Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2012, Article ID 406835, 13 pages
doi:10.1155/2012/406835
Research Article
Existence of Periodic Solutions to
Nonlinear Differential Equations of Third Order
with Multiple Deviating Arguments
Cemil Tunç
Department of Mathematics, Faculty of Sciences, Yüzüncü Yıl University, 65080 Van, Turkey
Correspondence should be addressed to Cemil Tunç, cemtunc@yahoo.com
Received 12 May 2012; Accepted 19 July 2012
Academic Editor: D. D. Ganji
Copyright q 2012 Cemil Tunç. 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 establish certain new sufficient conditions which guarantee the existence of periodic solutions
for a nonlinear differential equation of the third order with multiple deviating arguments. Using
the Lyapunov functional approach, we prove a specific theorem and provide an example to
illustrate the theoretical analysis in this work and the effectiveness of the method utilized here.
1. Introduction
It is known that functional differential equations, in particular, that delay differential equations can be used as models to describe many physical, biological systems, and so forth. In
reality, many actual systems have the property aftereffect, that is, the future states depend
not only on the present, but also on the past history, and after effect is also known to occur in
mechanics, control theory, physics, chemistry, biology, medicine, economics, atomic energy,
information theory, and so forth Burton 1, Kolmanovskii and Myshkis 2. Therefore, it is
important to investigate the qualitative behaviors of functional differential equations.
In 1978, using the known theorem of Yoshizawa 3, Theorem 37.2, Chukwu 4 found
certain sufficient conditions that guarantee the existence of a periodic solution to nonlinearlinear differential of the third order with the constant deviating argument h >0:
x f x, x , x x g xt − h, x t − h ixt − h
p t, x, x , xt − h, x t − h, x .
1.1
2
International Journal of Differential Equations
Later, in 1992, Zhu 5 considered the nonlinear differential equation of the third order
with the constant deviating argument r >0:
x ax φ x t − r fx pt,
1.2
and he discussed the existence of periodic solutions for this equation when pt is a periodic
function of period T , T > 0.
In 2000, Tejumola and Tchegnani 6 considered the nonlinear differential equation of
the third order with the constant deviating argument τ >0:
x f t, x, x , x g t, xt − τ, x t − τ hxt − τ
P1 t, x, x , x , xt − τ, x t − τ .
1.3
The authors established certain sufficient conditions on the existence of periodic of solutions
of this equation.
In 2010, Tunç 7 established certain sufficient conditions for the existence of a periodic
solution for the nonlinear differential equation of the third order with the constant deviating
argument r >0:
x ψ x x g x t − r fx p t, x, xt − r, x , x t − r, x .
1.4
However, a review to date of the literature indicates that the existence of periodic
solutions to the nonlinear differential equation of the third order with multiple deviating
arguments has not been investigated. The paper considers the nonlinear differential equation
of the third order with multiple constant deviating arguments τi , i 1, 2, . . . , n:
n
x ψ x x gi x t − τi fx
i1
p t, x, xt − τ1 , . . . , x , . . . , x t − τn , x .
1.5
The equation 1.5 is stated in system form as follows:
x y,
y z,
n
n t
gi y − fx z −ψ y z −
i1
i1
t−τi
gi ys zsds
1.6
p t, x, xt − τ1 , . . . , yt − τn , z ,
where τi are positive constants, that is, τi are constant deviating arguments, which are
determined in Section 2. It is assumed that the functions ψ, gi , f, and p· are continuous
in their respective arguments on , , , and × 2n3 , 0, ∞, −∞, ∞,
respectively; gi 0 f0 0 and p· is periodic in t of period T , T ≥ τi , the derivatives
International Journal of Differential Equations
3
gi y ≡ d/dygi y exist and are also continuous; throughout what follows xt, yt, and
zt are abbreviated as x,y, and z, respectively.
The motivation for this paper is a result of the research mentioned regarding ordinary
differential equations with a deviating argument. Our aim is to achieve the results established
in 5, 7 to 1.5 with multiple deviating arguments. Our results generalize the results established on the existence of periodic solution in 5, 7. This paper is the first known publication
regarding the existence of periodic solution for differential equations of the third order with
multiple deviating arguments.
In order to reach our main result, this paper offers fundamental information regarding
the general nonautonomous delay periodic differential system. Consider the delay periodic
system:
ẋt Ft, xt ,
xt xt θ,
−r ≤ θ ≤ 0,
t ≥ 0,
1.7
where F : 0, ∞×CH → n is a continuous mapping, FtT, ϕ Ft, ϕ for all ϕ ∈ C and for
some constant T > 0. We assume that F takes closed bounded sets into bounded sets of n .
Here C, · is the Banach space of continuous function φ : −r, 0 → n with supremum
norm, r > 0; for H > 0, we define CH ⊂ C by CH {φ ∈ C : φ
< H}, CH is the open H-ball
in C, C C−r, 0, n .
Theorem 1.1. Suppose that Ft, ϕ ∈ C0 ϕ and Ft, ϕ is periodic in t of period T, T ≥ r, and consequently for any α > 0 there exists an Lα > 0 such that ϕ ∈ Cα implies |Ft, ϕ| ≤ Lα. Suppose
that a continuous Lyapunov functional V t, ϕ exists, defined on t ∈ , ϕ ∈ S∗ , S∗ is the set of
ϕ ∈ C such that with |ϕ0| ≥ H (H may be large), and that V t, ϕ satisfies the following conditions.
i Continuous increasing functions as and bs exist, satisfying as > 0, bs > 0 for
s ≥ H and as → ∞ as s → ∞, such that
a φ0 ≤ V t, φ ≤ b φ ,
when ϕ0 ≥ H.
1.8
ii A continuous and positive function ws exists such that
V̇ t, φ ≤ −w ϕ0
for s ≥ H.
1.9
iii A constant H1 > 0, H1 > H, exists such that
rL γ ∗ < H1 − H,
1.10
where γ ∗ > 0 is a constant which is determined in the following way.
Using the condition on V t, ϕ, constants α > 0, β > 0, and γ > 0 exist such that bH1 ≤
aα, bα ≤ aβ, and bβ ≤ aγ, γ ∗ is defined by bγ ≤ aγ ∗ .
Under these conditions, a periodic solution of 1.7 of period T exists. In particular, the relation
rLγ ∗ < H1 − H is always satisfied if r is sufficiently small (see Yoshizawa [3]).
4
International Journal of Differential Equations
2. Main Result
The main result is the following theorem:
let τ max1≤i≤n τi .
Theorem 2.1. Suppose that positive constants a, bi , c, m, δ, Li , and τ exist such that the following
conditions hold:
sup f x c,
gi y
≥ bi ,
y
fx sgn x −→ ∞ as |x| −→ ∞,
gi 0 0,
/0 ,
y
g y ≤ Li , 0 ≤ ψ y − a ≤ δ ∀y, p· ≤ m.
i
ab − c > 0,
f0 0,
∀x /
0,
fx sgn x > 0
2.1
If
ab − c
ab − c
τ < min ,
,
2 μ bN1 4M1
2.2
then 1.5 has a periodic solution of period T , where μ abc/2b, b and N1 ni1 Li .
n
i1
bi , M1 n
i1 μLi /2
Proof. Define a Lyapunov functional V V xt , yt , zt by:
n
V xt , yt , zt V1 xt , yt , zt V2 x, y, z 1 Li
i1
0 t
−τi
|zθ|dθds,
2.3
ts
where
V1 xt , yt , zt μ
x
fξdξ fxy μ
0
n
1
γi
μyz z2 2
i1
y
n
ψ η ηdη 0
0 t
−τi
⎧ z
⎪
sgn x,
|x| ≥ 1,
⎪
⎪
M
⎪
⎪
⎪
⎪
⎨sgn z sgn x, |x| ≥ 1,
V2 x, y, z xz
⎪
⎪
,
|x| ≤ 1,
⎪
⎪
⎪
M
⎪
⎪
⎩
x sgn z,
|x| ≤ 1,
i1
y
gi η dη
0
z2 θdθds,
ts
|z| ≤ M
2.4
|z| ≥ M
|z| ≤ M
|z| ≥ M,
M, M > 1, and γi are certain positive constants; the constants γi will be determined later in
the proof.
International Journal of Differential Equations
5
It follows that V1 0, 0, 0 0. In view of the assumptions ψy ≥ a, gi y/y ≥ bi ,
y /
0, f0 0, fx sgn x > 0, x /
0, and sup{f x} c, obtain
V1 xt , yt , zt ≥ μ
x
0
n
1
fξdξ fxy μay2 2
i1
n
1
μyz z2 γi
2
i1
0 t
−τi
y
0
gi η
ηdη
η
z2 θdθds
ts
y
x
2
1
1
fξ
4
≥
ηdη
dξ
μb
−
f
by fx ξ
2b
2by2
0
0
0 t
n
2 1 1
γi
z2 θdθds.
μy z μ a − μ y2 2
2
−τ
ts
i
i1
2.5
Using the assumptions of Theorem 2.1, have
a−μa−
ab c
> 0,
2b
μb − f x ≥
ab − c
>0
2
2.6
so that
V1 xt , yt , zt ≥ D1 x2 D2 y2 D3 z2 ≥ D4 x2 y2 z2 ,
2.7
where D4 min{D1 , D2 , D3 }.
It is also clear that the function V2 is continuous and satisfies
|V2 | ≤ 1.
2.8
In view of 2.3, 2.7, 2.8, and the assumptions of Theorem 2.1, it can be shown that
V satisfies the condition i of Theorem 1.1.
Using a basic calculation, the time derivative of V1 along solutions of 1.6 results in
n
V̇1 xt , yt , zt f xy2 μz2 − μy
gi y − ψ y z2
i1
μy z
n t
i1
−
n
i1
t
γi
t−τi
t−τi
n
2
gi ys zsds γi τi z
z2 sds μy z p·.
i1
2.9
6
International Journal of Differential Equations
The assumption |g y| ≤ Li and the estimate 2|m||n| ≤ m2 n2 imply
μy
n t
i1
t−τi
n
gi ys zsds ≤ μy
i1
n
≤ μy
z
n t
i1
t−τi
gi
g ys |zs|ds
i
t−τi
t
Li |zs|ds
t−τi
i1
≤
t
n
n
μ
μ
Li
Li τi y2 2 i1
2 i1
ys zsds ≤ |z|
n t
≤ |z|
n t
z2 sds,
t−τi
g ys |zs|ds
2.10
i
t−τi
i1
t
Li |zs|ds
t−τi
i1
n
n
1
1
Li
≤
Li τi z2 2 i1
2 i1
t
z2 sds
t−τi
so that
n
V̇1 xt , yt , zt ≤ f xy2 μz2 − μy
gi y − ψ y z2
i1
n
i1
μ
γi τi z2 2
n
Li τi y2 i1
n 1
i1
1
Li μLi − γi
2
2
t
n
1
Li τi z2
2 i1
2.11
z2 sds μy z p·.
t−τi
Using the assumptions sup{f x} c > 0, ψy ≥ a and ab − c > 0, and the estimation
μ ab c/2b, it follows that
V̇1 xt , yt , zt
n g y
n
n
n
2
μ
1
i
2
−c−
≤− μ
γi τi z
Li τi y − a − μ −
Li τi −
y
2 i1
2 i1
i1
i1
n 1
i1
1
Li μLi − γi
2
2
t
z2 sds μy z p·
t−τi
n g y
n
n
μ
ab − c 1 i
2
≤− μ
−c−
−
Li 2γi τi z2
Li τi y −
y
2
2b
2
i1
i1
i1
n
1
1 μ Li − 2γi
2 i1
t
z2 sds μy z p·.
t−τi
2.12
International Journal of Differential Equations
If we choose γi 1/2
n
i1 1
7
μLi and use the assumption |p·| ≤ m, then
V̇1 xt , yt , zt
n g y
n
μ
i
−c−
≤− μ
Li τi y2
y
2 i1
i1
n
ab − c 1 −
−
2 μ Li τi z2
2b
2 i1
μmy m|z|.
2.13
An easy calculation from V2 x, y, z and 1.6 leads to
⎧ n
n t
⎪
1
⎪
⎪
−ψ y z− gi y −fx
gi ys zsdsp· sgn x,
⎪
⎪
⎪
M
⎪
i1
i1 t−τi
⎪
⎪
⎨0,
V̇2 · n
n t
⎪
yz
x
⎪
⎪
−ψ y z− gi y −fx
gi ys zsdsp· ,
⎪
⎪
⎪
M M
⎪
i1
i1 t−τi
⎪
⎪
⎩y sgn z,
|x| ≥ 1, |z| ≤ M
|x| ≥ 1, |z| ≥ M
|x| ≤ 1, |z| ≤ M
|x| ≤ 1, |z| ≥ M
2.14
so that
⎧
t
n n
⎪
1
⎪
⎪
− fx sgn x a δ m gi y Li
|zs|ds ,
⎪
⎪
⎪
M
⎪
t−τi
i1
i1
⎪
⎪
⎨0,
t
V̇2 · ≤
n n
⎪
⎪
⎪
y aδm
g i y Li
|zs|ds,
⎪
⎪
⎪
t−τi
⎪
i1
i1
⎪
⎪
⎩y,
|x| ≥ 1, |z| ≤ M
|x| ≥ 1, |z| ≥ M
|x| ≤ 1, |z| ≤ M
|x| ≤ 1, |z| ≥ M,
2.15
using the assumptions of Theorem 2.1.
First, we consider V in the domain max{|y| − K, |z| − M} ≥ 0, where the constants K
and M are large enough, which will be determined later. We have to discuss the following
two cases.
Case 10 |y| ≥ K ≥ 1, and x, z are arbitrary. In this case, it follows that:
n n
V̇2 · ≤ a δ m y gi y Li
i1
i1
t
t−τi
|zs|ds.
2.16
8
International Journal of Differential Equations
By the estimates 2.3, 2.13, 2.16, and τ max1≤i≤n τi , we get
V̇ xt , yt , zt
n g y
n
n μ
i
−c−
≤− μ
Li τi y2 gi y y
2 i1
i1
i1
n
2μ ab − c
−
−
Li τi z2 μm 1 y m|z|
2b
2
i1
a δ m n
t
Li
|zs|ds
t−τi
i1
t
n
n
Li
Li τi |z| −
|zs|ds
i1
2.17
t−τi
i1
n g y
n
n μ
i
−c−
≤− μ
Li τi y2 gi y y
2 i1
i1
i1
n
2μ ab − c
−
−
Li τi z2 μm 1 y m|z|
2b
2
i1
n
a δ my Li τi |z|.
i1
We now consider the term
μ
n
n n
μ
gi y y − gi y − cy2 −
Li τi y2
2 i1
i1
i1
2.18
and define
h
ab 3c
< 1.
2ab c
2.19
Then, there exists a constant K1 , K1 > 1, satisfying 1 − 1/μ|y| ≥ h for |y| ≥ K1 , so that,
when |y| ≥ K1 ,
μ
n
n n
μ
gi y y − gi y − cy2 −
Li τi y2
2
i1
i1
i1
n g y
n
μLi τi
1
i
2
y − c−
1−
y2
μ
y
2
μy
i1
i1
n
μLi τi
ab cb ab 3c 2
y − c−
y2
≥
2b
2ab c
2
i1
International Journal of Differential Equations
n
μLi τi
ab − c −
y2
−
4
2
i1
ab − c
− M1 τ y 2 .
≤−
4
9
2.20
Hence
V̇ xt , yt , zt
2 μ N1
ab − c
ab − c
2
− M1 τ y −
−
τ z2
≤−
4
2b
2
μm 1 a δ m y m N1 τ|z|,
2.21
where
M1 n
μLi
N1 n
Li .
2.22
ab − c
ab − c
,
τ < min ,
2 μ bN1 4M1
2.23
i1
2
,
i1
If
then the above estimate implies
V̇ xt , yt , zt ≤ −δ1 y2 z2 μm a δ m 1 y m N1 τ|z|
2.24
for a positive constant δ1 .
Let
ρ1 max μm δ a m 1, m N1 τ
2.25
so that
V̇ xt , yt , zt ≤ − δ1 y2 z2 ρ1 y |z|
δ
δ1 2
1
−
y z2 −
2
2
≤−
δ1 2
y z2
2
√
provided that |y| ≥ 2 1ρ1 δ1−1 .
ρ1
y −
δ1
2
ρ1
|z| −
δ1
2
−2
ρ12
δ12
2.26
10
International Journal of Differential Equations
√
Let K max{ 2 1ρ1 δ1−1 , K1 }. If |y| ≥ K, then
δ1 2
V̇ xt , yt , zt ≤ −
y z2 .
2
2.27
Case 20 |z| ≥ M, and x, y are arbitrary. Then
V̇2 x, y, z ≤ y.
2.28
By following a similar method shown in the first case, select γi 1/2
n
i1 1 μLi
and take
ab − c
ab − c
τ < min ,
,
2 μ bN1 4M1
2.29
one can easily obtain
V̇ xt , yt , zt ≤ −δ2 y2 z2 μm 1 y m N1 τ|z|
≤ −δ2 y2 z2 ρ2 y |z|
≤−
2.30
δ2 2
y z2
2
for some positive constants δ2 and ρ2 provided that |z| ≥ M K.
At the end, we consider V in max{|y| − K, |z| − M} ≤ 0.
Let |x| ≥ H > 1, where the constant H will be determined later. Hence
t
n
n 1
gi y a δ m Li
V̇2 · ≤ − fx sgn x max
|zs|ds .
M
|y|≤K i1
t−τi
i1
It follows from 2.3, 2.13, and 2.31 that
V̇ xt , yt , zt ≤ −
2 μ N1
ab − c
ab − c
2
− M1 τ y −
−
τ z2
4
2b
2
n 1
fx sgn x max gi y μmy m N1 τ|z| −
M
|y|≤K i1
aδm
n
i1
t
Li
t−τi
|zs|ds −
n
i1
t
Li
t−τi
|zs|ds
2.31
International Journal of Differential Equations
≤−
−
11
2 μ N1
ab − c
ab − c
2
− M1 τ y −
−
τ z2
4
2b
2
n 1
fx sgn x max gi y M
|y|≤K i1
a δ m N1 Mτ μmK mM .
2.32
Since fx sgn x → ∞ as |x| → ∞ and |x| ≥ H > 1, then we have
n fx sgn x ≥ 2M max gi y a δ m N1 Mτ μmK mM ,
|y|≤K i1
2.33
n fx sgn x
gi y a δ m N1 Mτ μmK mM ≤ 0.
max
−
2M
|y|≤K i1
2.34
so that
In view of the above discussion, it follows that:
V̇ xt , yt , zt ≤ −
−
2 μ N1
ab − c
ab − c
2
− M1 τ y −
−
τ z2
4
2b
2
2.35
1
fx sgn x.
2M
Subject to the evidence thus far, we can conclude that there exists a positive constant R, which
is large enough, such that
V̇ xt , yt , zt ≤ −wu for u2 ≥ R2 ,
2.36
where u x2 y2 z2 1/2 . Thus, the Lyapunov functional V xt , yt , zt satisfies all the assumptions of Theorem 1.1. The proof for Theorem 2.1 is complete.
Example 2.2. Consider the following nonlinear differential equation of the third order with
two constant deviating arguments, τ1 > 0, τ2 > 0:
x 4
1
1 x 2
x 4x t − τ1 sin x t − τ1 4x t − τ2 sin x t − τ2 11x
sin t cos t
,
3 sin t x2 · · · x2 t − τ2 x2
which is a special case of 1.5.
2.37
12
International Journal of Differential Equations
This equation can be written in the system form as follows:
x y,
y z,
z − 4 1
z − 8y 2 sin y − 11x
2
1y
t
4 cos ys zsds 4 cos ys zsds
t
t−τ1
2.38
t−τ2
sin t cos t
.
3 sin t · · · y2 t − τ2 z2
x2
When we compare the system described to this point with 1.6, it follows the existence of the
following estimates:
ψ y 4
a 4,
1
, 0 ≤ ψ y − 4 ≤ 1,
2
1y
g1 y 4y sin y,
g1 0 0,
δ 1,
g1 y
sin y
4
≥ 3,
b1 3,
y
y
g1 y 4 cos y,
g y 4 cos y ≤ 5 L1 ,
1
g2 y 4y sin y,
g2 0 0,
g2 y
sin y
4
≥ 3,
b2 3,
y
y
g2 y 4 cos y,
g y 4 cos y ≤ 5 L2 ,
2
fx 11x,
f0 0,
fx sgn x 11x sgn x > 0,
x / 0,
fx sgn x 11x sgn x −→ ∞ as |x| −→ ∞,
f x 11,
c 11,
ab − c 13 > 0,
57
ab 3c
< 1,
2ab c 70
p t, x, xt − τ1 , . . . , yt − τ2 , z
h
sin t cos t
≤ 1 m,
3 sin t x2 · · · y2 t − τ2 z2
2.39
International Journal of Differential Equations
p t 2π, x, xt − τ1 , . . . , yt − τ2 , z
13
sin t cos t
sint 2π cost 2π
3 sint 2π · · · z2
3 sin t · · · z2
p t, x, xt − τ1 , . . . , yt − τ2 , z ,
T 2π.
2.40
In view of the above estimates, it is seen that all the assumptions of Theorem 2.1 hold. This
discussion verifies that 2.37 has a periodic solution of period T , T 2π.
Acknowledgment
The author would like to expresses his sincere thanks and best regards to the anonymous
referees for their many helpful comments, corrections, and suggestions on the paper.
References
1 T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic
Press, Orlando, Fla, USA, 1985.
2 V. Kolmanovskii and A. Myshkis, Introduction To the Theory and Applications of Functional Differential
Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
3 T. Yoshizawa, Stability Theory by Liapunov’s Second Method, The Mathematical Society of Japan, Tokyo,
Japan, 1966.
4 E. N. Chukwu, “On the boundedness and the existence of a periodic solution of some nonlinear third
order delay differential equation,” Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, vol. 64, no. 5, pp. 440–447, 1978.
5 Y. F. Zhu, “On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system,” Annals of Differential Equations, vol. 8, no. 2, pp. 249–259, 1992.
6 H. O. Tejumola and B. Tchegnani, “Stability, boundedness and existence of periodic solutions of some
third and fourth order nonlinear delay differential equations,” Journal of the Nigerian Mathematics Society, vol. 19, pp. 9–19, 2000.
7 C. Tunç, “On the existence of periodic solutions to nonlinear third order ordinary differential equations
with delay,” Journal of Computational Analysis and Applications, vol. 12, no. 1, pp. 191–201, 2010.
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International Journal of
Differential Equations
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International Journal of
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Mathematical Physics
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Journal of
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Stochastic Analysis
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Discrete Dynamics in
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Applied Mathematics
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Function Spaces
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Optimization
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