Differential Equations, Vol. 38, No. 10, 2002, pp. 1423–1434. Translated from Differentsial’nye Uravneniya, Vol. 38, No. 10, 2002, pp. 1338–1347.
Original Russian Text Copyright c 2002 by Sedova.
ORDINARY
DIFFERENTIAL EQUATIONS
A Remark on the Lyapunov–Razumikhin Method
for Equations with Infinite Delay
N. O. Sedova
Ul’yanovsk State University, Ul’yanovsk, Russia
Received July 22, 1999
The choice of an appropriate phase space plays an important role in the construction of the
theory of equations with unbounded (infinite) delay, since it affects even the results concerning
the existence and uniqueness of solutions (see the survey [1]). An axiomatic approach to this
problem was developed in [2], where a number of axioms dealing with the phase space and the
right-hand side of the equation were introduced. Any space and functional satisfying these axioms
necessarily have specific properties; in particular, they satisfy conditions guaranteeing the existence
and uniqueness of solutions. This approach (later used in numerous papers related to the investigation of various properties of such equations, e.g., see [3–7]) permits one to define conditions
under which it is possible to construct limit equations with properties similar to those obtained for
ordinary equations in [8–10] and for functional-differential equations with finite delay in [11, 12].
The method of limit equations allows one to derive sufficient conditions for Lyapunov asymptotic
stability via finite-dimensional functions; for some class of nonautonomous equations, these conditions are more general than the well-known Lyapunov–Razumikhin results for equations with
unbounded delay [13–15].
1. MAIN DEFINITIONS, ASSUMPTIONS, AND AUXILIARY ASSERTIONS
Let B be a real vector space of one of the following types:
(i) a space of continuous functions on (−∞, 0] ranging in Rn ; two functions ϕ, ψ ∈ B are equal
(ϕ = ψ) if ϕ(s) = ψ(s) for all s ∈ (−∞, 0];
(ii) a space of measurable functions on (−∞, 0] ranging in Rn ; two functions ϕ, ψ ∈ B are equal
(ϕ = ψ) if ϕ(s) = ψ(s) for almost all s ∈ (−∞, 0] and ϕ(0) = ψ(0).
We denote the norm in Rn by | · |. We also assume that B is equipped with a norm | · |B such
that (B, | · |B ) is a Banach space. For a function x : (−∞, A) → Rn , 0 < A ≤ +∞, we introduce a
function xt : (−∞, 0] → Rn by setting xt (s) = x(t + s), s ≤ 0, for each t ∈ [0, A).
Definition 1 [13]. A space B is said to be admissible if there exist constants K, J > 0 and
a continuous function M : [0, ∞) → [0, ∞) such that the following conditions are satisfied. Let
0 ≤ a < A ≤ ∞. If x : (−∞, A) → Rn is continuous on [a, A) and xa ∈ B, then, for all t ∈ [a, A),
(B1) xt ∈ B, and xt is continuous with respect to t in | · |B ;
(B2) |xt |B ≤ K maxa≤s≤t |x(s)| + M (t − a) |xa |B ;
(B3) |ϕ(0)| ≤ J|ϕ|B for all ϕ ∈ B;
(B4) M (t) → 0 as t → ∞.
Let B be an admissible separable space, and let BH = {ϕ ∈ B : |ϕ|B < H}. Suppose that if ϕ
is bounded and continuous on (−∞, 0], then ϕ ∈ B and all functions ϕ−t , t ≥ 0, are bounded in
the norm of B, i.e., |ϕ−t |B ≤ L for some L > 0 [3] (here ϕ−t (s) = ϕ(−t + s), s ∈ (−∞, 0]).
Consider the system of functional-differential equations
ẋ(t) = X (t, xt ) ,
(SI)
0
where X : R × BH → R is a continuous mapping bounded on each set R × D , that is,
|X(t, ϕ)| ≤ m (D0 ) for all (t, ϕ) ∈ R+ × D 0 , whenever D0 is a closed bounded subset of BH . Then
+
n
0012-2661/02/3810-1423$27.00 c 2002 MAIK “Nauka/Interperiodica”
+
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SEDOVA
for each initial point (α, ϕ) ∈ R+ ×BH , there exists a nonextendable solution x(t; α, ϕ) of system (SI)
defined for t ∈ [−∞, β) with some β > α, that is, a continuous solution satisfying Eq. (SI) on [α, β)
and such that xα = ϕ; moreover, if x(t) is a nonextendable solution of (SI) on [α, β) such that
{xt : α < t < β} lies in a closed bounded subset of BH , then β = ∞ [13].
Suppose that the following assumption holds.
Assumption 1. The functional X(t, ϕ) is uniformly continuous on each set R+ × K, where
K ⊂ BH is a compact set; that is, for each ε > 0, there exists a δ = δ(ε, K) such that for all
(t1 , ϕ1 ) , (t2 , ϕ2 ) ∈ R+ × K, the inequalities |t2 − t1 | < δ and |ϕ2 − ϕ1 |B < δ imply that
|X (t2 , ϕ2 ) − X (t1 , ϕ1 )| < ε.
In this case, the family {Xτ (t, ϕ) = X(τ + t, ϕ), τ ∈ R+ } of translates is precompact in the
space of continuous functions defined on R+ × BH , and to system (SI), one can assign the family
of limit systems [16, 17]
ẋ(t) = X ∗ (t, xt ) ,
(1)
∗
where X (t, ϕ) is the limit functional of X determined by the sequence tk → +∞ as follows:
X ∗ (t, ϕ) = limk→∞ X (t + tk , ϕ), (t, ϕ) ∈ R+ × BH . (Here the convergence is uniform on each set
[0, T ] × K, where T > 0 and K is a compact set in BH .)
Assumption 2. The functional X(t, ϕ) satisfies the Lipschitz condition; i.e., for each compact
set K ⊂ BH , there exists an l = l(K) > 0 such that the inequality
|X (t, ϕ2 ) − X (t, ϕ1 )| ≤ l |ϕ2 − ϕ1 |B
(2)
holds for all ϕ1 , ϕ2 ∈ K.
By condition (2), the solutions of Eqs. (SI) and (1) are unique for each point (α, ϕ) ∈ R+ × BH .
We need the following theorems.
Theorem 1 [2]. Let xk (t) be a solution of the system ẋ(t) = X k (t, xt ) on [α, γ]. Suppose that
there exists a function x(t) defined on (−∞, γ] and such that xk (t) → x(t), xkt → xt on [α, γ], and
X k (t, ϕ) → X(t, ϕ) uniformly on a set Γ0 ⊂ BH containing t, xkt : t ∈ [α, γ], k ≥ 1 . Then x(t)
is a solution of system (SI) on [α, γ].
Theorem 2 [3]. Let x : (−∞, +∞) → Rn be a uniformly continuous function bounded on
[α, +∞), and let xα ∈ B. The positive orbit of the solution {xt : t ≥ α} of Eq. (SI) is precompact in B, and if x (tn + s) → ϕ(s) uniformly with respect to s on each closed interval [−T, 0]
as tn → +∞, then ϕ ∈ B and |xtn − ϕ|B → 0 as tn → +∞.
The following assertion clarifies the relationship between the solutions of Eqs. (SI) and (1).
Lemma 1. Suppose that tn → +∞, {ϕn } ∈ K ⊂ BH , ϕn → ϕ ∈ BH as n → ∞, and the functions x (t; tn + α, ϕn ) are solutions of system (SI). Then the sequence {xn (t) = x (t + tn ; tn + α, ϕn )}
lies in some compact set Kx ⊂ BH . If X ∗ (t, xt ) is the limit functional of X (t, xt ) with respect to the
sequence tn → +∞ and x∗ (t; α, ϕ) is a solution of system (1) on (−∞, β), then {xn (t)} converges
to x∗ (t; α, ϕ), and xnt → x∗t uniformly with respect to t ∈ [α, γ] for each γ ∈ (α, β).
Proof. We take an arbitrary γ ∈ (α, β). It follows from the assumptions about the function
X (t, xt ) that the sequence {xn (t) : t ∈ [α, γ]} is uniformly bounded and equicontinuous. Therefore,
extracting a subsequence if necessary, one can suppose that xn (t) converges to some function x̄(t)
uniformly on [α, γ]; moreover, xnα = ϕn → ϕ. It follows from assumption (B3) that ϕn (0) → ϕ(0),
and so x̄(α) = ϕ(0). We introduce the function
x̄(t)
if t ≥ α
0
x (t) =
ϕ(t − α) if t < α.
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The function x0 (t) is continuous on [α, γ], and ϕ ∈ B; consequently, x0t ∈ B for all t ∈ [α, γ]. From
assumption (B2), for α ≤ t ≤ γ, we obtain the estimate
xnt − x0t
B
≤ K max xn (s) − x0 (s) + M (t − α) xnα − x0α
α≤s≤t
B
= K max x (s) − x (s) + M (t − α)|ϕn − ϕ|B .
n
0
α≤s≤t
Since M (t) is a bounded function, it follows that for each ε > 0, there exists an N1 such that
|xnt − x0t |B < ε for n ≥ N1 and for all t ∈ [α, γ]. Consequently, Kx = {xnt , x0t : α ≤ t ≤ γ, n ≥ 1}
is a compact set. Let X (t + tn , xt ) converge to X ∗ (t, xt ) uniformly on the set [α, γ] × Kx . Then
all assumptions of Theorem 2 are valid, and x0 (t) is a solution of system (1); moreover, x0α = ϕ.
By the uniqueness of the solution of the limit system, we have x∗ (t) ≡ x0 (t). Therefore, xn (t)
converges to x∗ (t), and xnt → x∗t uniformly on [α, γ], where γ ∈ (α, β) is arbitrary, which completes
the proof of the lemma.
For a solution x(t) = x(t; α, ϕ) of system (SI), the positive limit set ω + (xt (α, ϕ)) is defined as
\
ω + (xt (α, ϕ)) =
Cl {xs : s ≥ t}
t≥α
and consists of the limits of the sequences {xtk } as tk → +∞. It follows from Theorem 1 that the
limit set ω + (xt (α, ϕ)) is nonempty and compact for a bounded solution x(t; α, ϕ). We need the following property of the limit set of a bounded solution of system (SI).
Lemma 2. Let a solution x(t; α, ϕ) of system (SI) be bounded for all t. Then the set ω + (xt (α, ϕ))
is semiinvariant with respect to the family of limit systems (1); i.e., for each point ψ ∈ ω + (xt (α, ϕ)) ,
there exists a limit equation and a solution x∗ (t; 0, ψ) of the limit equation such that
∗
xt (0, ψ) : t ∈ R+ ⊂ ω + .
This assertion, which is valid for equations with bounded delay (see [1]), can be proved for
Eq. (SI) in a similar way.
Let V = V (t, x),V ∈ C 1 (R+×GH , R+ ), be a Lyapunov function, where GH = {x ∈ Rn : |x| < H}.
Its derivative along the trajectories
of Eq. (SI) is the functional V 0 : R+ × BH → R [13] given by
Pn
0
V (t, xt ) = ∂V (t, x)/∂t + i=1 (∂V (t, x)/∂xi ) Xi (t, xt ).
On the set R+ ×BH , we introduce a functional W ranging in R+ . We use the following definition
in [13].
Definition 2. The pair (V, W ) is referred to as a Lyapunov–Razumikhin pair if the following
condition is satisfied.
If % > 0, t ≥ %, ϕ ∈ BH , ϕ−% ∈ BH , and ϕ is continuous on [−%, 0], then
V (t, ϕ(0)) ≤ W (t, ϕ) ≤ max max V (t + s, ϕ(s)), W (t − %, ϕ−% ) ,
(LR1)
−%≤s≤0
if
0 < V (t, ϕ(0)) = W (t, ϕ),
then
V 0 (t, ϕ) ≤ 0.
(LR2)
Assumption 3. The function V (t, x) is uniformly continuous and bounded on each set R+ × Ḡr ,
where Ḡr = {x ∈ Rn : |x| ≤ r < H}.
Assumption 4. The functionals W (t, ϕ) and U (t, ϕ) = V 0 (t, ϕ) are uniformly continuous on
each set R+ × K, where K ⊂ BH is a compact set.
Under these assumptions, the family {Vτ (t, x) = V (τ + t, x), τ ∈ R+ } of translates is precompact in the space C (R+ × GH , R+ ) equipped with the compact-open topology, and the families
Uτ (t, ϕ) = U (τ + t, ϕ), τ ∈ R+ ,
Wτ (t, ϕ) = W (τ + t, ϕ), τ ∈ R+
are precompact in the spaces C (R+ × BH , R) and C (R+ × BH , R+ ), respectively.
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Definition 3. A function V ∗ : R × GH → R is referred to as a limit function for V if there
exists a sequence {tn }, limn→∞ tn = +∞, such that the sequence {Vn (t, x) = V (tn + t, x)} uniformly
converges to V ∗ on the sets [0, T ] × Ḡr .
Limit functionals for W (t, ϕ) and U (t, ϕ) are defined by analogy with X ∗ (t, ϕ).
Definition 4. Functionals X ∗, V ∗, W ∗, and U ∗ are said to form a limit set, denoted by
(X , V ∗ , W ∗ , U ∗ ), if they are limit functionals associated with the same sequence tn → +∞.
∗
We assume that V and W satisfy the following assumption [13].
Assumption 5. For each c > 0, there exists a T = T (c) > 0 such that
W ∗ (t, ϕ) = max V ∗ (t + s, ϕ(s))
−T ≤s≤0
for any uniformly continuous function ϕ ∈ BH and any number t ∈ R such that
sup V ∗ (t + s, ϕ(s)) ≤ W ∗ (t, ϕ) = c.
s≤0
For c0 ∈ R and for a sequence tn → +∞, we introduce the sets
∗
∗
N (t, c0 , T, V ) = ϕ ∈ BH : max V (t + s, ϕ(s)) = c0 ,
−T ≤s≤0
∗
M (t, c0 , T, V ) = {ϕ ∈ N (t, c0 , T, V ∗ ) : V ∗ (t, ϕ(0)) = c0 } ,
L (t, U ∗ ) = {ϕ ∈ BH : U ∗ (t, ϕ) = 0} .
Remark. In the preceding constructions, it is important that the phase space is separable.
However, if B is not separable, then the families of translates of the functionals X, W , and U are
precompact in C (R+ × K → Rn ) for each compact set K ⊂ BH , and the corresponding limit functional [as well as the set L (t, U ∗ )] is determined by the sequence tk → +∞ and the compact set K.
Since the positive orbit of a bounded solution of system (SI) is precompact in BH , it follows that
all results of Sections 2 and 3 remain valid for a nonseparable space B with obvious modifications
in the statements and proofs.
2. AN ASYMPTOTIC STABILITY THEOREM
Prior to proceeding to stability, we consider the localization problem for the limit set of a
bounded solution of system (SI) with the use of limit equations as well as Lyapunov functions.
Suppose that system (SI) has a Lyapunov–Razumikhin pair (V, W ).
Theorem 3. Suppose that the following conditions are satisfied :
(1) the solution of system (SI) is defined and bounded for all t ≥ α; |x(t; α, ϕ)| ≤ r;
(2) Assumptions 1–5 are valid.
Then there exists a value c = c0 = const such that for each ψ ∈ ω + (xt (α, ϕ)) , there exists a
limit set (X ∗ , V ∗ , W ∗ , U ∗ ) and a solution x∗ (t, 0, ψ) of the limit system ẋ(t) = X ∗ (t, xt ) such that
x∗t ∈ ω + (xt (α, ϕ)) , x∗t ∈ N (t, c0 , T, V ∗ ) for all t; moreover, U ∗ (t, x∗t ) = 0 for each t ∈ R+ such
that x∗t ∈ M (t, c0 , T, V ∗ ) , i.e., x∗t ∈ L (t, U ∗ ).
Proof. Since the solution x(t; α, ϕ) is bounded, it follows that its positive orbit lies in a
compact set K ⊂ BH . From Definition 2 and the boundedness of a solution, we find that the
functional W (t, xt ) is defined for all t ≥ 0 and is not increasing along the solution. Indeed, if
V (t, x(t)) < W (t, xt ), then, by the continuity of the function V , we have V (t1 , x (t1 )) < W (t, xt )
for some δ > 0 and all t1 ∈ [t, t + δ]. Using relation (LR1), we obtain
W (t1 , xt1 ) ≤ max
max V (t1 + s, x (t1 + s)) , W (t, xt ) = W (t, xt ) .
t−t1 ≤s≤0
If V (t, x(t)) = W (t, xt ), then condition (LR2) implies that
V (t1 , x (t1 )) ≤ V (t, x(t))
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A REMARK ON THE LYAPUNOV–RAZUMIKHIN METHOD
for all t1 ∈ [t, t + δ] (δ > 0). Then
W (t1 , xt1 ) ≤ max
1427
max V (t1 + s, x (t1 + s)) , W (t, xt )
t−t1 ≤s≤0
= max {V (t, x(t)), W (t, xt )} = W (t, xt ) .
This, together with the lower boundedness of the functional W , implies the existence of the limit
lim W (t, xt (α, ϕ)) = c0 ≥ 0.
t→+∞
(3)
Let a point ψ ∈ ω + (xt (α, ϕ)) be determined by a sequence tn → +∞, ψ (n) = xtn (α, ϕ) → ψ
as n → ∞. Since the families {Xτ (t, x), τ ∈ R+ }, {Vτ (t, x), τ ∈ R+ }, and {Wτ (t, ϕ), τ ∈ R+ } of
translates are precompact, it follows that there exists a sequence nk → ∞ such that Vtnk → V ∗
on R+ × GH , Xtnk → X ∗ , Wtnk → W ∗ uniformly on compact sets. Then, by Lemma 1, the sequence
(k)
x (t) = x tnk + t; tnk , ψ (nk ) = {x (tnk + t; α, ϕ)} converges to the solution x∗ (t; 0, ψ) of the
equation ẋ(t) = X ∗ (t, x) uniformly on each closed interval [0, T ]. Moreover, each point x∗t (0, ψ)
is a limit point of the sequence {xtn +t (α, ϕ)} as n → ∞. Consequently, x∗t ∈ ω + (xt (α, ϕ)) for
all t ∈ R+ .
It follows from (3) that limk→+∞ W t + tnk , x(k)
= W ∗ (t, x∗t ) = c0 . Furthermore, from condit
tion (LR1), we obtain V ∗ (t, x∗ (t)) ≤ W ∗ (t, x∗t ) = c0 for all t ∈ R, and consequently,
sup V ∗ (t + s, x∗ (t + s)) ≤ c0 = W ∗ (t, x∗t ) .
s≤0
Now Proposition 5 implies that
max V ∗ (t + s, x∗ (t + s)) = c0 = const
−T ≤s≤0
(4)
for some T > 0. Therefore, x∗t ∈ N (t, c0 , T, V ∗ ) for all t ≥ 0.
Let x∗t ∈ M (t, c0 , T, V ∗ ), i.e., V ∗ (t, x∗ (t)) = limk→∞ V tnk + t, x(k) (t) = c0 . Then
V ∗ (t, x∗ (t)) = max V ∗ (t + s, x∗ (t + s)) .
−T ≤s≤0
(5)
(k)
Suppose that U ∗ (t, x∗t ) = −2ε < 0. Then U tnk + t, xt
− U ∗ (t, x∗t ) < −ε for some N1 > 0 and
(k)
for all k ≥ N1 ; therefore, U tnk + t, xt
< −ε.
Since the functional U (t, ϕ) is uniformly continuous, it follows that there exists a δ > 0 such
that U (t2 , ϕ2 ) − U (t1 , ϕ1 ) < ε/2 for all (t1 , ϕ1 ) , (t2 , ϕ2 ) ∈ R+ × K such that |t2 − t1 | < δ and
|ϕ2 − ϕ1 |B < δ. The uniform continuity of the function x(t), Theorem 2, and Lemma 1 imply that
(k)
(k)
xt2 − xt1
< δ for some δ1 ∈ (0, δ] and N ≥ N1 provided that |t2 − t1 | < δ1 and k ≥ N ; then,
B
n
o
(k)
(k)
(k)
since xt : t ≥ 0 is a compact set, we have U tnk + t2 , xt2 − U tnk + t1 , xt1
< ε/2 for
(k)
each k ≥ N . Consequently, U tnk + t1 , xt1 < −ε/2 < 0 for all t1 ∈ (t − δ1 , t], k ≥ N . We choose
an s0 = t∗ − t, t∗ ∈ (t − δ1 , t), such that s0 ∈ [−T, 0]. Then from the estimate of the derivative for
sufficiently large values of k, we have V tnk + t, x(k) (t) < V tnk + t + s0 , x(k) (t + s0 ) − ε |s0 | /2,
which, after the passage to the limit as k → ∞, implies that V ∗ (t, x∗ (t)) < V ∗ (t + s0 , x∗ (t + s0 )),
which in turn contradicts (5).
Assuming that U ∗ (t, x∗t ) = 2ε > 0 and performing similar considerations, we obtain
V 0 tnk + t1 , x(k)
> ε/2 > 0
t1
for all t1 ∈ [t, t + δ1 ), k ≥ N , and consequently,
V tnk + t + s∗ , x(k) (t + s∗ ) − V tnk + t, x(k) (t) > εs∗ /2
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for some s∗ > 0. By passing to the limit as k → ∞, we obtain
c0 = V ∗ (t, x∗ (t)) < V ∗ (t + s∗ , x∗ (t + s∗ )) ,
which contradicts (4). Therefore, U ∗ (t, x∗t ) = 0.
This result generalizes the corresponding theorem for equations with finite delay [12] and the
localization theorem for the positive limit set of an autonomous equation with infinite delay [13].
Suppose that X(t, 0) ≡ 0; then system (SI) has the zero solution. We consider the problem on
the Lyapunov asymptotic stability of the zero solution of system (SI).
Suppose that for system (SI), there exists a Lyapunov–Razumikhin pair (V, W ) satisfying Propositions 3 and 4. We assume that the following assumption is also satisfied.
Assumption 6. One has V (t, 0) = 0, and there exists a δ > 0 such that V (t, x) ≥ a(|x|)
on R+ × Gδ , where a(u) ∈ K , and Assumption 5 is valid for all (t, ϕ) belonging to R+ × Bδ . (Here
K = {σ ∈ C [R+ , R+ ] , σ(u) is strictly increasing, and σ(0) = 0}.)
We say that the set M (t, c0 , T, V ∗ ) ∩ L (t, U ∗ ) does not contain solutions of the system if for
each bounded solution x(t; α, ϕ) of this system, there exists a t∗ ≥ α such that xt (α, ϕ) does not
belong to M (t, c0 , T, V ∗ ) ∩ L (t, U ∗ ) for all t ∈ [t∗ , t∗ + T ], where T = T (c0 ) is the number given
by Proposition 5. Such a definition is caused by the fact that it is not the function V itself but its
maximum value on an interval of length T (depending, in general, on this maximum value) that
is constant along the solution; and the relation x∗t ∈ M (t, c0 , T, V ∗ ) ∩ L (t, U ∗ ) must be valid only
for some t on each interval of length T rather than on the entire real line. Therefore, the standard
meaning of the phrase “a set does not contain solutions of a system” (in which it is sufficient that
there exists at least one t for which the part xt of the solution does not lie in the set) must be
replaced by the requirement that the function x∗t ∈ B [where x∗ (t) is some solution of the limit
system] does not lie in the corresponding limit set for all t from some interval of length T . However,
this proves not to be important in many examples.
In view of the last definition and the above assumptions, we have the following assertion.
Theorem 4. Suppose that for some sequence tk → +∞, the set M (t, c0 , T, V ∗ ) ∩ L (t, U ∗ ) does
not contain solutions of the corresponding limit system ẋ(t) = X ∗ (t, xt ) for c0 > 0. Then the zero
solution of system (SI) is asymptotically stable.
Proof. The assumptions of the theorem, together with Proposition 6, imply the stability of
the zero solution of system (SI). Consequently, the solutions x(t; α, ϕ), ϕ ∈ Bδ , are bounded
for some δ = δ(α) > 0. By Theorem 3, there exists a c0 ≥ 0 such that for each limit point
ψ ∈ ω + (xt (α, ϕ)), ϕ ∈ Bδ , there exists a limit set (X ∗ , V ∗ , W ∗ , U ∗ ) such that the solution x∗ (t; 0, ψ)
of the corresponding limit system satisfies the condition x∗t1 ∈ M (t1 , c0 , T, V ∗ ) ∩ L (t1 , K, U ∗ ) for
some t1 from any interval of length T . Under the assumptions of the theorem, this implies that
V ∗ (t, x∗ (t)) = 0 for all t, where the limit function corresponds to the sequence tk → +∞ occurring
in the assumptions of the theorem. But then limt→+∞ V (t, x(t)) = 0, where x(t) = x(t; α, ϕ) is a
solution of system (SI) and ϕ ∈ Bδ . Hence it follows that x(t; α, ϕ) → 0 as t → +∞.
Example 1. Let k : (−∞, 0] → (0, ∞) be a measurable function such that
Z0
k(s − T )
k(s)ds < ∞,
sup
≤ L(T )
(6)
k(s)
s≤0
−∞
for some continuous function L : (0, ∞) → (0, ∞). Let M0 be the space of measurable functions
on (−∞, 0] ranging in Rn ; we say that ϕ is equivalent to ψ in nM0 if ϕ(s) = ψ(s) for almost all
o
R0
s ∈ (−∞, 0] and ϕ(0) = ψ(0). We introduce the spaces Mk = ϕ ∈ M0 : −∞ k(s)|ϕ(s)|ds < ∞
n
o
and Mkr = ϕ ∈ Mk : ϕ is continuous on [−r, 0] for r > 0 with the norm


Z0


kϕkMkr = kϕkrk = max
max |ϕ(s)|,
k(s)|ϕ(s)|ds .
−r≤s≤0

−∞
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A REMARK ON THE LYAPUNOV–RAZUMIKHIN METHOD
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The space Mkr is admissible [13]; moreover, the condition M (t) → 0, t → ∞, is equivalent to the
condition that limT →∞ L(T ) = 0 in (6). Let us now consider the scalar equation
Z0
ẋ(t) = a(t)x(t) + b(t)x(t − h) +
p(t, s)x(t + s)ds,
(7)
−∞
where a(t) and b(t) are uniformly continuous bounded functions, p(t, s) is uniformly continuous
R0
with respect to t, 0 < p(t, s) ≤ p1 (s), −∞ p1 (s)ds = 1, a(t) + b(t) ≤ −1;
Z0
sup p(t, s − %)/p(t − %, s) +
s≤0
p(t, s)ds ≤ 1
−%
for all % ≥ 0 and t ≥ %,
sup p1 (s − T )/p1 (s) ≤ L(T ),
lim L(T ) = 0,
T →∞
s≤0
and there exists a sequence τk → +∞ and numbers ε > 0 and δ(ε) > 0 such that a(t)+b(t)+1 < −δ
for t ∈ [τk , τk + ε].
As the space B, we choose Mph1 . Let V (t, x) = V (x) = x2 and

 0
2 


Z


2


W (t, ϕ) = max
max ϕ (s),
.
p(t, s)|ϕ(s)|ds


−h≤s≤0

−∞
Let us verify that (V, W ) is a Lyapunov–Razumikhin pair.
(1) We have

 0
2 


Z


2
2
V (x(t)) = x (t) ≤ max x (t), 
p(t, s)|x(t + s)|ds




−∞

 −%
2 
0


Z
Z


2


= max x (t),
p(t, s)|x(t + s)|ds + p(t, s)|x(t + s)|ds




−∞
(
Z0
−%
p(t, s − %)
p(t − %, s)|x(t − % + s)|ds
p(t − %, s)
≤ max x2 (t),
−∞
!2 )
Z0
+ max |x(t + s)|
−%≤s≤0



p(t, s)ds
−%



2
2


≤ max x (t),
p(t − %, s) |x−% (t + s)| ds , max x (t + s)
−%≤s≤0




−∞
= max max V (x(t + s)), W (t − %, xt−% ) .

2
Z0
−%≤s≤0
(2) If V (ϕ(0)) = W (t, ϕ), then we obtain

ϕ2 (0) ≥ 
i.e., |ϕ(0)| ≥
R0
−∞
Z0
2
p(t, s)|ϕ(s)|ds ;
−∞
p(t, s)|ϕ(s)|ds and |ϕ(0)| ≥ max−h≤s≤0 |ϕ(s)|.
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Then
Z0
0
p(t, s)ϕ(s)ds ≤ 2ϕ2 (0)(a(t) + b(t) + 1) ≤ 0.
2
V (t, ϕ) = 2a(t)ϕ (0) + 2b(t)ϕ(0)ϕ(−h) + 2ϕ(0)
−∞
Let us verify Proposition 5. Indeed, the limit functions of V and W have the form V ∗ (x) = x2
and

 0
2 


Z


∗
2
∗


W (t, ϕ) = max
,
max ϕ (s),
p (t, s)|ϕ(s)|ds



−h≤s≤0
−∞
where p∗ (t, s) = limn→∞ p (tn + t, s) for some sequence tn → +∞. Let sups≤0 V ∗ (ϕ(s)) ≤ W ∗ (t, ϕ),
where ϕ ∈ B is a uniformly continuous function. Then
 −T

Z
Z0
W ∗ (t, ϕ) 
p∗ (t, s)ds + p∗ (t, s)ds
−∞
−T
= W ∗ (t, ϕ)
Z0

p∗ (t, s)ds ≤ W ∗ (t, ϕ) ≤ 
−∞

Z0
≤ sup ϕ (s) 
2
s≤0
≤ W ∗ (t, ϕ)
2
p∗ (t, s)|ϕ(s)|ds
−∞
2
Z0
p (t, s)ds ≤ sup ϕ (s)
∗
2
s≤0
−∞
Z−T
Z0
−∞
p∗ (t, s)ds + max V ∗ (ϕ(s))
Z0
−T ≤s≤0
−∞
p∗ (t, s)ds
p∗ (t, s)ds.
−T
Hence it follows that W ∗ (t, ϕ) ≤ max−T ≤s≤0 V ∗ (ϕ(s)) for all T > 0 (in particular, for T = ε).
Finally, the relation
∗
∗
Z0
∗
2
U (t, ϕ) = 2a (t)ϕ (0) + 2b (t)ϕ(0)ϕ(−h) + 2ϕ(0)
p∗ (t, s)ϕ(s)ds
−∞
∗
∗
≤ 2ϕ (0)(a (t) + b (t) + 1) ≤ 0
2
is valid on the set M (c0 , ε, V ∗ ). Therefore, U ∗ (t, ϕ) ≤ −2ϕ2 (0)δ < 0 for t ∈ [0, ε] on the set
M (c0 , ε, V ∗ ) corresponding to the sequence {τk }; consequently, the set M (c0 , ε, V ∗ ) ∩ L (t, U ∗ )
does not contain solutions of the corresponding limit system for c0 > 0. Therefore, by Theorem 4,
the zero solution of Eq. (7) is asymptotically stable.
Therefore, well-known asymptotic stability theorems (e.g., see [14, 15]) cannot be used in this
example, since the derivative of the Lyapunov function is not sign-definite.
3. SOME GENERALIZATIONS
Here we analyze the attraction and stability properties of solutions of system (SI) whose righthand side is completely continuous but does not satisfy the requirement of uniform continuity
with respect to t. The assumptions of Propositions 3 and 4 providing the precompactness of the
Lyapunov function and the corresponding functionals are also omitted. In this case, the conditions
imposed on the corresponding limit sets prove to be more restrictive.
We use the same definition of the Lyapunov–Razumikhin pair as in Section 1 and replace Assumption 5 by the following assumption.
DIFFERENTIAL EQUATIONS
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A REMARK ON THE LYAPUNOV–RAZUMIKHIN METHOD
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Assumption 7. For each c ≥ 0, there exists a T = T (c) > 0 such that the condition
limn→∞ W (tn + t, ϕn ) = limn→∞ max−T ≤s≤0 V (tn + t + s, ϕn (s)) is valid for each uniformly continuous function ϕ ∈ BH , for sequences ϕn → ϕ (ϕn ∈ BH ), tn → +∞, and for t ∈ R such that
limn→∞ sups≤0 V (tn + t + s, ϕn (s)) ≤ limn→∞ W (tn + t, ϕn ) = c.
Let tn → +∞ be some sequence, and let t ∈ R and c ∈ R. We introduce the following sets:
N∞ (t, c0 , T, tn ) = ϕ ∈ BH : ∃ {ϕn ∈ BH } : ϕn → ϕ, lim max V (t + tn + s, ϕn (s)) = c0 ,
n→∞ −T ≤s≤0
o
n
M∞ (t, c0 , T, tn ) = ϕ ∈ N∞ (t, c0 , T, tn ) : ∃ {ϕn ∈ BH } : ϕn → ϕ, lim V (t + tn , ϕn (0)) = c0 ,
n→∞
n
o
−1
U∞ (t, 0, tn ) = ϕ ∈ BH : ∃ {ϕn ∈ BH } : ϕn → ϕ, lim U (t + tn , ϕn ) = 0 .
n→∞
Suppose that for system (SI), there exists a Lyapunov–Razumikhin pair (V, W ). Then under Assumption 7, we have the following theorem on the localization of the positive limit set ω + (xt (α, ϕ)),
which can be proved by analogy with Theorem 3.
Theorem 5. Let the following conditions be satisfied :
(1) the solution of system (SI) is defined and bounded for all t ≥ α, and |x(t; α, ϕ)| ≤ r;
(2) the functional W (t, ϕ) is bounded below in each domain R+ × Br , 0 < r ≤ H, and the
derivative admits the estimate |V 0 (t, ϕ)| ≥ U (t, ϕ) ≥ 0 for all (t, ϕ) ∈ R+ × Bδ ;
(3) U (t) = U (t, xt (α, ϕ)) is a function uniformly continuous with respect to t ∈ [α, +∞).
Then there exists a value c = c0 = const such that the set ω + (xt (α, ϕ)) is the union, over
sequences tn → +∞, of continuous mappings u : R → Br such that u(t) ∈ N∞ (t, c0 , T, tn ) for all t
−1
and u(t) ∈ U∞
(t, 0, tn ) for each t ∈ R such that u(t) ∈ M∞ (t, c0 , T, tn ).
Let us now present an asymptotic stability theorem related to Theorem 5. To this end, we suppose that X(t, 0) ≡ 0; then system (SI) has the zero solution. Suppose that for system (SI), there
exists a Lyapunov–Razumikhin pair with the following properties: V (t, 0) = 0, there exists a δ > 0
such that V (t, x) ≥ a(|x|) for (t, x) ∈ R+ × Ḡδ , where a(u) ∈ K , and Assumption 7 is valid for all
(t, ϕ) ∈ R+ × B̄δ .
Theorem 6. Let N∞ (t, c0 , T, tn ) = {0} for sufficiently small c0 ≥ 0 and for an arbitrary sequence tn → +∞. Then the zero solution of system (SI) is asymptotically stable.
Proof. It follows from the conditions imposed on the function V and from the definition of
a Lyapunov–Razumikhin pair that the zero solution of system (SI) is stable. Consequently, the
solutions x(t; α, ϕ), ϕ ∈ Bδ , are bounded for some δ > 0. Therefore, the assumptions of the theorem
provide the existence (for all t ≥ α) and boundedness of the functional W along the solution
x(t; α, ϕ), ϕ ∈ Bδ . By Theorem 5, there exists a c0 ≥ 0 such that the set ω + (xt (α, ϕ)), ϕ ∈ Bδ ,
consists of continuous mappings u(t) : R → Bε lying in the corresponding set N∞ (t, c0 , T, tn ) for
each t. This, together with the assumptions of the theorem, implies that u(t) ≡ 0 for all t; therefore,
ω + (xt (α, ϕ)) = {0}. Hence it follows that x(t; α, ϕ) → 0 as t → +∞. The proof of the theorem is
complete.
Remark. If we suppose that the lower bound of the absolute value of the derivative V 0 is
a functional U (t, ϕ) uniformly continuous with respect to (t, ϕ) ∈ R+ × K, where K ⊂ BH is a
compact set, then for the asymptotic stability, it is sufficient that there exists a sequence tk → +∞
−1
such that M∞ (t, c0 , T, tn ) ∩ U∞
(t, 0, tn ) = {0} for sufficiently small c0 ≥ 0. (This, together with
Theorem 5, implies that each bounded solution converges to zero.)
The results of this section generalize the theorems proved in [18] for finite-delay systems.
Example 2. Consider the following class of admissible spaces [13]. Let
g : (−∞, 0] → [1, ∞)
DIFFERENTIAL EQUATIONS
is a continuous nonincreasing function, and g(0) = 1.
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(8)
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By Cg we denote the space of continuous functions ϕ on (−∞, 0] ranging in Rn and such that
sups≤0 |ϕ(s)|/g(s) < ∞. Then Cg is a Banach space with respect to the norm |ϕ|g = |ϕ|Cg =
sups≤0 |ϕ(s)|/g(s). Consider the following subspace of Cg :
n
o
U Cg = ϕ ∈ Cg : ϕ/g is uniformly continuous on (−∞, 0] .
Let g satisfy condition (8), and let
[g(s + u)/g(s)] → 1
uniformly on
(−∞, 0]
as
u →0−.
(9)
Then U Cg is an admissible space; moreover, the condition M (t) → 0, t → ∞, is equivalent to the
condition limT →∞ sups≤0 [g(s)/g(s − T )] = 0 (see [13]).
Consider the scalar equation
Zt
(t + 1)ẋ(t) = −a(t)x(t) + b(t)
C(t − s)x(s)ds,
(10)
−∞
where a(t) and b(t) are continuous functions, 1/2 < A ≤ a(t) ≤ A1 , 0 ≤ |b(t)| ≤ B, C : [0, +∞) → R
R0
is a continuous function, and B −∞ |C(−s)|ds = m < A − 1/2. We choose an m∗ such that
m < m∗ ≤ q(A − 1/2), q < 1. Then there exists [19] a function g : (−∞, 0] → [1, +∞) satisfying
conditions (8) and (9) and such that
g(s)
lim sup
= 0,
T →∞ s≤0 g(s − T )
Z0
B
|C(−s)|g(s)ds = m∗ ≤ q(A − 1/2).
−∞
As an admissible phase space B, we choose U Cg . Let V (t, x) = (t + 1)x2 and
2
W (t, ϕ) = (t + 1) sup |ϕ(s)|/g(s) .
s≤0
Then (V, W ) is a Lyapunov–Razumikhin pair. Indeed, condition (LR1) is satisfied:
2
2
2
2
(t + 1)ϕ (0) ≤ (t + 1) max ϕ (0), max (ϕ(s)/g(s)) , sup (|ϕ(s)|/g(s))
−%≤s≤0
s≤−%
2
2
≤ (t + 1) max max ϕ (s), sup(|ϕ(s − %)|/g(s − %))
−%≤s≤0
s≤0
2
2
≤ (t + 1) max max ϕ (s), sup (|ϕ−% (s)| /g(s))
−%≤s≤0
s≤0
= max max V (ϕ(s)), W (t − %, ϕ−% ) .
−%≤s≤0
If V (t, ϕ(0)) = W (t, ϕ), then we obtain |ϕ(0)| ≥ sups≤0 |ϕ(s)|/g(s). Then


Z0
V 0 (t, ϕ) = ϕ2 (0) − 2 a(t)ϕ2 (0) + ϕ(0)b(t) C(−s)ϕ(s)ds
−∞
Z0
≤ ϕ2 (0) − 2Aϕ2 (0) + 2|ϕ(0)|B
|C(−s)|g(s)
−∞
1
≤ −2ϕ (0) A −
2
2
|ϕ(s)|
ds
g(s)
Z0
2
+ 2Bϕ (0)
|C(−s)|g(s)ds ≤ 0.
−∞
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A REMARK ON THE LYAPUNOV–RAZUMIKHIN METHOD
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Let us verify the validity of Assumption 7. We must show that if
lim sup (tn + t + 1 +
n→∞ s≤0
then
s) ϕ2n (s)
2
≤ lim (tn + t + 1) sup |ϕn (s)| /g(s) = c,
n→∞
s≤0
2
lim (tn + t + 1) sup |ϕn (s)| /g(s) = lim max (tn + t + 1 + s) ϕ2n (s),
n→∞
n→∞ −T ≤s≤0
s≤0
where {tn }, t, and {ϕn } satisfy Proposition 7. Note that
2
lim (tn + t + 1) sup |ϕn (s)| /g(s) = ls(t);
n→∞
s≤0
here and in the following,
2
ls(t) ≡ lim sup (tn + t + 1 + s){|ϕn (s)| /g(s)} .
n→∞ s≤0
By the assumptions for the function g(s), there exists a T > 0 such that g(s) ≥ 1 for all s ≤ −T .
Obviously,
2
sup(tn + t + 1 + s){|ϕn (s)|/g(s)}
s≤0
2
2
= max
sup (tn + t + 1 + s){|ϕn (s)|/g(s)} , max (tn + t + 1 + s){|ϕn (s)|/g(s)} .
−T ≤s≤0
−∞≤s≤−T
Then, taking account of the above assumptions, we find that if
ls(t) = lim
2
sup
n→∞ −∞≤s≤−T
(tn + t + 1 + s){|ϕn (s)| /g(s)} ,
then
ls(t) < lim
sup
n→∞ −∞≤s≤−T
(tn + t + 1 + s) ϕ2n (s) ≤ ls(t),
i.e., we have arrived at a contradiction. Therefore,
2
ls(t) = lim max (tn + t + 1 + s){|ϕn (s)| /g(s)} ,
n→∞ −T ≤s≤0
ls(t) ≤ lim max (tn + t + 1 + s) ϕ2n (s) ≤ ls(t),
n→∞ −T ≤s≤0
which is the desired assertion. The absolute value of the derivative of the function V can be
estimated below by the uniformly continuous functional
U (ϕ) = (2A − 1) ϕ2 (0) − q|ϕ(0)||ϕ|g .
It remains to note that M∞ (t, c0 , T, tn ) ∩ {U (ϕ) = 0} = {0} for each c0 ≥ 0. This (see the remark
to Theorem 6) implies that the zero solution of Eq. (10) is asymptotically stable.
The theorems in [14, 15] cannot be applied to this example, since the function V does not admit
an infinitesimal upper bound.
ACKNOWLEDGMENTS
The work was financially supported by the Russian Foundation for Basic Research (grant
no. 96-01-01067).
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