Journal of Inequalities in Pure and
Applied Mathematics
BOUNDEDNESS OF SOLUTIONS OF A THIRD-ORDER
NONLINEAR DIFFERENTIAL EQUATION
volume 6, issue 1, article 3,
2005.
CEMIL TUNÇ
Department of Mathematics
Faculty Arts and Sciences
Yüzüncü Yıl University,
65080, VAN-TURKEY.
Received 19 December, 2003;
accepted 04 January, 2005.
Communicated by: S.S. Dragomir
EMail: cemtunc@yahoo.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
173-03
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Abstract
Sufficient conditions are established for the boundedness of all solutions of
(1.1), and we also present some sufficient conditions, which ensure that the
limits of first and second order derivatives of the solutions of (1.1) tend to zero
as t → ∞. Our results improve and include those results obtained by previous
authors ([3], [5]).
2000 Mathematics Subject Classification: Primary: 34C11; Secondary: 34B15.
Key words: Differential equations of third order, Boundedness.
The author would like to express sincere thanks to the anonymous referees for
his/her valuable corrections, comments and suggestions.
Boundedness of Solutions of a
Third-Order Nonlinear
Differential Equation
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Proof of Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
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1.
Introduction
We consider the third order non-linear and non-autonomous ordinary differential equation
(1.1)
...
.
.. ..
.
.
..
.
..
x + f (x, x, x)x + g(x, x) + h(x, x, x) = p (t, x, x, x)
or its equivalent system
.
.
x = y, y = z,
(1.2)
.
z = −f (x, y, z)z − g(x, y) − h(x, y, z) + p(t, x, y, z).
It is assumed that f, g, h and p are continuous functions which depend only
on the arguments displayed explicitly, and the dots denote differentiation with
respect to t. The derivatives ∂f (x,y,z)
≡ fx (x, y, z), ∂f (x,y,z)
≡ fz (x, y, z),
∂x
∂z
∂h(x,y,z)
∂h(x,y,z)
∂h(x,y,z)
≡ hx (x, y, z), ∂y ≡ hy (x, y, z), ∂z ≡ hz (x, y, z) and ∂g(x,y)
≡
∂x
∂x
gx (x, y) exist and are continuous. Moreover, the existence and the uniqueness
of the solutions of (1.1) will be assumed.
In recent years, the boundedness properties of solutions of certain non-linear
differential equations of the third order have been investigated by a large number of mathematicians, and they have obtained many results for some special
cases of the equation (1.1) with h ≡ h(x), see ([1], [4], [5], [7] – [10]) and
. ..
references therein. However, in the case h ≡ h(x, x, x), the results about third
order nonlinear differential equations are relatively scarce.
In ([5], [6]), Ezeilo discussed the ultimate boundedness and the existence of
the periodic solutions of equations of the form
...
. ..
.
.
..
x + ψ(x)x + ϕ(x)x + v(x, x, x) = p(t).
Boundedness of Solutions of a
Third-Order Nonlinear
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Later, in a recent paper, Bereketoğlu and Györi [3] considered the differential
equation described as follows
(1.3)
...
. ..
.
.
..
.
..
x + f (x, x)x + g(x, x) + h(x, x, x) = p(t, x, x, x),
and the author established sufficient conditions under which all solutions of the
non-autonomous differential equation (1.3) are bounded and the limits of first
and second order derivatives of the solutions of (1.3) tend to zero as t → ∞. In
this paper, we shall be concerned with the boundedness results of the solutions
of third-order non-linear differential equations of the form (1.1).
The motivation for the present work has come from the papers of Ezeilo ([5],
[6]), Bereketoğlu and Györi [3] and the paper mentioned above. The results
obtained herein are comparable in generality to the works of Bereketoğlu and
Györi [3] and Ezeilo [5], and our results also include and improve the results
in ([3], [5]). It should also be noted that the first result obtained here is proved
. ..
without using the boundedness of h(x, x, x).
Boundedness of Solutions of a
Third-Order Nonlinear
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2.
Main Results
The main results of this paper are the following.
Theorem 2.1. Further to the basic assumptions on the functions f, g, h and
p assume that 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;
(ii)
(iii)
g(x,y)
y
Boundedness of Solutions of a
Third-Order Nonlinear
Differential Equation
≥ b for all x, y 6= 0;
h(x,0,0)
x
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≥ c for all x 6= 0;
(iv) 0 < hx (x, y, 0) ≤ c for all x, y;
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(v) hy (x, y, 0) ≥ 0 for all x, y;
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(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;
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(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, where
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Rt
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) of (1.2) which is determined
by the initial conditions
(2.1)
x(0) = x0 ,
y(0) = y0 ,
z(0) = z0
|x(t)| ≤ D,
|y(t)| ≤ D,
|z(t)| ≤ D
satisfies
for all t ≥ 0.
Theorem 2.2. Let all the conditions of Theorem 2.1 be satisfied; in addition,
we assume that e(t) is bounded for t ≥ 0 , that is, there is a positive constant
M such that |e(t)| ≤ M for all t ≥ 0. Then every solution x(t) of (1.2)
determined by the initial conditions (2.1) satisfies
.
Boundedness of Solutions of a
Third-Order Nonlinear
Differential Equation
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..
x(t) → 0, x(t) → 0 as → ∞.
Remark 1. Theorem 2.1 and Theorem 2.2 contain far less restrictive conditions than those established in Bereketoğlu and Györi [3, Theorem 1, Theorem
2]. Because the result established in [3] can be proved here without the assumptions hy (x, y, 0) ≥ 14 > 0 and ab + al4 > a2 m + c.
Remark 2. It should be noted that the function h satisfying conditions (iii)-(vi)
essentially reduces to something like h(x, y, z) = cx + h0 (x, z). For example,
the function h(x, y, z) := cx + z(x2 + m) satisfies the above conditions.
The proofs of Theorem 2.1 and Theorem 2.2 depend on some certain fundamental properties of a continuously differentiable Lyapunov function V =
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V (x, y, z) defined by:
Z
(2.2) V (x, y, z) = a
0
x
h(ξ, 0, 0)dξ + h(x, 0, 0)y
Z y
Z y
1
+
g(x, η)dη + a
f (x, η, 0)ηdη + ayz + z 2 .
2
0
0
Namely, this function and its time derivative satisfy some fundamental inequalities.
In the subsequent discussion we require the following lemmas.
Lemma 2.3. Subject to the assumptions (i)-(vi) of Theorem 2.1, V (0, 0, 0) = 0
and there is a positive constant K depending only on a, b and c such that
(2.3)
Boundedness of Solutions of a
Third-Order Nonlinear
Differential Equation
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V (x, y, z) ≥ K(x2 + y 2 + z 2 )
for all x, y, z.
Proof. It is clear that V (0, 0, 0) = 0. Since hx (x, y, z) ≤ c, g(x,y)
≥ b (y 6= 0)
y
and f (x, y, z) ≥ a, the function V (x, y, z) can be rearranged as follows (for
y 6= 0):
(2.4) V (x, y, z)
Z x
b
1
1
≥a
h(ξ, 0, 0)dξ + h(x, 0, 0)y + y 2 + a2 y 2 + ayz + z 2
2
2
2
0
1
1
[by + h(x, 0, 0)]2 + [ay + z]2
=
2b
2 Z
Z x
y
1
+
4
h(ξ, 0, 0)
(ab − hξ (ξ, 0, 0))ηdη dξ
2by 2
0
0
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≥
1
1
[by + h(x, 0, 0)]2 + [ay + z]2
2b
2 Z
Z x
y
1
+
4
h(ξ, 0, 0)
(ab − c)ηdη dξ ,
2by 2
0
0
(for y 6= 0) .
Now, it is obvious from (2.4) that the function V (x, y, z) defined in (2.2) is a
positive definite function which has infinite inferior limit and infinitesimal upper
limit. Hence, there is a positive constant K such that
V (x, y, z) ≥ K(x2 + y 2 + z 2 ).
The proof of this lemma is now complete.
Lemma 2.4. Under the assumptions of Theorem 2.1, there are positive constants D1 and D2 depending only on a and m such that, if (x(t), y(t), z(t)) is
any solution of (1.2), then
d
V = V (x(t), y(t), z(t)) ≤ −D1 (y 2 + z 2 ) + D2 (|y| + |z|)e(t).
dt
.
(2.5)
Proof. An easy calculation from (2.2) and (1.2) yields that
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Third-Order Nonlinear
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Z
.
y
(2.6) V = y 2 hx (x, 0, 0) + y
gx (x, η)dη
0
Z y
+ ay
fx (x, η, 0)ηdη + az 2 − f (x, y, z)z 2 − ayg(x, y)
0
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− W1 − W2 − W3 + (ay + z)p(t, x, y, z),
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where
W1 = af (x, y, z)yz − af (x, y, 0)yz,
W2 = −h(x, 0, 0)z + h(x, y, z)z,
W3 = −ayh(x, 0, 0) + ayh(x, y, z).
By (vii), we get
Z
y
y
Z
gx (x, η)dη ≤ 0,
0
y
fx (x, η, 0)ηdη ≤ 0.
y
Boundedness of Solutions of a
Third-Order Nonlinear
Differential Equation
0
It also follows from (vii), for z 6= 0, that
2 f (x, y, z) − f (x, y, 0)
= yz 2 fz (x, y, θ1 z) ≥ 0,
W1 = ayz
z
0 ≤ θ1 ≤ 1,
but W1 = 0 when z = 0. Hence
W1 ≥ 0 for all x, y, z.
Similarly, it is clear that
W2 = yzhy (x, θ2 y, 0) + z 2 hz (x, y, θ3 z) , 0 ≤ θ2 ≤ 1, 0 ≤ θ3 ≤ 1
W3 = ay 2 hy (x, θ4 y, 0) + ayzhz (x, y, θ5 z), 0 ≤ θ4 ≤ 1, 0 ≤ θ5 ≤ 1.
Then, combining the estimates for W1 , W2 , W3 with (2.6) we obtain
.
V ≤ y 2 hx (x, 0, 0) − yzhy (x, θ2 y, 0)
− z 2 hz (x, y, θ3 z) − ay 2 hy (x, θ4 y, 0) − ayzhz (x, y, θ5 z) + az 2
− f (x, y, z)z 2 − ayg(x, y) + (ay + z)p(t, x, y, z).
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The assumption (viii) shows that
.
(2.7) V ≤ −ay 2 hy (x, θ4 y, 0) + y 2 hx (x, 0, 0) − z 2 hz (x, y, θ3 z)
+ az 2 − f (x, y, z)z 2 − ayg(x, y) + (ay + z)p(t, x, y, z).
Also under the assumptions of the theorem we have
−ay 2 hy (x, θ4 y, 0) ≤ 0 for all x, y;
y 2 hx (x, 0, 0) ≤ cy 2 for all x, y;
−z 2 hz (x, y, θ3 z) ≤ −mz 2 for all x, y, z;
−f (x, y, z)z 2 ≤ −az 2 for all x, y, z;
−ayg(x, y) ≤ −aby 2 for all x, y;
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(ay + z)p(t, x, y, z) ≤ |ay + z| |p(t, x, y, z)|
≤ (a |y| + |z|)e(t)
≤ max {a, 1} (|y| + |z|)e(t).
Now, let D1 = min {ab − c, m} and D2 = max {a, 1} .
From the estimates just stated above and (2.7) we obtain
.
2
2
V ≤ −(ab − c)y − mz + max {a, 1} (|y| + |z|)e(t)
≤ −D1 (y 2 + z 2 ) + D2 (|y| + |z|)e(t).
This completes the proof of the lemma.
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Lemma 2.5. Let f be a non-negative function defined on [0, ∞) such that f is
integrable on [0, ∞) and uniformly continuous on [0, ∞). Then
lim f (t) = 0.
t→∞
Proof. See ([2]).
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Third-Order Nonlinear
Differential Equation
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3.
Proof of Theorems
Proof of Theorem 2.1. Consider the Lyapunov function V (x, y, z) defined by
(2.2). By Lemma 2.3, it is obvious that
V (x, y, z) = 0, at x2 + y 2 + z 2 = 0,
V (x, y, z) > 0, if x2 + y 2 + z 2 6= 0,
V (x, y, z) → ∞, as x2 + y 2 + z 2 → ∞.
Next suppose (x(t), y(t), z(t)) is any solution of (1.2) which satisfies the initial
conditions
x(0) = x0 , y(0) = y0 , z(0) = z0 .
Boundedness of Solutions of a
Third-Order Nonlinear
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Set
V (t) ≡ V (x(t), y(t), z(t)).
Then just as in Lemma 2.4,
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.
V ≤ −D1 (y 2 + z 2 ) + D2 (|y| + |z|)e(t),
so that
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V ≤ D2 (|y| + |z|)e(t).
It follows from the obvious inequalities
|y| < 1 + y 2 ,
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and
y2 + z2 ≤
1
V (x, y, z)
K
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that
.
V (t) ≤ D2 (2 + y 2 + z 2 )e(t)
D2
≤
e(t)V (t) + 2D2 e(t).
K
Integrating both sides of this inequality between 0 and t (t ≥ 0) and using
Gronwall-Reid-Bellman inequality, we obtain
Z t
1
V (0) + 2D2
χ(s)e(s)ds ,
V (t) ≤
χ(t)
0
where
Boundedness of Solutions of a
Third-Order Nonlinear
Differential Equation
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χ(t) = exp −
D2
K
Z
t
e(s)ds .
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0
Since χ(t) ≤ 1, and using (ix) we have
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V (t) ≤ (V (0) + 2D2 A) exp
D2
A
K
for t ≥ 0.
As V (0) = V (x0 , y0 , z0 ), this completes the proof.
Proof of Theorem 2.2. The proof of this theorem is similar to that of Bereketoğlu and Györi [3, Theorem 2] and hence it is omitted.
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References
...
..
[1] J. ANDRES, Boundedness results of solutions to the equation x + ax +
.
g(x)x + h(x) = p(t) without the hypothesis h(x)sgnx ≥ 0 for |x| > R,
Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 80(8) (1986), 533–
539.
[2] I. BARBALAT, Systeme d’equations differentielles d’oscillations non lineaires, Rev. Math. Pures Appl., 4 (1959), 267–270.
[3] H. BEREKETOĞLU AND I. GYÖRI,On the boundedness of the solutions
of a third -order nonlinear differential equation, Dynam. Systems Appl.,
6(2) (1997), 263–270.
[4] E.N. CHUKWU, On boundedness of solutions of third order differential
equations, Ann. Mat. Pur. Appl., 104(4) (1975), 123–149.
[5] J.O.C. EZEILO, A generalization of a theorem of Reissig for a certain third
order different equation, Ann. Mat. Pura. Appl., 87(4) (1970), 349–356.
[6] J.O.C. EZEILO, A furthet result on the existence of periodic solutions of
...
. ..
.
. ..
the equation x + ψ(x)x + ϕ(x)x + v(x, x, x) = p(t) with a bound v, Atti.
Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 55 (1978), 51–57.
[7] T. HARA, On the uniform ultimate boundedness of the solutions of certain
third order differential equations, J. Math. Anal. Appl., 80(2) (1981), 533–
544.
[8] M. HARROW, Further results for the solution of certain third order differential equations, J. London Math. Soc., 43 (1968), 587–592.
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Third-Order Nonlinear
Differential Equation
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[9] R. REISSIG, G. SANSONE AND R. CONTI, Nonlinear Differential Equations of Higher Order, Noordhoff, Groningen, 1974.
[10] 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.
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