ASYMPTOTIC STABILITY FOR FUNCTIONAL DIFFERENTIAL EQUATIONS

T. A. Burton and G. Makay

Department of Mathematics, Southern Illinois University at Carbondale Carbondale, Illinois 62901-4408

1. Introduction

We consider a system of functional differential equations with finite delay written as

x′(t) = f(t, x_{t}),  ′ = d/dt,  (1)

where f : [0, ∞) × C_{H} → R^{m} is continuous and takes bounded sets into bounded sets and f(t, 0) = 0. Here, (C, ∥ · ∥) is the Banach space of continuous functions ϕ : [−h, 0] → R^{m} with the supremum norm, h is a non-negative constant, C_{H} is the open H-ball in C, and x_{t}(s) = x(t + s) for −h ≤ s ≤ 0. Standard existence theory shows that if ϕ ∈ C_{H} and t ≥ 0, then there is at least one continuous solution x(t, t_{0}, ϕ) on [t_{0}, t_{0} + α) satisfying (1) for t > t_{0}, x_{t}(t_{0}, ϕ) = ϕ and α some positive constant; if there is a closed subset B ⊂ C_{H} such that the solution remains in B, then α = ∞. Also, | · | will denote the norm in R^{m} with |x| =max_{1≤i≤m}|x_{i}|.

We are concerned here with asymptotic stability in the context of Liapunov’s direct method. Thus, we are concerned with continuous, strictly increasing functions W_{i} : [0, ∞) → [0, ∞) with W_{i}(0) = 0, called wedges, and with Liapunov functionals V.

Definition: A continuous functional V : [0, ∞) × C_{H} → [0, ∞) which is locally Lipschitz in ϕ is called a Liapunov functional for (1) if there is a wedge W with

(i) W (|ϕ(0)|) ≤ V (t, ϕ), V (t, 0) = 0, and (ii) V′_{(1)}(t, x_{t}) = lim sup_{δ→0} \frac{1}{δ} \{V (t + δ, x_{t+δ}(t_{0}, ϕ)) − V (t, x_{t}(t_{0}, ϕ))\} ≤ 0.

Remark: A standard result states that if there is a Liapunov functional for (1), then x = 0 is stable. Definitions will be given in the next section.

Asymptotic Stability of Functional Differential Equations 2

The classical result on asymptotic stability may be traced back to Marachkov [9] through Krasovskii [7;pp. 151-154]. It may be stated as follows.

Theorem MK: Suppose there are a constant M, wedges W_{i}, and a Liapunov functional V (so W_{1}(|ϕ(0)|) ≤ V (t, ϕ) and V (t, 0) = 0) with

(i) V′_{(1)}(t, x_{t}) ≤ −W_{2}(|x(t)|) and (ii) |f(t, ϕ)| ≤ M if t ≥ 0 and ∥ϕ∥ < H. Then x = 0 is asymptotically stable.

Condition (ii) is troublesome, since it excludes many examples of considerable interest. And there are several results which reduce or eliminate (ii). For example, we showed [2] that if (iii) V (t, ϕ) ≤ W_{3}(|x_{t}|_{2}), where | · |_{2} is the L^{2}-norm, then uniform asymptotic stability would result. Other alternatives may be found in [3,4,5,6], for example.

We reduce (ii) in a variety of ways and obtain results on asymptotic stability, partial stability, and uniform asymptotic stability. We give an example in which we show that the zero solution of

x′′ + tx′ + x = 0  (2)

is uniformly asymptotically stable.

The following is a simplified corollary to our results and is stated here to focus the paper.

Theorem A: Suppose there is a Liapunov functional V, wedges W_{i}, positive constants K and J, a sequence {t_{n}} ↑ ∞ with t_{n} − t_{n−1} ≤ K such that

(i) V (t_{n}, ϕ) ≤ W_{2}(∥ϕ∥), (ii) V′_{(1)}(t, x_{t}) ≤ −W_{3}(|x(t)|) if t_{n} − h ≤ t ≤ t_{n}, and (iii) |f(t, ϕ)| ≤ J(t + 1)ln(t + 2) for t ≥ 0 and ∥ϕ∥ < H. Then x = 0 is AS.

2. Statement of results and examples

We now define the terminology to be used here.

3 T. A. Burton & G. Makay

Definition: The solution x = 0 of (1) is:

(a) stable if for each ε > 0 and t_{0} ≥ 0 there is a δ > 0 such that [∥ϕ∥ < δ, t ≥ t_{0}] imply that |x(t, t_{0}, ϕ)| < ε;

(b) uniformly stable (US) if for each ε > 0 there is a δ > 0 such that [t_{0} ≥ 0, ∥ϕ∥ < δ, t ≥ t_{0}] imply that |x(t, t_{0}, ϕ)| < ε;

(c) asymptotically stable (AS) if it is stable and if for each t_{0} ≥ 0 there is a γ > 0 such that ∥ϕ∥ < γ implies that x(t, t_{0}, ϕ) → 0 as t → ∞;

(d) uniformly asymptotically stable (UAS) if it is US and if there is a γ > 0 and for each μ > 0 there is a T > 0 such that [t_{0} ≥ 0, ∥ϕ∥ < γ, t ≥ t_{0} + T] imply that |x(t, t_{0}, ϕ)| < μ. In preparation for our main result we remind the reader that if V is a Liapunov

functional, then W_{1}(|ϕ(0)|) ≤ V (t, ϕ), V (t, 0) = 0, and V′_{(1)}(t, x_{t}) ≤ 0. So that our result applies also to ODE’s we introduce a positive number k which will replace h found in (1).

Theorem 1: Let k > 0, k ≥ h, let V be a Liapunov functional for (1) (so that W_{1}(|ϕ(0)|) ≤ V (t, ϕ), V (t, 0) = 0, and V′_{(1)}(t, x_{t}) ≤ 0) and x = (x_{1}, ..., x_{m}). Consider the following conditions for a given i (1 ≤ i ≤ m) and a given sequence {t_{n}} with t_{n} ↑ ∞:

(i) there are wedges W_{i}, U_{i}, Q_{i}, (ii) there is a sequence {λ_{n}^{(i)}} with λ_{n}^{(i)} ≥ λ > 0, λ is a constant, and that (iii) there are locally integrable functions M_{i}, P_{i} : [0, ∞) → [0, ∞) such that (iv) either M_{i} ≡ 0 or for each D > 0 with D/λ_{n}^{(i)} ≤ k there is a sequence {c_{n}^{(i)}}, c_{n}^{(i)} >

0, such that if a, b ∈ [t_{n} − k, t_{n}] with a < b, then ∫_{a}^{b} M_{i}(t)dt ≤ λ_{n}^{(i)}(b − a) and ∫_{s_{n}}^{s_{n}+D/λ_{n}^{(i)}} P_{i}(s)ds ≥ c_{n}^{(i)} for all s_{n} ∈ [t_{n} − k, t_{n} − D/λ_{n}^{(i)}],

(v) V′_{(1)}(t, x_{t}) ≤ −P_{i}(t)U_{i}(|x_{i}|) for ∥x_{t}∥ < H and t ∈ [t_{n} − k, t_{n}], and (vi) V′_{(1)}(t, x_{t}) ≤ −Q_{i}(|x′_{i}|) + M_{i}(t) for ∥x_{t}∥ < H and t ∈ [t_{n} − k, t_{n}] with Q_{i} convex

downward. We then have the following conclusions:

(I) If (i)-(vi) hold for all i satisfying 1 ≤ i ≤ m and for some {t_{n}} ↑ ∞ with c_{n}^{(i)} ≥ c_{0} > 0 for all n and all i, if t_{n} − t_{n−1} is bounded, and if V (t, ϕ) ≤ W (∥ϕ∥), then x = 0 is UAS.

(II) If (i)-(vi) hold for an arbitrary sequence {t_{n}} ↑ ∞ and for some i satisfying 1 ≤ i ≤ m,

Asymptotic Stability of Functional Differential Equations 4

if c_{n}^{(i)} ≥ c_{0} > 0 for all n then any solution x(t) which remains in C_{H} satisfies x_{i}(t) → 0 as t → ∞.

(III) If (i)-(vi) hold for all i satisfying 1 ≤ i ≤ m and for some sequence {t_{n}} ↑ ∞, if V (t_{n}, ϕ) ≤ W (∥ϕ∥), if c_{n}^{(i)} ≥ c_{n} for 1 ≤ i ≤ m and some c_{n} with ∑_{n=1}^{∞} c_{n} = ∞, then x = 0 is AS. Remark: Theorem 1 is long because it is stated in terms of separate components

of x; but an example will show that it is well worth the detail. However, to grasp the significance we will now state some useful corollaries.

Corollary 1: Suppose there is a Liapunov functional V, a locally integrable function M : [0, ∞) → [0, ∞) and a monotone increasing function λ : [0, ∞) → (1, ∞) such that if 0 < b − a < h then

(i) ∫_{a}^{b} M(t)dt ≤ λ(b)(b − a) and ∫_{1}^{∞} \frac{dt}{λ(t)} = ∞. Suppose also that there are wedges, a constant K > 0, and a sequence {t_{n}} ↑ ∞ with t_{n} − t_{n−1} ≤ K such that (ii) V (t_{n}, ϕ) ≤ W (∥ϕ∥) and if t_{n} − h ≤ t ≤ t_{n} then (iii) V′_{(1)}(t, x_{t}) ≤ −W_{2}(|x(t)|) and (iv) V′_{(1)}(t, x_{t}) ≤ −W_{3}(|x′(t)|) + M(t), W_{3} is convex downward. Then x = 0 is AS.

Corollary 2: Suppose there is a Liapunov functional V, wedges W_{i}, positive constants K and J, a sequence {t_{n}} ↑ ∞ with t_{n} − t_{n−1} ≤ K such that

(i) V (t_{n}, ϕ) ≤ W_{2}(∥ϕ∥), (ii) V′_{(1)}(t, x_{t}) ≤ −W_{3}(|x(t)|) if t_{n} − h ≤ t ≤ t_{n}, and (iii) |f(t, ϕ)| ≤ J(t + 1)ln(t + 2) for t ≥ 0 and ∥ϕ∥ < H. Then x = 0 is AS.

Corollary 3: Suppose there are a Liapunov functional V and a wedge W_{2} with (i) V (t, ϕ) ≤ W_{2}(∥ϕ∥). In addition, suppose there are locally integrable functions M, P : [0, ∞) → [0, ∞), a positive constant K, sequences {t_{n}} ↑ ∞ and {λ_{n}} with t_{n} − t_{n−1} ≤ K, such that if 0 < b − a < h and if t_{n} − h ≤ t ≤ t_{n} with b ≤ t_{n}, then for each D > 0 there is a c > 0 with

5 T. A. Burton & G. Makay

(ii) ∫_{a}^{b} M(s)ds ≤ λ_{n}(b − a) and ∫_{t}^{t+D/λ_{n}} P (s)ds ≥ c, (iii) V′_{(1)}(t, x_{t}) ≤ −P (t)W_{3}(|x(t)|) for t_{n} − h ≤ t ≤ t_{n}, and (iv) V′_{(1)}(t, x_{t}) ≤ −W_{4}(|x′(t)|) + M(t), W_{4} is convex downward. Then x = 0 if UAS.

Corollary 4: (Marachkov-Krasovskii) If there is a Liapunov functional V, wedges W_{i}, and a constant M such that

(i) V (t, ϕ) ≤ W_{2}(∥ϕ∥), (ii) V′_{(1)}(t, x_{t}) ≤ −W_{3}(|x(t)|), (iii) |f(t, ϕ)| ≤ M if t ≥ 0 and ∥ϕ∥ < H, then x = 0 is UAS.

We now give an example of Corollary 2.

Example 1: Let a, b : [0, ∞) → R be continuous and suppose there are constants c_{1} ≥ 1, c_{2} > 0, c_{3} > 0, c_{4} > 0 with

(a) a(t) − c_{1}|b(t + 1)| =: α(t) ≥ c_{3},

(b) there is a sequence {t_{n}} ↑ ∞ and K > 0 with 1 ≤ t_{n+1}−t_{n} ≤ K and ∫_{t_{n}−1}^{t_{n}} |b(s+1)|ds ≤ c_{2}.

(c) a(t) + |b(t)| ≤ c_{4}(t + 1)ln(t + 2). Then the zero solution of

x′(t) = −a(t)x + b(t)x(t − 1)  (3)

is AS.

Proof: Define

V (t, x_{t}) = |x(t)| + c_{1} ∫_{t−1}^{t} |b(s + 1)||x(s)|ds

so that

\begin{aligned}V′_{(3)}(t, x_{t}) & ≤ −a(t)|x| + |b(t)||x(t − 1)| + c_{1}|b(t + 1)||x| − c_{1}|b(t)||x(t − 1)| \\ & ≤ −[a(t) − c_{1}|b(t + 1)|]|x| ≤ −α(t)|x|. \end{aligned}

Take H = 1 and W (r) = r. Then for ∥ϕ∥ < H we have

|ϕ(0)| ≤ V (t, ϕ), V (t, 0) = 0
V (t_{n}, ϕ) ≤ |ϕ(0)| + c_{1}c_{2}∥ϕ∥

Asymptotic Stability of Functional Differential Equations 6

and

V′(t, x_{t}) ≤ −c_{3}|x(t)|.

The conditions of Corollary 2 are satisfied.

Examples of a(t) and b(t) are easily constructed so that this equation is not uniformly stable. Let m(t) = −[t]sin2πt, r(t) = −[t](cos2πt − 1)/2π and c(t) = |sinπt| − sinπt, where [·] stands for the greatest integer function. Consider the scalar equation

x′ = (m(t) − 1 − e^{2}ln(t + 1))x(t) + \frac{1}{2}c(t)(lnt)x(t − 1)

for t ≥ 1. Note that

∫_{n}^{n+1}[t]sin2πtdt = n ∫_{n}^{n+1} sin2πtdt = −\frac{n}{2π} (cos2π(n + 1) − cos2πn) = 0

so that if n ≤ t < n + 1 then

r(t) = ∫_{0}^{t} −[s]sin2πsds = \frac{n}{2π} (cos2πt − 1) = \frac{[t]}{2π} (cos2πt − 1).

Let

V (t) = V (t, x_{t}) = e^{2r(t)}x^{2} + \frac{1}{2} ∫_{t−1}^{t} e^{2r(s+1)}ln(s + 1)c(s + 1)x^{2}(s)ds

so that

V′(t) ≤ (−2m(t) + 2m(t) − 2 − 2e^{2}ln(t + 1))e^{2r(t)}x^{2} + c(t)(lnt)x(t)x(t − 1)e^{2r(t)}

+ \frac{1}{2}e^{2r(t+1)}ln(t + 1)c(t + 1)x^{2} − \frac{1}{2}e^{2r(t)}(lnt)c(t)x^{2}(t − 1)

≤ −(2 + 2e^{2}ln(t + 1))e^{2r(t)}x^{2} + (lnt)e^{2r(t)}x^{2} + \frac{c(t)}{2}(lnt)e^{2r(t)}x^{2}(t − 1)

+ e^{2r(t+1)}ln(t + 1)x^{2}(t) − \frac{1}{2}e^{2r(t)}(lnt)c(t)x^{2}(t − 1).

Now

e^{2r(t+1)} = e^{−2([t]+1)(cos2πt−1)/2π} ≤ e^{2}e^{2r(t)}

so V′(t) ≤ −2x^{2}(t). Also, V (t) ≥ x^{2}(t). Finally, when n is even

V (n) = x^{2} + \frac{1}{2} ∫_{n−1}^{n} e^{2r(s+1)}ln(s + 1)(|sinπ(s + 1)| − sinπ(s + 1))x^{2}(s)ds = x^{2}.

Hence, the conditions of Corollary 2 are satisfied and x = 0 is AS.

7 T. A. Burton & G. Makay

Remark: This result will not follow from the work of Busenberg and Cooke [6] because they require that for each η > 0 there exists τ > 0 such that ∫_{t}^{t+η} a(s)ds ≤ τ. It will not follow from Burton [2] because that result requires that V (t, ϕ) ≤ W_{2}(|ϕ(0)|) + W_{3}(|ϕ|_{2}), where | · |_{2} is the L^{2}-norm. It will not follow from Burton-Hatvani [5] for the same reason. It will not follow from Makay [8] because he requires V (t, ϕ) ≤ W (∥ϕ∥). It will not follow from Wang [12] because he requires uniform stability.

In the next example it is very easy to show AS by a variety of classical techniques [1,10,11]. But it requires all of the flexibility of Theorem 1 to show UAS.

Example 2: The zero solution of

x′′ + tx′ + x = 0,  t ≥ 1  (4)

is UAS.

Proof: Write (4) as

\begin{aligned}x′ & = −\frac{x + y}{t} \\ y′ & = \frac{2x}{t} + (\frac{2}{t} − t) y. \end{aligned}

Let (x_{1}, x_{2}) = (x, y) and define V = (x^{2} + y^{2})/2 so that

V′ = −\frac{1}{t}x^{2} − \frac{1}{t}xy + \frac{2}{t}xy + (\frac{2}{t} − t) y^{2}

≤ −\frac{1}{t}x^{2} + \frac{1}{2t}(x^{2} + y^{2}) + (\frac{2}{t} − t) y^{2}

= −\frac{1}{2t}x^{2} + (\frac{1}{2t} + \frac{2}{t} − t) y^{2}.

Thus, for t ≥ 4 we have (v) of Theorem 1 satisfied:

(v) V′ ≤ −\frac{1}{2t}x^{2} − \frac{t}{2}y^{2} =: −P_{1}(t)U_{1}(|x_{1}|) − P_{2}(t)U_{2}(|x_{2}|). (We remark that at this point we have V′ ≤ −\frac{k}{t} V so the zero solution is AS.)

Again for t ≥ 4 we have

V′ ≤ −\frac{1}{2t}(x^{2} + y^{2}) ≤ −\frac{1}{4t}|x + y|^{2} = −\frac{t}{4}|x′(t)|^{2}

or (vi) V′ ≤ −\frac{1}{4}|x′(t)|^{2} + 0 = −Q_{1}(|x′_{1}|) + M_{1}(t) so that for i = 1 we satisfy (vi) with M_{1}(t) = 0. Likewise, for t ≥ 4 we have

\begin{aligned}V′ & ≤ −\frac{1}{2t}|x| − \frac{t}{2}|y| + \frac{1}{2t} + \frac{t}{2} for |(x, y)| ≤ 1 \\ & ≤ −\frac{1}{4}|y′| + t; \end{aligned}

thus, for t ≥ 4 and |(x, y)| ≤ 1 we have (vi) V′ ≤ −\frac{1}{4}|y′| + t =: −Q_{2}(|x′_{2}|) + M_{2}(t). We see that (iv) is satisfied for i = 1, while for i = 2 we have

\begin{aligned}∫_{a}^{b} M_{2}(t)dt & = ∫_{a}^{b} tdt = \frac{t^{2}}{2} |_{a}^{b} = \frac{b + a}{2}(b − a) \\ & ≤ b(b − a); \end{aligned}

thus, if k = 1, t_{n} = n, b ≤ t_{n}, then we have

∫_{a}^{b} M_{2}(t)dt ≤ n(b − a) =: λ_{n}^{(2)}(b − a)

so that if D > 0, then for n − 1 ≤ s_{n} ≤ n − \frac{D}{n} we have

∫_{s_{n}}^{s_{n}+D/n} P_{2}(t)dt ≥ ∫_{s_{n}}^{s_{n}+D/n} \frac{t}{2}dt ≥ \frac{n − 1}{2} \frac{D}{n} ≥ \frac{D}{4} .

We have (ii) satisfied with t_{n} = n, λ_{n}^{(2)} = n, and (iv) satisfied for i = 2 with c_{n}^{(2)} = D/4. As V is autonomous, it is a Liapunov function and conditions of Theorem 1 (I) are satisfied. This completes the proof.

3. Proof of Theorem 1

We prove (I) first. Since V is a Liapunov functional we have W_{1}(|ϕ(0)|) ≤ V (t, ϕ) and V′_{(1)}(t, x_{t}) ≤ 0. The additional assumptions that V (t, ϕ) ≤ W (∥ϕ∥) yields US. For ε_{1} = H find δ_{1} of US and take γ = δ_{1} in the definition of UAS. Let μ > 0 be given and find the δ_{2} of US so that [∥ϕ∥ < δ_{2}, t_{0} ≥ 0, t ≥ t_{0}] imply that |x(t, t_{0}, ϕ)| < μ.

We will find T > 0 such that if ϕ ∈ C_{γ} and t_{0} ≥ 0, then |x(t, t_{0}, ϕ)| < μ if t ≥ t_{0} + T. Let x(t) = x(t, t_{0}, ϕ) and V (t) = V (t, x_{t}(t_{0}, ϕ)).

Consider the intervals S_{n} = [t_{n} − k, t_{n}], where we may suppose, by renumbering, that t_{n} − k ≥ t_{n−1}. For a given n, suppose that ∥x_{t}∥ ≥ δ_{2}. Then there is an r_{n} ∈ S_{n} with |x_{i}(r_{n})| ≥ δ_{2} for some i. Let −α_{n} = V (t_{n}) − V (t_{n} − k).

(a) If |x_{i}(t)| ≥ δ_{2}/2 for t ∈ S_{n}, then by (v) we have V′(t) ≤ −P_{i}(t)U_{i}(δ_{2}/2) on S_{n}. Let D = kλ, so that

−α_{n} = V (t_{n}) − V (t_{n} − k) ≤ −U_{i}(δ_{2}/2) ∫_{t_{n}−k}^{t_{n}} P_{i}(s)ds ≤ −c_{n}^{(i)}U_{i}(δ_{2}/2).

(b) If (a) fails, the there are p_{n} < q_{n} with [p_{n}, q_{n}] ⊂ S_{n} and with |x_{i}(t)| between δ_{2}/2 and δ_{2} on [p_{n}, q_{n}]; to be definite, say |x_{i}(p_{n})| = δ_{2}/2 and |x_{i}(q_{n})| = δ_{2}. To simplify arithmetic in Jensen’s inequality, let k ≤ 1. Then we integrate (vi), use Jensen’s inequality, and have

\begin{aligned}−α_{n} & ≤ V (q_{n}) − V (p_{n}) ≤ −Q_{i} (∫_{p_{n}}^{q_{n}} |x′_{i}(s)|ds) \\ & + ∫_{p_{n}}^{q_{n}} M_{i}(s)ds ≤ −Q_{i}(δ_{2}/2) + (q_{n} − p_{n})λ_{n}^{(i)}. \end{aligned}

If M_{i} = 0, then α_{n} ≥ Q_{i}(δ_{2}/2). (bi) If α_{n} ≥ Q_{i}(δ_{2}/2)/2, this will suffice for our proof. (bii) If α_{n} < Q_{i}(δ_{2}/2)/2, then D := Q_{i}(δ_{2}/2)/2 ≤ (q_{n} − p_{n})λ_{n}^{(i)}. We then integrate (v) and

have

\begin{aligned}−α_{n} & ≤ V (q_{n}) − V (p_{n}) ≤ −U_{i}(δ_{2}/2) ∫_{p_{n}}^{q_{n}} P_{i}(s)ds \\ & ≤ −U_{i}(δ_{2}/2) ∫_{p_{n}}^{p_{n}+D/λ_{n}^{(i)}} P_{i}(s)ds ≤ −c_{n}^{(i)}U_{i}(δ_{2}/2). \end{aligned}

From (a), (b), (bi) and (bii) we find

α_{n} ≥ min_{i}[c_{n}^{(i)}U_{i}(δ_{2}/2), Q_{i}(δ_{2}/2)/2] ≥ min_{i}[c_{0}U_{i}(δ_{2}/2), Q_{i}(δ_{2}/2)/2] =: α.

If t > t_{n}, then

0 ≤ V (t) ≤ V (t_{0}) − nα ≤ W (δ_{1}) − nα,

a contradiction if n > W (δ_{1})/α. Now there is a k > 0 with t_{n} − t_{n−1} ≤ k so we may select N > W (δ_{1})/α and then T = Nk. This completes the proof of (I).

The other proofs are parallel. We must only change t_{n} for (II), while in (III) we need to change t_{n} and c_{n}^{(i)}.

To prove (II) we first note that it is not vacuous. The zero solution is stable so there are solutions remaining in C_{H}. Suppose that x(t) remains in C_{H} and x_{i}(t) ↛ 0 as t → ∞.

Asymptotic Stability of Functional Differential Equations 10

Then there is an ε > 0 and a sequence {t_{n}} ↑ ∞ with t_{n+1} ≥ t_{n} + k and |x_{i}(t_{n})| ≥ ε. Let S_{n} = [t_{n} − k, t_{n}] and −α_{n} = V (t_{n}) − V (t_{n} − k) where V (t) = V (t, x_{t}). Using the same proof as in (I) we have

α_{n} ≥ min_{i}[c_{0}U_{i}(ε/2), Q_{i}(ε/2)/2] =: α.

If t > t_{n}, then 0 ≤ V (t) ≤ V (t_{0}) − nα, a contradiction for large n. This proves (II).

To prove (III), we note again that it is not vacuous, as in (II), and we consider a solution x(t) remaining in C_{H} on an interval [t_{0}, ∞). Suppose that x(t) ↛ 0 and note that V′(t, x_{t}) ≤ 0 so that if t ≥ t_{n} then W_{1}(|x(t)|) ≤ V (t, x_{t}) ≤ V (t_{n}, x_{t_{n}}) ≤ W (∥x_{t_{n}}∥); thus there is an ε > 0 with ∥x_{t_{n}}∥ ≥ ε and so there is an i for each n with |x_{i}(r_{n})| ≥ ε, where r_{n} ∈ [t_{n} − h, t_{n}]. Let S_{n} = [t_{n} − k, t_{n}]. Once again the same proof gives

α_{n} ≥ min_{i}[c_{n}^{(i)}U_{i}(ε/2), Q_{i}(ε/2)/2] ≥ min_{i}[c_{n}U_{i}(ε/2), Q_{i}(ε/2)/2].  (∗)

Since t > t_{n} yields

\begin{aligned}0 ≤ V (t, x_{t}) & ≤ V (t_{1}, x_{t_{1}}) − ∑_{i=2}^{n} α_{i} \\ & ≤ W (∥x_{t_{1}}∥) − ∑_{i=2}^{n} α_{i}, \end{aligned}  (∗∗)

the second choice in (*) can hold only for finitely many n. Since ∑_{n=0}^{∞} c_{n} = ∞, a contradiction results in (**) for large n. This completes the proof.

4. Proofs of the corollaries

First, note that Corollary 1 is just a statement of Theorem 1 (III) without a separate statement for each component. Also, λ_{n} = λ(t_{n}) will suffice, since P (t) = 1 and so

∫_{s_{n}}^{s_{n}+D/λ_{n}} 1dt = ∫_{s_{n}}^{s_{n}+D/λ(t_{n})} dt = \frac{D}{λ(t_{n})} =: c_{n}

and ∑ c_{n} diverges since ∫_{1}^{∞} \frac{dt}{λ(t)} diverges and λ is increasing.

Corollary 2 follows from Corollary 1 when we note that (iv) of Corollary 1 is satisfied, because for ∥ϕ∥ < 1 we have

V′(t, x_{t}) ≤ −W_{2}(|x(t)|) ≤ −|f(t, x_{t})| + J(t + 1)ln(t + 2)

11 T. A. Burton & G. Makay

and M(t) = J(t + 1)ln(t + 2) satisfies condition (iv) of Corollary 1.

Corollary 3 plays the role for Theorem 1 (I) that Corollary 1 plays for Theorem 1 (III). It merely avoids the component conditions.

References

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