Available at
http://pvamu.edu/aam
Appl. Appl. Math.
ISSN: 1932-9466
Applications and Applied
Mathematics:
An International Journal
(AAM)
Vol. 10, Issue 2 (December 2015), pp. 772 – 782
A Boundedness and Stability Results for a Kind
of Third Order Delay Differential Equations
Moussadek Remili and Djamila Beldjerd
Department of Mathematics
University of Oran1, Algeria
remilimous@gmail.com
Received: December 28, 2014; Accepted: June 3, 2015
Abstract
The objective of this study was to get some sufficient conditions which guarantee the asymptotic
stability and uniform boundedness of the null solution of some differential equations of third order
with the variable delay. The most efficient tool for the study of the stability and boundedness of
solutions of a given nonlinear differential equation is provided by Lyapunov theory. However the
construction of such functions which are positive definite with corresponding negative definite
derivatives is in general a difficult task, especially for higher-order differential equations with
delay. Such functions and their time derivatives along the system under consideration must satisfy
some fundamental inequalities. Here the Lyapunov second method or direct method is used as a
basic tool. By defining an appropriate Lyapunov functional, we prove two new theorems on the
asymptotic stability and uniform boundedness of the null solution of the considered equation.
Our results obtained in this work improve and extend some existing well-known related results
in the relevant literature which were obtained for nonlinear differential equations of third order
with a constant delay. We also give an example to illustrate the importance of the theoretical
analysis in this work and to test the effectiveness of the method employed.
Keywords: Stability, Boundedness, Lyapunov functional, Third-order Delay differential equations
MSC 2010 No.: 34C11, 34D20, 34K20
772
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
1.
773
Introduction
The qualitative behavior of solutions of various differential equations of third order with and
without delay have been extensively studied. In the relevant literature, a good deal of work has
been done and many interesting results have been obtained. We refer readers to the papers of Hara
(1971), Zhu (1992), Omeike (2009) and Tunç (2009), to mention a few as well as the references
cited therein for some works on the subject, where the Lyapunov function or functional approach
have been the most effective method to determine the stability and boundedness of solutions.
Omeike (2009) considered the following nonlinear differential equation of third order, with a
constant deviating argument r,
x000 (t) + a(t)x00 (t) + b(t)f (x0 (t)) + c(t)h(x(t − r)) = e(t).
He studied the stability and boundedness of solutions of this equation when e(t) = 0 and e(t) 6= 0.
In Tunç (2009), the author discussed sufficient conditions which ensure the boundedness of the
delay differential equation of the form
x000 (t) + a(t)ψ(x0 (t))x00 (t) + b(t)f (x0 (t)) + c(t)h(x(t − r))
= e(t, x(t), x(t − r), x0 (t), x0 (t − r), x00 (t)).
Our objective in this paper is to extend the results verified by Tunç (2009) to obtain sufficient
conditions for the stability and boundedness of solutions of the following delay differential
equation:
00
[φ(x(t))x0 (t)] + a(t)ψ(x0 (t))x00 (t) + b(t)f (x0 (t)) + c(t)h(x(t − r(t)))
= e(t, x(t), x(t − r(t)), x0 (t), x0 (t − r(t)), x00 (t)),
(1)
where φ is a twice differentiable function and ρ is a positive constant with 0 ≤ r(t) ≤ ρ, which
will be determined later. The functions a(t), b(t), c(t), f (x0 ), h(x), ψ(x0 ), and
e(t, x(t), x(t − r(t)), x0 (t), x0 (t − r(t)), x00 (t)) are continuous in their respective arguments; the
derivatives a0 (t), b0 (t), c0 (t), h0 (x), f 0 (x0 ) are continuous for all x, y with h(0) = f (0) = 0. In
addition, it is also assumed that the functions e(t, x, x0 , x(t − r(t)), x0 (t − r(t)), x00 ), f (x0 (t)) and
h(x(t − r(t))) satisfy a Lipschitz condition in x, x0 , x00 , x(t − r(t)), and x0 (t − r(t)).
Remark 1.1.
Clearly the equation discussed in Tunç (2009) is a special case of equation (1) when φ(x) = 1
and r(t) = r. Moreover, if ψ(x0 ) = 1 and f (x0 ) = x0 , then (1) reduces to the case studied by
Remili and Oudjedi (2014).
2.
Preliminaries
First we will give some basic definitions and important stability criteria for the general nonautonomous delay differential system. We consider
774
M. Remili & D. Beldjerd
x0 = f (t, xt ), xt (θ) = x(t + θ) ,
−r ≤ θ ≤ 0, t ≥ 0,
(2)
where f : I × CH → Rn is a continuous mapping,
f (t, 0) = 0, CH := {φ ∈ (C[−r, 0], Rn ) : kφk ≤ H},
and for H1 < H, there exist L(H1 ) > 0, with |f (t, φ)| < L(H1 ) when kφk < H1 .
Definition 2.1. Burton (2005)
An element ψ ∈ C is in the ω − limit set of φ, say Ω(φ), if x(t, 0, φ) is defined on [0, +∞)
and there is a sequence {tn }, tn → ∞, as n → ∞, with kxtn (φ) − ψk → 0 as n → ∞ where
xtn (φ) = x(tn + θ, 0, φ) for −r ≤ θ ≤ 0.
Definition 2.2 Burton (2005)
A set Q ⊂ CH is an invariant set if for any φ ∈ Q, the solution of (2.1), x(t, 0, φ), is defined on
[0, ∞) and xt (φ) ∈ Q for t ∈ [0, ∞).
Lemma 2.3. Burton (1985)
If φ ∈ CH is such that the solution xt (φ) of (2.1) with x0 (φ) = φ is defined on [0, ∞) and
kxt (φ)k ≤ H1 < H for t ∈ [0, ∞), then Ω(φ) is a non-empty, compact, invariant set and
dist(xt (φ), Ω(φ)) → 0 as t → ∞.
Lemma 2.4. Krasovskii (1963)
If there is a continuous functional V (t, φ) : [0, +∞] × CH → [0, +∞] locally Lipschitz in φ and
wedges Wi such that:
(i)
0
(t, φ) ≤ 0
If W1 (kφk) ≤ V (t, φ), V (t, 0) = 0 and V(2,1)
then the zero solution of (2.1) is stable. If in addition V (t, φ) ≤ W2 (kφk) then the zero solution
of (2.1) is uniformly stable.
(ii)
If W1 (kφk) ≤ V (t, φ) ≤ W2 (kφk) and
0
V(2,1)
(t, φ) ≤ −W3 (kφk),
then the zero solution of (2.1) is uniformly asymptotically stable.
3.
Assumptions and main results
We assume that there are positive constants a, b, c, δ, δ0 , δ1 , δ2 , ψ1 , φ0 , φ1 , ρ, and σ, such
that the following conditions hold
i)
0 < a ≤ a(t) ≤ A, 0 < b ≤ b(t) ≤ B, 0 < c ≤ c(t) ≤ C,
ii)
1 ≤ ψ(y) ≤ ψ1 , 0 < φ0 ≤ φ(x) ≤ φ1 ,
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
775
h(x)
≥ δ > 0 (x 6= 0), and |h0 (x)| ≤ δ1 for all x,
x
f (y)
δ2 ≥
≥ δ0 > 0 (y 6= 0),
y
iii)
iv)
Z
+∞
|φ0 (u)| du < ∞,
v)
Z
−∞
∞
vi)
|c0 (s)| ds ≤ N1 < ∞ and c0 (t) → 0 as t → ∞,
0
vii) for some ρ ≥ 0 and 0 < σ < 1, 0 ≤ r(t) ≤ ρ, r0 (t) ≤ σ.
Before stating theorems, we introduce the following notation ∆ = µbδ0 − Cδ1 φ1 .
For the case e ≡ 0, the following result is introduced.
Theorem 3.1.
In addition to the assumptions (i)-(vii), assume that the following conditions are satisfied
H1)
H2)
φ1 δ1 C
< µ < a,
bδ0
µψ1 0
δ1
δ2
∆
|a (t)| + |b0 (t)| − c0 (t) < 2 .
2
φ0
φ0
µ
φ1
Then the zero solution of (1) is uniformly asymptotically stable, provided that
ρ < min
φ30 ∆(1 − σ)
2(a − µ)
,
φ1 Cδ1 Cδ1 φ21 (µ + φ0 + µφ20 (1 − σ))
.
Proof:
Equation (1) can be expressed as the following system
1
y
φ(x)
= z
y
a(t)φ0 (x)
y
y
a(t)
2
= −
ψ(
)z +
ψ(
)y − b(t)f
φ(x) φ(x)
φ3 (x)
φ(x)
φ(x)
Z t
y(s) 0
−c(t)h(x) + c(t)
h (x(s))ds.
t−r(t) φ(x(s))
x0 =
y0
z0
(3)
We define the following Lyapunov functional W = W (t, xt , yt , zt ):
W (t, xt , yt , zt ) = e−β(t) V (t, xt , yt , zt ) = e−β(t) V,
(4)
776
M. Remili & D. Beldjerd
where
V
y
1
µ
= µc(t)H(x) + c(t)h(x)y + b(t)φ(x)F (
) + z2 +
yz
φ(x)
2
φ(x)
Z y
Z 0 Z t
φ(x)
y 2 (ξ)dξds,
+µa(t)
ψ(u)udu + λ
−r(t)
0
and
t
Z
[
β(t) =
0
(5)
t+s
|α(s)| |c0 (s)|
+
]ds,
ω
c
such that
x
Z
Z
h(u)du, F (y) =
H(x) =
0
0
y
φ0 (x(t)) 0
f (u)du, and α(t) = 2
x (t).
φ (x(t))
ω and λ are positive constants which will be specified later in the proof. The above Lyapounov
functional V can be rewritten as the following
Z y
φ(x)
1
δ1 2
µ
V = µc(t) H(x) + h(x)y + 2 y + µa(t)
[ψ(u) −
]udu
µ
2µ
a(t)
0
µ
y
δ1 c(t) 2
1
y)2 + b(t)φ(x)F (
)−
y
+ (z +
2
φ(x)
φ(x)
2µ
Z 0 Z t
+λ
y 2 (ξ)dξds.
−r(t)
t+s
Letting
G(x, y) = H(x) +
δ1
1
yh(x) + 2 y 2 ,
µ
2µ
assumption (iii) implies that
δ1
µ
1 2
G(x, y) = H(x) + 2 (y + h(x))2 −
h (x)
2µ
δ1
2δ1
Z x
h0 (u)
≥
1−
h(u)du ≥ 0.
δ1
0
It is clear from (iv) that
F
y
φ(x)
Z
=
0
y
φ(x)
f (u)du ≥
δ0 y 2
.
2 φ2 (x)
Hence, using the above estimate, the assumption (ii), (3.4), and the fact that the integral
R0 Rt 2
y (ξ)dξds is positive, we deduce that
−r(t) t+s
V
1
µ
y)2
≥ µc(t)G(x, y) + (z +
2
φ(x)
1 µa
µ
δ0 b Cδ1
+
1−
+
−
y2.
2
2 φ (x)
a
φ1
µ
(6)
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
Condition (H1) implies that 1 −
k small enough such that
777
µ
δ0 b Cδ1
> 0 and
−
> 0. Thus there exist a positive constant
a
φ1
µ
V ≥ µcG(x, y) + k(y 2 + z 2 ).
(7)
From (ii), (v), and (vi), we get
Z t
|α(s)| |c0 (s)|
β(t) =
+
ds,
ω
c
0
Z
1 θ2 (t) |φ0 (u)|
N1
≤
du
+
ω θ1 (t) φ2 (u)
c
Z +∞
N1
N2
N1
1
|φ0 (u)| du +
≤
+
= N < ∞,
≤
2
2
ωφ0 −∞
c
ωφ0
c
where θ1 (t) = min{x(0), x(t)} and θ2 (t) = max{x(0), x(t)}. Therefore we can find a continuous
function W1 (kXk), where X = (x, y, z), such that
W1 (kXk) ≥ 0
and
W1 (kXk) ≤ W.
The existence of a continuous function W2 (kXk) which satisfies the inequality W ≤ W2 (kXk)
is easily verified.
0
0
denote the time derivative of the Lyapunov functional V (t, xt , yt , zt )
(t, xt , yt , zt ) = V(3)
Let V(3)
along the trajectories of the system (3). An easy computation shows that
y
0
V(3)
= µc0 (t)H(x) + c0 (t)yh(x) + b0 (t)φ(x)F (
)
φ(x)
µ 2 a(t)
y
y
y
+
z −
Ψ(
)z 2 − µb(t)
f(
)
φ(x)
φ(x) φ(x)
φ(x) φ(x)
Z y
φ(x)
c(t)h0 (x) 2
0
+µa (t)
Ψ(u)udu +
y + λr(t)y 2
φ(x)
0
y
y
y
2
) − b(t)yφ(x)f (
) + (a(t)Ψ(
) − µ)zy
+α(t) b(t)φ (x)F (
φ(x)
φ(x)
φ(x)
Z t
Z t
µ
h0 (x(s))
0
+c(t)(
y + z)
y(s)
ds − λ(1 − r (t))
y 2 (ξ)dξ.
φ(x)
φ(x(s))
t−r(t)
t−r(t)
When we apply the hypotheses of the theorem we obtain
c(t)h0 (x) 2
y
y
0
0
2
V(3.1) ≤ µc (t)G(x, y) +
y − µb(t)
f(
) + λρy
φ(x)
φ(x) φ(x)
"
#
Z y
φ(x)
y
δ
1
+ µa0 (t)
Ψ(u)udu + b0 (t)φ(x)F (
)−
c0 (t)y 2
φ(x)
2µ
0
µ − a(t) 2
3
2
+
z + |α(t)| δ2 By + (Aψ1 − µ)|zy|
φ(x)
2
Z t
Z t
µ
h0 (x(s))
+c(t)(
y + z)
y(s)
ds − λ(1 − σ)
y 2 (ξ)dξ.
φ(x)
φ(x(s))
t−r(t)
t−r(t)
778
M. Remili & D. Beldjerd
Using the Schwartz inequality |uv| ≤ 12 (u2 + v 2 ), we have
3δ2 2
1
|α(t)|
By + (Aψ1 − µ)|zy| ≤
|α(t)| 3δ2 By 2 + (Aψ1 − µ)(y 2 + z 2 )
2
2
≤ k1 |α(t)| (y 2 + z 2 ),
1
where k1 = (Aψ1 − µ + 3δ2 B). Since |h0 (x)| ≤ δ1 , we get the following inequalities
2
Z t
Z
y(s) 0
µc(t)
Cδ1 µρ 2 Cµδ1 t
y 2 (ξ)dξ,
y
h (x(s))ds ≤
y +
φ(x) t−r(t) φ(x)
2φ0
2φ30 t−r(t)
and
Z
t
c(t)z
t−r(t)
y(s) 0
Cδ1 ρ 2 Cδ1
h (x(s))ds ≤
z + 2
φ(x)
2
2φ0
Z
t
y 2 (ξ)dξ.
t−r(t)
It can be easily deduced from the above estimates that
1 µψ1 0
δ2 0
δ1 0
0
0
V(3) ≤ µc (t)G(x, y) +
|a (t)| + |b (t)| − c (t) y 2
2
2 φ0
φ
µ
0
µCδ1
∆
a − µ Cδ1 ρ 2
2
− 2 − (λ +
)ρ y −
−
z
φ1
2φ0
φ1
2
Z t
µ
Cδ1
(1 + ) − λ(1 − σ)
y 2 (ξ)dξ + k1 |α(t)| (y 2 + z 2 ).
+
2φ20
φ0
t−r(t)
If we take
Cδ1 (φ0 + µ)
= λ, by using (H2) the last inequality becomes
2φ30 (1 − σ)
0
V(3)
Taking ω =
Cδ1 φ0 + µ
µbδ0 − Cδ1 φ1
−
( 2
+ µ)ρ y 2
≤ µc (t)G(x, y) −
2
2φ1
2φ0 φ0 (1 − σ)
a − µ Cδ1 ρ 2
−
−
z + k1 |α(t)| (y 2 + z 2 ).
φ1
2
0
k
and using the inequalities (3.6), (3.5), and (3.2) we obtain
k1
0
W(3)
k1 |α(t)| |c0 (t)|
= e
−(
+
)V
k
c0
a − µ Cδ1 ρ
−β(t)
0
≤ e
µc (t)G(x, y) −
−
z2
φ1
2
∆
Cδ1
φ0 + µ
−
−
+ µ ρ y 2 + k1 |α(t)| (y 2 + z 2 )
2φ21
2φ0 φ20 (1 − σ)
k1 |α(t)| |c0 (t)|
2
2
−
+
µcG(x, y) + k(y + z ) .
k
c
−β(t)
0
V(3)
(8)
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
779
Since c0 (t) − |c0 (t)| ≤ 0 we have
0
W(3)
a − µ Cδ1 ρ
≤ e
−
−
z2
φ1
2
∆
Cδ1
φ0 + µ
−
+ µ ρ y2 .
−
2φ21
2φ0 φ20 (1 − σ)
−β(t)
Therefore if we choose
ρ < min
2(a − µ)
φ30 ∆(1 − σ)
,
φ1 Cδ1 Cδ1 φ21 (µ + φ0 + µφ20 (1 − σ))
,
then
0
W(3)
(t, xt , yt , zt ) ≤ −γ(y 2 + z 2 ), for some γ > 0,
From (3), W3 (kXk) = γ(y 2 + z 2 ) is positive definite function. Hence, Lemma 2.4 guarantees that
the trivial solution of Equation (1) is uniformly asymptotically stable and completes the proof of
the theorem. We will now state our main results for the case e 6= 0.
Theorem 3.2.
If all the assumptions of Theorem 3.1 are satisfied then all solutions of equation (1) are bounded
provided
|e(t, x, x(t − r(t)), y, y(t − r(t)), z)| ≤ |q(t)|, and
Z t
|q(s)|ds < ∞, for all t ≥ 0.
0
Proof:
The proof of this theorem is similar to that of the proof of the theorem in Tunç (2009) and hence
it is omitted. 4.
Example
We consider the following fourth order non-autonomous differential equation
2
00 2
sin (x)
1 −t
cos (y)
0
+3 x
+
e +3
+ 8 x00
2
2
1+x
4
1+y
0
1
193
2x
+
cos t +
x0 +
3
3
(1 + x02 )
1
15
x(t − r(t))
+
sin t +
x(t − r(t)) +
4
4
1 + x2 (t − r(t))
1
=
.
1 + t2 + x2 + x2 (t − r(t)) + y 2 + y 2 (t − r(t)) + z 2
(9)
780
M. Remili & D. Beldjerd
Take
φ(x) =
sin2 (x)
cos2 (y)
2y
x
,
ψ(y)
=
+x ,
+
3
,
f
(y)
=
y
+
+ 8 , h(x) = 2
2
2
2
1+x
(1 + y )
1+y
x +1
1
1
193
1
15
a(t) = e−t + 3 , b(t) = cos t +
, c(t) = sin t +
4
3
3
4
4
and
e(t, x, x(t − r(t)), y, y(t − r(t)), z) =
1+
t2
+
x2
+
x2 (t
1
.
− r(t)) + y 2 + y 2 (t − r(t)) + z 2
It can be seen that
13
1
1
1
3 = a ≤ a(t) = e−t + 3 ≤ , |a0 (t)| = | − e−t | ≤ , t ≥ 0,
4
4
4
4
1
193
194
1
1
64 = b ≤ b(t) = cos t +
≤
, 0 ≤ |b0 (t| = | sin t| ≤ , t ≥ 0,
3
3
3
3
3
1
15
1
1
1
7
0
= c ≤ c(t) = sin t +
≤ 4 = C, − ≤ c (t) = cos t ≤ , t ≥ 0,
2
4
4
4
4
4
h(x)
1
1=δ≤
=1+
with x 6= 0, |h0 (x)| ≤ δ1 = 2 and µ = 1,
x
1 + x2
f (y)
2
=1+
≤ δ2 = 3
y
(1 + y 2 )
cos2 (y)
1 < 8 = ψ0 ≤ ψ(y) =
+ 8 ≤ ψ1 = 9,
1 + y2
sin2 (x)
3 = φ0 ≤ φ(x) =
+ 3 ≤ φ1 = 4.
1 + x2
1 = δ0 ≤
Easy computations show that conditions (H1) and (H2) are satisfied.
Indeed,
φ1 δ1 C
1
= < µ < a = 3. We have also
bδ0
2
µψ1 0
δ2
δ1
µ 1
1
13
µbδ0 − φ1 Cδ1
|a (t)| + |b0 (t)| − c0 (t) ≤ + +
=
<
= 2.
2
φ0
φ0
µ
4 3 2µ
12
φ21
It is straightforward to verify that
Z
+∞
−∞
+∞
2 cos u sin u 2u sin2 u |φ (u)| du ≤
1 + u2 + (1 + u2 )2 du
−∞
≤ 2π + 2.
0
Z
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
781
We have
e(t, x, x(t − r(t)), y, y(t − r(t)), z) =
t2
1+ +
1
≤
,
1 + t2
and
Z
0
+∞
x2
+
x2 (t
1
− r(t)) + y 2 + y 2 (t − r(t)) + z 2
1
dt < ∞.
1 + t2
All assumptions of Theorems 3.1 and 3.2 hold true, thus their conclusions also follow.
5.
Conclusion
The problem of the stability and boundedness of solutions of differential equations is very
important in the theory and applications of differential equations. In the present work, conditions
were obtained for the stability and boundedness for certain third order non-linear non-autonomous
differential equations with the variable delay. Using Lyapunov second or direct method, a Lyapunov functional was defined and used to obtain our results.
REFERENCES
Ademola, A.T. and Arawomo, P.O. (2013). Uniform stability and boundedness of solutions of
nonlinear delay differential equations of third order, Math. J. Okayama Univ., Vol. 55, pp.
157-166.
Ademola, A.T., Arawomo, P.O., Ogunlaran, O.M. and Oyekan, E.A. (2013). Uniform stability,
boundedness and asymptotic behavior of solutions of some third order nonlinear delay
differential equations, Differential Equations and Control Processes, N4.
Bai, Y. and Guo, C. (2013). New results on stability and boundedness of third order nonlinear
delay differential equations, Dynam. Systems Appl., Vol. 22 , No. 1, pp. 95-104.
Burton, T.A. (1985). Stability and periodic solutions of ordinary and functional differential
equations, Mathematics in Science and Engineering, Vol. 178. Academic Press, Inc., Orlando,FL.
Burton, T.A. (2005). Volterra Integral and Differential Equations, Mathematics in Science and
Engineering V (202), second edition.
Greaf, J.R. and Remili, M. (2012). Some properties of monotonic solutions of x000 + p(t)x0 +
q(t)f (x) = 0, Pan. American Mathematical Journal, Vol. 22, No. 2, pp. 31-39.
Hara, T. (1971). On the asymptotic behavior of solutions of certain of certain third order
ordinary differential equations, Proc. Japan Acad., Vol. 47, pp. 03-908.
Krasovskii, N.N. (1963). Stability of Motion. Stanford Univ. Press.
Omeike, M.O. (2010). New results on the stability of solution of some non-autonomous delay
differential equations of the third order, Differential Equations and Control Processes, Vol.
2010, No. 1, pp. 18–29.
782
M. Remili & D. Beldjerd
Omeike, M.O. (2009). Stability and boundedness of solutions of some non-autonomous delay
differential equation of the third order, An. Stiint. Univ. Al. I. Cuza Iasi. Mat.(N.S.), Vol.
55, No. 1, pp. 49-58.
Remili, M. and Oudjedi, D. L. (2014). stability and boundedness of the solutions of non
autonomous third order differential equations with delay, Acta Univ. Palacki. Olomuc., Fac.
rer. nat., Mathematica, Vol. 53, No. 2, pp. 137-149.
Remili, M. and Beldjerd, D. (2014). On the asymptotic behavior of the solutions of third order
delay differential equations, Rend. Circ. Mat. Palermo, Vol. 63, No. 3 , pp. 447-455.
Sadek, A. I. (2005). On the Stability of Solutions of Some Non-Autonomous Delay Differential
Equations of the Third Order, Asymptot. Anal., Vol. 43, No. (1-2), pp. 1-7.
Sadek, A. I. (2003). Stability and Boundedness of a Kind of Third-Order Delay Differential
System, Applied Mathematics Letters, Vol. 16, No. 5, pp. 657-662.
Swick, K. (1969). On the boundedness and the stability of solutions of some non-autonomous
differential equations of the third order, J. London Math. Soc., Vol. 44 , pp. 347–359.
Tunç, C.(2007). Stability and boundedness of solutions of nonlinear differential equations of
third order with delay, Journal Differential Equations and Control Processes, No. 3, pp. 1-13.
Tunç, C.(2009). Boundedness in third order nonlinear differential equations with bounded delay.
Bol. Mat. Vol. 16, No. 1, pp. 1-10.
Tunç, C. (2010). On some qualitative behaviors of solutions to a kind of third order nonlinear
delay differential equations, Electron. J. Qual. Theory Differ. Equ., No. 12, pp. 1-9.
Tunç, C. (2010). Some stability and boundedness conditions for non-autonomous differential
equations with deviating arguments, E. J. Qualitative Theory of Diff. Equ., No. 1., pp. 1–12.
Tunç, C. (2010). On the stability and boundedness of solutions of nonlinear third order differential equations with delay, Filomat, Vol. 24, No. 3, pp. 1-10.
Tunç, C. (2013). Stability and boundedness of the nonlinear differential equations of third order
with multiple deviating arguments, Afr. Mat., Vol. 24, No. 3, pp. 381-390.
Yoshizawa, T. (1966). Stability theory by Liapunov’s second method, The Mathematical Society
of Japan, Tokyo.
Zhu, Y.F. (1992). On stability, boundedness and existence of periodic solution of a kind of
third order nonlinear delay differential system, Ann. Differential Equations, Vol. 8, No. 2,
pp. 249-259.