Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 863178, 12 pages
doi:10.1155/2009/863178
Research Article
On the Convergence of Solutions of Certain
Third-Order Differential Equations
Ercan Tunç
Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpaşa University, 60240-Tokat, Turkey
Correspondence should be addressed to Ercan Tunç, ercantunc72@yahoo.com
Received 29 December 2008; Revised 24 March 2009; Accepted 5 April 2009
Recommended by Leonid Shaikhet
We establish sufficient conditions for the convergence of solutions of a certain third-order nonlinear
differential equations. By constructing a Lyapunov function as the basic tool, some results which
exist in the relevant literature are generalized.
Copyright q 2009 Ercan 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.
1. Introduction
As well known, in the investigation of qualitative behaviors of solutions, stability,
convergence, boundedness, oscillation, and so forth of solutions are very important problems
in theory and applications of differential equations. For example, in applied sciences, some
practical problems concerning mechanics, the engineering technique fields, economy, control
theory, physics, chemistry, biology, medicine, atomic energy, information theory, and so forth
are associated with certain higher-order linear or nonlinear differential equations. Ever since
Lyapunov 1 proposed his famous theory on the stability of motion, For some papers
published on the qualitative behaviors of solutions of nonlinear second-and third-order
differential equations, the readers can referee to the papers of Afuwape and Omeike 2, 3,
Ezeilo 4, 5, Meng 6, Tejumola 7, 8, Tunç 9–11, Omeike 12, and the references listed
in these papers as well as one can refer to the books of Reissig et al. 13, 14. The motivation
for the present work has been inspired basically by the paper of Afuwape and Omeike 2
and the papers listed above. Our aim here is to extend the results established by Afuwape
and Omeike 2 to nonlinear differential equation 1.4 for the convergence of all solutions of
this equation. In 2008, Afuwape and Omeike 2 considered third-order nonlinear differential
equations of the form
...
x aẍ gẋ hx pt, x, ẋ, ẍ,
1.1
2
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and by introducing a Lyapunov function they discussed the convergence of solutions for
this equation. During establishment of the results, Afuwape and Omeike 2 defined the
following relations with respect to the functions g and h:
g y2 − g y1
≤ b0 < ∞,
y2 − y1
0<b≤
1.2
for any pair of constants y2 , y1 y2 /
y1 and
0<δ≤
hx2 − hx1 ≤ kab,
x2 − x1
1.3
for any pair of constants x2 , x1 x2 /
x1 , where k < 1 is a positive constant.
In this paper, we consider nonlinear differential equation of the form
...
x fẍ gẋ hx pt, x, ẋ, ẍ,
1.4
where the functions f, g, h, and p are continuous in their respective arguments, with the
functions f, g, and h are not necessarily differentiable. In addition to 1.2 and 1.3 we
assume that
0<a≤
fz2 − fz1 ≤ a0 < ∞,
z2 − z1
1.5
for any pair of constants z2 , z1 z2 /
z1 .
By convergence of solutions we mean, any two solutions x1 t, x2 t of 1.4 are said
to converge to each other if
x2 t − x1 t −→ 0,
ẋ2 t − ẋ1 t −→ 0,
ẍ2 t − ẍ1 t −→ 0
1.6
as t → ∞.
2. Main Results
The following results are established.
Theorem 2.1. Suppose that f0 g0 h0, and that
i there are constants a > 0, a0 > 0 such that fz satisfies inequalities 1.5,
ii there are constants b > 0,
b0 > 0 such that gy satisfies inequalities 1.2,
iii there are constants δ > 0, k < 1 such that for any ξ, η η /
0, the incrementary ratio for h
satisfies
h ξ η − hξ
lies in I0
η
with I0 δ, kab,
2.1
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3
iv there is a continuous function φt such that
p t, x2 , y2 , z2 − p t, x1 , y1 , z1 ≤ φt |x2 − x1 | y2 − y1 |z2 − z1 |
2.2
holds for arbitrary t, x1 , y1 , z1 , x2 , y2 , z2 , and satisfies
t
2.3
φν τdτ ≤ D1 t
0
for some constant D1 > 0, where ν is a constant in the range 1 ≤ ν ≤ 2.
Then all solutions of 1.4 converge.
A very important step in the proof of Theorem 2.1 will be to give estimate for any two
solutions of 1.4. This in itself, being of independent interest, is giving as follows.
Theorem 2.2. Let x1 t, x2 t be any two solutions of 1.4. Suppose that all the conditions of
Theorem 2.1 are satisfied, then for each fixed ν, in the range 1 ≤ ν ≤ 2, there exist constants D2 , D3 ,
and D4 such that for t2 ≥ t1 ,
St2 ≤ D2 St1 exp −D3 t2 − t1 D4
t2
φ τdτ ,
ν
2.4
t1
where
St x2 t − x1 t2 ẋ2 t − ẋ1 t2 ẍ2 t − ẍ1 t2 .
2.5
We have the following corollaries when x1 t 0 and t1 0.
Corollary 2.3. Suppose that p 0 in 1.4 and suppose further that conditions (i), (ii), and (iii) of
Theorem 2.1 hold, then the trivial solution of 1.4 is exponentially stable in the large.
Also, if we put ξ 0 in 2.1 with η η / 0 arbitrary, we get the following.
Corollary 2.4. If p /
0 and hypotheses (i), (ii), and (iii) of Theorem 2.1 hold for arbitrary ηη /
0 ,
and ξ 0, then there exists a constant D5 > 0 such that every solution xt of 1.4 satisfies
|xt| ≤ D5 ,
|ẋt| ≤ D5 ,
|ẍt| ≤ D5 .
2.6
3. Preliminary Results
On setting ẋ y, ẏ z, 1.4 can be replaced by an equivalent system
ẋ y,
ẏ z,
ż −fz − g y − hx p t, x, y, z .
3.1
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Let xi t, yi t, zi t, i 1, 2, be any two solutions of 3.1 such that
fz2 − fz1 ≤ a0 z2 /
z1 ,
z −z
2 1 g y2 − g y1
y2 ≤ b0
b≤
/ y1 ,
y2 − y1
hx2 − hx1 δ≤
≤ kab x2 / x1 ,
x2 − x1
a≤
3.2
where a0 , a, b0 , b, δ, and k are finite constants, and k will be determined later.
Our investigation rests mainly on the properties of the function, W Wx2 − x1 , y2 −
y1 , z2 − z1 defined by
2
2
2W β 1 − β b2 x2 − x1 2 βb y2 − y1 αba−1 y2 − y1
2
αa−1 z2 − z1 2 z2 − z1 a y2 − y1 1 − β bx2 − x1 ,
3.3
where 0 < β < 1 and α > 0 are constants.
Following the argument used in 5, we can easily verify the following for W.
Lemma 3.1. (i) W0, 0, 0 0.
(ii) There exist finite positive constants D6 , D7 such that
2
2
D6 x2 − x1 2 y2 − y1 z2 − z1 2 ≤ W ≤ D7 x2 − x1 2 y2 − y1 z2 − z1 2 ,
3.4
where
D6 1
min β 1 − β b2 , b β αa−1 , αa−1 ,
2
3.5
and using the inequality |x||y| ≤ 1/2x2 y2 ,
D7 1
max b 1 − β 1 b a, b β αa−1 a 1 a b 1 − β , 1 αa−1 a b 1 − β .
2
3.6
If we define the function Wt by
W x2 t − x1 t, y2 t − y1 t, z2 t − z1 t
3.7
and using the fact that the solutions xi , yi , zi , i 1, 2, satisfy 3.1, then St as defined in
2.5 becomes
2
St x2 t − x1 t2 y2 t − y1 t z2 t − z1 t2 .
3.8
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5
Lemma 3.2. Assume that the conditions (i), (ii), and (iii) of Theorem 2.1 are satisfied. Then, there
exist positive finite constants D8 and D9 such that
dW
≤ −2D8 S D9 S1/2 |θ|,
dt
3.9
where θ pt, x2 , y2 , z2 − pt, x1 , y1 , z1 .
Proof of Lemma 3.2
Differentiating the function W in 3.3 along the system 3.1 we obtain
Ẇ dW
−W1 − W2 − W3 − W4 − W5 − W6 − W7 W8 ,
dt
3.10
in which
2
W1 γ1 b 1 − β Hx2 , x1 x2 − x1 2 η1 a G y2 , y1 − b 1 − β
y2 − y1
ξ1 αa−1 Fz2 , z1 z2 − z1 2 Fz2 , z1 − az2 − z1 2 ,
W2 γ2 b 1 − β Hx2 , x1 x2 − x1 2 ξ2 αa−1 Fz2 , z1 z2 − z1 2
1 αa−1 x2 − x1 z2 − z1 Hx2 , x1 ,
2
W3 γ3 b 1 − β Hx2 , x1 x2 − x1 2 η2 a G y2 , y1 − b 1 − β
y2 − y1
ax2 − x1 y2 − y1 Hx2 , x1 ,
W4 γ4 b 1 − β Hx2 , x1 x2 − x1 2 ξ3 αa−1 Fz2 , z1 z2 − z1 2
b 1 − β x2 − x1 z2 − z1 Fz2 , z1 − a ,
2
W5 γ5 b 1 − β Hx2 , x1 x2 − x1 2 η3 a G y2 , y1 − b 1 − β y2 − y1
b 1 − β x2 − x1 y2 − y1 G y2 , y1 − b ,
2
W6 ξ4 αa−1 Fz2 , z1 z2 − z1 2 η4 a G y2 , y1 − b 1 − β
y2 − y1
1 αa−1 y2 − y1 z2 − z1 G y2 , y1 − b ,
2
W7 ξ5 αa−1 Fz2 , z1 z2 − z1 2 η5 a G y2 , y1 − b 1 − β
y2 − y1
a y2 − y1 z2 − z1 Fz2 , z1 − a ,
W8 b 1 − β x2 − x1 a y2 − y1 1 αa−1 z2 − z1 θt,
3.11
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with
fz2 − fz1 z1 ,
z2 /
z2 − z1
g y2 − g y1
G y2 , y1 y2 / y1 ,
y2 − y1
Fz2 , z1 Hx2 , x1 hx2 − hx1 x2 − x1
3.12
x1 ,
x2 /
and ξi , ηi , and γi , i 1, 2, 3, 4, 5 are strictly positive constants such that
5
ξi 1,
i1
5
ηi 1,
i1
5
γi 1.
3.13
i1
Also, let us denote Fz2 , z1 , Gy2 , y1 , and Hx2 , x1 simply by F, G, and H, respectively. For
strictly positive constants k1 , k2 , k3 , k4 , k5, and k6 conveniently chosen later, we get
1 αa−1 x2 − x1 z2 − z1 H
2
1/2
1/2
1 −1 −1
1/2
−1
1/2
H x2 − x1 k1 1 αa
H z2 − z1 k1 1 αa
2
1
− k12 1 αa−1 Hx2 − x1 2 − k1−2 1 αa−1 Hz2 − z1 2 ,
4
ax2 − x1 y2 − y1 H
2
1
k2 a1/2 H 1/2 x2 − x1 k2−1 a1/2 H 1/2 y2 − y1
2
2
1
− k22 aHx2 − x1 2 − k2−2 aH y2 − y1 ,
4
b 1 − β x2 − x1 z2 − z1 F − a
1/2
1 −1 1/2
k3 b 1 − β1/2 F − a1/2 x2 − x1 k3 b1/2 1 − β
F − a1/2 z2 − z1 2
2
1
− k3−2 b 1 − β F − ax2 − x1 2 − k32 b 1 − β F − az2 − z1 2 ,
4
b 1 − β x2 − x1 y2 − y1 G − b
2
1/2
1/2
1
k4 b1/2 1 − β
G − b1/2 x2 − x1 k4−1 b1/2 1 − β
G − b1/2 y2 − y1
2
2
1
− k42 b 1 − β G − bx2 − x1 2 − k4−2 b 1 − β G − b y2 − y1 ,
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1 αa−1 y2 − y1 z2 − z1 G − b
7
2
1/2
1/2
1
k5 1 αa−1
G − b1/2 y2 − y1 k5−1 1 αa−1
G − b1/2 z2 − z1 2
2 1
− k52 1 αa−1 G − b y2 − y1 − k5−2 1 αa−1 G − bz2 − z1 2 ,
4
a y2 − y1 z2 − z1 F − a
1 −1 1/2
k6 a F − a1/2 y2 − y1 k6 a1/2 F − a1/2 z2 − z1 2
2
2
1
− k6−2 aF − a y2 − y1 − k62 aF − az2 − z1 2
4
3.14
Thus,
W2 2
1/2
1/2
1
k1 1 αa−1
H 1/2 x2 − x1 k1−1 1 αa−1
H 1/2 z2 − z1 2
γ2 b 1 − β H − k12 1 αa−1 H x2 − x1 2
1
ξ2 αa−1 F − k1−2 1 αa−1 H z2 − z1 2 ,
4
2
1 −1 1/2 1/2
1/2 1/2
W3 k2 a H x2 − x1 k2 a H y2 − y1 2
γ3 b 1 − β H − k22 aH x2 − x1 2
1 −2
2
η2 a G − b 1 − β − k2 aH y2 − y1 ,
4
W4 W5 1/2
1/2
1 −1 1/2 k3 b
F − a1/2 z2 − z1 1−β
F − a1/2 x2 − x1 k3 b1/2 1 − β
2
1 −2 γ4 b 1 − β H − k3 b 1 − β F − a x2 − x1 2
4
ξ3 αa−1 F − k32 b 1 − β F − a z2 − z1 2 ,
2
2
1
k4 b1/2 1 − β1/2 G − b1/2 x2 − x1 k4−1 b1/2 1 − β1/2 G − b1/2 y2 − y1 2
γ5 b 1 − β H − k42 b 1 − β G − b x2 − x1 2
1 −2 2
η3 a G − b 1 − β − k4 b 1 − β G − b y2 − y1 ,
4
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W6 2
1/2
1/2
1
k5 1 αa−1
G − b1/2 y2 − y1 k5−1 1 αa−1
G − b1/2 z2 − z1 2
2
η4 a G − b 1 − β − k52 1 αa−1 G − b y2 − y1
1
ξ4 αa−1 F − k5−2 1 αa−1 G − b z2 − z1 2 ,
4
2
1 −1 1/2
k6 a F − a1/2 y2 − y1 k6 a1/2 F − a1/2 z2 − z1 W7 2
1
2
η5 a G − b 1 − β − k6−2 aF − a y2 − y1
4
ξ5 αa−1 F − k62 aF − a z2 − z1 2 .
3.15
Moreover, in view of 3.2, we obtain for all xi , zi i 1, 2 in R,
W2 ≥ 0,
3.16
if
k12 ≤
γ2 1 − β ab
α a
with H ≤
4ξ2 γ2 α 1 − β a2 b
α a2
,
3.17
and for all xi , yi i 1, 2 in R,
W3 ≥ 0,
3.18
if
k22 ≤
γ3 1 − β b
a
with H ≤
4η2 γ3 β 1 − β b2
.
a
3.19
Combining all the inequalities in 3.16 and 3.18, we have for all xi , yi , zi i 1, 2 in R,
W2 ≥ 0,
W3 ≥ 0,
3.20
4ξ2 γ2 α 1 − β a 4η2 γ3 β 1 − β b
< 1.
with k min
,
a2
α a2
3.21
if
H ≤ kab
Also, for all xi , zi i 1, 2in R,
W4 ≥ 0,
3.22
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9
if
ξ3 α
a0 − a
≤ k32 ≤ ,
4γ4 δ
1 − β ba0 − a
3.23
W5 ≥ 0,
3.24
for all xi , yi i 1, 2 in R,
if
1 − β b0 − b
δγ5
,
≤ k42 ≤
4βaη3
b0 − b
3.25
for all yi , zi i 1, 2 in R,
W6 ≥ 0,
3.26
η4 βba2
α ab0 − b
≤ k52 ≤
,
4ξ4 αa
α ab0 − b
3.27
if
and for all yi , zi i 1, 2 in R,
W7 ≥ 0,
3.28
a0 − a
ξ5 α
≤ k62 ≤
.
4η5 βb
aa0 − a
3.29
if
Further
2
W1 ≥ 2D10 x2 − x1 2 y2 − y1 z2 − z1 2 ,
3.30
where 2D10 min{γ1 bδ1 − β, η1 abβ, ξ1 α}, on the other hand
1/2
2
W8 ≤ D11 x2 − x1 2 y2 − y1 z2 − z1 2
|θt|,
where D11 2max{b1 − β, a, 1 αa−1 }.
3.31
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Bringing together the estimates just obtained for W1 , W2 , W3 , W4 , W5 , W6 , W7 , and W8
in 3.10 and using 3.8, we have
dW
≤ −2D10 St D11 S1/2 t|θt|.
dt
3.32
This completes the proof of Lemma 3.2.
4. Proof of Theorem 2.2
This follows directly from 5, on using inequality 3.32. Let ν be any constant in the range
1 ≤ ν ≤ 2. Set 2μ 2 − ν, so that 0 ≤ μ ≤ 1/2. We rewrite 3.32 in the form
dW
D10 S ≤ −D10 S D11 S1/2 |θ| D11 Sμ W ∗ ,
dt
4.1
where
W ∗ |θ| − D12 S1/2 S1/2−μ ,
4.2
−1
, considering the two cases
with D12 D10 D11
i |θ| < D12 S1/2 and
ii |θ| ≥ D12 S1/2
separately. If |θ| < D12 S1/2 , then W ∗ < 0. On the other hand, if |θ| ≥ D12 S1/2 , then the
definition of W ∗ in 4.2 gives at least
W ∗ ≤ S1/2−μ |θ|
4.3
and also S1/2 ≤ |θ|/D12 . This implies that
S1/21−2μ ≤
|θ| 1−2μ
.
D12
4.4
Therefore
S1/21−2μ |θ| ≤
|θ| 1−2μ
× |θ|,
D12
4.5
from which together with W ∗ , we obtain
W ∗ ≤ D13 |θ|21−μ ,
4.6
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2μ−1
where D13 D12
11
. Again due to 4.1 and using the estimate on W ∗ for W ∗ , we have
dW
D10 S ≤ D11 D13 Sμ |θ|21−μ
dt
μ 21−μ
≤ D14 S φ
4.7
S1−μ ,
where D14 31−μ D11 D13 , which follows from
|θ| p t, x2 , y2 , z2 − p t, x1 , y1 , z1 ≤ φt |x2 − x1 | y2 − y1 |z2 − z1 | .
4.8
In view of the fact that ν 21 − μ, we obtain
dW
≤ −D10 S D14 φν S,
dt
4.9
and on using inequality 3.4, we have
dW D15 − D16 φν t W ≤ 0
dt
4.10
for some positive constants D15 and D16 . On integrating 4.10 from t1 to t2 t2 ≥ t1 , we have
Wt2 ≤ Wt1 exp −D15 t2 − t1 D16
t2
φ τdτ .
ν
4.11
t2
Again, using Lemma 3.1, we obtain 2.4, with D2 D7 D6−1 , D3 D15 , and D4 D16 . This
completes the proof of Theorem 2.2.
5. Proof of Theorem 2.1
This follows from the estimate 2.4 and the condition 2.3 on φt. Choose D1 D3 D4−1 in
2.3. From the estimate 2.4, if
t2
t1
φν τdτ ≤ D3 D4−1 t2 − t1 ,
5.1
then the exponential index remains negative for all t2 − t1 ≥ 0. Then, as t t2 − t1 → ∞, we
have St → 0, and this gives
x2 − x1 −→ 0,
y2 − y1 −→ 0,
as t → ∞. This completes the proof of Theorem 2.1.
z2 − z1 −→ 0,
5.2
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Acknowledgment
The author would like to express sincere thanks to the anonymous referees for their
invaluable corrections, comments, and suggestions.
References
1 A. M. Liapunov, Stability of Motion, Academic Press, New York, NY, USA, 1966.
2 A. U. Afuwape and M. O. Omeike, “Convergence of solutions of certain non-homogeneous third
order ordinary differential equations,” Kragujevac Journal of Mathematics, vol. 31, pp. 5–16, 2008.
3 A. U. Afuwape and M. O. Omeike, “Further ultimate boundedness of solutions of some system
of third order nonlinear ordinary differential equations,” Acta Universitatis Palackianae Olomucensis.
Facultas Rerum Naturalium. Mathematica, vol. 43, pp. 7–20, 2004.
4 J. O. C. Ezeilo, “A note on the convergence of solutions of certain second order differential equations,”
Portugaliae Mathematica, vol. 24, pp. 49–58, 1965.
...
5 J. O. C. Ezeilo, “New properties of the equation x aẍ bẋ hx P t, x, ẋ, ẍ for certain special
−1
values of the incrementary ratio y {hx y − hx},” in Équations différentielles et fonctionnelles
non linéaires (Actes Conférence Internat. “Equa-Diff 73”, Brussels/Louvain-la-Neuve, 1973), P. Janssens, J.
Mawhin, and N. Rouche, Eds., pp. 447–462, Hermann, Paris, France, 1973.
6 F. W. Meng, “Ultimate boundedness results for a certain system of third order nonlinear differential
equations,” Journal of Mathematical Analysis and Applications, vol. 177, no. 2, pp. 496–509, 1993.
7 H. O. Tejumola, “On the convergence of solutions of certain third-order differential equations,” Annali
di Matematica Pura ed Applicata, vol. 78, no. 1, pp. 377–386, 1968.
8 H. O. Tejumola, “Convergence of solutions of certain ordinary third order differential equations,”
Annali di Matematica Pura ed Applicata, vol. 94, no. 1, pp. 247–256, 1972.
9 C. Tunç, “Boundedness of solutions of a third-order nonlinear differential equation,” Journal of
Inequalities in Pure and Applied Mathematics, vol. 6, no. 1, article 3, pp. 1–6, 2005.
10 C. Tunç and E. Tunç, “New ultimate boundedness and periodicity results for certain third-order
nonlinear vector differential equations,” Mathematical Journal of Okayama University, vol. 48, pp. 159–
172, 2006.
11 C. Tunç and E. Tunç, “On the asymptotic behavior of solutions of certain second-order differential
equations,” Journal of the Franklin Institute, vol. 344, no. 5, pp. 391–398, 2007.
12 M. O. Omeike, “New result in the ultimate boundedness of solutions of a third-order nonlinear
ordinary differential equation,” Journal of Inequalities in Pure and Applied Mathematics, vol. 9, no. 1,
article 15, pp. 1–8, 2008.
13 R. Reissig, G. Sansone, and R. Conti, Non-Linear Differential Equations of Higher Order, Noordhoff
International, Leyden, The Netherlands, 1974.
14 T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Publications of the Mathematical Society
of Japan, no. 9, The Mathematical Society of Japan, Tokyo, Japan, 1966.