NEW RESULT IN THE ULTIMATE BOUNDEDNESS
OF SOLUTIONS OF A THIRD-ORDER NONLINEAR
ORDINARY DIFFERENTIAL EQUATION
Ultimate Boundedness
of Solutions
M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
M.O. OMEIKE
Department of Mathematics
University of Agriculture
Abeokuta, Nigeria.
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EMail: moomeike@yahoo.com
Received:
26 March, 2007
Accepted:
15 January, 2008
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
Primary: 34C11; Secondary: 34B15.
Key words:
Differential equations of third order, Boundedness.
Abstract:
Sufficient conditions are established for the ultimate boundedness of solutions
of certain third-order nonlinear differential equations. Our result improves on
Tunc’s [C. Tunc, Boundedness of solutions of a third-order nonlinear differential
equation, J. Inequal. Pure and Appl. Math., 6(1) Art. 3,2005,1-6].
Acknowledgements:
The author would like to express sincere thanks to the anonymous referees for
their invaluable corrections, comments and suggestions.
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Contents
1
Introduction
3
2
Preliminaries
6
3
Proof of Theorem 1.1
13
Ultimate Boundedness
of Solutions
M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
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1.
Introduction
We consider the third-order nonlinear ordinary differential equation,
...
x + f (x, ẋ, ẍ)ẍ + g(x, ẋ) + h(x, ẋ, ẍ) = p(t, x, ẋ, ẍ)
(1.1)
or its equivalent system
(1.2)
ẋ = y,
ẏ = z,
ż = −f (x, y, z)z − g(x, y) − h(x, y, z) + p(t, x, y, z),
Ultimate Boundedness
of Solutions
M.O. Omeike
where f, g, h and p are continuous in their respective arguments, and the dots denote
differentiation with respect to t. The derivatives
∂f (x, y, z)
≡ fx (x, y, z),
∂x
∂h(x, y, z)
≡ hy (x, y, z),
∂y
∂f (x, y, z)
∂h(x, y, z)
≡ fz (x, y, z),
≡ hx (x, y, z),
∂z
∂x
∂h(x, y, z)
∂g(x, y)
≡ hz (x, y, z) and
≡ gx (x, y)
∂z
∂x
exist and are continuous. Moreover, the existence and the uniqueness of solutions of
(1.1) will be assumed. It is well known that the ultimate boundedness is a very important problem in the theory and applications of differential equations, and an effective
method for studying the ultimate boundedness of nonlinear differential equations is
still the Lyapunov’s direct method (see [1] – [8]).
Recently, Tunc [7] discussed the ultimate boundedness results of Eq. (1.1) and
the following result was proved.
Theorem A (Tunc [7]). Further to the assumptions on the functions f, g, h and
p assume the following conditions are satisfied (a, b, c, l, m and A− some positive
constants):
(i) f (x, y, z) ≥ a and ab − c > 0 for all x, y, z;
vol. 9, iss. 1, art. 15, 2008
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(ii)
(iii)
g(x,y)
y
≥ b for all x, y 6= 0;
h(x,0,0)
x
≥ c for all x 6= 0;
(iv) 0 < hx (x, y, 0) < c, for all x, y;
(v) hy (x, y, 0) ≥ 0 for all x, y;
Ultimate Boundedness
of Solutions
(vi) hz (x, y, 0) ≥ m for all x, y;
(vii) yfx (x, y, z) ≤ 0, yfz (x, y, z) ≥ 0 and gx (x, y) ≤ 0 for all x, y, z;
M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
(viii) yzhy (x, y, 0) + ayzhz (x, y, z) ≥ 0 for all x, y, z;
(ix) |p(t, x, y, z)| ≤ e(t) for all t ≥ 0, x, y, z,
Rt
where 0 e(s)ds ≤ A < ∞.
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Then, given any finite numbers x0 , y0 , z0 there is a finite constant D = D(x0 , y0 , z0 )
such that the unique solution (x(t), y(t), z(t)) of (1.2) which is determined by the
initial conditions
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x(0) = x0 ,
y(0) = y0 ,
z(0) = z0
|x(t)| ≤ D,
|y(t)| ≤ D,
|z(t)| ≤ D
satisfies
for all t ≥ 0.
Theoretically, this is a very interesting result since (1.1) is a rather general thirdorder nonlinear differential equation. For example, many third order differential
equations which have been discussed in [5] are special cases of Eq. (1.1), and some
known results can be obtained by using this theorem. However, it is not easy to apply
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Theorem A to these special cases to obtain new or better results since Theorem A
has some hypotheses which are not necessary for the stability of many nonlinear
equations. The Lyapunov function used in the proof of Theorem A is not complete
(see [2]). Furthermore, the boundedness result considered in [7] is of the type in
which the bounding constant depends on the solution in question.
Our aim in this paper is to further study the boundedness of solutions of Eq. (1.1).
In the next section, we establish a criterion for the ultimate boundedness of solutions
of Eq. (1.1), which extends and improves Theorem A.
Our main result is the following theorem.
Ultimate Boundedness
of Solutions
M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
Theorem 1.1. Further to the basic assumptions on the functions f, g, h and p assume that the following conditions are satisfied (a, b, c, ν and A− some positive constants):
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(i) f (x, y, z) > a and ab − c > 0 for all x, y, z;
(ii)
(iii)
g(x,y)
y
≥ b for all x, y 6= 0;
h(x,y,z)
x
≥ ν for all x 6= 0;
(v) yfx (x, y, z) ≤ 0, yfz (x, y, z) ≥ 0 and gx (x, y) ≤ 0 for all x, y, z;
(vi) |p(t, x, y, z)| ≤ A < ∞ for all t ≥ 0.
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|ẋ(t)| ≤ D,
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Then every solution x(t) of (1.1) satisfies
|x(t)| ≤ D,
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(iv) hx (x, 0, 0) ≤ c, hy (x, y, 0) ≥ 0 and hz (x, 0, z) ≥ 0 for all x, y, z;
(1.3)
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|ẍ(t)| ≤ D
for all sufficiently large t, where D is a constant depending only on a, b, c, A and ν.
2.
Preliminaries
It is convenient here to consider, in place of the equation (1.1), the system (1.2). It is
to be shown then, in order to prove the theorem, that, under the conditions stated in
the theorem, every solution (x(t), y(t), z(t)) of (1.2) satisfies
|x(t)| ≤ D,
(2.1)
|y(t)| ≤ D,
|z(t)| ≤ D
for all sufficiently large t, where D is the constant in (1.3).
Our proof of (2.1) rests entirely on two properties (stated in the lemma below) of
the function V = V (x, y, z) defined by
Ultimate Boundedness
of Solutions
M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
V = V1 + V2 ,
(2.2)
where V1 , V2 are given by
Z x
Z
h(ξ, 0, 0)dξ + 2
(2.3a) 2V1 = 2
0
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y
Z
ηf (x, η, 0)dη + 2δ
0
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y
g(x, η)dη
0
2
2
+ δz + 2yz + 2δyh(x, 0, 0) − αβy ,
2
Z
x
Z
y
(2.3b) 2V2 = αβbx + 2a
h(ξ, 0, 0)dξ + 2a
ηf (x, η, 0)dη
0
0
Z y
+2
g(x, η)dη + z 2 + 2aαβxy + 2αβxz + 2ayz + 2yh(x, 0, 0),
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where
1
a
< δ < cb , and α, β are some positive constants such that








ab − c
1 aδ − 1
ν(aδ − 1)
α < min
2 ; a ; abδ ; β[f (x, y, z) − a]2 




 β a + ν −1 g(x,y)
−b
y
Close
and β will be fixed to advantage later.
Lemma 2.1. Subject to the conditions of Theorem 1.1, V (0, 0, 0) = 0 and there is a
positive constant D1 depending only on a, b, c, α and δ such that
V (x, y, z) ≥ D1 (x2 + y 2 + z 2 )
(2.4)
for all x, y, z. Furthermore, there are finite constants D2 > 0, D3 > 0 dependent
only on a, b, c, A, ν, δ, and α such that for any solution (x(t), y(t), z(t)) of (1.2),
M.O. Omeike
d
V̇ ≡ V (x(t), y(t), z(t)) ≤ −D2 ,
dt
(2.5)
Ultimate Boundedness
of Solutions
vol. 9, iss. 1, art. 15, 2008
provided that x2 + y 2 + z 2 ≥ D3 .
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Proof of Lemma 2.1. To verify (2.4) observe first that the expressions (2.3) defining
2V1 , 2V2 may be rewritten in the forms
Z x
2
δ 2
h(x, 0, 0)
2V1 = 2
h(ξ, 0, 0)dξ − h (x, 0, 0) + δb y +
b
b
0
Z y
+ 2
ηf (x, η, 0)dη − δ −1 y 2 − αβy 2 + δ(z + δ −1 y)2
0
Z y
2
+δ 2
g(x, η)dη − by
0
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and
Z
2V2 = αβ(b − αβ)x + a 2
x
h (x, 0, 0)
−1 2
2
h(ξ, 0, 0)dξ − β
n 1
o2 Z y
−2
−1 12
−1 2
+ β a y + β a h(x, 0, 0) + 2
g(x, η)dη − βa y
0
0
Z
+a 2
y
ηf (x, η, 0)dη − ay
2
+ (αβx + ay + z)2 .
0
The term 2
equal to
Rx
0
h(ξ, 0, 0)dξ − δb h2 (x, 0, 0) in the rearrangement for 2V1 is evidently
Z
2
0
x
δ
δ
1 − hξ (ξ, 0, 0) h(ξ, 0, 0)dξ − h2 (0, 0, 0).
b
b
By conditions (iii) and (iv) of Theorem 1.1 and h(0, 0, 0) = 0, we have
Z x
δ 2
δ
δ
2
1 − hξ (ξ, 0, 0) h(ξ, 0, 0)dξ − h (0, 0, 0) ≥ 1 − c νx2 .
b
b
b
0
In the same way, using (iii) and (iv), it can be shown that the term
Z x
−1 2
2
h(ξ, 0, 0)dξ − β h (x, 0, 0)
0
appearing in the rearrangement for 2V2 satisfies
Z x
c
−1 2
2
h(ξ, 0, 0)dξ − β h (x, 0, 0) ≥ 1 −
νx2 ,
β
0
for all x.
Since h(x,y,z)
≥ ν (x 6= 0), g(x,y)
≥ b, (y 6= 0) and f (x, y, z) > a, and
x
y
combining all these with (2.2), we have
δ
c
x2
2V ≥ ν 1 − c + αβ(b − αβ) + aν 1 −
b
β
2
β
1
1
2
+
a − − αβ + b −
y + δ z + y + (αβx + ay + z)2
δ
a
δ
Ultimate Boundedness
of Solutions
M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
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for all x, y and z. Hence if we choose β = ab the constants 1 − δb c, b − αβ, 1 − βc ,
a − 1δ − αβ and b − βa are either zero or positive. This implies that there exists a
constant D1 small enough such that (2.4) holds.
To deal with the other half of the lemma, let (x(t), y(t), z(t)) be any solution of
(1.2) and consider the function
V (t) ≡ V (x(t), y(t), z(t)) .
By an elementary calculation using (1.2), (2.2) and (2.3), we have that
Z y
Z y
(2.6) V̇ = (1 + δ)y
gx (x, η)dη + (1 + a)y
ηfx (x, η, 0)dη
0
By (v), we get
y
Z
gx (x, η)dη ≤ 0,
y
0
M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
0
{f (x, y, z) − f (x, y, 0)} 2
{h(x, y, z) − h(x, 0, 0)} 2
yz − (1 + a)
y
− (1 + a)
z
y
h(x, y, z) 2 g(x, y) 2
{h(x, y, z) − h(x, 0, 0)} 2
− (1 + δ)
z − αβ
x −
y
z
x
y
g(x, y) 2
−a
y + δhx (x, 0, 0)y 2 + hx (x, 0, 0)y 2 + aαβy 2
y
g(x, y)
2
2
2
− δf (x, y, z)z − [f (x, y, z) − a]z + z − αβ
− b xy
y
− αβ{f (x, y, z) − a}xz + {αβx + (1 + a)y + (1 + δ)z}p(t, x, y, z).
Z
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y
fx (x, η, 0)ηdη ≤ 0.
y
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0
It follows from (v), for z 6= 0 that
{f (x, y, z) − f (x, y, 0)} 2
W1 = a
yz = afz (x, y, θ1 z)yz 2 ≥ 0,
z
0 ≤ θ1 ≤ 1 but W1 = 0 when z = 0. Hence
W1 ≥ 0 for all x, y, z.
Similarly, it is clear that
W2 =
{h(x, y, z) − h(x, 0, 0)} 2
y = hy (x, θ2 y, 0)y 2 ≥ 0,
y
0 ≤ θ2 ≤ 1 but W2 = 0 when y = 0. Hence
Ultimate Boundedness
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M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
W2 ≥ 0 f or all x, y.
Also,
{h(x, y, z) − h(x, 0, 0)} 2
W3 =
z = hz (x, 0, θ3 z)z 2 ≥ 0,
z
0 ≤ θ3 ≤ 1 but W3 = 0 when z = 0. Hence
W3 ≥ 0 for all x, z.
Then, combining the estimates W1 , W2 , W3 and (iii) with (2.6) we obtain
V̇ ≤ −αβνx2 − (ab − c − αβa)y 2 − (b − δc)y 2 − (aδ − 1)z 2
g(x, y)
2
− az − αβ
− b xy − αβ{f (x, y, z) − a}xz
y
+ {αβx + (1 + a)y + (1 + δ)z}p(t, x, y, z)
(
"
2 #)
1
g(x,
y)
−b
y2
= − αβνx2 − ab − c − αβ a + ν −1
2
y
− (b − δc)y 2 − aδ − 1 − αβν −1 [f (x, y, z) − a]2 z 2 − az 2
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1
− αβν
4
(
)
2
g(x,
y)
2
x + 2ν −1
− b y + x + 2ν −1 (f (x, y, z) − a)z
y
+ {αβx + (1 + a)y + (1 + δ)z}p(t, x, y, z).
If we choose
α < min








ab − c
ν(aδ − 1)
1 aδ − 1
; ;
,
;
2

a abδ β[f (x, y, z) − a]2 
g(x,y)


−1
β a + ν

−b
y
it follows that
Ultimate Boundedness
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M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
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1
V̇ ≤ − αβνx2 − (b − δc)y 2 − az 2 + {αβx + (1 + a)y + (1 + δ)z}p(t, x, y, z)
2
≤ −D4 (x2 + y 2 + z 2 ) + D5 (|x| + |y| + |z|),
where
D4 = min
1
αβν; b − δc; a ,
2
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D5 = A max{αβ; 1 + a; 1 + δ}.
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Moreover,
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1
2
V̇ ≤ −D4 (x2 + y 2 + z 2 ) + D6 (x2 + y 2 + z 2 ) ,
(2.7)
1
where D6 = 3 2 D5 .
1
If we choose (x2 + y 2 + z 2 ) 2 ≥ D7 = 2D6 D4−1 , inequality (2.7) implies that
1
V̇ ≤ − D4 (x2 + y 2 + z 2 ).
2
Close
We see at once that
V̇ ≤ −D8 ,
provided that x2 + y 2 + z 2 ≥ 2D8 D4−1 ; and this completes the verification of (2.5).
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of Solutions
M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
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3.
Proof of Theorem 1.1
Let (x(t), y(t), z(t)) be any solution of (1.2). Then there is evidently a t0 ≥ 0 such
that
x2 (t0 ) + y 2 (t0 ) + z 2 (t0 ) < D3 ,
where D3 is the constant in the lemma; for otherwise, that is if
2
2
2
x (t) + y (t) + z (t) ≥ D3 ,
t ≥ 0,
Ultimate Boundedness
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M.O. Omeike
then, by (2.5),
vol. 9, iss. 1, art. 15, 2008
V̇ (t) ≤ −D2 < 0,
t ≥ 0,
and this in turn implies that V (t) → −∞ as t → ∞, which contradicts (2.4). Hence
to prove (1.3) it will suffice to show that if
(3.1)
2
2
2
x (t) + y (t) + z (t) < D9
for
t = T,
where D9 ≥ D3 is a finite constant, then there is a constant D10 > 0, depending on
a, b, c, δ, α and D9 , such that
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(3.2)
2
2
2
x (t) + y (t) + z (t) ≤ D10
for
t ≥ T.
Our proof of (3.2) is based essentially on an extension of an argument in the proof
of [8, Lemma 1]. For any given constant d > 0 let S(d) denote the surface: x2 +y 2 +
z 2 = d. Because V is continuous in x, y, z and tends to +∞ as x2 + y 2 + z 2 → ∞,
there is evidently a constant D11 > 0, depending on D9 as well as on a, b, c, δ and α,
such that
(3.3)
min
(x,y,z)∈S(D11 )
V (x, y, z) >
max
(x,y,z)∈S(D9 )
V (x, y, z).
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It is easy to see from (3.1) and (3.3) that
(3.4)
x2 (t) + y 2 (t) + z 2 (t) < D11
t ≥ T.
for
For suppose on the contrary that there is a t > T such that
x2 (t) + y 2 (t) + z 2 (t) ≥ D11 .
Then, by (3.1) and by the continuity of the quantities x(t), y(t), z(t) in the argument
displayed, there exist t1 , t2 , T < t1 < t2 such that
(3.5a)
x2 (t1 ) + y 2 (t1 ) + z 2 (t1 ) = D9 ,
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M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
(3.5b)
x2 (t2 ) + y 2 (t2 ) + z 2 (t2 ) = D11
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and such that
(3.6)
D9 ≤ x2 (t) + y 2 (t) + z 2 (t) ≤ D11 ,
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t1 ≤ t ≤ t2 .
But, writing V (t) ≡ V (x(t), y(t), z(t)) , since D9 ≥ D3 , (3.6) obviously implies [in
view of (2.5)] that
V (t2 ) < V (t1 )
and this contradicts the conclusion [from (3.3) and (3.5)]:
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V (t2 ) > V (t1 ).
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Hence (3.4) holds. This completes the proof of (1.3), and the theorem now follows.
Remark 1. Clearly, our theorem is an improvement and extension of Theorem A.
In particular, from our theorem we see that (viii) assumed in Theorem A is not
necessary, and (iv) and (ix) can be replaced by hx (x, 0, 0) ≤ c and (vi) of Theorem
1.1 respectively, for the ultimate boundedness of the solutions of Eq. (1.1).
Remark 2. Clearly, unlike in [7], the bounding constant D in Theorem 1.1 does not
depend on the solution of (1.1).
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References
[1] H. BEREKETOGLU AND I. GYORI, On the boundedness of the solutions of a
third-order nonlinear differential equation, Dynam. Systems Appl., 6(2) (1997),
263–270.
[2] E.N. CHUKWU, On the boundedness of solutions of third order differential
equations, Ann. Mat. Pur. Appl., 104(4) (1975), 123–149.
[3] J.O.C. EZEILO, An elementary proof of a boundedness theorem for a certain
third order differential equation, J. Lond. Math. Soc., 38 (1963), 11–16.
[4] J.O.C. EZEILO, A further result on the existence of periodic solutions of the
...
equation x + Ψ(ẋ)ẍ + φ(x)ẋ + v(x, ẋ, ẍ) = p(t) with a bound ν, Atti. Accad.
Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 55 (1978), 51–57.
[5] R. REISSSIG, G. SANSONE AND R. CONTI, Nonlinear Differential Equations
of Higher Order, Noordhoff, Groningen, 1974.
[6] K.E. SWICK, Boundedness and stability for a nonlinear third order differential
equation, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 56(6) (1974),
859–865.
[7] C. TUNC, Boundedness of solutions of a third-order nonlinear differential
equation, J. Inequal. Pure and Appl. Math., 6(1) (2005), Art. 3. [ONLINE:
http://jipam.vu.edu.au/article.php?sid=472].
[8] T. YOSHIZAWA, On the evaluation of the derivatives of solutions of y” =
f (x, y, x0 ), Mem. Coll. Sci., Univ. Kyoto Ser. A. Math., 28 (1953), 27–32. (1953),
133–141.
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of Solutions
M.O. Omeike
vol. 9, iss. 1, art. 15, 2008
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