On the uniform asymptotic stability of functional differential equations of the neutral type
Authors: Hale J. K. , Izé A. F. 
Published: June 2010

Consider the functional equations of neutral type (1)

(
d

/

d
t
)
D
(
t
,

x
t

)
=
f
(
t
,

x
t

)

(d/dt)D(t,{x_t}) = f(t,{x_t})

and (2)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d slash d t right-parenthesis left-bracket upper D left-parenthesis t comma x Subscript t Baseline right-parenthesis minus upper G left-parenthesis t comma x Subscript t Baseline right-parenthesis right-bracket equals f left-parenthesis t comma x Subscript t Baseline right-parenthesis plus upper F left-parenthesis t comma x Subscript t Baseline right-parenthesis">

(
d

/

d
t
)
[
D
(
t
,

x
t

)
−
G
(
t
,

x
t

)
]
=
f
(
t
,

x
t

)
+
F
(
t
,

x
t

)

(d/dt)[D(t,{x_t}) - G(t,{x_t})] = f(t,{x_t}) + F(t,{x_t})

where

D
,
f

D,f

are bounded linear operators from

C
[
a
,
b
]

C[a,b]

into

R
n

{R^n}

or

C
n

{C^n}

for each fixed

t
t

in
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma normal infinity right-parenthesis comma upper F equals upper F 1 plus upper F 2 comma upper G equals upper G 1 plus upper G 2 comma StartAbsoluteValue upper F 1 left-parenthesis t comma phi right-parenthesis EndAbsoluteValue less-than-over-equals v left-parenthesis t right-parenthesis StartAbsoluteValue phi EndAbsoluteValue comma StartAbsoluteValue upper G 1 left-parenthesis t comma phi right-parenthesis EndAbsoluteValue less-than-over-equals pi left-parenthesis t right-parenthesis StartAbsoluteValue phi EndAbsoluteValue comma pi left-parenthesis t right-parenthesis">

[
0
,
∞
)
,
F
=

F
1

+

F
2

,
G
=

G
1

+

G
2

,

|

F
1

(
t
,
ϕ
)

|

≦
v
(
t
)

|

ϕ

|

,

|

G
1

(
t
,
ϕ
)

|

≦
π
(
t
)

|

ϕ

|

,
π
(
t
)

[0,\infty ),F = {F_1} + {F_2},G = {G_1} + {G_2},|{F_1}(t,\phi )| \leqq v(t)|\phi |,|{G_1}(t,\phi )| \leqq \pi (t)|\phi |,\pi (t)

, bounded and for any

ε
>
0

\varepsilon > 0

, there exists

δ
(
ε
)
>
0

\delta (\varepsilon ) > 0

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper F 2 left-parenthesis t comma phi right-parenthesis EndAbsoluteValue less-than-over-equals epsilon StartAbsoluteValue phi EndAbsoluteValue comma StartAbsoluteValue upper G 2 left-parenthesis t comma phi right-parenthesis EndAbsoluteValue less-than-over-equals epsilon StartAbsoluteValue phi EndAbsoluteValue comma t greater-than-over-equals 0 comma StartAbsoluteValue phi EndAbsoluteValue greater-than delta left-parenthesis epsilon right-parenthesis">

|

F
2

(
t
,
ϕ
)

|

≦
ε

|

ϕ

|

,

|

G
2

(
t
,
ϕ
)

|

≦
ε

|

ϕ

|

,
t
≧
0
,

|

ϕ

|

>
δ
(
ε
)

|{F_2}(t,\phi )| \leqq \varepsilon |\phi |,|{G_2}(t,\phi )| \leqq \varepsilon |\phi |,t \geqq 0,|\phi | > \delta (\varepsilon )

. The authors prove that if (1) is uniformly asymptotically stable, then there is a

ζ
0

,
0
>

ζ
0

>
1

{\zeta _0},0 > {\zeta _0} > 1

, such that for any

p
>
0
,
0
>
ζ
>

ζ
0

p > 0,0 > \zeta > {\zeta _0}

, there are constants

v
0

>
0
,

M
0

>
0
,

s
0

>
0

{v_0} > 0,{M_0} > 0,{s_0} > 0

, such that if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi left-parenthesis t right-parenthesis greater-than upper M 0 comma t greater-than-over-equals s 0 comma left-parenthesis 1 slash p right-parenthesis integral Subscript t Superscript t plus p Baseline v left-parenthesis s right-parenthesis d s greater-than zeta v 0 comma t greater-than 0">

π
(
t
)
>

M
0

,
t
≧

s
0

,
(
1

/

p
)

∫
t

t
+
p

v
(
s
)
d
s
>
ζ

v
0

,
t
>
0

\pi (t) > {M_0},t \geqq {s_0},(1/p)\int _t^{t + p} {v(s)ds > \zeta {v_0}} ,t > 0

, then the solution

x
=
0

x = 0

of (2) is uniformly asymptotically stable. The result generalizes previous results which consider only terms of the form

F
1

,

G
1

{F_1},{G_1}

or

F

2
,

G
2

{F_{2,}}{G_2}

but not both simultaneously, and the stronger hypothesis

lim

t
→
∞

π
(
t
)
=
0

{\lim _{t \to \infty }}\pi (t) = 0

.