: σ^{∗}.

Let y(t) = x(t) + ∫_{0}^{t} G(t − s)x(s)ds. By Lemmas 2.1 and 2.2, we have

|x(t)| = |y(t) − ∫_{0}^{t} Z(t − s)y(s)ds|
≤ σ^{∗} + ∫_{0}^{t_{N}} |Z(t − s)||y(s)|ds + ∫_{t_{N}}^{t} |Z(t − s)||y(s)|ds
≤ σ^{∗} + 2B ∫_{t_{N}}^{t} |Z(u)|du + σ^{∗} ∫_{0}^{+∞} |Z(u)|du

for t ≥ t_{N}. Note that t_{N} ≤ t_{0} + N(h + L). We also choose N large enough so that

2B ∫_{N(h+L)}^{+∞} |Z(u)|du < σ^{∗}.

Define Γ = ∫_{0}^{+∞} |Z(u)|du. Then |x(t)| < (2 + Γ)σ^{∗} for all t ≥ t_{N}. For any ε > 0, choose σ > 0 so that σ^{∗} = ε/(2 + Γ). Letting T = N(h + L), we get |x(t)| < ε for all t ≥ T + t_{0}. This proves the zero solution of (1.1) is uniformly asymptotically stable.

Remark 2.1. It is clear from the proof that Theorem 2.1 can be extended to nonconvolution type with modifications on G and Φ. For more discussions on such extension, we refer the reader to Lakshmikantham, Wen, and Zhang [20].

3 Construction of a Liapunov functional

Consider the linear scalar Volterra integrodifferential equation

x′(t) = Ax(t) + ∫_{0}^{t} C(t − s)x(s)ds  (3.1)

where A is a constant and C : R^{+} → R is continuous. We will construct a Liapunov functional that is more general than those exist in the literature (see [6], [14]) and provides a stability region that is closer to the one given by the characteristic equation of (3.1). Our method here is inspired by the work of Knyazhishche and Shcheglov [15] who investigated the linear scalar differential equations with finite delay.

Theorem 3.1. Suppose there exists a constant α ≤ 0 such that G_{α}(t) = ∫_{t}^{+∞} C(u)e^{αu}due^{−αt} exists with ∫_{0}^{+∞} |G_{α}(u)|du < 1 and ∫_{0}^{+∞} u|G_{α}(u)|du < +∞. If

A + ∫_{0}^{+∞} C(u)e^{αu}du + |α| ∫_{0}^{+∞} |G_{α}(u)|du < 0,  (3.2)

then the zero solution of (3.1) is uniformly asymptotically stable.

Proof. Let x(t) = x(t, t_{0}, ϕ) be a solution of (3.1) and define

V_{1}(t) = \frac{1}{2}(x(t) + ∫_{0}^{t} G_{α}(t − s)x(s)ds)^{2}  (3.3)

for t ≥ t_{0}. Taking the derivative along the solution, we obtain

V′_{1}(t) = (x(t) + ∫_{0}^{t} G_{α}(t − s)x(s)ds)(Ax(t) + ∫_{0}^{t} C(t − s)x(s)ds  (3.4)
+ ∫_{0}^{+∞} C(u)e^{αu}du x(t) − ∫_{0}^{t} C(t − s)x(s)ds − α ∫_{0}^{t} G_{α}(t − s)x(s)ds)
= (x(t) + ∫_{0}^{t} G_{α}(t − s)x(s)ds)(a_{1}x(t) − α ∫_{0}^{t} G_{α}(t − s)x(s)ds)
= a_{1}x^{2}(t) + (a_{1} − α)x(t) ∫_{0}^{t} G_{α}(t − s)x(s)ds − α( ∫_{0}^{t} G_{α}(t − s)x(s)ds)^{2}

where a_{1} = A + ∫_{0}^{+∞} C(u)e^{αu}du. It follows from Cauchy’s inequality

( ∫_{0}^{t} G_{α}(t − s)x(s)ds)^{2} ≤ ∫_{0}^{t} |G_{α}(t − s)|ds ∫_{0}^{t} |G_{α}(t − s)|x^{2}(s)ds.

Thus, for any constant δ > 0 we have

V′_{1}(t) ≤ a_{1}x^{2}(t) + (a_{1} − α)x(t)∫_{0}^{t}G_{α}(t−s)x(s)ds
+|α| ∫_{0}^{t}|G_{α}(t−s)|ds∫_{0}^{t}|G_{α}(t−s)|x^{2}(s)ds
≤ a_{1}x^{2}(t) + \frac{(a_{1} − α)^{2}}{4δ} x^{2}(t)
+(|α| + δ) ∫_{0}^{t} |G_{α}(t − s)|ds ∫_{0}^{t} |G_{α}(t − s)|x^{2}(s)ds.  (3.5)

Next, define

V_{2}(t) = p ∫_{0}^{t} ∫_{t−s}^{+∞} |G_{α}(u)|dux^{2}(s)ds  (3.6)

where p is a positive constant and set

V (t) = V_{1}(t) + V_{2}(t).

Observe that

V′_{2}(t) = p ∫_{0}^{+∞} |G_{α}(u)|dux^{2}(t) − p ∫_{0}^{t} |G_{α}(t − s)|x^{2}(s)ds.

Combining with (3.5), this yields

V′(t) ≤ (a_{1} + \frac{(a_{1} − α)^{2}}{4δ} + p ∫_{0}^{+∞} |G_{α}(u)|du)x^{2}(t)  (3.7)
+ [(|α| + δ) ∫_{0}^{+∞} |G_{α}(u)|du − p] ∫_{0}^{t} |G_{α}(t − s)|x^{2}(s)ds.

We will choose the best δ and p so that

a_{1} + \frac{(a_{1} − α)^{2}}{4δ} + p ∫_{0}^{+∞} |G_{α}(u)|du < 0  (3.8)

and

(|α| + δ) ∫_{0}^{+∞} |G_{α}(u)|du − p ≤ 0.  (3.9)

Notice that if ∫_{0}^{+∞} |G_{α}(u)|du ≠ 0, then (3.8) and (3.9) hold if and only if

(|α| + δ)( ∫_{0}^{+∞} |G_{α}(u)|du)^{2} ≤ p ∫_{0}^{+∞} |G_{α}(u)|du < |a_{1}| − \frac{(a_{1} − α)^{2}}{4δ} .

This implies that we choose δ > 0 so that

(|α| + δ)( ∫_{0}^{+∞} |G_{α}(u)|du)^{2} < |a_{1}| − \frac{(a_{1} − α)^{2}}{4δ} .  (3.10)

Let Q = (∫_{0}^{+∞} |G_{α}(u)|du)^{2}. We may assume that G_{α}(u) ≢ 0 since G_{α}(u) ≡ 0 for all u ∈ R^{+} implies C(u) ≡ 0 for all u ∈ R^{+}. Thus we assume that Q > 0. It is clear that (3.10) is equivalent to

4Qδ^{2} + 4(|α|Q − |a_{1}|)δ + (a_{1} − α)^{2} < 0.  (3.11)

The quadratic function of δ on the left-hand side of (3.11) has its minimum at

δ_{∗} = \frac{|a_{1}| − |α|Q}{2Q}.

Since ∫_{0}^{+∞} |G_{α}(u)|du < 1, we have 0 < Q < 1 and

|a_{1}| − |α|Q = −A − ∫_{0}^{+∞} C(u)e^{αu}du − |α|Q
≥ −(A + ∫_{0}^{+∞} C(u)e^{αu}du + |α| ∫_{0}^{+∞} |G_{α}(u)|du) > 0

by (3.2). Thus δ_{∗} > 0 and

4Qδ_{∗}^{2} + 4(|α|Q − |a_{1}|)δ_{∗} + (a_{1} − α)^{2}
= \frac{(|a_{1}| − |α|Q)^{2}}{Q} − \frac{2(|a_{1}| − |α|Q)^{2}}{Q} + (a_{1} − α)^{2}
= \frac{Q − 1}{Q}[|a_{1}|^{2} − Q|α|^{2}]
= \frac{Q − 1}{Q}(|a_{1}| + \sqrt{Q}|α|)(|a_{1}| − \sqrt{Q}|α|)
= \frac{1 − Q}{Q}(|a_{1}| + \sqrt{Q}|α|)(A + ∫_{0}^{+∞} C(u)e^{αu}du + |α| ∫_{0}^{+∞} |G_{α}(u)|du) < 0.

Let

γ_{1} =: (|α| + δ_{∗})Q = (|α| + \frac{|a_{1}| − |α|Q}{2Q})Q = \frac{|a_{1}| + |α|Q}{2}

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 10

and

γ_{2} =: |a_{1}| − \frac{(a_{1} − α)^{2}}{4δ_{∗}} = \frac{|a_{1}|^{2} − α^{2}Q + (1 − Q)|a_{1}|^{2}}{2(|a_{1}| − |α|Q)} .

Define

p = \frac{γ_{1} + γ_{2}}{2\sqrt{Q}} = \frac{(1 − Q)|a_{1}|^{2} + 2|a_{1}|^{2} − Q(1 + Q)|α|^{2}}{4(|a_{1}| − |α|Q)\sqrt{Q}} .  (3.12)

We now obtain

a_{1} + \frac{(a_{1} − α)^{2}}{4δ_{∗}} + p\sqrt{Q} = −\frac{(1 − Q)(|a_{1}|^{2} − α^{2}Q)}{4(|a_{1}| − |α|Q)}

and

(|α| + δ_{∗})Q − p\sqrt{Q} = −\frac{(1 − Q)(|a_{1}|^{2} − α^{2}Q)}{4(|a_{1}| − |α|Q)}.

Define

β = \frac{(1 − Q)(|a_{1}|^{2} − α^{2}Q)}{4(|a_{1}| − |α|Q)} min\{1, \frac{1}{\sqrt{Q}}\}.

Combining (3.3), (3.6), and (3.7), we have

V (t) = \frac{1}{2}(x(t) + ∫_{0}^{t} G_{α}(t − s)x(s)ds)^{2} + p ∫_{0}^{t} ∫_{t−s}^{+∞} |G_{α}(u)|dux^{2}(s)ds
≤ x^{2}(t) + ∫_{0}^{t} (|G_{α}(t − s)| + p ∫_{t−s}^{+∞} |G_{α}(u)|du)x^{2}(s)ds
=: x^{2}(t) + ∫_{0}^{t} Φ(t − s)x^{2}(s)ds  (3.13)

and

V′(t) ≤ −βx^{2}(t) − β ∫_{0}^{t} |G_{α}(t − s)|x^{2}(s)ds  (3.14)

where p is given in (3.12) and

Φ(t) = |G_{α}(t)| + p ∫_{t}^{+∞} |G_{α}(u)|du.

One can show by interchanging the order of integration that

∫_{0}^{+∞} ∫_{s}^{+∞} |G_{α}(u)|duds = ∫_{0}^{+∞} u|G_{α}(u)|du < +∞.  (3.15)

Thus, Φ ∈ L^{1}(R^{+}). By Theorem 2.1, the zero solution of (3.1) is uniformly asymptotically stable. This completes the proof.

When α = 0, Theorem 3.1 takes the following form.

Corollary 3.1. Suppose that

A + ∫_{0}^{+∞} C(u)du < 0,
∫_{0}^{∞} | ∫_{t}^{∞} C(u)du|dt < 1,

and

∫_{0}^{∞} s |∫_{s}^{+∞} C(u)du|ds < +∞.  (3.16)

Then the zero solution of (3.1) is uniformly asymptotically stable.

In ([6], p.162) Burton and Mahfoud proved the following theorem. Brauer [2], Jordan [13], and Krisztin [18] also investigated the same problem in one or higher dimensions using the characteristic equation of (3.1).

Theorem B_{1}. Suppose that

A + ∫_{0}^{+∞} C(u)du < 0,  (3.17)
∫_{0}^{∞} | ∫_{t}^{∞} C(u)du|dt < 1,  (3.18)

and

∫_{0}^{∞} u|C(u)|du < +∞.  (3.19)

Then the zero solution of (3.1) is uniformly asymptotically stable.

Although (3.19) is weaker than (3.16) for C(u) with constant sign, one can show that (3.19) implies (3.16) for α < 0. In fact, if (3.19) holds, we have

∫_{0}^{+∞} u|G_{α}(u)|du ≤ ∫_{0}^{+∞} u ∫_{u}^{+∞} |C(s)|e^{α(s−u)}dsdu

= ∫_{0}^{+∞} ∫_{0}^{s} ue^{−αu}du|C(s)|e^{αs}ds
= ∫_{0}^{+∞} (\frac{1}{|α|}se^{−αs} − \frac{1}{α^{2}} e^{−αs} + \frac{1}{α^{2}} )|C(s)|e^{αs}ds
≤ α^{∗} ∫_{0}^{+∞}(s + 1)|C(s)|ds < +∞

where α^{∗} = max\{\frac{1}{|α|}, \frac{1}{α^{2}} \}.

Corollary 3.2. Suppose (3.19) holds and there exists a constant α < 0 such that

∫_{0}^{+∞} |G_{α}(u)|du < 1  (3.20)

and

A + ∫_{0}^{+∞} C(u)e^{αu}du + |α| ∫_{0}^{+∞} |G_{α}(u)|du < 0,  (3.21)

then the zero solution of (3.1) is uniformly asymptotically stable.

It is known (see [10],[21]) that the zero solution of (3.1) is uniformly asymptotically stable if and only if

|z − A − C^{∗}(z)| ≠ 0

for Re(z) ≥ 0, where z is a complex number and C^{∗} is the Laplace transform of C defined by C^{∗}(z) = ∫_{0}^{+∞} e^{−zt}C(t)dt. Thus, condition (3.17) is necessary for uniform asymptotic stability. The following example shows that (3.18) does not hold, but (3.20) and (3.21) are satisfied for an appropriate α. Consider

x′(t) = 0.06x(t) − ∫_{0}^{t} e^{−0.9(t−s)}x(s)ds.  (3.22)

Let A = 0.06 and C(t) = −e^{−0.9t}. Then

∫_{0}^{+∞} | ∫_{t}^{+∞} C(u)du|dt = ∫_{0}^{+∞} \frac{1}{0.9}e^{−0.9t}dt = \frac{1}{(0.9)^{2}} > 1.

Thus, (3.18) fails. Letting α = −0.8, we have

G_{α}(t) = ∫_{t}^{+∞} C(u)e^{αu}due^{−αt} = −\frac{1}{1.7}e^{−0.9t},

∫_{0}^{+∞} |G_{α}(u)|du = \frac{1}{1.7} ∫_{0}^{+∞} e^{−0.9t}dt = \frac{1}{1.53} < 1,

and

A + ∫_{0}^{+∞} C(u)e^{αu}du + |α| ∫_{0}^{+∞} |G_{α}(u)|du
= 0.06 − ∫_{0}^{+∞} e^{−1.7u}du + \frac{0.8}{1.53}
< −0.005.

It is obvious that ∫_{0}^{+∞} u|G_{α}(u)|du < +∞. By Theorem 3.1, the zero solution of (3.22) is uniformly asymptotically stable.

Generalizations of Theorem B_{1} to systems and noncovolution type of equations can be found in Burton and Mahfoud [6] and Zhang [24]. The author has not yet obtained a satisfactory extension of Theorem 3.1 to systems and will pursue further investigation. Finally we present a theorem on asymptotic stability.

Theorem 3.2. Suppose there exists a constant α ≤ 0 such that (3.20) and (3.21) hold. Then the zero solution of (3.1) is asymptotically stable.

Proof. It is clear that the zero solution of (3.1) is stable by (3.13) and (3.14). Let x(t) = x(t, t_{0}, ϕ) be a solution of (3.1). We will show that x(t) → 0 as t → +∞. Indeed, by (3.14) and Cauchy’s inequality, we have

V′(t) ≤ −\frac{β}{2} (x(t) + ∫_{0}^{t} G_{α}(t − s)x(s)ds)^{2}.  (3.23)

This implies that

γ(t) =: (x(t) + ∫_{0}^{t} G_{α}(t − s)x(s)ds)^{2} ∈ L^{1}(R^{+}).  (3.24)

By (3.4), we have

|γ′(t)| = 2|a_{1}x^{2}(t) + (a_{1}−α)x(t) ∫_{0}^{t}G_{α}(t − s)x(s)ds − α( ∫_{0}^{t}G_{α}(t − s)x(s)ds)^{2}|
≤ (2|a_{1}| + |a_{1} − α|)x^{2}(t) + (2|α| + |a_{1} − α|) ∫_{0}^{t} |G_{α}(t − s)|x^{2}(s)ds. (3.25)

Combining (3.24) and (3.25), we get |γ′(t)| ∈ L^{1}(R^{+}). Therefore, lim_{t→+∞} γ(t) = 0. Let y(t) = x(t) + ∫_{0}^{t} G_{α}(t − s)x(s)ds. By Lemmas 2.1 and 2.2, we have

|x(t)| = |y(t) − ∫_{0}^{t} Z(t − s)y(s)ds|

≤ |y(t)| + | ∫_{0}^{t} Z(t − s)y(s)ds| → 0

since Z ∈ L^{1}(R^{+}) and y(t) → 0 as t → +∞. This completes the proof.

Corollary 3.3. If

A + ∫_{0}^{+∞} C(u)du < 0

and

∫_{0}^{∞} | ∫_{t}^{∞} C(u)du|dt < 1,

then the zero solution of (3.1) is asymptotically stable.

Corollary 3.4. If

A + ∫_{0}^{+∞} |C(u)|du < 0,  (3.26)

then the zero solution of (3.1) is asymptotically stable.

Proof. If (3.26) holds, we can choose α sufficiently large so that conditions of Theorem 3.2 will be satisfied. In fact, for α < 0 we have

|α| ∫_{0}^{+∞} |G_{α}(u)|du ≤ −α ∫_{0}^{+∞} ∫_{t}^{+∞} |C(u)|e^{αu}due^{−αt}dt
= ( ∫_{t}^{+∞} |C(u)|e^{αu}du)e^{−αt}|_{0}^{+∞} + ∫_{0}^{+∞} |C(u)|du
= − ∫_{0}^{+∞} |C(u)|e^{αu}du + ∫_{0}^{+∞} |C(u)|du.

This yields

∫_{0}^{+∞} |G_{α}(u)|du ≤ \frac{1}{|α|} ∫_{0}^{+∞} |C(u)|du

for α < 0. If ∫_{0}^{+∞} |C(u)|du < |α|, then ∫_{0}^{+∞} |G_{α}(u)|du < 1. By (3.26), we also have

A + ∫_{0}^{+∞} C(u)e^{αu}du + |α| ∫_{0}^{+∞} |G_{α}(u)|du

≤ A + ∫_{0}^{+∞} C(u)e^{αu}du − ∫_{0}^{+∞} |C(u)|e^{αu}du + ∫_{0}^{+∞} |C(u)|du
≤ A + ∫_{0}^{+∞} |C(u)|du < 0.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 30, p. 15

By Theorem 3.2, the zero solution of (3.1) is asymptotically stable.

Burton and Mahfoud ([6], p.146) showed that (3.26) yields uniform asymptotic stability by proving that x(t) ∈ L^{1}(R^{+}, R).

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