CUBO A Mathematical Journal
Vol.14, No¯ 03, (63–69). October 2012

Uniformly boundedness of a class of non-linear differential
equations of third order with multiple deviating arguments

Cemil Tunç, Hilmi Ergören

Department of Mathematics, Faculty of Sciences,

Yüzüncü Yıl University,

65080, Van, TURKEY

email: cemtunc@yahoo.com, hergoren@yahoo.com

ABSTRACT

This paper deals with a certain third-order non-linear differential equation with multiple

deviating arguments. Some sufficient conditions are set up for all solutions and their

derivatives to be uniformly bounded.

RESUMEN

En este art́ıculo se estudia un tipo de ecuaciones diferenciales no lineales de tercer orden

con argumentos de desviación múltiple. Se establecen algunas condiciones suficientes

para que todas las soluciones y sus derivadas sean uniformemente acotadas.

Keywords and Phrases: Non-linear differential equation; third order; multiple deviating argu-

ments; bounded solutions.

2010 AMS Mathematics Subject Classification: 34C25; 34K13; 34K25.



64 Cemil Tunç and Hilmi Ergören CUBO
14, 3 (2012)

1 Introduction

Consider the following third order nonlinear differential equation with multiple deviating arguments

x
′′′

(t) + f1(t, x(t))x
′′

(t) + f2(t, x(t))x
′

(t) + g0(t, x(t)) +

n∑

i=1

gi(t, x(t − τi(t)) = p(t), (1.1)

where f1, f2 and gi(i = 0, 1, 2, ..., n) are continuous functions on R
+ × R, τi(t) ≥ 0 (i = 1, 2, ..., n)

and p(t) are bounded continuous functions on R and R+ = [0, +∞), respectively.

Define y(t) =
dx(t)

dt
+α1x(t) and z(t) =

dy(t)

dt
+α2y(t), where α1 and α2 are some constants.

Then, we can transform (1.1) into the following system

dx(t)

dt
= −α1x(t) + y(t),

dy(t)

dt
= −α2y(t) + z(t),

dz(t)

dt
= −(f1(t, x(t)) − α1 − α2)z(t)

+((f1(t, x(t)) − α1)(α1 + α2) − f2(t, x(t)) − α
2

2
)y(t)

+((α1 − f1(t, x(t)))α
2

1
+ f2(t, x(t))α1)x(t) − g0(t, x(t))

−

n∑

i=1

gi(t, x(t − τi(t)) + p(t). (1.2)

In applied science some practical problems are associated with higher-order nonlinear differ-

ential equations, such as nonlinear oscillations (Afuwape et al.[1], Andres[2] and Fridedrichs[3]),

electronic theory (Rauch[4]), biological models and other models (Cronin[5] and Gopalsamy[6]).

Just as above, in the past few decades, the study for third order differential equations has been

paid attention by many scholars. Many results relative to the stability, boundedness of solutions

of third order differential equations with delays or without delays have been obtained (see Li[7],

Murakami[8], Ademola et al.[9], Tunç and Ergören[10], Tunç[11−13] and references therein). How-

ever, to the best of our knowledge, no authors have considered the boundedness of solutions of

third order differential equations with multiple deviating arguments in non-Liapunov sense, in

spite of the fact that some authors (see Afuwape and Castellanos[14], Gao and Liu[15], and Yu

and Zhao[16]) have obtained some results for the third order ones with a deviating argument and

second order ones with multiple deviating arguments. Thus, it is worthwhile to continue to the

investigation of the boundedness of solutions of (1.1) in this case.

The main objective of this paper is to study the uniformly boundedness of solutions of

(1.1). We will establish some sufficient conditions satisfying the solutions of (1.1) to be uni-

formly bounded. Our result is new and complement to previously known results. In particular, an

example is also given to illustrate the effectiveness of the new result.



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Uniformly boundedness of a class of non-linear ... 65

2 Definitions and Assumptions

We assume that h = max
1≤i≤n

{

sup
t∈R

τi(t)

}

≥ 0. Let C ([−h, 0], R) denote the space of continuous

functions φ : [−h, 0] → R with the supremum norm ‖ .‖. It is known in Burton[17], Hale[18]

and Kuang[19] that there exists a solution of (1.2) on an interval [0, T) satisfying the initial

condition and (1.1) on [0, T) for gi(i = 0, 1, 2, ..., n), φ, f1, f2, p and τi(t)(i = 1, 2, ..., n) con-

tinuous, given a continuous initial function φ ∈ C ([−h, 0], R) and a vector (y0, z0) ∈ R
2. If

the solution remains bounded, then T = +∞. We denote such a solution by (x(t), y(t), z(t)) =

(x(t, φ, y0, z0), y(t, φ, y0, z0), z(t, φ, y0, z0)), where y(s) = y(0) and z(s) = z(0) for all s ∈ [−h, 0].

Then, it follows that (x(t), y(t), z(t)) can be defined on [−h, +∞).

Definition. Solutions of (1.2) are called uniformly bounded (UB) if for each B1 > 0 there

is a B2 > 0 such that (φ, y0, z0) ∈ C ([−h, 0], R) × R
2 and ‖φ‖ + ‖y0‖ + ‖z0‖ ≤ B1 imply that

|x(t, φ, y0, z0)| + |y(t, φ, y0, z0)| + |z(t, φ, y0, z0)| ≤ B2 for all t ∈ R
+.

In this work, we also assume that the following conditions hold:

i)
∣

∣((α1 − f1(t, u))α
2

1
+ f2(t, u)α1)u − g0(t, u)

∣

∣ ≤ L0 |u| + q0 for all u ∈ R and t ≥ 0,

ii) |g1(t, u)| ≤ L1 |u| + q1, |g2(t, u)| ≤ L2 |u| + q2, ..., |gn(t, u)| ≤ Ln |u| + qn for all u ∈ R and

t ≥ 0,

iii) α3 = inf
t≥0

(f1(t, u) − α1 − α2)

−sup
t≥0

∣

∣(f1(t, u) − α1)(α1 + α2) − f2(t, u) − α
2

2

∣

∣ >
n∑

i=0

Li, where α1 > 1, α2 > 1, α3 > 0

are some constants, and Li and qi (i = 0, 1, 2, ..., n) are nonnegative constants.

3 Main Result

Theorem 1. Suppose (i)-(iii) hold. Then, solutions of (1.2) are uniformly bounded.

Proof. Let (x(t), y(t), z(t)) = (x(t, φ, y0, z0), y(t, φ, y0, z0), z(t, φ, y0, z0)) be a solution of system

(1.2) defined on [0, T). We may assume that T = +∞ since the following estimates give a priori

bound on (x(t), y(t), z(t)) .

Calculating the upper right derivative of |x(s)| , |y(s)| and |z(s)| , in view of (i) − (iii), we have

D+(|x(s)|)s=t = sgn(x(t)){−α1x(t) + y(t)}

≤ −α1 |x(t)| + |y(t)| , (3.1)

D+(|y(s)|)s=t = sgn(y(t)){−α2y(t) + z(t)}

≤ −α2 |y(t)| + |z(t)| (3.2)



66 Cemil Tunç and Hilmi Ergören CUBO
14, 3 (2012)

and

D+(|z(s)|)s=t = sgn(z(t)){−(f1(t, x(t)) − α1 − α2)z(t)

+((f1(t, x(t)) − α1)(α1 + α2) − f2(t, x(t)) − α
2

2
)y(t)

+((α1 − f1(t, x(t)))α
2

1
+ f2(t, x(t))α1)x(t) − g0(t, x(t))

−

n∑

i=1

gi(t, x(t − τi(t)) + p(t)}

≤ −inf
t≥0

(f1(t, x(t)) − α1 − α2) |z(t)|

+sup
t≥0

∣

∣(f1(t, x(t)) − α1)(α1 + α2) − f2(t, x(t)) − α
2

2

∣

∣ |y(t)|

+L0 |x(t)| +

n∑

i=1

Li |x(t − τi(t))| +

n∑

i=0

qi + |p(t)| . (3.3)

Let

M(t) = max
−h≤s≤t

{max {|x(s)| , |y(s)| , |z(s)|}} , (3.4)

where y(s) = y(0), z(s) = z(0) for all −h ≤ s ≤ 0. It is clear that max {|x(t)| , |y(t)| , |z(t)|} ≤ M(t)

and M(t) is non-decreasing for t ≥ −h. Now, we consider the following two cases:

Case I):

M(t) > max {|x(t)| , |y(t)| , |z(t)|} (3.5)

for all t ≥ 0, then we claim that

M(t) ≡ M(0) (3.6)

is a constant for all t ≥ 0.

By contrapositive, assume (3.6) does not hold, then, there exists t1 > 0 such that M(t1) >

M(0).

Here max {|x(t)| , |y(t)| , |z(t)|} ≤ M(0) for all −h ≤ t ≤ 0 and there exists β ∈ (0, t1) such that

max {|x(β)| , |y(β)| , |z(β)|} = M(t1) ≥ M(β) which contradicts (3.5). This contradiction implies

that (3.6) holds. It follows that there exists t2 > 0 such that max {|x(t)| , |y(t)| , |z(t)|} ≤ M(t) =

M(0) for all t ≥ t2.

Case II): There is a point t0 ≥ 0 such that M(t0) = max {|x(t0)| , |y(t0)| , |z(t0)|}. Let η =

min

{

α1 − 1, α2 − 1, α3 −
n∑

i=0

Li

}

> 0 and θ =
n∑

i=0

qi + sup
t∈R+

|p(t)| + 1 be constants, where t ≥ 0.

Then, if M(t0) = max {|x(t0)| , |y(t0)| , |z(t0)|} = |x(t0)|, then we obtain

D+(|x(s)|)s=t0 ≤ −α1 |x(t0)| + |y(t0)|

≤ (−α1 + 1)M(t0)

< −ηM(t0) + θ. (3.7)



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Uniformly boundedness of a class of non-linear ... 67

If M(t0) = max {|x(t0)| , |y(t0)| , |z(t0)|} = |y(t0)|, then we have

D+(|y(s)|)s=t0 ≤ −α2 |y(t)| + |z(t)|

≤ (−α2 + 1)M(t0)

< −ηM(t0) + θ. (3.8)

If M(t0) = max {|x(t0)| , |y(t0)| , |z(t0)|} = |z(t0)|, then we get

D+(|z(s)|)s=t0 ≤ −inf
t≥0

(f1(t, x(t)) − α1 − α2) |z(t0)|

+sup
t≥0

∣

∣(f1(t, x(t)) − α1)(α1 + α2) − f2(t, x(t)) − α
2

2

∣

∣ |y(t0)|

+L0 |x(t0)| +

n∑

i=1

Li |x(t0 − τi(t0))| +

n∑

i=0

qi + |p(t)|

≤ (
n∑

i=0

Li − α3)M(t0) +

n∑

i=0

qi + |p(t)|

< −ηM(t0) + θ. (3.9)

In addition, if M(t0) ≥
θ

η
, then (3.7), (3.8) and (3.9) imply that M(t) is strictly decreasing

in a small neighborhood (t0, t0 + δ0). This contradicts that M(t) is non-decreasing. Therefore,

M(t0) <
θ

η
and

max {|x(t0)| , |y(t0)| , |z(t0)|} <
θ

η
. (3.10)

For ∀t > t0, by the same approach used in the proof of (3.10), we have

max {|x(t)| , |y(t)| , |z(t)|} <
θ

η
, if M(t) = max {|x(t)| , |y(t)| , |z(t)|} .

On the other hand, if M(t) > max {|x(t)| , |y(t)| , |z(t)|} , t > t0, we can choose t0 ≤ t3 < t

such that M(t3) = max {|x(t3)| , |y(t3)| , |z(t3)|} <
θ

η
and M(s) > max {|x(s)| , |y(s)| , |z(s)|} for all

s ∈ (t3, t]. Using a similar argument as in the proof of Case (I), we can show that M(s) ≡

M(t3) is a constant, for all s ∈ (t3, t], which implies max {|x(t)| , |y(t)| , |z(t)|} < M(t) = M(t3) =

max {|x(t3)| , |y(t3)| , |z(t3)|} <
θ

η
.

To sum up, the solutions of (1.2) are uniformly bounded. The proof is complete.

4 An example

Consider the following equation



68 Cemil Tunç and Hilmi Ergören CUBO
14, 3 (2012)

x
′′′

(t) + (11 −
1

1 + t + x2(t)
)x

′′

(t) + (31 −
4

1 + t + x2(t)
)x

′

(t) + (26 −
1

1 + t + x2(t)
)x(t)

+
1

1 + t + x2(t)
x(t − |sin t|) + (sin t) sin x(t − e|sin t|) =

1

1 + t2
. (4.1)

Setting y(t) =
dx(t)

dt
+ 2x(t) and z(t) =

dy(t)

dt
+ 2y(t), we can transform (4.1) into

dx(t)

dt
= −2x(t) + y(t),

dy(t)

dt
= −2y(t) + z(t),

dz(t)

dt
= −(7 −

1

1 + t + x2(t)
)z(t) + y(t) +

1

1 + t + x2(t)
x(t)) (4.2)

−
1

1 + t + x2(t)
x(t − |sin t|) − (sin t) sin x(t − e|sin t|) +

1

1 + t2
.

Then, we can satisfy the following assumptions:

i)
∣

∣((α1 − f1(t, u))α
2

1
+ f2(t, u)α1)u − g0(t, u)

∣

∣ =

∣

∣

∣

∣

1

1 + t + u2
u

∣

∣

∣

∣

≤ L0 |u| + q0,

ii)

∣

∣

∣

∣

1

1 + t + u2
u

∣

∣

∣

∣

≤ L1 |u| + q1and |sin t sin u| ≤ L2 |u| + q2 ,

iii) α3 = inf
t≥0

(7−
1

1 + t + u2
)−1 >

2∑

i=0

Li by taking suitable Li and qi such as L0 = L1 = L2 = 1

for appropriate qi (i = 0, 1, 2). Hence, all solutions of the system (4.2) are uniformly bounded.

Received: April 2011. Revised: February 2012.

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