Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 195916, 14 pages
doi:10.1155/2010/195916
Research Article
Fixed Points and Stability in Nonlinear
Equations with Variable Delays
Liming Ding,1, 2 Xiang Li,1 and Zhixiang Li1
1
Department of Mathematics and System Science, College of Science,
National University of Defence Technology, Changsha 410073, China
2
Air Force Radar Academy, Wuhan 430010, China
Correspondence should be addressed to Liming Ding, limingding@sohu.com
Received 9 July 2010; Accepted 18 October 2010
Academic Editor: Hichem Ben-El-Mechaiekh
Copyright q 2010 Liming Ding et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We consider two nonlinear scalar delay differential equations with variable delays and give some
new conditions for the boundedness and stability by means of the contraction mapping principle.
We obtain the differences of the two equations about the stability of the zero solution. Previous
results are improved and generalized. An example is given to illustrate our theory.
1. Introduction
Fixed point theory has been used to deal with stability problems for several years. It
has conquered many difficulties which Liapunov method cannot. While Liapunov’s direct
method usually requires pointwise conditions, fixed point theory needs average conditions.
In this paper, we consider the nonlinear delay differential equations
x t −atxt − r1 t btgxt − r2 t,
1.1
x t −atfxt − r1 t btgxt − r2 t,
1.2
where r1 t, r2 t : 0, ∞ → 0, ∞, r max{r1 0, r2 0}, a, b : 0, ∞ → R, f, g : R → R
are continuous functions. We assume the following:
A1 r1 t is differentiable,
A2 the functions t − r1 t, t − r2 t : 0, ∞ → −r, ∞ is strictly increasing,
A3 t − r1 t, t − r2 t → ∞ as t → ∞.
2
Fixed Point Theory and Applications
Many authors have investigated the special cases of 1.1 and 1.2. Since Burton
1 used fixed point theory to investigate the stability of the zero solution of the equation
x t −atxt − r, many scholars continued his idea. For example, Zhang 2 has studied
the equation
x t −atxt − rt,
1.3
Becker and Burton 3 have studied the equation
x t −atfxt − rt,
1.4
Jin and Luo 4 have studied the equation
x t −atxt − r1 t btx1/3 t − r2 t.
1.5
Burton 5 and Zhang 6 have also studied similar problems. Their main results are the
following.
Theorem 1.1 Burton 1. Suppose that rt r, a constant, and there exists a constant α < 1 such
that
t
t
|as r|ds t−r
|as r|e
−
t
s
aurdu
s
0
|au r|du ds ≤ α,
1.6
s−r
∞
for all t ≥ 0 and 0 asds ∞. Then, for every continuous initial function ψ : −r, 0 → R, the
solution xt xt, 0, ψ of 1.3 is bounded and tends to zero as t → ∞.
Theorem 1.2 Zhang 2. Suppose that r is differentiable, the inverse function ht of t−rt exists,
and there exists a constant α ∈ 0, 1 such that for t ≥ 0
i
t
lim inf
t→∞
ahs > −∞,
1.7
0
ii
t
t−rt
|ahs|ds t
e
−
t
s
ahudu
|ahs|
0
s
|ahv|dv ds θs,
1.8
s−rs
t t
where θt 0 e− s ahudu |as||r s|ds. Then, the zero solution of 1.3 is
asymptotically stable if and only if
iii
t
0
ahsds −→ ∞,
as t −→ ∞.
1.9
Fixed Point Theory and Applications
3
Theorem 1.3 Burton 7. Suppose that rt r, a constant. Let f be odd, increasing on 0, L,
and satisfies a Lipschitz condition, and let x − fx be nondecreasing on 0, L. Suppose also that for
each L1 ∈ 0, L, one has
L1 − fL1 sup
t
t≥0
fL1 sup
t≥0
t
e−
s
aurdu
|as r|ds fL1 sup
t≥0
0
t
e
−
t
s aurdu
|as r|
0
s
t
|au r|du
t−r
1.10
|au r|du ds < L1 ,
s−r
and there exists J > 0 such that
−
t
as rds ≤ J
for t ≥ 0.
1.11
0
Then, the zero solution of 1.4 is stable.
Theorem 1.4 Becker and Burton 3. Suppose f is odd, strictly increasing, and satisfies a
Lipschitz condition on an interval −l, l and that x − fx is nondecreasing on 0, l. If
t
sup
t ≥ t1
audu <
t−rt
1
,
2
1.12
where t1 is the unique solution of t − rt 0, and if a continuous function a : 0, ∞ → R exists
such that
at at 1 − r t ,
1.13
on 0, ∞, then the zero solution of 1.5 is stable at t 0. Furthermore, if f is continuously
differentiable on −l, l with f 0 / 0 and
t
audu −→ ∞ as t −→ ∞,
1.14
0
then the zero solution of 1.4 is asymptotically stable.
In the present paper, we adopt the contraction mapping principle to study the
boundedness and stability of 1.1 and 1.2. That means we investigate how the stability
property will be when 1.3 and 1.4 are added to the perturbed term btgxt − r2 t.
We obtain their differences about the stability of the zero solution, and we also improve and
generalize the special case r1 t r1 . Finally, we give an example to illustrate our theory.
2. Main Results
From existence theory, we can conclude that for each continuous initial function ψ : −r, 0 →
R there is a continuous solution xt, 0, ψ on an interval 0, T for some T > 0 and
4
Fixed Point Theory and Applications
xt, 0, ψ ψt on −r, 0. Let CS1 , S2 denote the set of all continuous functions φ : S1 → S2
and ψ max{|ψt| : t ∈ −r, 0}. Stability definitions can be found in 8.
Theorem 2.1. Suppose that the following conditions are satisfied:
i g0 0, and there exists a constant L > 0 so that if |x|, |y| ≤ L, then
gx − g y ≤ x − y,
2.1
ii there exists a constant α ∈ 0, 1 and a continuous function h : −r, ∞ → R such that
t
t−r1 t
|hs|ds t
e−
t
s
hudu
|hs|
s
s−r1 s
0
t
e
−
t
s hudu
|hu|du ds
hs − r1 s 1 − r s − as |bs| ds ≤ α,
2.2
1
0
iii
t
lim inf
t→∞
hsds > −∞.
2.3
0
Then, the zero solution of 1.1 is asymptotically stable if and only if
iv
t
hsds −→ ∞,
as t −→ ∞.
2.4
0
Proof. First, suppose that iv holds. We set
t
J sup − hsds .
t≥0
2.5
0
Let S {φ | φ ∈ C−r, ∞, R, φ supt≥−r |φt| < ∞}, then S is a Banach space.
t
Multiply both sides of 1.1 by e 0 hsds , and then integrate from 0 to t to obtain
xt x0 e−
−
t
0
t
0
hsds
t
e−
t
s
hudu
hsxsds
0
e−
t
s
hudu
asxs − r1 sds t
0
e−
t
s
2.6
hudu
bsgxs − r2 sds.
Fixed Point Theory and Applications
5
By performing an integration by parts, we have
xt x0 e
−
t
0
hsds
t
e
−
t
s
hudu
s−r1 s
0
t
e−
t
s
e−
t
s
huxudu
ds
hudu
hs − r1 s 1 − r1 s − as xs − r1 sds
hudu
bsgxs − r2 sds,
0
t
s
2.7
0
or
xt x0 e−
−
t
t
0
hsds
e−
t
s
− e−
hudu
t
0
hsds
−r1 0
0
t
hsxsds t
t−r1 t
hsxsds
s
hs
0
t
0
s−r1 s
huxudu ds
e− s hudu hs − r1 s 1 − r1 s − as xs − r1 sds
t
e−
t
s
hudu
2.8
bsgxs − r2 sds.
0
Let
φ | φ ∈ S, supφt ≤ L, φt ψt for t ∈ −r, 0, φt −→ 0 as t −→ ∞ .
M
2.9
t≥−r
Then, M is a complete metric space with metric supt≥0 |φt − ηt| for φ, η ∈ M. For all φ ∈ M,
define the mapping P
P φ t ψt,
t ∈ −r, 0,
t
t
P φ t ψ0e− 0 hsds − e− 0 hsds
−
t
e−
t
s
hudu
t
e−
t
s
hudu
0
t
0
e−
t
s
hudu
−r1 0
hsψsds t
t−r1 t
hsφsds
s
hs
0
0
s−r1 s
huφudu ds
hs − r1 s 1 − r1 s − as φs − r1 sds
bsg φs − r2 s ds,
t ≥ 0.
2.10
6
Fixed Point Theory and Applications
By i and g0 0,
0
P φ t ≤ ψ 1 −r1 0
L
t
t−r1 t
t
0
|hs|ds e−
|hs|ds t
e−
t
s
t
0
hsds
hudu
|hs|
s
0
s−r1 s
|hu|du ds
− s hudu hs − r1 s 1 − r1 s − as |bs| ds
e
t
0
≤ ψ 1 2.11
−r1 0
|hs|ds eJ αL.
0
Thus, when ψ ≤ δ 1 − αL/1 −r1 0 |hs|dseJ , |P φt| ≤ L.
We now show that P φt → 0 as t → ∞. Since φt → 0 and t − r1 t → ∞ as
t → ∞, for each ε > 0, there exists a T1 > 0 such that t > T1 implies |φt − r1 t| < ε. Thus, for
t ≥ T1 ,
t
t
hsφsds ≤ ε
|hs|ds ≤ αε.
|I1 | t−r1 t
t−r1 t
2.12
Hence, I1 → 0 as t → ∞. And
s
t t
− s hudu
hs
huφudu ds
|I2 | e
0
s−r1 s
≤
T1
e−
t
s
hudu
|hs|
s
s−r1 s
0
t
e−
t
s
hudu
|hs|
s
s−r1 s
T1
≤L
T1
e−
t
s
hudu
|hu|φudu ds
|hs|
|hu|φudu ds
s
0
s−r1 s
|hu|du ds ε
t
e−
t
s
hudu
|hs|
T1
s
s−r1 s
|hu|du ds,
2.13
By ii and iv, there exists T2 > T1 such that t ≥ T2 implies
T1
L
0
e−
t
s
hudu
|hs|
s
s−r1 s
|hu|du ds < ε.
2.14
Apply ii to obtain |I2 | < ε 2ε < 2ε. Thus, I2 → 0 as t → ∞. Similarly, we can show that the
rest term in 2.10 approaches zero as t → ∞. This yields P φt → 0 as t → ∞, and hence
P φ ∈ M.
Fixed Point Theory and Applications
7
Also, by ii, P is a contraction mapping with contraction constant α. By the contraction
mapping principle, P has a unique fixed point x in M which is a solution of 1.1 with xs ψs on −r, 0 and xt → 0 as t → ∞.
In order to prove stability at t 0, let ε > 0 be given. Then, choose m > 0 so that
m < min{ε, L}. Replacing L with m in M, we see there is a δ > 0 such that ψ < δ implies
that the unique continuous solution x agreeing with ψ on −r, 0 satisfies |xt| ≤ m < ε for all
t ≥ −r. This shows that the zero solution of 1.1 is asymptotically stable if iv holds.
Conversely, suppose iv fails. Then, by iii, there exists a sequence {tn }, tn → ∞ as
t
n → ∞ such that limn → ∞ 0n hsds l for some l ∈ R. We may choose a positive constant N
satisfying
−N ≤
tn
hsds ≤ N,
2.15
0
for all n ≥ 1. To simplify the expression, we define
ωs hs − r1 s 1 − r1 s − as |bs| hs
s
s−r1 s
|hu|du,
2.16
for all s ≥ 0. By ii, we have
tn
e−
tn
s
hudu
ωsds ≤ α.
2.17
0
This yields
tn
s
tn
e 0 hudu ωsds ≤ αe 0
hudu
≤ αeN .
2.18
0
t s
The sequence { 0n e 0 hudu ωsds} is bounded, so there exists a convergent subsequence. For
brevity of notation, we may assume that
tn
lim
n→∞
s
e 0 hudu ωsds γ,
2.19
0
for some γ ∈ R and choose a positive integer k so large that
tn
s
e 0 hudu ωsds <
tk
for all n ≥ k, where δ0 > 0 satisfies 2δ0 JeN α < 1.
δ0
,
4N
2.20
8
Fixed Point Theory and Applications
By iii, J in 2.5 is well defined. We now consider the solution xt xt, tk , ψ of
1.1 with ψtk δ0 and |ψs| ≤ δ0 for s ≤ tk . We may choose ψ so that |xt| ≤ L for t ≥ tk
and
ψ tk −
t
k
tk −r1 tk hsψsds ≥
1
δ0 .
2
2.21
It follows from 2.10 with xt P xt that for n ≥ tk ,
tn tn
t
1
tn
− t n hudu
hsxsds ≥ δ0 e k
−
e− s hudu ωsds
xtn −
2
tn −r1 tn tk
−
1
δ0 e
2
e
≥e
≥
−
−
tn
t
k
tn
t
k
tn
t
k
hudu
hudu
hudu
−
1
δ0 e
4
tn
t
k
−e
−
tn
0
hudu
tn
s
e 0 hudu ωsds
tk
tk
1
δ0 − e− 0 hudu
2
1
δ0 − N
2
hudu
≥
tn
e
s
0
tn
s
e0
hudu
tk
hudu
ωsds
2.22
ωsds
tk
1
δ0 e−2N > 0.
4
On the other hand, if the solution of 1.1 xt xt, tk , ψ → 0 as t → ∞, since tn − r1 tn →
∞ as n → ∞ and ii holds, we have
xtn −
tn
tn −r1 tn hsxsds −→ 0 as n −→ ∞,
2.23
which contradicts 2.22. Hence, condition iv is necessary for the asymptotically stability of
the zero solution of 1.1. The proof is complete.
When r1 t r1 , a constant, ht at r1 , we can get the following.
Corollary 2.2. Suppose that the following conditions are satisfied:
i g0 0, and there exists a constant L > 0 so that if |x|, |y| ≤ L, then
gx − g y ≤ x − y,
2.24
Fixed Point Theory and Applications
9
ii there exists a constant α ∈ 0, 1 such that for all t ≥ 0, one has
t
|as r1 |ds t−r1
t
e−
t
s
aur1 du
|as r1 |
0
t
e
−
t
s aur1 du
s
|au r1 |du ds
s−r1
2.25
|bs|ds ≤ α,
0
iii
t
lim inf
t→∞
as r1 ds > −∞.
2.26
0
Then, the zero solution of 1.1 is asymptotically stable if and only if
iv
t
as r1 ds −→ ∞,
as t −→ ∞.
2.27
0
Remark 2.3. We can also obtain the result that xt is bounded by L on −r, ∞. Our results
generalize Theorems 1.1 and 1.2.
Theorem 2.4. Suppose that a continuous function a : 0, ∞ → R exists such that at at1 −
r1 t and that the inverse function ht of t − r1 t exists. Suppose also that the following conditions
are satisfied:
i there exists a constant J > 0 such that supt≥0 {−
t
0
ahsds} < J,
ii there exists a constant L > 0 such that fx, x − fx, gx satisfy a Lipschitz condition
with constant K > 0 on an interval −L, L,
iii f and g are odd, increasing on 0, L. x − fx is nondecreasing on 0, L,
iv for each L1 ∈ 0, L, one has
L1 − fL1 sup
t
t≥0
gL1 sup
t≥0
e−
t
s
ahudu
ahs|ds fL1 sup
|
t≥0
0
t
e
−
t
hudu
sa
0
Then, the zero solution of 1.2 is stable.
|bs|ds < L1 .
t
t−r1 t
ahs|ds
|
2.28
10
Fixed Point Theory and Applications
Proof. By iv, there exists α ∈ 0, 1 such that
L − fLsup
t
t≥0
e−
t
ahudu
s
ahs|ds fLsup
|
t≥0
0
gLsup
t≥0
t
t
e
−
t
hudu
sa
t−r1 t
ahs|ds
|
2.29
|bs|ds ≤ αL.
0
Let S be the space of all continuous functions φ : −r, ∞ → R such that
t
φ : sup e−dK2 0 |ahs||bs|ds φt : t ∈ −r, ∞ < ∞,
K
2.30
where d > 3 is a constant. Then, S, | · |K is a Banach space, which can be verified with
Cauchy’s criterion for uniform convergence.
The equation 1.2 can be transformed as
ahtfxt x t −
d
dt
t
t−r1 t
ahsfxsds btgxt − r2 t
−
ahtxt aht xt − fxt
d t
ahsfxsds btgxt − r2 t.
dt t−r1 t
2.31
By the variation of parameters formula, we have
xt x0 e
−
t
0
ahsds
t
t−r1 t
−e
−
t
0
ahsds
0
−r1 0
ahsfxsds t
e−
ahsfxsds t
e−
t
s
ahudu
xs − fxs ds
0
t
s
ahudu
bsgxs − r2 sds.
0
2.32
Let
M
φ | φ ∈ S, sup φt ≤ L, φt ψt, t ∈ −r, 0 ,
t≥−r
2.33
Fixed Point Theory and Applications
11
then M is a complete metric space with metric |φ − η|K for φ, η ∈ M. For all φ ∈ M, define
the mapping P
P φ t ψt,
t ∈ −r, 0,
P φ t ψ0e
t
−
e−
t
0
ahsds
t
−e
ahudu
s
−
t
0
ahsds
−r1 0
ahsf φs ds t−r1 t
ahsf ψs ds
2.34
φs − f φs ds
0
t
0
t
e−
t
s
ahudu
bsg φs − r2 s ds.
0
By i, iii, and 2.29, we have
J
P φ t ≤ ψ e eJ f ψ fLsup
t
t≥0
t−r1 t
0
−r1 0
ahs|ds L − fLsup
|
t≥0
ahs|ds gLsup
|
t≥0
≤ ψ eJ eJ f ψ 0
−r1 0
t
t
e
−
t
s−r1 s
ahudu
ahsds
0
t
e s ahudu |bs|ds
0
ahs|ds αL.
|
2.35
0
Thus, there exists δ ∈ 0, L such that eJ 1 Kδ −r1 0 |
ahs|ds < 1 − αL and |P φt| ≤ L.
Hence, P φ ∈ M.
We now show that P is a contraction mapping in M. For all φ, η ∈ M,
P φ t − P η t ≤
t
e−
t
s
ahudu
ahs|φs − f φs − ηs f ηs ds
|
0
t
t−r1 t
t
0
e−
ahs|f φs − f ηs ds
|
t
s
ahudu
|bs|g φs − r2 s − g ηs − r2 s ds.
2.36
12
Fixed Point Theory and Applications
Since
e−dK2
≤
t
ahs||bs|ds
0 |
t
t
e−
t
s
ahudu
ahs|φs − f φs − ηs f ηs ds
|
0
e−
t
s
ahudu
s
ahs|K φs − ηse−dK2 0 |ahu||bu|du
|
0
t
× e−dK2 s |ahu||bu|du ds
t t
t
ahs|Ke−dK2 s |ahu||bu|du dsφ − ηK
≤ e− s ahudu |
0
≤
1 φ − η K ,
d
e−dK2
≤
≤
≤
e
t−r1 t
t−r1 t
ahs|Ke−dK2
|
1 φ − η K ,
d
t
ahs||bs|ds
0 |
t
t
e−
t
s
t
ahu||bu|du
s |
ahudu
dsφ − ηK
|bs|g φs − r2 s − g ηs − r2 s ds
0
e−
t
s
ahudu
0
s
s−r2 s
|
ahu||bu|du
|bs|K φs − r2 s − ηs − r2 se−dK2 0
ahu||bu|du
s−r2 s |
t
ahs|f φs − f ηs ds
|
s
t
ahs|K φs − ηse−dK2 0 |ahu||bu|du e−dK2 s |ahu| |bu|du ds
|
e−
t
s
ahudu
0
≤
t
t−r1 t
t
−dK2
≤
ahs||bs|ds
0 |
t
e−dK2
≤
t
1 φ − η K ,
d
t
e−dK2
t
ahu||bu|du
s |
|bs|Ke−dK2
t
ds
ahu||bu|du
s |
dsφ − ηK
2.37
we have e−dK2 0 |ahs||bs|ds |P φt − P ηt| ≤ 3/d|φ − η|K . That means |P φ − P η| ≤
3/d|φ − η|K . Hence, P is a contraction mapping in M with constant 3/d. By the contraction
mapping principle, P has a unique fixed point x in M, which is a solution of 1.2 with
xs ψs on −r, 0 and supt≥−r |xt| ≤ L.
In order to prove stability at t 0, let ε > 0 be given. Then, choose m > 0 so that
m < min{ε, L}. Replacing L with m in M, we see there is a δ > 0 such that ψ < δ implies
that the unique continuous solution x agreeing with ψ on −r, 0 satisfies |xt| ≤ m < ε for all
t ≥ −r. This shows that the zero solution of 1.2 is stable. That completes the proof.
Fixed Point Theory and Applications
13
When r1 t r1 , a constant, we have the following.
Corollary 2.5. Suppose that the following conditions are satisfied:
t
i there exists a constant J > 0 such that supt≥0 {− 0 as r1 ds} < J,
ii there exists a constant L > 0 such that fx, x − fx, gx satisfy a Lipschitz condition
with constant K > 0 on an interval −L, L,
iii f and g are odd, increasing on 0, L. x − fx is nondecreasing on 0, L,
iv for each L1 ∈ 0, L, one has
L1 − fL1 sup
t
t≥0
e−
t
s
aur1 du
|as r1 |ds fL1 sup
t≥0
0
gL1 sup
t
e
t≥0
−
t
s aur1 du
t
|au r1 |du
t−r1
2.38
|bs|ds < L1 .
0
Then, the zero solution of the equation
x t −atfxt − r1 btgxt − r2 t
2.39
is stable.
Corollary 2.6. Suppose that the following conditions are satisfied:
t
i there exists a constant J > 0 such that supt≥0 {− 0 asds} < J,
ii there exists a constant L > 0 such that fx, x − fx, gx satisfy a Lipschitz condition
with constant K > 0 on an interval −L, L,
iii f and g are odd, increasing on 0, L. x − fx is nondecreasing on 0, L,
iv for each L1 ∈ 0, L, one has
L1 − fL1 sup
t≥0
t
0
e−
t
s
audu
|as|ds gL1 sup
t≥0
t
e−
t
s
audu
|bs|ds < L1 .
2.40
0
Then, the zero solution of
x t −atfxt btgxt − rt
2.41
is stable.
Remark 2.7. The zero solution of 1.2 is not as asymptotically stable as that of 1.1. The key
is that M is not complete under the weighted metric when added the condition to M that
φt → 0 as t → ∞.
Remark 2.8. Theorem 2.4 makes use of the techniques of Theorems 1.3 and 1.4.
14
Fixed Point Theory and Applications
3. An Example
We use an example to illustrate our theory. Consider the following differential equation:
x t −atxt − r1 t btgxt − r2 t.
3.1
where r1 t 0.281t, r2 ∈ CR , R, gx x3 , at 1/0.719t 1, and bt μ sin t/t 1,
μ > 0. This equation comes from 4.
Choosing ht 1.2/t 1, we have
t
t−r1 t
|hs|ds t
0.719t
t
e−
t
s
0
e−
t
s
hudu
t
hs − r1 s 1 − r1 s − asds
hudu 0
t
t1
1.2
ds 1.2 ln
< 0.396,
s1
0.719t 1
|hs|
s
s−r1 s |hu|du
ds < 0.396,
1 − 1.2 × 0.719
ds
0 e− s 1.2/u1du
0.719s 1
1 − 1.2 × −0.719 t − t 1.2/u1du 1.2
<
ds < 0.1592,
e s
0
0.719s 1
s1
t t
μ
.
e− s hudu |bs|ds ≤
1.2
0
t
3.2
Let α : 0.396 0.396 0.1592 μ/1.2, when μ is sufficiently small, α < 1. Then, the condition
ii of Theorem√2.1 is satisfied.
Let L 3/3, then
t the condition i of Theorem 2.1 is satisfied.
t
And 0 hsds 0 1.2/s 1ds 1.2 lnt 1, then the condition iii and iv of
Theorem 2.1 are satisfied.
According to Theorem 2.1, the zero solution of 3.1 is asymptotically stable.
References
1 T. A. Burton, “Stability by fixed point theory or Liapunov theory: a comparison,” Fixed Point Theory,
vol. 4, no. 1, pp. 15–32, 2003.
2 B. Zhang, “Fixed points and stability in differential equations with variable delays,” Nonlinear Analysis:
Theory, Methods and Applications, vol. 63, no. 5–7, pp. e233–e242, 2005.
3 L. C. Becker and T. A. Burton, “Stability, fixed points and inverses of delays,” Proceedings of the Royal
Society of Edinburgh. Section A, vol. 136, no. 2, pp. 245–275, 2006.
4 C. Jin and J. Luo, “Stability in functional differential equations established using fixed point theory,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 11, pp. 3307–3315, 2008.
5 T. A. Burton, “Stability and fixed points: addition of terms,” Dynamic Systems and Applications, vol. 13,
no. 3-4, pp. 459–477, 2004.
6 B. Zhang, “Contraction mapping and stability in a delay-differential equation,” in Dynamic Systems and
Applications, vol. 4, pp. 183–190, Dynamic, Atlanta, Ga, USA, 2004.
7 T. A. Burton, “Stability by fixed point methods for highly nonlinear delay equations,” Fixed Point
Theory, vol. 5, no. 1, pp. 3–20, 2004.
8 J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York,
NY, USA, 1993.