On the convergence of solutions of certain inhomogeneous fourth-order differential equations
The main purpose of this paper is to give sufficient conditions for the convergence of solutions of a certain class of fourth-order nonlinear differential equations using Lyapunov’s second method. Nonlinear functions involved are not necessarily differentiable, but a certain incrementary ratio for a...
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2010
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irk-123456789-1661582020-02-19T01:26:42Z On the convergence of solutions of certain inhomogeneous fourth-order differential equations Tunc, E. Короткі повідомлення The main purpose of this paper is to give sufficient conditions for the convergence of solutions of a certain class of fourth-order nonlinear differential equations using Lyapunov’s second method. Nonlinear functions involved are not necessarily differentiable, but a certain incrementary ratio for a function h lies in a closed subinterval of the Routh–Hurwitz interval. Головною метою статті є наведення достатніх умов для збіжності розв'язків деякого класу нелінійних диференціальних рівнянь четвертого порядку з використанням другого методу Ляпунова. Розглядувані нелінійні функції необов'язково диференційовні, але функція h задовольняє деяке відношення приростів, що лежать у замкненому підінтервалі інтервалу Рута-Гурвіца. 2010 Article On the convergence of solutions of certain inhomogeneous fourth-order differential equations / E. Tunc // Український математичний журнал. — 2010. — Т. 62, № 5. — С. 714–721. — Бібліогр.: 8 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166158 517.9 en Український математичний журнал Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Короткі повідомлення Короткі повідомлення |
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Короткі повідомлення Короткі повідомлення Tunc, E. On the convergence of solutions of certain inhomogeneous fourth-order differential equations Український математичний журнал |
| description |
The main purpose of this paper is to give sufficient conditions for the convergence of solutions of a certain class of fourth-order nonlinear differential equations using Lyapunov’s second method. Nonlinear functions involved are not necessarily differentiable, but a certain incrementary ratio for a function h lies in a closed subinterval of the Routh–Hurwitz interval. |
| format |
Article |
| author |
Tunc, E. |
| author_facet |
Tunc, E. |
| author_sort |
Tunc, E. |
| title |
On the convergence of solutions of certain inhomogeneous fourth-order differential equations |
| title_short |
On the convergence of solutions of certain inhomogeneous fourth-order differential equations |
| title_full |
On the convergence of solutions of certain inhomogeneous fourth-order differential equations |
| title_fullStr |
On the convergence of solutions of certain inhomogeneous fourth-order differential equations |
| title_full_unstemmed |
On the convergence of solutions of certain inhomogeneous fourth-order differential equations |
| title_sort |
on the convergence of solutions of certain inhomogeneous fourth-order differential equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2010 |
| topic_facet |
Короткі повідомлення |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/166158 |
| citation_txt |
On the convergence of solutions of certain inhomogeneous fourth-order differential equations / E. Tunc // Український математичний журнал. — 2010. — Т. 62, № 5. — С. 714–721. — Бібліогр.: 8 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT tunce ontheconvergenceofsolutionsofcertaininhomogeneousfourthorderdifferentialequations |
| first_indexed |
2025-07-14T20:49:16Z |
| last_indexed |
2025-07-14T20:49:16Z |
| _version_ |
1837656867533225984 |
| fulltext |
UDC 517.9
E. Tunç (Gaziosmanpas.a Univ., Turkey)
ON THE CONVERGENCE OF SOLUTIONS
OF CERTAIN NON-HOMOGENEOUS FOURTH ORDER
DIFFERENTIAL EQUATIONS
ПРО ЗБIЖНIСТЬ РОЗВ’ЯЗКIВ ДЕЯКИХ
НЕОДНОРIДНИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
ЧЕТВЕРТОГО ПОРЯДКУ
The main purpose of this paper is to give sufficient conditions for the convergence of solutions of a certain
class of fourth order nonlinear differential equations with the use of Lyapunov’s second method. Nonlinear
functions involved are not necessarily differentiable, but a function h satisfies a certain incremental ratio that
lie in the closed sub-interval of the Routh – Hurwitz interval.
Головною метою статтi є наведення достатнiх умов для збiжностi розв’язкiв деякого класу нелiнiйних
диференцiальних рiвнянь четвертого порядку з використанням другого методу Ляпунова. Розглядуванi
нелiнiйнi функцiї необов’язково диференцiйовнi, але функцiя h задовольняє деяке вiдношення прирос-
тiв, що лежать у замкненому пiдiнтервалi iнтервалу Рута – Гурвiца.
1. Introduction. The convergence of solutions is very important in the theory and
applications of differential equations. In the recent years, the convergence problem has
been the subject of investigation by a number of authors for various forms and orders
of equations (see, for example, [1 – 8]). In this connection, Afuwape [2] discussed the
convergence of the solutions of the differential equations of the form
x(iv) + a
...
x + b
..
x+ g(
.
x) + h(x) = p(t, x,
.
x,
..
x,
...
x)
with p(t, x,
.
x,
..
x,
...
x) is equals to q(t) + r(t, x,
.
x,
..
x,
...
x). During establishment of the
results, Afuwape [2] assumed that h was not necessarily differentiable but satisfied an
incremental ratio η−1 (h(ξ + η)− h(ξ)) , η 6= 0, which lies in a closed subinterval I0 of
the Routh – Hurwitz interval
(
0,
(ab− c) c
a2
)
, where
I0 ≡
[
∆0,
K (ab− c) c
a2
]
, (1)
∆0 > 0 and K < 1.
In this work, we shall be concerned here with equation of the form
x(iv) + f(
...
x) + b
..
x+ g(
.
x) + h(x) = p(t, x,
.
x,
..
x,
...
x), (2)
where b is a positive constant, the functions f, g, h and p are real-valued and continuous
for values of their respective arguments and dots denote differentiation with respect to
t. Moreover, f(0) = g(0) = h(0) = 0. Using Lyapunov’s second method, our results
assert the existence of convergence of solutions with the functions f, g and h are not
necessarily differentiable.
Definition. Any two solutions x1(t), x2(t) of the equation (2) are said to converge
to each other if
x2(t)− x1(t)→ 0,
.
x2(t)− .
x1(t)→ 0,
..
x2(t)− ..
x1(t)→ 0,
...
x2(t)− ...
x1(t)→ 0 as t→∞.
c© E. TUNÇ , 2010
714 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
ON THE CONVERGENCE OF SOLUTIONS OF CERTAIN NON-HOMOGENEOUS FOURTH . . . 715
2. Main results. The main results of this paper are the following.
Theorem 1. In addition to the fundamental assumptions imposed on f, g, h and
p, we assume that
(i) there are positive constants a, a0 such that
a ≤ f(w2)− f(w1)
w2 − w1
≤ a0, w2 6= w1; (3)
(ii) there are positive constants c, c0 such that
c ≤ g(y2)− g(y1)
y2 − y1
≤ c0, y2 6= y1, (4)
and
abc > c20;
(iii) there are constants ∆0 > 0, K < 1 such that for any ξ, η (η 6= 0), the
incremental ratio for h satisfies
(h(ξ + η)− h(ξ))
η
∈ I0 (5)
with I0 as defined (1);
(iv) there is a continuous function φ(t) such that∣∣p(t, x2, y2, z2, w2)− p(t, x1, y1, z1, w1)
∣∣ ≤
≤ φ(t)
{
|x2 − x1|+ |y2 − y1|+ |z2 − z1|+ |w2 − w1|
}
holds for arbitrary t, x1, y1, z1, w1, x2, y2, z2 and w2.
Then if there exists a constant D1 such that if
t∫
0
φν(τ)dτ ≤ D1t (6)
for some ν, with 1 ≤ ν ≤ 2, then all solutions of (2) converge.
Theorem 2. Assume the conditions of Theorem 1 are satisfied. Let x1(t), x2(t) be
any two solutions of (2). Then for each fixed ν, 1 ≤ ν ≤ 2, there are constants D2, D3
and D4 such that for t2 ≥ t1,
S(t2) ≤ D2S(t1) exp
−D3(t2 − t1) +D4
t2∫
t1
φν(τ)dτ
, (7)
where
S(t) =
{
[x2(t)−x1(t)]2 +[
.
x2(t)− .
x1(t)]2 +[
..
x2(t)− ..
x1(t)]2 +[
...
x2(t)− ...x1(t)]2
}
. (8)
We have the following corollaries when x1(t) = 0 and t1 = 0.
Corollary 1. Suppose that p = 0 in (2) and suppose further that conditions (i), (ii)
and (iii) of Theorem 1 hold, then the trivial solution of (2) is exponentially stable in the
large.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
716 E. TUNÇ
Also, if we put ξ = 0 in (5) with η (η 6= 0) arbitrary, we get:
Corollary 2. If p = 0 and hypotheses (i), (ii) and (iii) of Theorem 1 hold for
arbitrary η (η 6= 0), and ξ = 0, then there exists a constant D5 > 0 such that every
solution x(t) of (2) satisfies
|x(t)| ≤ D5,
∣∣ .x(t)
∣∣ ≤ D5,
∣∣..x(t)
∣∣ ≤ D5,
∣∣...x(t)
∣∣ ≤ D5.
Proof of Theorem 2. It is convenient here to consider (2) as the equivalent system
.
x = y,
.
y = z,
.
z = w,
.
w = −f(w)− bz − g(y)− h(x) + p(t, x, y, z, w).
(9)
Let (xi(t), yi(t), zi(t), wi(t)) , i = 1, 2, be any two solutions of (9) such that inequali-
ties (3), (4) and
∆0 ≤
h(x2)− h(x1)
x2 − x1
≤ K(ab− c)c
a2
are satisfied. The basic tool in the proofs of the convergence theorems will be the function
2V = c2ε(1− ε)x2 + ac
[
(D − 1) + ε
]
y2 + 2c
[
ε+ (D − 1)
]
yz+
+εDw2 + b(D − 1)z2 + 2εaDzw + εa2Dz2+
+
[
(1− ε)D − 1
]
[az + w]2 +
[
c(1− ε)x+ by + az + w
]2
, (10)
where 0 < ε < 1, ab − c > δ > 0, δ = abε and D − 1 =
δ + cε
ab− c− δ
. Indeed we can
rearrange the terms in (10) to obtain
2V = 2V1 + 2V2 + 2V3,
where
2V1 = ac[(D − 1) + ε]y2 + 2c[ε+ (D − 1)]yz + b(D − 1)z2,
2V2 = εa2Dz2 + 2εaDzw + εDw2
and
2V3 = c2ε(1− ε)x2 + [(1− ε)D − 1][az + w]2 + [c(1− ε)x+ by + az + w]2.
We note that V3 obviously positive definite. Also Vi, i = 1, 2, regarded as quadratic
forms in y and z, z and w respectively is positive and non-negative. Let us recall that a
real 2× 2 matrix (
a1 a2
a3 a4
)
is positive definite if and only if it is symmetric, and the elements a1, a4 and a1a4−a2a3
are non-negative. Thus we can rearrange the terms in V1 as
(y, z)
(
ac[(D − 1) + ε] c[ε+ (D − 1)]
c[ε+ (D − 1)] b(D − 1)
)(
y
z
)
,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
ON THE CONVERGENCE OF SOLUTIONS OF CERTAIN NON-HOMOGENEOUS FOURTH . . . 717
from which we have as a condition for the positive definite.
Similarly, V2 is non-negative. Hence V is positive definite. We can therefore find a
constant D6 > 0, such that
D6
(
x2 + y2 + z2 + w2
)
≤ V. (11)
Furthermore, by using Schwartz inequality |y| |z| ≤ 1
2
(
y2 + z2
)
, it can be easily obtai-
ned that
V ≤ D7
(
x2 + y2 + z2 + w2
)
, (12)
where D7 is a positive constant.
Using inequalities (11) and (12), we have
D6
(
x2 + y2 + z2 + w2
)
≤ V ≤ D7
(
x2 + y2 + z2 + w2
)
.
The following result can be easily verified for W ≡ V.
Lemma 1. Let the function W (t) = W
(
x2 − x1, y2 − y1, z2 − z1, w2 − w1
)
be
defined by
2W = c2ε(1− ε) (x2 − x1)2 + ac[(D − 1) + ε] (y2 − y1)2 +
+2c[ε+ (D − 1)] (y2 − y1) (z2 − z1) + εD (w2 − w1)2 +
+b(D − 1) (z2 − z1)2 + 2εaD (z2 − z1) (w2 − w1) + εa2D (z2 − z1)2 +
+[(1− ε)D − 1][a (z2 − z1) + (w2 − w1)]2+
+[c(1− ε) (x2 − x1) + b (y2 − y1) + a (z2 − z1) + (w2 − w1)]2,
where 0 < ε < 1 and W (0, 0, 0, 0) = 0, then there exist finite constants D6 > 0, D7 > 0
such that
D6
{
(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 + (w2 − w1)2
}
≤W ≤
≤ D7
{
(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 + (w2 − w1)2
}
. (13)
If we denote the function W (t) by W
(
x2(t) − x1(t), y2(t) − y1(t), z2(t) − z1(t),
w2(t)−w1(t)
)
, and using the fact that the solutions (xi, yi, zi, wi) , i = 1, 2, satisfy the
system (9), then S(t) as defined in (8) becomes
S(t) =
{
[x2(t)− x1(t)]2 + [y2(t)− y1(t)]2 + [z2(t)− z1(t)]2 + [w2(t)− w1(t)]2
}
.
Next we prove a result on the derivative of W (t) with respect to t.
Lemma 2. Let the hypotheses (i), (ii) and (iii) of Theorem 1 hold, then there exist
positive finite constants D8 and D9 such that
dW
dt
≤ −2D8S +D9S
1/2 |θ| , (14)
where θ = p(t, x2, y2, z2, w2)− p(t, x1, y1, z1, w1).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
718 E. TUNÇ
Proof. Using the system (9), a direct computation of
dW
dt
gives after simplification
dW
dt
= −W1 +W2, (15)
where
W1 = c(1− ε)H(x2, x1)(x2 − x1)2 + bcε(y2 − y1)2+
+δD(z2 − z1)2 +D(F − a)(w2 − w1)2+
+
(
G(y2, y1)− c
)[
c(1− ε)(x2 − x1) + b(y2 − y1)+
+aD(z2 − z1) +D(w2 − w1)
]
(y2 − y1) +H(x2, x1)
[
b(y2 − y1)+
+aD(z2 − z1) +D(w2 − w1)
]
(x2 − x1)+
+(F (w2, w1)− a)
[
c(1− ε)(x2 − x1) + b(y2 − y1) + aD(z2 − z1)
]
(w2 − w1),
W2 = θ(t)
[
c(1− ε)(x2 − x1) + b(y2 − y1) + aD(z2 − z1) +D(w2 − w1)
]
with
F (w2, w1) =
f(w2)− f(w1)
w2 − w1
, w2 6= w1,
G(y2, y1) =
g(y2)− g(y1)
y2 − y1
, y2 6= y1,
H(x2, x1) =
h(x2)− h(x1)
x2 − x1
, x2 6= x1.
Let χ1 = G(y2, y1) − c ≥ 0 for y2 6= y1 and χ2 = F (w2, w1) − a ≥ 0 for w2 6= w1.
Furthermore let H (x2, x1) be denote simply by H, and define
6∑
i=1
αi = 1,
6∑
i=1
βi = 1,
4∑
i=1
γi = 1,
6∑
i=1
ξi = 1,
where αi > 0, βi > 0, γi > 0 and ξi > 0. Then W1 re-arranged as
W1 = W11 +W12 +W13 +W14 +W15 +W16 +W17 +W18 +W19 +W21,
where
W11 =
{
α1c(1− ε)H (x2 − x1)2 + b (β1cε+ χ1) (y2 − y1)2 +
+γ1δD (z2 − z1)2 + ξ1Dχ2 (w2 − w1)2
}
,
W12 =
{
β2bcε (y2 − y1)2 + χ1c(1− ε) (x2 − x1) (y2 − y1) +
+α2c(1− ε)H (x2 − x1)2
}
,
W13 =
{
β3bcε (y2 − y1)2 + χ1aD (y2 − y1) (z2 − z1) + γ2δD (z2 − z1)2
}
,
W14 =
{
β4bcε (y2 − y1)2 + χ1D (y2 − y1) (w2 − w1) + ξ2Dχ2 (w2 − w1)2
}
,
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
ON THE CONVERGENCE OF SOLUTIONS OF CERTAIN NON-HOMOGENEOUS FOURTH . . . 719
W15 =
{
α3c(1− ε)H (x2 − x1)2 + bH (x2 − x1) (y2 − y1) + β5bcε (y2 − y1)2
}
,
W16 =
{
α4c(1− ε)H (x2 − x1)2 + aDH (x2 − x1) (z2 − z1) + γ3δD (z2 − z1)2
}
,
W17 =
{
α5c(1− ε)H (x2 − x1)2 +
+DH (x2 − x1) (w2 − w1) + ξ3Dχ2 (w2 − w1)2
}
,
W18 =
{
α6c(1− ε)H (x2 − x1)2 + χ2c(1− ε) (x2 − x1) (w2 − w1) +
+ξ4Dχ2 (w2 − w1)2
}
,
W19 =
{
β6bcε (y2 − y1)2 + χ2b (y2 − y1) (w2 − w1) + ξ5Dχ2 (w2 − w1)2
}
and
W21 =
{
γ4δD (z2 − z1)2 + χ2aD (z2 − z1) (w2 − w1) + ξ6Dχ2 (w2 − w1)2
}
.
Since each W1i, i = 1, . . . , 9, and W21 are quadratic in their respective variables, then
by using the fact that any quadratic of the form Ar2 + Brs + Cs2 is non-negative if
4AC −B2 ≥ 0, it follows that
W12 ≥ 0 if χ2
1 ≤
4α2β2bε∆0
1− ε
,
W13 ≥ 0 if χ2
1 ≤
4γ2β3bcεδ
a2D
,
W14 ≥ 0 if χ2
1 ≤
4β4ξ2bcεχ2
D
,
W15 ≥ 0 if H ≤ 4α3β5c
2ε(1− ε)
b
,
W16 ≥ 0 if H ≤ 4α4γ3c(1− ε)δ
a2D
,
W17 ≥ 0 if H ≤ 4α5ξ3c(1− ε)χ2
D
,
W18 ≥ 0 if χ2 ≤
4α6ξ4D∆0
c(1− ε)
,
W19 ≥ 0 if χ2 ≤
4β6ξ5cεD
b
,
and
W21 ≥ 0 if χ2 ≤
4γ4ξ6δ
a2
.
Thus W1 ≥W11 provided that above inequalities are satisfied in addition to
0 ≤ χ2
1 ≤ 4 min
{
α2β2bε∆0
1− ε
,
γ2β3bcεδ
a2D
,
β4ξ2bcεχ2
D
}
,
0 ≤ χ2 ≤ 4 min
{
α6ξ4D∆0
c(1− ε)
,
β6ξ5cεD
b
,
γ4ξ6δ
a2
}
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
720 E. TUNÇ
and H lying in
I0 ≡
[
∆0,
K (ab− c) c
a2
]
,
where I0 is a closed sub-interval of the Routh – Hurwitz interval
(
0,
(ab− c) c
a2
)
, and
K =
(
4
ab− c
)
min
{
α3β5a
2cε(1− ε)
b
,
α4γ3(1− ε)δ
D
,
α5γ3a
2(1− ε)χ2
D
}
< 1.
On choosing 2D8 = min {c(1− ε)∆0, bcε, δD,Dχ2} , we have
W1 ≥W11 ≥ 2D8S (16)
and if D9 = 2 max {c(1− ε), b, aD,D} then
W2 ≤ D9S
1/2 |θ| . (17)
Combining (16) and (17) in (15), inequality (14) is obtained. At last the conclusion to
the proof of Theorem 2 will now be given. For this purpose, let ν be any constant in the
range 1 ≤ ν ≤ 2 and set σ = 1− ν
2
, so that 0 ≤ σ ≤ 1
2
. We re-write (14) in the form
dW
dt
+D8S ≤ D9S
σW ∗, (18)
where
W ∗ = S1/2−σ
(
|θ| −D10S
1/2
)
(19)
with D10 =
D8
D9
. If |θ| < D10S
1/2, then W ∗ < 0. On the other hand, if |θ| ≥ D10S
1/2,
then the definition of W ∗ in (19) gives at least
W ∗ ≤ S(1/2−σ) |θ|
and also S1/2 ≤ |θ|
D10
. Thus
S(1−2σ)/2 ≤
[
|θ|
D10
](1−2σ)
and from this together with W ∗ follows
W ∗ ≤ D11 |θ|2(1−σ)
,
where D11 = D
(σ−1)
10 . On using the estimate on W ∗ in inequality (18), we obtain
dW
dt
+D8S ≤ D9D11S
σ |θ|2(1−σ) ≤ D12S
σφ2(1−σ)S(1−σ),
where D12 > 2D9D11, which follows from∣∣p(t, x2, y2, z2, w2)− p(t, x1, y1, z1, w1)
∣∣ ≤
≤ φ(t)
{
|x2 − x1|+ |y2 − y1|+ |z2 − z1|+ |w2 − w1|
}
.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 5
In view of the fact that ν = 2(1− σ), we obtain
dW
dt
≤ −D8S +D12φ
νS,
and on using inequalities (13), we have
dW
dt
+
(
D13 −D14φ
ν(t)
)
W ≤ 0 (20)
for some constants D13 and D14. On integrating the estimate (20) from t1 to t2, t2 ≥ t1,
we have
W (t2) ≤W (t1) exp
−D13(t2 − t1) +D14
t2∫
t1
φν(τ)dτ
.
Again, using Lemma 1, we obtain (7), with D2 = D7D
−1
6 , D3 = D13 and D4 = D14.
Theorem 2 is proved.
Proof of Theorem 1. The proof follows from the estimate (7) and the condition (6)
on φ(t). On Choosing D1 = D3D
−1
4 in (6). Then, as t = (t2 − t1) → ∞, S(t) → 0,
which proves that as t→∞,
x2(t)− x1(t)→ 0,
.
x2(t)− .
x1(t)→ 0,
..
x2(t)− ..
x1(t)→ 0,
...
x2(t)− ...
x1(t)→ 0.
The theorem is proved.
Remark. If φ(t) ≡ D15 (a constant), our results will still remain valid.
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Received 05.08.09
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