Stability for retarded functional differential equations
It is known that retarded functional differential equations can be regarded as Banach-space-valued generalized ordinary differential equations (GODEs). In this paper, some stability concepts for retarded functional differential equations are introduced and they are discussed using known stability re...
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Federson, M. Schwabik, S. 2020-02-09T16:31:53Z 2020-02-09T16:31:53Z 2008 Stability for retarded functional differential equations / M. Federson, S. Schwabik // Український математичний журнал. — 2008. — Т. 60, № 1. — С. 107–126. — Бібліогр.: 9 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164507 517.9 It is known that retarded functional differential equations can be regarded as Banach-space-valued generalized ordinary differential equations (GODEs). In this paper, some stability concepts for retarded functional differential equations are introduced and they are discussed using known stability results for GODEs. Then the equivalence of the different concepts of stabilities considered here are proved and converse Lyapunov theorems for a very wide class of retarded functional differential equations are obtained by means of the correspondence of this class of equations with GODEs. Відомо, що функціональні диференціальні рівняння із запізненням можна розглядати як узагальнені звичайні диференціальні рівняння (УЗДР) зі значеннями у банаховому просторі. У статті введено деякі концепції стабільності для функціональних диференціальних рівнянь із запізненням, а також обговорено ці концепції з використанням відомих результатів щодо стабільності УЗДР. Доведено еквівалентність різних концепцій стабільності, що розглянуті у даній роботі. Для дуже широкого класу функціональних диференціальних рівнянь із запізненням одержано зворотні теореми Ляпунова на основі того факту, що цей клас рівнянь відповідає УЗДР. Supported by Grant n. IAA100190702 of the Grant Agency of the Acad. Sci. of the Czech Republic. en Інститут математики НАН України Український математичний журнал Статті Stability for retarded functional differential equations Стабільність для функціональних диференціальних рівнянь iз запiзненням Article published earlier |
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Stability for retarded functional differential equations |
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Stability for retarded functional differential equations Federson, M. Schwabik, S. Статті |
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Stability for retarded functional differential equations |
| title_full |
Stability for retarded functional differential equations |
| title_fullStr |
Stability for retarded functional differential equations |
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Stability for retarded functional differential equations |
| title_sort |
stability for retarded functional differential equations |
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Federson, M. Schwabik, S. |
| author_facet |
Federson, M. Schwabik, S. |
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Статті |
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2008 |
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Стабільність для функціональних диференціальних рівнянь iз запiзненням |
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It is known that retarded functional differential equations can be regarded as Banach-space-valued generalized ordinary differential equations (GODEs). In this paper, some stability concepts for retarded functional differential equations are introduced and they are discussed using known stability results for GODEs. Then the equivalence of the different concepts of stabilities considered here are proved and converse Lyapunov theorems for a very wide class of retarded functional differential equations are obtained by means of the correspondence of this class of equations with GODEs.
Відомо, що функціональні диференціальні рівняння із запізненням можна розглядати як узагальнені звичайні диференціальні рівняння (УЗДР) зі значеннями у банаховому просторі. У статті введено деякі концепції стабільності для функціональних диференціальних рівнянь із запізненням, а також обговорено ці концепції з використанням відомих результатів щодо стабільності УЗДР. Доведено еквівалентність різних концепцій стабільності, що розглянуті у даній роботі. Для дуже широкого класу функціональних диференціальних рівнянь із запізненням одержано зворотні теореми Ляпунова на основі того факту, що цей клас рівнянь відповідає УЗДР.
|
| issn |
1027-3190 |
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https://nasplib.isofts.kiev.ua/handle/123456789/164507 |
| citation_txt |
Stability for retarded functional differential equations / M. Federson, S. Schwabik // Український математичний журнал. — 2008. — Т. 60, № 1. — С. 107–126. — Бібліогр.: 9 назв. — англ. |
| work_keys_str_mv |
AT federsonm stabilityforretardedfunctionaldifferentialequations AT schwabiks stabilityforretardedfunctionaldifferentialequations AT federsonm stabílʹnístʹdlâfunkcíonalʹnihdiferencíalʹnihrívnânʹizzapiznennâm AT schwabiks stabílʹnístʹdlâfunkcíonalʹnihdiferencíalʹnihrívnânʹizzapiznennâm |
| first_indexed |
2025-11-27T05:58:45Z |
| last_indexed |
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1850800520326807552 |
| fulltext |
UDC 517.9
M. Federson (Inst. Ciê. Mat. de Comput., Univ. São Paulo-Campus São Carlos, Brazil),
Š. Schwabik (Math. Inst. Acad. Sci. Czech Republic, Praha, Czech Republic)
STABILITY FOR RETARDED
FUNCTIONAL DIFFERENTIAL EQUATIONS∗
СТАБIЛЬНIСТЬ ДЛЯ ФУНКЦIОНАЛЬНИХ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ IЗ ЗАПIЗНЕННЯМ
It is known that retarded functional differential equations can be regarded as Banach-space valued generali-
zed ordinary differential equations (GODEs). In this paper some stability concepts for retarded functional
differential equations are introduced and they are discussed using known stability results for GODEs. Then
the equivalence of the different concepts of stabilities considered here are proved and converse Lyapunov
theorems for a very wide class of retarded functional differential equations are obtained by means of the
correspondence of this class of equations with GODEs.
Вiдомо, що функцiональнi диференцiальнi рiвняння iз запiзненням можна розглядати як узагальненi
звичайнi диференцiальнi рiвняння (УЗДР) зi значеннями у банаховому просторi. У статтi введено
деякi концепцiї стабiльностi для функцiональних диференцiальних рiвнянь iз запiзненням, а також
обговорено цi концепцiї з використанням вiдомих результатiв щодо стабiльностi УЗДР. Доведено
еквiвалентнiсть рiзних концепцiй стабiльностi, що розглянутi у данiй роботi. Для дуже широ-
кого класу функцiональних диференцiальних рiвнянь iз запiзненням одержано зворотнi теореми
Ляпунова на основi того факту, що цей клас рiвнянь вiдповiдає УЗДР.
Notations. Let X be a Banach space and I ⊂ R be an interval of the real line.
We denote by G(I,X) be the space of locally regulated functions f : I → X, that
is, for each compact interval [a, b] ⊂ I, the lateral limits f(t+) = limρ→0+ f(t + ρ),
t ∈ [a, b), and f(t−) = limρ→0− f(t + ρ), t ∈ (a, b], exist and are finite. When
I = [a, b] we write G([a, b], X) which is a Banach space when endowed with the usual
supremum norm. In G(I,X) we consider the topology of locally uniform convergence.
By G−(I,X), we mean the subspace of G(I,X) of left continuous functions for which
f(t−) = limρ→0− f(t+ ρ) = f(t), t ∈ I, except for the left endpoint of I.
We denote by BV (I,X) the space of functions f : I → X which are locally of
bounded variation, that is, for each compact interval [a, b] ⊂ I, the restriction of f
to [a, b], f
∣∣
[a,b]
, is of bounded variation. In BV ([a, b], X), we consider the variation
norm given by ‖f‖ = ‖f(a)‖+ varbaf, where varbaf stands for the variation of f in the
interval [a, b]. Then BV ([a, b], X) is a Banach space and BV ([a, b], X) ⊂ G([a, b], X).
When f ∈ BV (I,X) is also left continuous (f ∈ BV (I,X) ∩ G−(I,X)), we write
f ∈ BV −(I,X).
We write C(I,X) to denote the space of continuous functions f : I → X. We
consider the Banach space C([a, b], X) equipped with the usual supremum norm and in
C(I,X) we consider the topology of locally uniform convergence.
It is clear that C(I,X) ⊂ G−(I,X) and BV −(I,X) ⊂ G−(I,X).
For simplifying our considerations we restrict ourselves to the case of left continuous
functions everywhere, when some discontinuities can occur.
1. Retarded functional differential equations. Let us consider the initial value
problem for a retarded functional differential equation:
∗ Supported by Grant n. IAA100190702 of the Grant Agency of the Acad. Sci. of the Czech Republic.
c© M. FEDERSON, Š. SCHWABIK, 2008
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1 107
108 M. FEDERSON, Š. SCHWABIK
ẏ(t) = f(yt, t),
yt0 = φ,
(1.1)
where φ ∈ G−
(
[−r, 0],Rn
)
, r ≥ 0, and f(φ, t) maps an open subset Ω of
G−([−r, 0],Rn)×[t0,+∞) to Rn. Given a function y : [t0−r,+∞) → Rn, we consider
yt : [−r, 0] → Rn defined, as usual, by
yt (θ) = y (t+ θ) , θ ∈ [−r, 0], t ∈ [t0,+∞).
Let us recall the concept of a solution of problem (1.1).
Definition 1.1. Let σ > 0. A function y ∈ G−([t0 − r, t0 + σ],Rn) such that
(yt, t) ∈ G−
(
[−r, 0],Rn
)
× [t0, t0 + σ] for all t ∈ [t0, t0 + σ], yt0 = φ and
ẏ(t) = f (yt, t)
for almost all t ∈ [t0, t0 + σ] is called a (local) solution of (1.1) in [t0, t0 + σ] (or
sometimes also in [t0 − r, t0 + σ]) with initial condition (φ, t0).
The system (1.1) is known to be equivalent to the “integral” equation
y(t) = y(t0) +
t∫
t0
f (ys, s) ds, t ∈ [t0,+∞),
yt0 = φ,
(1.2)
when the integral exists in the Lebesgue sense (cf. [1]). In fact we will use (1.2) for
the concept of the initial value problem (1.1). This makes it clear that if a solution y is
defined on some interval [t0, t0 + σ] with σ > 0 then y, being an indefinite integral of
a Lebesgue integrable function, is necessarily absolutely continuous on [t0, t0 + σ] (we
write y ∈ AC
(
[t0, t0 + σ],Rn)
)
.
Let G1 ⊂ G−
(
[t0 − r,+∞),Rn) with the following property: if y = y(t), t ∈
∈ [t0 − r,+∞), is an element of G1 and t̄ ∈ [t0 − r,+∞), then ȳ given by
ȳ(t) =
y(t), t0 − r 6 t 6 t̄,
y (t̄) , t̄ < t < +∞,
also belongs to G1.
Let L1(I,X) denote the space of locally Bochner integrable functions f : I → X,
integrable in each compact of I, where I ⊂ R is an interval and X is a Banach space.
If X is finite-dimensional then we have in mind the Lebesgue integral.
Let | · | be a norm in Rn.
We consider f(φ, t) : G−
(
[−r, 0],Rn
)
× [t0,+∞) → Rn, the right-hand side of
the differential equation in (1.1), such that the mapping t 7→ f (yt, t) belongs to
L1
(
[t0,+∞), Rn
)
for y ∈ G1 and the following conditions are fulfilled:
(A) there is M ∈ L1
(
[t0,+∞),R
)
such that for all x ∈ G1 and all u1, u2 ∈
∈ [t0,+∞), ∣∣∣∣∣∣
u2∫
u1
f (xs, s) ds
∣∣∣∣∣∣ 6
u2∫
u1
M(s)ds;
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
STABILITY FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS 109
(B) there is L ∈ L1
(
[t0,+∞),R
)
such that for all x, y ∈ G1 and all u1, u2 ∈
∈ [t0,+∞), ∣∣∣∣∣∣
u2∫
u1
[f (xs, s)− f (ys, s)] ds
∣∣∣∣∣∣ 6
u2∫
u1
L(s) ‖xs − ys‖ ds,
the norm on the right-hand side is the norm in G−
(
[−r, 0],Rn
)
given by ‖φ‖ =
= supt∈[−r,0] |φ(t)| for φ ∈ G−
(
[−r, 0],Rn
)
.
Of course the functions M and L above depend on the choice of t0.
If f(0, t) = 0 for every t ∈ R, then y ≡ 0 is a solution of (1.1). The next
definitions concern stability concepts for the solution y ≡ 0 of (1.1). The following
three definitions are the classical definitions for Lyapunov stability, uniform (Lyapunov)
stability and uniform asymptotic stability of the trivial solution of (1.1). See [1], for
instance.
Definition 1.2. The trivial solution of system (1.1) is called (Lyapunov) stable if
for every ε > 0, there exists δ = δ(ε, t0) > 0 such that if φ ∈ G− ([−r, 0],Rn) and
y : [γ, v] → Rn, with [γ, v] ⊂ [t0 − r,+∞) and [γ, v] 3 t0, is a solution of (1.1) such
that yt0 = φ and
‖φ‖ < δ,
then ∥∥yt(t0, φ)
∥∥ < ε, t ∈ [t0, v].
Definition 1.3. The trivial solution of system (1.1) is called uniformly stable if the
number δ in Definition 1.2 is independent of t0.
Definition 1.4. The solution y ≡ 0 of (1.1) is called uniformly asymptotically
stable if there exists a δ0 > 0 and for every ε > 0, there exists a T = T (ε, δ0) > 0
such that if φ ∈ G− ([−r, 0],Rn) , and y : [γ, v] → Rn, with [γ, v] ⊂ [t0 − r,+∞) and
[γ, v] 3 t0, is solution of (1.1) such that yt0 = φ and
‖φ‖ < δ0,
then ∥∥yt(t0, φ)
∥∥ < ε, t ∈ [γ, v] ∩ [γ + T,+∞).
The next definition of stability of the solution y ≡ 0 of (1.1) is borrowed from [2].
Definition 1.5. The solution y ≡ 0 of (1.1) is said to be integrally stable if for
every ε > 0 there is a δ = δ(ε) > 0 such that if φ ∈ G− ([−r, 0],Rn) , ‖φ‖ < δ and
p ∈ L1([t0, t1],Rn) with
∫ t1
t0
∣∣p(s)∣∣ds < δ, then
∣∣y(t; t0, φ)
∣∣ < ε for every t ∈ [t0, t1],
where y(t; t0, φ) is a solution of the perturbed equation
ẏ(t) = f (yt, t) + p(t),
yt0 = φ.
(1.3)
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
110 M. FEDERSON, Š. SCHWABIK
The solution of equation (1.3) has to be interpreted as a solution of the “integral”
equation
y(t) = y(t0) +
t∫
t0
f (ys, s) ds+
t∫
t0
p(s)ds,
yt0 = φ,
(1.4)
where the integral is considered in the Lebesgue sense. The solution of (1.3), when it
exists, is absolutely continuous on [t0, t1] (i.e., y(·; t0, φ) ∈ AC
(
[t0, t1],Rn)
)
.
Now we introduce a concept of stability of the trivial solution of (1.1) which generali-
zes Definition 1.5 and will be essential to our purposes.
Definition 1.6. The solution y ≡ 0 of (1.1) is said to be variationally stable if for
every ε > 0 there is a δ = δ(ε) > 0 such that if φ ∈ G−
(
[−r, 0],Rn
)
, ‖φ‖ < δ and
P ∈ BV −
(
[t0, t1],Rn
)
with vart1t0P < δ, then∣∣y(t; t0, φ)
∣∣ < ε for every t ∈ [t0, t1],
where y(t; t0, φ) is a solution of
y(t) = y(t0) +
t∫
t0
f (ys, s) ds+ P (t)− P (t0), t ∈ [t0, t1],
yt0 = φ.
(1.5)
It can be seen immediately that the solution y of (1.5) is of bounded variation and
left continuous, that is, y ∈ BV −
(
[t0, t1],Rn
)
⊂ G−
(
[t0, t1],Rn
)
.
Note that (1.4) is a particular case of (1.5) for P (t) =
∫ t
t0
p(s)ds, t ≥ t0. If
p ∈ L1
(
[t0, t1],Rn
)
, then we have P ∈ AC([t0, t1],Rn) ⊂ BV −
(
[t0, t1],Rn
)
, the
derivative Ṗ (s) =
dP
ds
exists almost everywhere in [t0, t1] and
vart1t0P =
t1∫
t0
∣∣Ṗ (s)
∣∣ds =
t1∫
t0
∣∣p(s)∣∣ds.
Having this in mind we can easily see that the variational stability of the trivial solution
of (1.1) is a concept which is more general than that of integral stability. Therefore we
consider the variational stability only.
Definition 1.7. The solution y ≡ 0 of (1.1) is called variationally attracting if
there is a δ̃ > 0 and for every ε > 0, there exist a T = T (ε) > 0 and a ρ = ρ(ε) > 0
such that if
‖φ‖ < δ̃ and vart1t0P < ρ
with P ∈ BV −
(
[t0, t1],Rn
)
, then∣∣y(t; t0, φ)
∣∣ < ε for all t ≥ t0 + T, t ∈ [t0, t1],
where y(t; t0, φ) is a solution of the equation (1.5) satisfying yt0 = φ.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
STABILITY FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS 111
Definition 1.8. The solution y ≡ 0 of (1.1) is called variationally asymptotically
stable if it is variationally stable and variationally attracting.
It is clear by the definitions that if the solution y ≡ 0 of (1.1) is variationally stable
then it is also Lyapunov stable. Similarly also for the asymptotic stabilities.
Maybe at this moment the reader is wondering why Definitions 1.6 to 1.8 are
presented for RFDEs. One reason is that stability with respect to permanently acti-
ng perturbations is of interest for technology. The second is a pragmatic one, since we
have results on stability for GODEs at our disposal which can be used in this context.
See [3, 4] and the development of the theory in the next section.
To the first reason we add that the perturbation in the case of integral stability can be
large enough as long as its integral is small. One could also consider perturbations of the
form p(t, y, yt) and the same technique would apply. However, the theory around would
be more complicated technically. In the case of variational stability, we can think about
the possibility of perturbing the original equation (1.1) by an integrable function plus a
Dirac sum acting on a countable set and then interpret the solution appropriately. In this
case, the solution is a left continuous function. It is clear that (1.5) can be interpreted
as an equation with impulses acting at points of discontinuity of the function P and
described in the form given e.g. in the book [5] and, of course, in numerous papers of
the Kiev ODE group concentrated around this two personalities.
2. The GODE corresponding to (1.5). Let X be a Banach space and consider
Ω ⊂ X×R. Assume that G : Ω → X is a given X-valued function with G(x, t) defined
for each (x, t) ∈ Ω.
Having the concept of Kurzweil integrability in mind (see for instance [3, 6, 7] or
[4]), we now present the concept of generalized differential equation.
Definition 2.1. A function x : [α, β] → X is called a solution of the generalized
ordinary differential equation
dx
dτ
= DG(x, t) (2.1)
on the interval [α, β] ⊂ R if (x(t), t) ∈ Ω for all t ∈ [α, β] and if the equality
x(v)− x(γ) =
v∫
γ
DG(x(τ), t) (2.2)
holds for every γ, v ∈ [α, β], where the integral is considered in the sense of Kurzweil.
Let us mention that the theory of generalized ordinary differential equations presented
e.g. in [7] is for the case when X = Rn, but it is easy to check that all the basic results
hold also for the case of a Banach space.
Given an initial condition (z0, t0) ∈ Ω the following definition of the solution of the
initial value problem for the equation (2.1) will be used.
Definition 2.2. A function x : [α, β] → X is a solution of the generalized ordinary
differential equation (2.1) with the initial condition x(t0) = z0 on the interval [α, β] ⊂ R
if t0 ∈ [α, β], (x(t), t) ∈ Ω for all t ∈ [α, β] and if the equality
x(v)− z0 =
v∫
t0
DG(x(τ), t) (2.3)
holds for every v ∈ [α, β].
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
112 M. FEDERSON, Š. SCHWABIK
Now we consider Ω = G1 × [t0,+∞) and we define a special class of functions
F : Ω → X.
Definition 2.3. We say that a function G : Ω → X belongs to the class F(Ω, h),
if there exists a nondecreasing, left continuous function h : [t0,+∞) → R such that∥∥G(x, s2)−G(x, s1)
∥∥ ≤ |h(s2)− h(s1)| (2.4)
for all (x, s2), (x, s1) ∈ Ω and∥∥G(x, s2)−G(x, s1)−G(y, s2) +G(y, s1)
∥∥ ≤ ‖x− y‖|h(s2)− h(s1)| (2.5)
for all (x, s2), (x, s1), (y, s2), (y, s1) ∈ Ω.
Suppose f(φ, t) : G1 × [t0,+∞) → Rn such that for each y ∈ G1 the mapping
t 7→ f (yt, t) belongs to L1([t0,+∞),Rn) and f satisfies conditions (A) and (B).
Assume further that P ∈ BV −
(
[t0,+∞),Rn
)
.
For y ∈ G1 and t ∈ [t0 − r,+∞), define
F (y, t) (ϑ) =
0, t0 − r 6 ϑ 6 t0 or t0 − r 6 t 6 t0,
ϑ∫
t0
f (ys, s) ds, t0 6 ϑ 6 t < +∞,
t∫
t0
f (ys, s) ds, t0 6 t 6 ϑ < +∞.
(2.6)
and, for t ∈ [t0 − r,+∞), put
P (t) (ϑ) =
0, t0 − r 6 ϑ 6 t0 or t0 − r 6 t 6 t0,
P (ϑ)− P (t0), t0 6 ϑ 6 t < +∞,
P (t)− P (t0), t0 6 t 6 ϑ < +∞.
(2.7)
Then
G(y, t) = F (y, t) + P (t) (2.8)
defines an element G(y, t) of G−
(
[t0 − r,+∞),Rn
)
and G(y, t)(ϑ) ∈ Rn is the value
of G(y, t) at a point ϑ ∈ [t0 − r,+∞), that is,
G : G1 × [t0 − r,+∞) → G−([t0 − r,+∞),Rn).
The idea to construct the right-hand side of a GODE which corresponds to a functi-
onal differential equation of the form (1.1) is due to C. Imaz, F. Oliva and Z. Vorel from
their papers [8] and [9].
Let h : [t0,+∞) → R be defined by
h(t) =
t∫
t0
[
M(s) + L(s)
]
ds+ vartt0P, t ∈ [t0,+∞).
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STABILITY FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS 113
Then the function h is left continuous and nondecreasing, since M, L : [t0,+∞) → R
are nonnegative a.e. and P ∈ BV −
(
[t0,+∞),Rn
)
.
Under the given assumptions, it is a matter of routine to prove that the function G
given by (2.8) belongs to the class F(Ω, h), where Ω = G1 × [t0,+∞) (see e.g. [6]).
Consider G given by (2.8). If [α, β] ⊂ [t0,+∞) and x : [α, β] → G−([t0− r,+∞),
Rn) is a solution of (2.1) in [α, β], then x is of bounded variation in [α, β] and
varβα x ≤ h(β)− h(α) < +∞.
Moreover, every point in [α, β] at which the function h is continuous is a point of
continuity of the solution x : [α, β] → G−
(
[t0 − r,+∞),Rn
)
and we have
x(σ+)− x(σ) = lim
s→σ+
x(s)− x(σ) = G(x(σ), σ+)−G(x(σ), σ)
for σ ∈ [α, β) and
x(σ)− x(σ−) = x(σ)− lim
s→σ−
x(s) = G
(
x(σ), σ)−G(x(σ), σ−
)
for σ ∈ (α, β], where G(x, σ+) = lims→σ+G(x, s), for σ ∈ [α, β) and G(x, σ−) =
= lims→σ−G(x, s), for σ ∈ (α, β]. For a proof of these facts, the reader may want to
consult [7], for instance.
Now we present a result on the existence of the integral involved in the definition
of the solution of the generalized equation (2.1). This result is a particular case of
Corollary 3.16 and Proposition 3.6, both from [7].
Lemma 2.1. Let G ∈ F(Ω, h). Suppose x : [α, β] → X, [α, β] ⊂ [t0,+∞), is of
bounded variation in [α, β] and (x(s), s) ∈ Ω for every s ∈ [α, β]. Then the integral∫ β
α
DG(x(τ), t) exists and the function s 7→
∫ s
α
DG(x(τ), t) ∈ X is of bounded
variation for all s ∈ [α, β].
The next result concerns the existence of a solution of (2.1) (see [6], Theorem 2.15).
Theorem 2.1. Let G : Ω → X be an element of the class F(Ω, h), where the
function h is left continuous (i.e., h(t−) = h(t), t ∈ (a,+∞)). Then for every (x̃, t0) ∈
∈ Ω such that for x̃+ = x̃ + G(x̃, t0+) − G(x̃, t0) we have (x̃+, t0) ∈ Ω and there
exists a ∆ > 0 such that on the interval [t0, t0 + ∆] there exists a unique solution
x : [t0, t0 + ∆] → X of the generalized ordinary differential equation (2.1) for which
x(t0) = x̃.
Consider the generalized equation (2.1). We will work now with a specific initial
value problem for equation (2.3) with G given by (2.8).
Let φ ∈ G− ([−r, 0],Rn) and σ > 0 be given. A function x(t) defined on the
interval [t0− r, t0 +σ] and taking values in G−
(
[t0− r, t0 +σ],Rn
)
is a (local) solution
of the generalized ordinary differential equation (2.1) in the interval [t0, t0 + σ] (or in
[t0 − r, t0 + σ]), with initial condition x (t0) ∈ G1 given for φ ∈ G−
(
[−r, 0],Rn
)
by
x(t0)(ϑ) =
φ(ϑ− t0) for ϑ ∈ [t0 − r, t0],
x(t0)(t0) for ϑ ∈ [t0, t0 + σ]
if
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114 M. FEDERSON, Š. SCHWABIK
x(v) = x (t0) +
v∫
t0
DG (x (τ) , t)
for every v ∈ [t0, t0 + σ].
For a proof of the next result, see [6], Lemma 3.3.
Proposition 2.1. If x(t) is a solution of (2.1) in the interval [t0, t0 + σ] , then for
v ∈ [t0, t0 + σ] we have
x(v)(ϑ) = x(v)(v), ϑ > v, ϑ ∈
[
t0 − r, t0 + σ
]
,
and
x(v)(ϑ) = x (ϑ) (ϑ), v > ϑ, ϑ ∈ [t0 − r, t0 + σ].
Left continuous regulated functions with the properties of Proposition 2.1 are candi-
dates for considering them as solutions of the initial value problem described above
for (2.1).
The next result is the key to our approach to retarded functional differential equations
by the theory of generalized differential equations. It states the correspondence between
these equations by relating their solutions in a one-to-one manner. For a proof of it
see [6].
Proposition 2.2. (i) Consider equation (1.5), where f : G1 × [t0, t0 + σ] → Rn,
t 7→ f (yt, t) is Lebesgue integrable on [t0, t0 + σ], P ∈ BV −([t0, t0 + σ],Rn) and
(A), (B) are fulfilled. Let y(t) be a solution of problem (1.5) in the interval [t0, t0 + σ].
Given t ∈ [t0 − r, t0 + σ] , let
x(t)(ϑ) =
y(ϑ), ϑ ∈ [t0 − r, t],
y(t), ϑ ∈ [t, t0 + σ].
Then x(t) ∈ G− ([t0 − r, t0 + σ],Rn) is a solution of (2.1) in [t0 − r, t0 + σ] where the
right-hand side of (2.1) is given by (2.8).
(ii) Reciprocally, letG be given by (2.8) and x(t) be a solution of (2.1) in the interval
[t0 − r, t0 + σ] satisfying the initial condition
x(t0)(ϑ) =
φ(ϑ− t0), t0 − r ≤ ϑ ≤ t0,
x(t0)(t0), t0 ≤ ϑ ≤ t0 + σ.
(2.9)
For every ϑ ∈ [t0 − r, t0 + σ], define
y(ϑ) =
x(t0)(ϑ), t0 − r 6 ϑ 6 t0,
x(ϑ)(ϑ), t0 6 ϑ 6 t0 + σ.
Then y(ϑ) is a solution of (1.5) in [t0 − r, t0 + σ] and y(ϑ) = x (t0 + σ) (ϑ) for all
ϑ ∈ [t0 − r, t0 + σ] .
Proposition 2.2 gives a one-to-one correspondence between the solutions y of (1.5)
and the solutions x of (2.1). Thus given a solution y of (1.5), we have an x given by
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STABILITY FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS 115
Proposition 2.2, (i), which satisfies equation (2.1). Therefore taking t0 ≤ t1 ≤ t2 ≤
≤ t0 + σ we get∥∥x(t2)− x(t1)
∥∥ = sup
ϑ∈[t0−r,t0+σ]
∣∣x(t2)(ϑ)− x(t1)(ϑ)
∣∣ =
= sup
ϑ∈[t0,t0+σ]
∣∣x(t2)(ϑ)− x(t1)(ϑ)
∣∣ = sup
ϑ∈[t1,t2]
∣∣y(ϑ)− y(t1)
∣∣ ≤ vart2t1y
and taking t0 < t1 < t2 < . . . < tk = t0 + σ we get
k∑
i=1
∥∥x(ti)− x(ti−1)
∥∥ ≤ k∑
i=1
vartiti−1
y = vart0+σt0 y.
Hence
vart0+σt0 x ≤ vart0+σt0 y.
It has to be noted that a solution y of (1.5) is a function of bounded variation and
therefore the corresponding x is also of bounded variation.
Reciprocally, if G is given by (2.8) and x(t) is a solution of (2.1) with the initial
condition (2.9), then it can be shown by the procedure above that y given by Propositi-
on 2.2, (ii), satisfies
vart0+σt0 y ≤ vart0+σt0 x < +∞.
In this manner, we have the situation of a one-to-one correspondence between the
solutions of (1.5) and (2.1) and their variations (in different spaces) are the same and
finite.
Remark 2.1. Let us note that in our paper [6] a similar approach to impulsive
retarded functional equations was presented. Of course the definition in this case is
slightly more complicated by an additional term. The complication is technical only, the
reasoning of this note can be used similarly for this case, too. Again, the link between
GODEs and classical systems with impulses as they are described in the book [5] is
given in [7].
3. Concepts of stability for GODEs. In this section, Ω = Bc × [t0 − r,∞), where
Bc =
{
y ∈ X; ‖y‖ < c
}
, c > 0, and X is any Banach space. Let r > 0. In the sequel,
we assume that for F : Ω → X we have F ∈ F(Ω, h) and F (0, t) − F (0, s) = 0, for
t, s ∈ [t0 − r,+∞). Then for every [γ, v] ⊂ [t0 − r,+∞), we have
v∫
γ
DF (0, t) = F (0, v)− F (0, γ) = 0
and, therefore, x ≡ 0 is a solution of the generalized equation
dx
dτ
= DF (x, t) (3.1)
on [t0 − r,+∞).
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116 M. FEDERSON, Š. SCHWABIK
If F ∈ F(Ω, h) and x : [γ, v] → X is a solution of (3.1), where [γ, v] ⊂ [t0−r,+∞),
then x is of bounded variation in [γ, v]. Thus it is natural to measure the distance between
two solutions by the variation norm.
The next stability concepts are based on the variation of the solutions around x ≡ 0.
Definition 3.1. The solution x ≡ 0 of (3.1) is called variationally stable if for every
ε > 0, there exists δ = δ(ε) > 0 such that if x : [γ, v] → Bc, t0 − r 6 γ < v < +∞ is
a function of bounded variation on [γ, v] such that
‖x(γ)‖ < δ
and
varvγ
x(s)− s∫
γ
DF (x(τ), t)
< δ,
then ∥∥x(t)∥∥ < ε, t ∈ [γ, v].
Definition 3.2. The solution x ≡ 0 of (3.1) is called variationally attracting if there
exists a δ0 > 0 and for every ε > 0, there exist a T = T (ε) > 0 and a ρ = ρ(ε) > 0
such that if x : [γ, v] → Bc, t0 − r 6 γ < v < +∞, is a function of bounded variation
in [γ, v] such that ∥∥x(γ)∥∥ < δ0
and
varvγ
x(s)− s∫
γ
DF (x(τ), t)
< ρ,
then ∥∥x(t)∥∥ < ε, for t ∈ [γ, v] ∩ [γ + T,+∞) and γ > t0 − r.
Definition 3.3. The solution x ≡ 0 of (3.1) is called variationally asymptotically
stable if it is variationally stable and variationally attracting.
To Definitions 3.1 – 3.3, it should be noted that if x : [γ, v] → X is a solution of
(3.1) then:
(a) x is of bounded variation on [γ, v] and
(b) varvγ
(
x(s)−
∫ s
γ
DF (x(τ), t)
)
= 0.
Also, the conditions in Definition 3.1 mean that the function x of bounded variation
is close (in the variation norm:
∥∥x(γ)∥∥+var
(
x(s)−
∫ s
γ
DF (x(τ), t))
)
to the solution
x ≡ 0 of (3.1).
Besides the generalized differential equation (3.1), let us consider the perturbed
generalized equation
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STABILITY FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS 117
dx
dτ
= D
[
F (x, t) + P (t)
]
(3.2)
where P ∈ BV −
(
[t0 − r,∞), X
)
. It is easy to verify that for the function G(x, t) =
= F (x, t)+P (t) we have G ∈ F(Ω, hP ), where hP (t) = h(t)+vart−rP . Therefore the
solutions of (3.2) have good properties (existence, uniqueness, etc., see Theorem 2.1,
for instance).
Let us present now some other definitions.
Definition 3.4. The solution x ≡ 0 of (3.1) is called stable with respect to
perturbations if for every ε > 0, there exists δ = δ(ε) > 0 such that if ‖x0‖ < δ
and P ∈ BV −
(
[γ, v], X
)
is continuous from the left with varvγP < δ then∥∥x(t, γ, x0)
∥∥ < ε for every t ∈ [γ, v]
where x(t, γ, x0) is a solution of the perturbed generalized equation (3.2) with
x(γ, γ, x0) = x0 and [γ, v] ⊂ [t0 − r,+∞).
Definition 3.5. The solution x ≡ 0 of (3.1) is called attracting with respect to
perturbations if there is a δ̃ > 0 and for every ε > 0, there exist a T = T (ε) > 0 and a
ρ = ρ(ε) > 0 such that if
‖x0‖ < δ̃ and varvγP < ρ
with P ∈ BV −
(
[γ, v], X
)
, then∥∥x(t, γ, x0)
∥∥ < ε for all t ≥ γ + T, t ∈ [γ, v],
where x(t, γ, x0) is a solution of the perturbed generalized equation (3.2) with
x(γ, γ, x0) = x0 and [γ, v] ⊂ [t0 − r,+∞).
Definition 3.6. The solution x ≡ 0 of (3.1) is called asymptotically stable with
respect to perturbations if it is both stable and attracting with respect to perturbations.
It turns out that the respective definitions presented above are equivalent. Indeed,
we have the following result.
Proposition 3.1. The following statements hold.
(i) The solution x ≡ 0 of (3.1) is variationally stable if and only if it is stable with
respect to perturbations.
(ii) The solution x ≡ 0 of (3.1) is variationally attracting if and only if it is attracting
with respect to perturbations.
(iii) The solution x ≡ 0 of (3.1) is variationally asymptotically stable if and only if
it is asymptotically stable with respect to perturbations.
Proof. Let us prove (i). Assume that the solution x ≡ 0 of (3.1) is variationally
stable. Let for ε > 0 the quantity δ > 0 be given according to Definition 3.4. Then for
the solution x(t) = x(t, γ, x0) of the perturbed generalized equation (3.2) on [γ, v], we
have ‖x(γ)‖ = ‖x(γ, γ, x0)‖ < δ and for any s1, s2 ∈ [γ, v] we get
x(s2)− x(s1) =
s2∫
s1
DF (x(τ), t) + P (s2)− P (s1),
that is,
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118 M. FEDERSON, Š. SCHWABIK
x(s2)−
s2∫
γ
DF (x(τ), t)−
x(s1)− s1∫
γ
DF (x(τ), t)
= P (s2)− P (s1)
and this implies
varvγ
x(s)− s∫
γ
DF (x(τ), t)
= varvγP < δ.
Therefore the variational stability implies∥∥x(t)∥∥ =
∥∥x(t, γ, x0)
∥∥ < ε for t ∈ [γ, v]
and the trivial solution of (3.1) is stable with respect to perturbations.
Reciprocally, if the solution x ≡ 0 of (3.1) is stable with respect to perturbations,
take x : [γ, v] → Bc, −r 6 γ < v < +∞, a function of bounded variation on [γ, v]
such that ‖x(γ)‖ < δ and
varvγ
x(s)− s∫
γ
DF (x(τ), t)
< δ,
where δ > 0 corresponds to some ε > 0 from Definition 3.4.
For s ∈ [γ, v], let P (s) = x(s)−
∫ s
γ
DF (x(τ), t). Then for s1, s2 ∈ [γ, v], we have
P (s2)− P (s1) = x(s2)− x(s1)−
s2∫
s1
DF (x(τ), t).
Hence
x(s2)− x(s1) =
s2∫
s1
DF (x(τ), t) + P (s2)− P (s1), s1, s2 ∈ [γ, v],
which means that x is a solution of (3.2) in [γ, v]. Besides, varvγP < δ, P is left
continuous and
∥∥x(γ)∥∥ =
∥∥x(γ, γ, x0)
∥∥ =
∥∥P (γ)
∥∥ < δ. Therefore the stability with
respect to perturbations implies
∥∥x(t)∥∥ =
∥∥x(t, γ, x0)
∥∥ < ε, for all t ∈ [γ, v], and this
means that the solution x ≡ 0 of (3.1) is variationally stable.
Coming to the attractive part in item (ii), assume first that the solution x ≡ 0 of
(3.1) is variationally attracting. Then there is a δ0 > 0 and for every ε > 0, there exist
a T = T (ε) > 0 and a ρ = ρ(ε) > 0 such that if x : [γ, v] → Bc, −r 6 γ < v < +∞,
is a function of bounded variation in [γ, v] such that ‖x(γ)‖ < δ0 and
varvγ
x(s)− s∫
γ
DF (x(τ), t)
< ρ,
then
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STABILITY FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS 119∥∥x(t)∥∥ < ε, t ∈ [γ, v] ∩ [γ + T,+∞), γ > −r.
If ‖x0‖ < δ̃ and P ∈ BV −
(
[γ, v], X
)
is such that varvγP < ρ, then denote
x(t) = x(t, γ, x0) the solution of the perturbed generalized equation (3.2) satisfying
x(γ, γ, x0) = x0. It follows that ‖x(γ)‖ < δ̃ and we have
varvγ
x(s)− s∫
γ
DF (x(τ), t)
= varvγP < δ.
Hence by Definition 3.2, we get∥∥x(t, γ, x0)
∥∥ =
∥∥x(t)∥∥ < ε for all t ≥ γ + T, t ∈ [γ, v],
that is, the solution x ≡ 0 of (3.1) is attracting with respect to perturbations.
Reciprocally, if the solution x ≡ 0 of (3.1) is attracting with respect to perturbations,
suppose x : [γ, v] → Bc, −r 6 γ < v < +∞, is a left continuous function of bounded
variation in [γ, v] and such that
∥∥x(γ)∥∥ < δ0 and
varvγ
x(s)− s∫
γ
DF (x(τ), t)
< ρ.
As in the previous part of the proof, it is easy to see that x(t) is a solution of (3.2) on
[γ, v], where P (s) = x(s)−
∫ s
γ
DF (x(τ), t) for s ∈ [γ, v]. This function P belongs to
BV −([γ, v], X) and there exists a δ0 > 0 and for every ε > 0, there exist a T = T (ε) >
> 0 and a ρ = ρ(ε) > 0 such that varvγP < ρ. Definition 3.5 now yields∥∥x(t)∥∥ < ε, t ∈ [γ, v] ∩ [γ + T,+∞), γ > −r,
which means that we have the variational attractivity of the trivial solution of (3.1).
Item (iii) follows from (i) and (ii) and we finished the proof.
4. Stability relations between the equations. Consider the retarded system (1.1).
Let G1 ⊂ G([t0 − r,+∞),Rn) be defined as in the beginning of the paper.
We assume that f(φ, t) : G1 × [t0,+∞) → Rn is such that for each y ∈ G1 the
mapping t 7→ f (yt, t) belongs to L1
(
[t0 − r,+∞),Rn
)
and conditions (A) and (B) are
fulfilled. Suppose in addition that f(0, t) = 0 for every t ∈ [t0,+∞). Thus y ≡ 0 is a
solution of (1.1) in [t0 − r,+∞).
For y ∈ G1 and t ∈ [t0 − r,+∞), define F (y, t) as in (2.6). Then
F : G1 × [t0 − r,+∞) → C([t0 − r,+∞),Rn)
and by definition we have F (0, t) = 0, for all t ∈ [t0−r,+∞). Then x ≡ 0 is a solution
of the generalized differential equation
dx
dτ
= DF (x, t) (4.1)
in [t0 − r,+∞).
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120 M. FEDERSON, Š. SCHWABIK
By the results from Proposition 2.2, there is a well described one-to-one correspondence
between solutions of equations (1.1) and (4.1) with F given by (2.6).
We will also consider the perturbed retarded equation (1.5) and, again by Proposition
2.2, its corresponding perturbed generalized equation
dx
dτ
= DG(x, t) = D
[
F (x, t) + P (t)
]
, (4.2)
where F is given by (2.6) and P given by (2.7).
We have
P : [t0 − r,+∞) → G−
(
[t0 − r,+∞),Rn
)
and then
G : G1 × [t0 − r,+∞) → G−
(
[t0 − r,+∞),Rn
)
.
We are now able to present a result which relates the respective concepts of variational
stability and variational attractivity of the trivial solution of the retarded equation (1.1)
and the trivial solution of its corresponding generalized equation (4.1).
Theorem 4.1. The following statements hold.
(i) The solution y ≡ 0 of (1.1) is variationally stable if and only if the solution x ≡ 0
of (4.1) is variationally stable.
(ii) The solution y ≡ 0 of (1.1) is variationally attracting if and only if the solution
x ≡ 0 of (4.1) is variationally attracting.
(iii) The solution y ≡ 0 of (1.1) is variationally asymptotically stable if and only if
the solution x ≡ 0 of (4.1) is variationally asymptotically stable.
Proof. We start by proving (i). Suppose the trivial solution of (1.1) in [t0− r,+∞)
is variationally stable. Then given ε > 0, there exists δ = δ(ε) > 0 such that if
φ ∈ G−([−r, 0],Rn) is such that ‖φ‖ < δ and P (t) belongs to BV −
(
[t0, t1],Rn
)
with
vart1t0P < δ, then ∣∣y(t; t0, φ)
∣∣ < ε
2
, t ∈ [t0, t1],
where y(t; t0, φ) is a solution of (1.5).
We want to prove that the trivial solution of generalized equation (4.1), with F
given by (2.6), is stable with respect to perturbations. Then the result will follow by
Proposition 3.1.
Suppose δ = δ(ε) > 0 from Definition 1.6 is such that δ < ε/2. Let x(t; t0, x0) be
a solution of the perturbed generalized equation (4.2) with F given by (2.6), P given
by (2.7) and x(t0; t0, x0) = x0 and assume that ‖x0‖ < δ, where x0 ∈ G−
(
[t0 −
− r,+∞),Rn
)
, and P ∈ BV −
(
[t0, t1],Rn
)
with vart1t0P < δ.
We have
∥∥x(t0)∥∥ =
∥∥x(t0; t0, x0)
∥∥ =
∥∥x0
∥∥ < δ which means that
supθ∈[t0−r,+∞) |x(t0)(θ)| < δ and therefore supθ∈[t0−r,t0] |φ(θ − t0)| < δ. Thus
‖φ‖ < δ.
Since x is a solution of the perturbed generalized equation we have
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STABILITY FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS 121
x(s2)− x(s1) =
s2∫
s1
D
[
F (x(τ), t) + P (t)
]
=
s2∫
s1
DF (x(τ), t) + P (s2)− P (s1)
for s1, s2 ∈ [t0, t1].
Therefore
x(s2)−
s2∫
t0
DF (x(τ), t)− x(s1) +
s1∫
t0
DF (x(τ), t) = P (s2)− P (s1).
Hence
vart1t0
x(s)− s∫
t0
DF (x(τ), t)
= vart1t0P < δ.
Thus by the variational stability of the trivial solution of (1.1),
∣∣y(t)∣∣ < ε/2, for all
t ∈ [t0, t1].
Finally, we have∥∥x(t)∥∥ = sup
θ∈[t0−r,+∞)
∣∣x(t)(θ)∣∣ = sup
θ∈[t0−r,t]
∣∣y(θ)∣∣ 6
6 ‖φ‖+ sup
θ∈[t0,t]
∣∣y(θ)∣∣ 6 δ +
ε
2
< ε
and we have the sufficiency of item (i).
Now, using (i) from Proposition 3.1, we assume that the trivial solution of (4.1) is
stable with respect to perturbations. Thus given ε > 0, let δ = δ(ε) > 0 be the quantity
from Definition 3.4.
Let y(t; t0, φ) be a solution of the perturbed retarded equation (1.5). Suppose ‖φ‖ <
< δ and P ∈ BV −
(
[t0, t1],Rn
)
with vart1t0P < δ. We want to prove that y ≡ 0 is
variationally stable, that is,
∣∣y(t; t0, φ)
∣∣ < ε, t ∈ [t0, t1]. Then the reciprocal of item (i)
(necessity) will follow by Proposition 3.1.
Let x(t; t0, x0) be the solution of the perturbed generalized equation (4.2) with F
given by (2.6) and P given by (2.7), that is, x is the solution corresponding to y
obtained according to Proposition 2.2. We have vart1t0x 6 vart1t0y (see the comments
after Proposition 2.2) and, analogously, vart1t0P 6 vart1t0P < δ. Thus, from the stability
with respect to perturbations of the trivial solution of (4.1), ‖x(t)‖ < ε, that is,
sup
θ∈[t0−r,+∞)
∣∣x(t)(θ)∣∣ < ε.
Therefore the relation in Proposition 2.2 implies
sup
θ∈[t0−r,t]
∣∣y(θ)∣∣ < ε, t ∈ [t0, t1].
In particular,
sup
θ∈[t0,t1]
∣∣y(θ)∣∣ 6 sup
θ∈[t0−r,t1]
∣∣y(θ)∣∣ < ε.
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122 M. FEDERSON, Š. SCHWABIK
Now we will prove (ii).
At first, suppose the trivial solution of the retarded equation (1.1) is variationally
attracting. Thus there exists δ0 > 0 and for every ε > 0, let T = T (ε) > 0 and
ρ = ρ(ε) > 0 be from Definition 1.7.
Let x(t; t0, x0) be a solution of the perturbed generalized equation (4.2) with F given
by (2.6) and P given by (2.7) and let y(t; t0, φ) be the solution of the perturbed retarded
equation (1.5) obtained from x according to Proposition 2.2.
Let δ > 0 be such that ‖x0‖ < δ and suppose P ∈ BV −([t0, t1],Rn) with vart1t0P <
< ρ. We can suppose, without loss of generality, that δ < min{δ0, ρ, ε/2}. Then∥∥x(t0)∥∥ =
∥∥x0
∥∥ < δ0
and
vart1t0
x(s)− s∫
t0
DF (x(τ), t)
= vart1t0P < ρ.
But the variational attractivity of the trivial solution of (1.1) implies∣∣y(t; t0, φ)
∣∣ =
∣∣y(t)∣∣ < ε
2
, t > t0 + T, t ∈ [t0, t1].
Then, taking δ < ε/2, we obtain∥∥x(t)∥∥ = sup
θ∈[t0−r,+∞)
∣∣x(t)(θ)∣∣ = sup
θ∈[t0−r,t]
∣∣y(θ)∣∣ 6
6 ‖φ‖+ sup
θ∈[t0,t]
|y(θ)| < ‖x0‖+
ε
2
< ε
for t > t0 + T, t ∈ [t0, t1], where we applied the relations of Proposition 2.2 to get the
second equality and ‖φ‖ = ‖x0‖, since
‖x0‖ =
∥∥x(t0)∥∥ = sup
θ∈[t0−r,+∞)
∣∣x(t)(θ)∣∣ = sup
θ∈[t0−r,t0]
|φ(θ − t0)| = ‖φ‖.
Thus ∥∥x(t; t0, x0)
∥∥ =
∥∥x(t)∥∥ < ε, t > t0 + T, t ∈ [t0, t1],
and hence x is attracting with respect to perturbations. The sufficiency of (ii) follows
then by Proposition 3.1.
Now we will prove the reciprocal of item (ii). Suppose then that the trivial solution
of generalized equation (4.1) is attracting with respect to perturbations. Then there exists
δ0 > 0 and given ε > 0 let T = T (ε) > 0 and ρ = ρ(ε) > 0 be from Definition 3.5.
Let y(t; t0, φ) be a solution of the perturbed retarded equation (1.3), or equivalently,
of equation (1.5) with P (t) =
∫ t
t0
p(s)ds, t ≥ t0. Suppose ‖φ‖ < δ0 and P ∈
∈ BV −([t0, t1],Rn) with vart1t0P < ρ.
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STABILITY FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS 123
By Proposition 2.2, it follows that ‖x0‖ = ‖φ‖ < δ0. Also, for P given by (2.7),
we have vart1t0P 6 vart1t0P < ρ (see the comments after Proposition 2.2). Therefore the
attractivity with respect to perturbations of the trivial solution of (4.1) implies∥∥x(t)∥∥ =
∥∥x(t; t0, x0)
∥∥ < ε, t > t0 + T, t ∈ [t0, t1].
Therefore, for t > t0 + T, t ∈ [t0, t1], we have by Proposition 2.2,∣∣y(t)∣∣ =
∣∣y(t; t0, φ)
∣∣ =
∣∣x(t)(t)∣∣ 6
∥∥x(t)∥∥ < ε.
Assertion (iii) follows from (i) and (ii) and from Proposition 3.1.
5. Converse Lyapunov theorems. In the book [7] and in [4] direct Lyapunov-type
theorems for stability of a solution of a GODE are given. In [3] they are used for
equation (1.1).
Converse Lyapunov theorems are an interesting topic, we present them shortly in
this concluding section of the paper.
In order to obtain converse Lyapunov theorems for equation (1.1), we need the
following results, borrowed from [7] or [4], for the generalized differential equation (4.1).
Let us consider the general case where Ω = Bc × [t0 − r,∞), with Bc =
{
y ∈
∈ X; ‖y‖ < c
}
, c > 0, and X is a Banach space. Suppose F : Ω → X is such that
F ∈ F(Ω, h) and F (0, t) − F (0, s) = 0, for t, s ∈ [t0 − r,+∞) and consider the
generalized differential equation
dx
dτ
= DF (x, t). (5.1)
The following two results are respectively Theorems 10.23 and 10.24 from [7]. They
can also be found in [4].
Theorem 5.1. If the trivial solution x ≡ 0 of the generalized differential equati-
on (5.1) is variationally stable, then for every 0 < a < c, there exists a function
V : [t0 − r,+∞) × Ba → R, where Ba = {y ∈ X; ‖y‖ < a}, such that for every
x ∈ Ba, the function V (·, x) belongs to BV −([t0 − r,+∞),R) and the following
conditions hold:
(i) V (t, 0) = 0, t ∈ [t0 − r,+∞);
(ii) |V (t, z)− V (t, y)| 6 ‖z − y‖, t ∈ [t0 − r,+∞), z, y ∈ Ba;
(iii) V is positive definite along every solution x(t) of the generalized equation (5.1),
that is, there is a function b : [0,+∞) → R of Hahn class such that
V (t, x(t)) > b
(
‖x(t)‖
)
, (t, x(t)) ∈ [t0 − r,+∞)×Ba;
(iv) for all solutions x(t) of (5.1),
V̇ (t, x(t)) = lim sup
η→0+
V (t+ η, x(t+ η))− V (t, x(t))
η
6 0,
that is, the right derivative of V along every solution x(t) of (5.1) is non-positive.
Theorem 5.2. If the trivial solution x ≡ 0 of the generalized differential equati-
on (5.1) is variationally asymptotically stable, then for every 0 < a < c, there exists a
function V : [t0 − r,+∞) × Ba → R such that for every x ∈ Ba, the function V (·, x)
belongs to BV −
(
[t0 − r,+∞),R
)
and the following conditions hold:
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124 M. FEDERSON, Š. SCHWABIK
(i) V (t, 0) = 0, t ∈ [t0 − r,+∞);
(ii)
∣∣V (t, z)− V (t, y)
∣∣ 6 ‖z − y‖, t ∈ [t0 − r,+∞), z, y ∈ Ba;
(iii) V is positive definite along every solution x(t) of the generalized equation (5.1),
that is, there is a function b : [0,+∞) → R of Hahn class such that
V (t, x(t)) > b
(
‖x(t)‖
)
, (t, x(t)) ∈ [t0 − r,+∞)×Ba;
(iv) for all solutions x(s) of (5.1) defined for s > t, where x(t) = z ∈ Ba, the
relation
V̇ (t, x(t)) = lim sup
η→0+
V (t+ η, x(t+ η))− V (t, x(t))
η
6 V (t, z)
holds.
Now let us consider the more specialized equation (4.1), with F given by (2.6),
corresponding to the retarded system (1.1). We consider X = G−([t0 − r,+∞),Rn).
As in [3], we need to relate a Lyapunov functional for (4.1) to a Lyapunov functional
for (1.1)
Let y : [γ, v] → Rn be a solution of equation (1.1) on [γ, v] ⊂ [t0 − r,+∞),
[γ, v] 3 t0, such that yt = ψ for a given t > t0, that is, ψ ∈ G−([−r, 0],Rn) and
ψ(θ) = yt (θ) = y (t+ θ) = y(t)−
∫
[ t+θ,t]
f (ys, s) ds
for almost every θ ∈ [−r, 0]. In this case, we write yt+η = yt+η(t, ψ) for every η > 0.
Then if U : [t0 − r,+∞)×G−
(
[−r, 0],Rn
)
→ R, we define
D+U(t, ψ) = lim sup
η→0+
U(t+ η, yt+η(t, ψ))− U(t, yt(t, ψ))
η
,
for t > t0.
Let x be a solution of the generalized equation (4.1) on the interval [γ, v] ⊂ [t0 −
− r,+∞), [γ, v] 3 t0, with initial condition x(t0) = x0, where
x(t0)(ϑ) =
φ(ϑ− t0) for ϑ ∈ [t0 − r, t0],
x(t0)(t0) for ϑ ∈ [t0,+∞).
(5.2)
Then x(t)(t + θ) = y(t + θ), for all t ∈ [t0 − r,+∞) and all θ ∈ [−r, 0] and hence
(x(t))t = yt for all t.
On the other hand, given x(t) ∈ G−([t0−r,+∞),Rn), since x is locally of bounded
variation, we can consider x(t) as a solution on [γ, v] ⊂ [t0 − r,+∞), [γ, v] 3 t0, of
the generalized equation (4.1), with initial condition x(t0) = x0 given by (5.2). Then
Proposition 2.2 implies we can find a solution y(t; t0, φ) of (1.1) by means of the solution
x(t; t0, x0) of (4.1). Suppose (x(t))t = ψ. In this case, we write xψ(t) instead of x(t)
and we have yt = ψ.
Therefore (t, xψ(t)) 7→ (t, yt(t, ψ)) is a one-to-one mapping and we can define
V : [t0 − r,+∞)×G−([t0 − r,+∞),Rn) → R by
V (t, xψ(t)) = U(t, yt(t, ψ)), t > t0. (5.3)
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STABILITY FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS 125
Then we have
D+U(t, ψ) = lim sup
η→0+
V (t+ η, xψ(t+ η))− V (t, xψ(t))
η
for all t > t0. We write U̇(t, yt) = D+U(t, yt).
With the notation above, we now are able to present converse Lyapunov results for
equation (1.1).
Theorem 5.3. If the trivial solution y ≡ 0 of the retarded differential equation (1.1)
is variationally stable, then for every 0 < a < c, there exists a function U : [t0−r,+∞)×
×Ea → R, where Ea =
{
ψ ∈ G−([−r, 0],Rn); ‖ψ‖ < a
}
, such that for every x ∈ Ea,
the function U(·, ψ) belongs to BV −([t0 − r,+∞),R) and the following conditions
hold:
(i) U(t, 0) = 0, t ∈ [t0 − r,+∞);
(ii)
∣∣U(t, ψ)− U(t, ψ)
∣∣ 6 ‖ψ − ψ‖, t ∈ [t0 − r,+∞), ψ, ψ ∈ Ea;
(iii) U is positive definite along every solution y(t) of the retarded equation (1.1),
that is, there is a function b : [0,+∞) → R of Hahn class such that
U(t, yt) > b(‖yt‖), (t, yt) ∈ [t0 − r,+∞)× Ea;
(iv) for all solutions y(t) of (1.1),
U̇(t, yt) = lim sup
η→0+
U(t+ η, yt+η)− V (t, yt))
η
6 0,
that is, the right derivative of U along every solution y(t) of (1.1) is non-positive.
Proof. If the trivial solution of (1.1) is variationally stable, then by Theorem 4.1 the
trivial solution of the generalized equation (4.1) with F given by (2.6) and P is given
by (2.7) is also variationally stable. Then by Theorem 5.1, there exists a function V
satisfying all conditions in that theorem. Define U : [t0−r,+∞)×G−([−r, 0],Rn) → R
by the relation in (5.3). Then as in the proof of [3], Theorem 4.3, U has the properties
above and the proof is complete.
The proof of the next result follows as in the proof of Theorem 5.3, but applying [3],
Theorem 4.5 instead of [3], Theorem 4.3.
Theorem 5.4. If the trivial solution y ≡ 0 of the retarded differential equation (1.1)
is variationally asymptotically stable, then for every 0 < a < c, there exists a function
U : [t0 − r,+∞)×Ea → R such that for every x ∈ Ba, the function U(·, x) belongs to
BV −([t0 − r,+∞),R) and the following conditions hold:
(i) U(t, 0) = 0, t ∈ [t0 − r,+∞);
(ii)
∣∣U(t, ψ)− U(t, ψ)
∣∣ 6 ‖ψ − ψ‖, t ∈ [t0 − r,+∞), ψ, ψ ∈ Ea;
(iii) U is positive definite along every solution y(t) of the retarded equation (1.1),
that is, there is a function b : [0,+∞) → R of Hahn class such that
U(t, yt) > b(‖yt‖), (t, yt) ∈ [t0 − r,+∞)× Ea;
(iv) for all solutions y(s) of (1.1) defined for s > t, where y(t) = ψ ∈ Ea, the
relation
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126 M. FEDERSON, Š. SCHWABIK
U̇(t, yt) = lim sup
η→0+
U(t+ η, yt+η)− U(t, yt)
η
6 U(t, ψ)
holds.
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Received 15.10.07
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