1 Géza Makay

ON THE ASYMPTOTIC STABILITY OF THE SOLUTIONS

OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

GÉZA MAKAY

Bolyai Institute, Aradi vértanúk tere 1, Szeged, H-6720, Hungary Department of Mathematics, Southern Illinois University at Carbondale Carbondale, Illinois 62901-4408

1. Introduction

Denote by C the space of continuous functions ϕ : (−∞, 0] → R^{n} and BC ⊆ C the space of bounded continuous functions. In the function space BC we use the measure

∥ϕ∥_{g} := ∫_{−∞}^{0} |ϕ(s)|g(s)ds,

where g : R_{−} → R_{+} with ∫_{−∞}^{0} g(s)ds = 1 and | · | an arbitrary norm in R^{n}. If x : (−∞, T) → R^{n} is continuous and t ∈ (−∞, T), we define x_{t} ∈ C by x_{t}(s) := x(t + s) for s ≤ 0.

Consider the functional differential equation

ẋ = f(t, x_{t}),  (1.1)

where f : R × C → R^{n} is continuous and f(t, 0) = 0 for all t ∈ R. We suppose, that for each t_{0} ∈ R and each ϕ ∈ C there is a unique solution x(·; t_{0}, ϕ) defined on an interval (−∞, t_{0} + α), α > 0, and satisfying (1.1) on (t_{0}, α). We assume, that the function x continuously depends on the initial data t_{0}, ϕ. Sufficient conditions implying existence, uniqueness and continuous dependence can be found in [11].

Denote by K the set of strictly increasing continuous functions W : R_{+} → R_{+} with W (0) = 0. If V is a continuous functional from R×C into R, then V̇ denotes the derivative of functional V with respect to equation (1.1) defined by

V̇ (t, ϕ) = V̇_{(1.1)}(t, ϕ) = lim sup_{δ→0+} \frac{V (t + δ, x_{t+δ}(·; t, ϕ)) − V (t, ϕ)}{δ}.

On the Asymptotic Stability of Functional Differential Equations 2

In this paper we study the asymptotic stability of the zero solution of equation (1.1) using Lyapunov functionals. For the historical overview consider the following functional differential equations with finite delay

ẋ = f(t, x_{t}),  (1.2)

where x_{t}(s) = x(t + s) for s ∈ [−r, 0]. The first Lyapunov-type theorem for this equation can be found in the book of Krasovskii [8].

Theorem A: Suppose that there are a functional V and functions W_{1}, W_{2}, W_{3} ∈ K such that

W_{1}(|ϕ(0)|) ≤ V (t, ϕ) ≤ W_{2}(∥ϕ∥),
V̇_{(1.2)}(t, ϕ) ≤ −W_{3}(∥ϕ∥),

where t ∈ R, ϕ : [−r, 0] → R^{n} is continuous and ∥ϕ∥ = sup{|ϕ(s)| : s ∈ [−r, 0]}. Then the zero solution of equation (1.2) is uniformly asymptotically stable.

This theorem cannot be generalized to the case of equations with infinite delay, because if we have the supremum norm in the estimate of V̇, then V (t, x_{t}) has to tend to −∞ for every nonzero solution. Driver [5] proved the following theorem for equations with infinite delay using a pseudo norm |ϕ(0)|.

Theorem B: Suppose that for some M > 0 there exists H > 0 such that |f(t, ϕ)| ≤ H whenever t ∈ R and ϕ ∈ C_{M} := {ψ ∈ C : ∥ψ∥ ≤ M}. If, in addition, there are a continuous functional V : R × BC → R which is locally Lipschitz with respect to the second variable, and functions W_{1}, W_{2} ∈ K such that V (t, 0) = 0, W_{1}(|ϕ(0)|) ≤ V (t, ϕ) and V̇_{(1.1)}(t, ϕ) ≤ −W_{2}(|ϕ(0)|), then the zero solution of equation (1.1) is asymptotically stable.

The first condition of this theorem is commonly called ”the boundedness condition”. For equations with finite delay the following well known theorems are useful.

Theorem C ([6, Theorem 5.2.1]): Suppose that the boundedness condition holds for the function f and there are a continuous functional V and functions W_{1}, W_{2}, W_{3} ∈ K such that

W_{1}(|ϕ(0)|) ≤ V (t, ϕ) ≤ W_{2}(∥ϕ∥),  (1.3)

3 Géza Makay

V̇_{(1.2)}(t, ϕ) ≤ −W_{3}(|ϕ(0)|).  (1.4)

Then the zero solution of equation (1.2) is uniformly asymptotically stable.

Theorem D ([7, Theorem 4.1]): Suppose that all but condition (1.4) of Theorem C hold, and

V̇_{(1.2)}(t, ϕ) ≤ −W_{3}(|||ϕ|||),  (1.5)

where |||ϕ||| = \sqrt{∫_{−r}^{0} |ϕ(s)|ds}. Then the zero solution of equation (1.2) is uniformly asymptotically stable.

The main problem with these theorem is that the boundedness condition either holds or not for a given differential equation. It is an old problem (see [2, p. 252]), whether the boundedness condition can be dropped from Theorems C and D ([7],[10]). Burton and Hatvani ([3]) proved the following Theorem E and Burton [1] proved Theorem F which do not use this condition.

Theorem E: Suppose that there are a functional V and functions W_{1}, W_{2}, W_{3}, W_{4} ∈ K such that

W_{1}(|ϕ(0)|) ≤ V (t, ϕ) ≤ W_{2}(|ϕ(0)|) + W_{3}(|||ϕ|||),  (1.6)
V̇_{(1.2)}(t, ϕ) ≤ −W_{4}(|||ϕ|||).  (1.7)

Then the zero solution of equation (1.2) is uniformly asymptotically stable.

Theorem F: Suppose that all but condition (1.7) of Theorem E hold and

V̇_{(1.2)}(t, ϕ) ≤ −W_{4}(|ϕ(0)|)  (1.8)

holds. Then the zero solution of equation (1.2) is uniformly asymptotically stable.

In the following we generalize these theorems to the case of equations with infinite delay. The trouble with these theorems is the use of norm ||| · ||| which has no meaning in the space BC; therefore we will use the measure ∥ · ∥_{g}.

The other problem we investigate is how conditions in our theorems corresponding to (1.7) and (1.8) in Theorems E and F can be weakened. In practice we have a coefficient η(t) in the right-hand side of the estimate of V̇. We prove theorems based on such estimate for uniform asymptotic stability of functional differential equations with infinite delay.

2. Main results

First of all we prove a proposition needed by the proof our first theorem. This proposition states that if the ∥ · ∥_{g} measure of a solution is large at one moment, then it is large in a small interval as well.

Proposition 1: Suppose, that x ∈ C_{M} and g : R_{−} → R_{+} is continuous and bounded function. Then there exists W ∈ K such that for all ε > 0 there exists a δ_{0} > 0 such that ∥x_{t_{1}}∥_{g} ≥ ε implies ∥x_{t}∥_{g} ≥ W (ε) for all t ∈ (t_{1} − δ_{0}, t_{1} + δ_{0}).

Proof: Suppose that g(s) ≤ K and ε ≤ ∥x_{t_{1}}∥_{g}. There exists a constant S > 0 such that ∫_{−∞}^{−S} g(s)ds ≤ ε/3M (we can assume, that S as a function of ε is monotone decreasing). Then ∫_{−∞}^{0} |x_{t_{1}}(s)|g(s)ds ≤ ε/3 + ∫_{−S}^{0} |x_{t_{1}}(s)|g(s)ds, so 2ε/3 ≤ ∫_{−S}^{0} |x_{t_{1}}(s)|g(s)ds. Define the following constants and set:

\begin{matrix}r := ε/3S & Q_{r} := \{s ∈ [−S, 0] : |x_{t_{1}}(s)|g(s) ≥ r\}, \\ u := r/2M & R_{u} := \{s ∈ [−S, 0] : g(s) > u\}, \\ v := r/2K & S_{v}(t) := \{s ∈ [−S, 0] : |x_{t}(s)| > v\}. \end{matrix}

Then 2ε/3 ≤ ∫_{−S}^{0} |x_{t_{1}}(s)|g(s)ds = ∫_{Q_{r}} + ∫_{Q_{r}^{c}} ≤ μ(Q_{r})MK + μ(Q_{r}^{c})r, so μ(Q_{r}) ≥ (2ε/3 − Sr)/(MK − r) ≥ ε/3MK. Moreover, it can be proved easily that Q_{r} ⊆ R_{u} ∩ S_{v}(t_{1}) and μ(R_{u}∩S_{v}(t_{1})) ≥ μ(Q_{r}) ≥ ε/3MK. Since R_{u} is an open set, there are n > 0, a_{1}, b_{1}, ..., a_{n}, b_{n} such that (a_{i}, b_{i}) ⊆ R_{u} (a_{i} < b_{i} ≤ a_{i+1}) and μ((∪(a_{i}, b_{i})) ∩ S_{v}(t_{1})) ≥ ε/6MK. Let δ_{0} := ε/(12MKn) and −δ_{0} < δ < δ_{0} (δ_{0} depends only on ε, M, K and g). Obviously

\begin{aligned}S_{v}(t_{1}) − δ & = \{s − δ : s ∈ [−S, 0], |x_{t_{1}}(s)| > v\} \\ & = \{w ∈ [−S − δ, −δ] : |x_{t_{1}+δ}(w)| > v\}, \end{aligned}

so (S_{v}(t_{1}) − δ) ∩ [−S, 0] ⊆ S_{v}(t_{1} + δ). (a_{i}, b_{i}) ⊆ [−S, 0] implies, that

μ((a_{i}, b_{i}) ∩ S_{v}(t_{1} + δ)) ≥ μ((a_{i}, b_{i}) ∩ ((S_{v}(t_{1}) − δ) ∩ [−S, 0]))
= μ((a_{i}, b_{i}) ∩ (S_{v}(t_{1}) − δ))
≥ μ((a_{i}, b_{i}) ∩ S_{v}(t_{1})) − δ_{0},

so ∑ μ((a_{i}, b_{i}) ∩ S_{v}(t_{1} + δ)) ≥ ∑ μ((a_{i}, b_{i}) ∩ S_{v}(t_{1})) − nδ_{0} ≥ ε/12MK. If t = t_{1} + δ (t ∈ (t_{1} − δ_{0}, t_{1} + δ_{0})), then

\begin{aligned}∥x_{t}∥_{g} & = ∫_{−∞}^{0} |x_{t}(s)|g(s)ds ≥ ∫_{(∪(a_{i},b_{i}))∩S_{v}(t)} |x_{t}(s)|g(s)ds \\ & ≥ vuμ((∪(a_{i}, b_{i})) ∩ S_{v}(t)) ≥ ε^{3}/(432M^{2}S^{2}K^{2}) =: W (ε) \end{aligned}

5 Géza Makay

and the proof is complete.

Definition: Let S_{0} := inf{S ≥ 0 : ∫_{−S}^{0} g(s)ds > 0}.

Lemma 1: Let S > S_{0} and ν_{0} := ∫_{−S}^{0} g(s)ds > 0. If for some t′ ∈ R we have |x(t)| + ∥x_{t}∥_{g} ≥ ε for all t ∈ [t′ − S, t′], then there exists a t′′ ∈ [t′ − S, t′] such that ∥x_{t′′}∥_{g} ≥ εν_{0}/(1 + ν_{0}).

Proof: Suppose that ∥x_{t}∥_{g} < εν_{0}/(1 + ν_{0}) for all t ∈ [t′ − S, t′]. Then from |x(t)| + ∥x_{t}∥_{g} ≥ ε we have |x(t)| ≥ ε/(1 + ν_{0}), so

\begin{aligned}∥x_{t′}∥_{g} & = ∫_{−∞}^{0} |x_{t′}(s)|g(s)ds ≥ ∫_{−S}^{0} |x_{t′}(s)|g(s)ds \\ & ≥ ∫_{−S}^{0} ε/(1 + ν_{0})g(s)ds = εν_{0}/(1 + ν_{0}), \end{aligned}

which is a contradiction, and the proof is complete.

Now we can prove our first theorem.

Theorem 1: Assume that g : R_{−} → R_{+} is bounded and continuous. Suppose that there exists V : R × C → R a continuous functional, W_{1}, W_{2}, W_{3} ∈ K such that

(a) W_{1}(|x(t)|) ≤ V (t, x_{t}) ≤ W_{2}(|x(t)| + ∥x_{t}∥_{g}),

(b) V̇ (t, x_{t}) ≤ −W_{3}(∥x_{t}∥_{g}). Then the zero solution of equation (1.1) is uniformly asymptotically stable.

Proof: The uniform stability follows immediately. Let γ be the number for 1 in uniform stability, so if t_{0} ∈ R and ∥x_{t_{0}}∥ < γ then |x(t)| ≤ 1 for all t ≥ t_{0}. Moreover V (t_{0}, x_{t_{0}}) ≤ W_{2}(|x(t_{0})| + ∥x_{t_{0}}∥_{g}) ≤ W_{2}(2). Let S > S_{0} and 0 < ν_{0} := ∫_{−S}^{0} g(s)ds. For every ω > 0 we must find a T > 0 such that if t_{0} ∈ R, then ∥x_{t_{0}}∥ < γ implies |x(t)| < ω for all t ≥ t_{0} + T. Let N be a natural number, δ_{0} > 0 be a real number and T := (S + 2δ_{0})N. Suppose, for a contradiction, that there exists t′ ≥ t_{0} + T such that |x(t′)| ≥ ω. Now for t ∈ [t_{0}, t′] we have

\begin{aligned}|x(t)| + ∥x_{t}∥_{g} & ≥ W_{2}^{−1}(V (t, x_{t})) ≥ W_{2}^{−1}(V (t′, x_{t′})) \\ & ≥ W_{2}^{−1}(W_{1}(|x(t′)|)) ≥ W_{2}^{−1}(W_{1}(ω)) =: ε. \end{aligned}

From Lemma 1 we have t_{i} ∈ [t_{0} + δ_{0}(2i − 1) + (i − 1)S, t_{0} + (2i − 1)δ_{0} + iS] (i = 1, 2, ..., N) such that ∥x_{t_{i}}∥_{g} ≥ εν_{0}/(1 + ν_{0}). From Proposition 1 we have the above δ_{0} and W ∈ K such that ∥x_{t}∥_{g} ≥ W (εν_{0}/(1 + ν_{0})) for all t ∈ (t_{i} − δ_{0}, t_{i} + δ_{0}) (i = 1, 2, ..., N). Obviously the intervalls (t_{i} − δ_{0}, t_{i} + δ_{0}) are disjoint. Now

V (t_{0} + T, x_{t_{0}+T}) ≤ V (t_{0}, x_{t_{0}}) + ∫_{t_{0}}^{t_{0}+T} V̇ (s, x_{s})ds ≤ W_{2}(2) − ∑ ∫_{t_{i}−δ_{0}}^{t_{i}+δ_{0}} W_{3}(∥x_{s}∥_{g})ds
≤ W_{2}(2) − ∑ ∫_{t_{i}−δ_{0}}^{t_{i}+δ_{0}} W_{3}(W (εν_{0}/(1 + ν_{0})))ds
= W_{2}(2) − 2Nδ_{0}W_{3}(W (εν_{0}/(1 + ν_{0}))).

Now let N := [W_{2}(2)/(2δ_{0}W_{3}(W (εν_{0}/(1 + ν_{0})))] + 1, so V (t_{0} + T, x_{t_{0}+T}) < 0 which is a contradiction, and the proof is complete.

Theorem 1 can be generalized, if we use a function η in condition (b) in the following way: V̇ (t, x_{t}) ≤ −η(t)W_{3}(∥x_{t}∥_{g}). In this case we have the following inequality:

V (t_{0} + T, x_{t_{0}+T}) ≤ W_{2}(2) − W_{3}(W (εν_{0}/(1 + ν_{0}))) ∑ ∫_{t_{i}−δ_{0}}^{t_{i}+δ_{0}} η(s)ds,

so we need η to be ”large” in the following sense.

Definition: The function η : R → R_{+} is uniformly weakly integrally positive with parameters δ and S (UWIP(δ, S)) iff for all M > 0 there exists a Q_{M} > 0 such that if Q ≥ Q_{M}, and t_{i}, δ_{i} (i=1,2,...) satisfy δ_{i} ≥ δ, t_{i} + δ_{i} < t_{i+1} − δ_{i+1} < t_{i} + δ_{i} + S and I := ∪(t_{i} − δ_{i}, t_{i} + δ_{i}), then ∑ ∫_{I∩[t_{1}−δ_{1},t_{1}−δ_{1}+Q]} η(s)ds > M.

Theorem 2: Suppose that all but condition (b) are satisfied in Theorem 1. If we have a function η : R → R_{+} which is UWIP(δ, 2S) for every δ > 0 and for some S > S_{0}, and if

(b) V̇ (t, x_{t}) ≤ −η(t)W_{3}(∥x_{t}∥_{g}), then the zero solution of equation (1.1) is uniformly asymptotically stable.

Proof: The proof is left to the reader.

Conditions of Theorem 2 can be further weakened, if the function g satisfies a stronger condition.

Proposition 2: Suppose that the functions x : R → R_{+} and g : R_{−} → R_{+} are bounded, and there exists a positive, monotone increasing function g_{1} : R_{−} → R_{+} such

7 Géza Makay

that g_{1}(s) ≤ g(s). Then for every δ_{0} > 0 and ε > 0 there exists γ > 0 such that if ∥x_{t_{1}}∥_{g} ≥ ε for some t_{1} ≥ t_{0} then ∥x_{t}∥_{g} ≥ γ for all t ∈ [t_{1}, t_{1} + δ_{0}].

Proof: Let M, K > 0 be given such that |x(t)| ≤ M and g(s) ≤ K. Then there exists S > 0 such that ∫_{−∞}^{−S} g(s))ds ≤ ε/2M so (as in proof of Proposition 1) we have ∫_{−S}^{0} |x_{t_{1}}(s)|g(s)ds ≥ ε/2. If t ∈ [t_{1}, t_{1} + δ_{0}] then

∫_{−∞}^{0} |x_{t}(s)|g(s)ds ≥ ∫_{−S−(t−t_{1})}^{−(t−t_{1})} |x_{t}(s)|g(s)ds = ∫_{−S}^{0} |x_{t_{1}}(s)|g(s − (t − t_{1}))ds
= ∫_{−S}^{0} |x_{t_{1}}(s)|g(s)g(s − (t − t_{1}))/g(s)ds
≥ ∫_{−S}^{0} |x_{t_{1}}(s)|g(s)g_{1}(s − (t − t_{1}))/Kds
≥ ∫_{−S}^{0} |x_{t_{1}}(s)|g(s)g_{1}(−S − δ_{0})/Kds ≥ εg_{1}(−S − δ_{0})/2K =: γ

and the proof is complete.

This proposition can be used instead of Proposition 1 in the proof of Theorem 2, so we have proved the following

Theorem 3: Suppose that all but condition (b) of Theorem 2 are satisfied. Assume, that there exist a positive, monotone increasing function g_{1}, with g_{1}(s) ≤ g(s), and an η, which is UWIP(δ, 2S) for some δ > 0 and S > S_{0} such that

(b) V̇ (t, x_{t}) ≤ −η(t)W_{3}(∥x_{t}∥_{g}). Then the zero solution of equation (1.1) is uniformly asymptotically stable.

In the following we prove a theorem using the measure |ϕ(0)| and a function η in the above estimate of V̇. The following proposition corresponds to Proposition 1 in Theorems 1, 2 and 3.

Proposition 3: Suppose, that g(s) ≤ K and |x(t)| ≤ M. Let ε > 0 be given and let S′ > 0 be such that ∫_{−∞}^{−S′} g(s)ds ≤ ε/3M and S > S′. If there exists t_{1} ≥ t_{0} + S such that |x(t)| + ∥x_{t}∥_{g} ≥ ε for all t ∈ [t_{1} − S, t_{1}] then there exists Q ⊆ [t_{1} − S, t_{1}] such that μ(Q) ≥ ε/4MK and |x(t)| ≥ ε/4SK for all t ∈ Q.

Proof: If it is not the case, then we have t_{2} ∈ [t_{1} − ε/4MK, t_{1}] such that |x(t_{2})| < ε/4SK. We can suppose, that M and K are large enough that the inequality S ≥ S′ +

On the Asymptotic Stability of Functional Differential Equations 8

ε/4MK holds. Then from |x(t_{2})| + ∥x_{t_{2}}∥_{g} ≥ ε we have ∥x_{t_{2}}∥_{g} ≥ ε − ε/4SK. We can assume that 4SK ≥ 6 so ∥x_{t_{2}}∥_{g} ≥ 5ε/6. Then we can easily prove the inequality ε/2 ≤ ∫_{−S′}^{0} |x_{t_{2}}(s)|g(s)ds. Let r := ε/4S′ and

Q_{r} := {s ∈ [−S′, 0] : |x_{t_{2}}(s)|g(s) ≥ r}.

Now (as in Proposition 1) we have μ(Q_{r}) ≥ ε/4MK. Moreover if s ∈ Q_{r} then |x_{t_{2}}(s)| ≥ r/g(s) ≥ ε/4SK. Let Q = Q_{r} + t_{2} = {s + t_{2} : s ∈ Q_{r}} and the proof is complete.

Definition: The function η : R → R_{+} is uniformly positive in measure with parameter S (UPIM(S)) iff for every ε > 0 there exists a δ > 0 such that if t ≥ t_{0} +S, Q ⊆ [t−S, t] and μ(Q) ≥ ε then ∫_{Q} η(s)ds ≥ δ.

Theorem 4: Assume that g : R_{−} → R_{+} is bounded. Suppose that there exists V : R × C → R a continuous functional, W_{1}, W_{2}, W_{3} ∈ K such that

(a) W_{1}(|x(t)|) ≤ V (t, x_{t}) ≤ W_{2}(|x(t)| + ∥x_{t}∥_{g}),

(b) V̇ (t, x_{t}) ≤ −η(t)W_{3}(|x(t)|), where η is UPIM(S) for all S > 0. Then the zero solution of equation (1.1) is uniformly asymptotically stable.

Proof: The solution is uniformly stable. Let γ be the number for 1 in uniform stability, so if t_{0} ∈ R and ∥x_{t_{0}}∥ < γ then |x(t)| ≤ 1 for all t ≥ t_{0}. Moreover V (t_{0}, x_{t_{0}}) ≤ W_{2}(|x(t_{0})|+ ∥x_{t_{0}}∥_{g}) ≤ W_{2}(2). For all ω > 0 we must find a T > 0 such that if t_{0} ∈ R, then ∥x_{t_{0}}∥ < γ implies |x(t)| < ω for all t ≥ t_{0}+T. Let N be a natural number and T := SN, where S is given in Proposition 3. Suppose for a contradiction, that there exists t′ ≥ t_{0} + T such that |x(t′)| ≥ ω. Now for t ∈ [t_{0}, t′] we have

\begin{aligned}|x(t)| + ∥x_{t}∥_{g} & ≥ W_{2}^{−1}(V (t, x_{t})) ≥ W_{2}^{−1}(V (t′, x_{t′})) \\ & ≥ W_{2}^{−1}(W_{1}(|x(t′)|)) ≥ W_{2}^{−1}(W_{1}(ω)) =: ε. \end{aligned}

From Proposition 3 we have Q_{i} ⊆ [t_{0} + (i − 1)S, t_{0} + iS] (i = 1, 2, ..., N) such that |x(t)| ≥ ε/4SK for all t ∈ Q_{i} and μ(Q_{i}) ≥ ε/4MK. Now

\begin{aligned}V (t_{0} + T, x_{t_{0}+T}) & ≤ V (t_{0}, x_{t_{0}}) + ∫_{t_{0}}^{t_{0}+T} V̇ (s, x_{s})ds ≤ W_{2}(2) − ∑ ∫_{Q_{i}} η(s)W_{3}(|x(s)|)ds \\ & ≤ W_{2}(2) − W_{3}(ε/4SK) ∑ ∫_{Q_{i}} η(s)ds ≤ W_{2}(2) − W_{3}(ε/4SK)Nδ, \end{aligned}

where δ is in the definition of uniform positivity in measure for ε/4MK. Define

N := [W_{2}(2)/(W_{3}(ε/4SK)δ)] + 1,

so V (t_{0} + T, x_{t_{0}+T}) < 0, which is a contradiction. Then the proof is complete.

Note that this theorem for η ≡ 1 can be found in the papers of Burton ([1]) and Burton and Zhang ([4]).

At the end we state two propositions in order to generalize our theorems. First of all we generalize the measure ∥ · ∥_{g}.

Definition: ∥ϕ∥_{g}^{h} := ∫_{−∞}^{0} h(|ϕ(s)|)g(s)ds, where h : R_{+} → R_{+} and ∫_{−∞}^{0} g(s)ds < ∞.

Definition: The measures ∥ · ∥_{g_{1}}^{h_{1}} and ∥ · ∥_{g_{2}}^{h_{2}} are equivalent, iff for all M > 0 there exist W_{1}, W_{2} ∈ K such that W_{1}(∥ϕ∥_{g_{1}}^{h_{1}}) ≤ ∥ϕ∥_{g_{2}}^{h_{2}} ≤ W_{2}(∥ϕ∥_{g_{1}}^{h_{1}}) for all ϕ ∈ C_{M}.

It is obvious, that if two measures are equivalent, then they can be used interchangable in theorems using Lyapunov functionals for local stability. The following theorems prove equivalencies, which generalize our theorems.

Proposition 4: Let the functions h_{1}, h_{2} : R_{+} → R_{+} with W_{1}, W_{2} ∈ K be such that W_{1}(h_{1}(s)) ≤ h_{2}(s) ≤ W_{2}(h_{1}(s)) and g_{1}, g_{2} : R_{−} → R_{+} with W_{3}, W_{4} ∈ K such that W_{3}(g_{1}(s)) ≤ g_{2}(s) ≤ W_{4}(g_{1}(s)). Suppose that there exists K > 0 such that g_{1}(s), g_{2}(s) ≤ K and for all M > 0 there exists H(M) > 0 such that h_{1}(s), h_{2}(s) ≤ H for all s ∈ [0, M]. Then the measures ∥ · ∥_{g_{1}}^{h_{1}} and ∥ · ∥_{g_{2}}^{h_{2}} are equivalent.

Proof: Let M > 0 be given, ϕ ∈ C_{M}, H = H(M) and ε := ∥ϕ∥_{g_{2}}^{h_{2}}. If ε = 0 then we have ∥ϕ∥_{g_{1}}^{h_{1}} ≥ W (∥ϕ∥_{g_{2}}^{h_{2}}) for all W ∈ K, so we can assume ε > 0. There exists 0 < S = S(ε, M) such that ∫_{−∞}^{−S} g_{2}(s)ds ≤ ε/3H. (We can assume, that S(ε) is a monotone decreasing function.) Then (as in Proposition 1)

2ε/3 ≤ ∫_{−S}^{0} h_{2}(|ϕ(s)|)g_{2}(s)ds.

Let r := ε/3S and Q_{r} := {s ∈ [−S, 0] : h_{2}(|ϕ(s)|)g_{2}(s) ≥ r}. Then we have the inequality

μ(Q_{r}) ≥ (2ε/3 − Sr)/(HK − r) ≥ ε/3HK

On the Asymptotic Stability of Functional Differential Equations 10

and if s ∈ Q_{r}, then h_{2}(|ϕ(s)|) ≥ r/g_{2}(s) ≥ r/K and g_{2}(s) ≥ r/h_{2}(|ϕ(s)|) ≥ r/H. So

∥ϕ∥_{g_{1}}^{h_{1}} ≥ ∫_{Q_{r}} h_{1}(|ϕ(s)|)g_{1}(s)ds
≥ ∫_{Q_{r}} W_{2}^{−1}(h_{2}(|ϕ(s)|))W_{4}^{−1}(g_{2}(s))ds
≥ W_{2}^{−1}(r/K)W_{4}^{−1}(r/H)μ(Q_{r})
≥ W_{2}^{−1}(r/K)W_{4}^{−1}(r/H)ε/3HK =: W (ε)
= W (∥ϕ∥_{g_{2}}^{h_{2}}),

and the proof is complete.

Proposition 5: Measures ∥x_{t}∥_{g} + |x(t)| and W_{1}(∥x_{t}∥_{g}) + W_{2}(|x(t)|) are equivalent. Proof:

W_{1}(|x(t)|) + W_{2}(∥x_{t}∥_{g}) ≤ W (|x(t)|) + W (∥x_{t}∥_{g}) ≤ 2W (|x(t)| + ∥x_{t}∥_{g}),

where W (x) =max{W_{1}(x), W_{2}(x)}. On the other hand,

W_{1}(|x(t)|) + W_{2}(∥x_{t}∥_{g}) ≥ W (|x(t)|) + W (∥x_{t}∥_{g}) ≥ W ((|x(t)| + ∥x_{t}∥_{g})/2), where W (x) =min{W_{1}(x), W_{2}(x)} and the proof is complete.

It is left to the reader to state the generalized version of Theorems 1 through 4 using Propositions 4 and 5.

3. Example

Consider the following functional differential equation with infinite delay

ẋ(t) = −a(t)f(x(t)) + b(t) ∫_{−∞}^{t} C(t − s)g(x(s))ds,  (3.1)

where f, g : R^{n} → R, a, b : R → R and C : R_{+} → R. Define the functional

V (t, x_{t}) = |x(t)| + ∫_{−∞}^{t} ∫_{t}^{∞} β(u)|C(u − s)|du|g(x(s))|ds

where β : R → R_{+}.

First we prove condition (a) of our theorems. It is obvious that |x(t)| ≤ V (t, x_{t}). Suppose, that there exist functions β : R → R_{+} and W_{1}, W_{3} ∈ K such that

∫_{t}^{∞} β(u)|C(u − s)|du ≤ W_{1}(|C(t − s)|), and  (3.2)
|g(x)| ≤ W_{3}(|x|).  (3.3)

From Proposition 4 we have W_{2} ∈ K such that

V (t, x_{t}) ≤ |x(t)| + W_{2} (∫_{−∞}^{t} |C(t − s)||x(s)|ds) ,

and condition (a) is satisfied if

∫_{0}^{∞} |C(s)|ds < ∞.  (3.4)

For condition (b) we prove an inequality for the derivative of the functional V with respect to equation (3.1).

\begin{aligned}V̇_{(3.1)}(t, x_{t}) = & − a(t)|f(x(t))| + |b(t)| ∫_{−∞}^{t} |C(t − s)||g(x(s))|ds− \\ & ∫_{−∞}^{t} β(t)|C(t − s)||g(x(s))|ds + ∫_{t}^{∞} β(u)|C(u − t)|du|g(x(t))| \end{aligned}

Suppose, that

|g(x)| ≤ c|f(x)|  (3.5)

for some c ≥ 0, then

\begin{aligned}V̇_{(3.1)}(t, x_{t}) ≤ & − [a(t) − c ∫_{t}^{∞} β(u)|C(u − t)|du] |f(x(t))|− \\ & [β(t) − |b(t)|] ∫_{−∞}^{t} |C(t − s)||g(x(s))|ds \end{aligned}

Now we present an example for Theorem 2. Suppose, that there are functions β, W_{1} and W_{3}, W_{4} and a constant c such that conditions (3.2), (3.4) and (3.5) are hold,

|g(x)| ≥ W_{4}(|x|)  (3.6)

and [a(t) − c ∫_{t}^{∞} β(u)|C(u − t)|du] ≥ 0. Then

\begin{aligned}V̇_{(3.1)}(t, x_{t}) ≤ & − [β(t) − |b(t)|] ∫_{−∞}^{t} |C(t − s)||g(x(s))|ds ≤ \\ & − [β(t) − |b(t)|] W_{5} (∫_{−∞}^{t} |C(t − s)||x(s)|ds) \end{aligned}

and condition (b) is satisfied if the function [β(t) − |b(t)|] is UWIP(δ, 2S) for every δ > 0 and some S > S_{0}. So we proved the following

Corollary 1: Suppose, that there are β : R → R_{+}, c ≥ 0 and W_{1}, W_{3} ∈ K such that conditions (3.2), (3.3), (3.4), (3.5) and (3.6) are hold, [a(t) − c ∫_{t}^{∞} β(u)|C(u − t)|du] ≥ 0, and [β(t) − |b(t)|] is UWIP(δ, 2S) for some δ > 0 and S > S_{0}. Then the zero solution of equation (3.1) is uniformly asymptotically stable.

For Theorem 4 we assume that β(t) = |b(t)|, |f(x)| ≥ W_{2}(|x|) for some W_{2} ∈ K and the function [a(t) − c ∫_{t}^{∞} |b(u)||C(u − t)|du] is UPIM(S) for all S > 0. Then

V̇_{(3.1)}(t, x_{t}) ≤ − [a(t) − c ∫_{t}^{∞} |b(u)||C(u − t)|du] W_{2}(|x|),

and condition (b) of Theorem 4 is satisfied. Therefore

Corollary 2: If there are W_{1}, W_{2}, W_{3} ∈ K and c ≥ 0 such that conditions (3.1), (3.2), (3.3), (3.4) and (3.5) hold, |f(x)| ≥ W_{2}(|x|) for some W_{2} ∈ K and the function [a(t) − c ∫_{t}^{∞} |b(u)||C(u − t)|du] is UPIM(S) for all S > 0, then the zero solution of equation (3.1) is uniformly asymptotically stable.

Acknowledgement: The author wishes to thank Prof. T. Burton and Prof. L. Hatvani for their valuable suggestions.

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13 Géza Makay

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