Instability of solutions for certain nonlinear vector differential equations of fourth order
The main purpose of this paper is to give a result with an explanatory example which deals directly with the instability of the trivial solution of a certain nonlinear vector differential equation of fourth order. The result which will be established here improves and includes a well-known instabi...
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| Cite this: | Instability of solutions for certain nonlinear vector differential equations of fourth order / C. Tunç // Нелінійні коливання. — 2009. — Т. 12, № 1. — С. 120-129. — Бібліогр.: 15 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1783872025-02-09T20:47:06Z Instability of solutions for certain nonlinear vector differential equations of fourth order Нестійкість розв’язків деяких нелінійних векторних рівнянь четвертого порядку Неустойчивость решений некоторых нелинейных векторных уравнений четвёртого порядка Tunç, C. The main purpose of this paper is to give a result with an explanatory example which deals directly with the instability of the trivial solution of a certain nonlinear vector differential equation of fourth order. The result which will be established here improves and includes a well-known instability result in the literature established for a scalar nonlinear differential equation of fourth order. Отримано результат iз пояснювальним прикладом про нестiйкiсть тривiального розв’язку деякого нелiнiйного векторного диференцiального рiвняння четвертого порядку, що покращує вiдомий результат про нестiйкiсть для скалярного диференцiального рiвняння четвертого порядку. 2009 Article Instability of solutions for certain nonlinear vector differential equations of fourth order / C. Tunç // Нелінійні коливання. — 2009. — Т. 12, № 1. — С. 120-129. — Бібліогр.: 15 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/178387 517.9 en Нелінійні коливання application/pdf Інститут математики НАН України |
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The main purpose of this paper is to give a result with an explanatory example which deals directly with
the instability of the trivial solution of a certain nonlinear vector differential equation of fourth order. The
result which will be established here improves and includes a well-known instability result in the literature
established for a scalar nonlinear differential equation of fourth order. |
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Article |
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Tunç, C. |
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Tunç, C. Instability of solutions for certain nonlinear vector differential equations of fourth order Нелінійні коливання |
| author_facet |
Tunç, C. |
| author_sort |
Tunç, C. |
| title |
Instability of solutions for certain nonlinear vector differential equations of fourth order |
| title_short |
Instability of solutions for certain nonlinear vector differential equations of fourth order |
| title_full |
Instability of solutions for certain nonlinear vector differential equations of fourth order |
| title_fullStr |
Instability of solutions for certain nonlinear vector differential equations of fourth order |
| title_full_unstemmed |
Instability of solutions for certain nonlinear vector differential equations of fourth order |
| title_sort |
instability of solutions for certain nonlinear vector differential equations of fourth order |
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Інститут математики НАН України |
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2009 |
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https://nasplib.isofts.kiev.ua/handle/123456789/178387 |
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Instability of solutions for certain nonlinear vector differential equations of fourth order / C. Tunç // Нелінійні коливання. — 2009. — Т. 12, № 1. — С. 120-129. — Бібліогр.: 15 назв. — англ. |
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Нелінійні коливання |
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AT tuncc instabilityofsolutionsforcertainnonlinearvectordifferentialequationsoffourthorder AT tuncc nestíikístʹrozvâzkívdeâkihnelíníinihvektornihrívnânʹčetvertogoporâdku AT tuncc neustoičivostʹrešeniinekotoryhnelineinyhvektornyhuravneniičetvertogoporâdka |
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2025-11-30T15:37:05Z |
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| fulltext |
UDC 517 . 9
INSTABILITY OF SOLUTIONS FOR CERTAIN NONLINEAR VECTOR
DIFFERENTIAL EQUATIONS OF FOURTH ORDER
НЕСТIЙКIСТЬ РОЗВ’ЯЗКIВ ДЕЯКИХ НЕЛIНIЙНИХ
ВЕКТОРНИХ РIВНЯНЬ ЧЕТВЕРТОГО ПОРЯДКУ
C. Tunç
Yüzüncü Yil Univ.
65080, Van – Turkey
e-mail: cemtunc@yahoo.com
The main purpose of this paper is to give a result with an explanatory example which deals directly with
the instability of the trivial solution of a certain nonlinear vector differential equation of fourth order. The
result which will be established here improves and includes a well-known instability result in the literature
established for a scalar nonlinear differential equation of fourth order.
Отримано результат iз пояснювальним прикладом про нестiйкiсть тривiального розв’язку де-
якого нелiнiйного векторного диференцiального рiвняння четвертого порядку, що покращує
вiдомий результат про нестiйкiсть для скалярного диференцiального рiвняння четвертого по-
рядку.
1. Introduction. As is well-known, the area of differential equations is an old but durable subject
that remains alive and useful to a wide variety of engineers, scientists, and mathematicians.
The principles of differential equations are largely about the qualitative theory of ordinary
differential equations. Qualitative theory refers to the study of the behavior of solutions, for
example the investigation of stability, instability, boundedness of solutions and etc., without
determining explicit formulas for the solutions. It originated with Poincare at the beginning of
the twentieth century and has been the most important theme of ordinary differential equati-
ons in that century. During this period, very little attention is paid to techniques for finding
analytic formulas for solutions. That is to say, there are actually very few equations (beyond
linear equations with constant-coefficient and even there are difficulties if the order of the
equation or system is high) for which we can do this. Therefore, it is very important to interpret
qualitative behavior of solutions when there is no analytical expression for the solutions of any
differential equation under investigation. So far, in the relevant literature, some methods have
been improved to obtain information about qualitative behaviors of solutions of differential
equations without solving them. Among them the Lyapunov’s [1] second (or direct) method is
most well-known. Namely, Lyapunov’s second (or direct) method is one of the basic tools for
investigating the qualitative behavior of solutions; stability, instability, boundedness, existence
of periodic solutions and etc. of nonlinear differential equations of higher order. During the
past fifty year, by using Lyapunov’s [1] second (or direct) method, throughout hundreds of the
papers, qualitative behaviors of solutions, stability, instability, boundedness, existence of peri-
odic solutions and etc. for various ordinary scalar or vector nonlinear differential equations of
second-, third-, fourth-, fifth-, sixth-, seventh- and eighth-order have been investigated extensi-
vely in the relevant literature. However, we will not discuss details of these works here. Now,
c© C. Tunç, 2009
120 ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 1
INSTABILITY OF SOLUTIONS FOR CERTAIN NONLINEAR VECTOR DIFFERENTIAL . . . 121
we consider the constant-coefficient fourth order scalar differential equation
x(4) + a1
...
x + a2ẍ+ a3ẋ+ a4x = 0. (1)
As we know from the qualitative theory of ordinary differential equations that the trivial soluti-
on of the equation (1) is unstable if the associated (auxiliary) equation
ψ(r) = r4 + a1r
3 + a2r
2 + a3r + a4 = 0 (2)
has at the least one root with a positive real part. Naturally, the existence of such a root depends
on (though not always on all of) the coefficients a1, a2, a3 and a4 of equation (2). The correspon-
ding auxiliary equation (2) of equation (1) has no purely imaginary roots r = iλ (λ 6= 0) if
X1 ≡ λ4 − a2λ
2 + a4 6= 0
or if
X2 ≡ a1λ
2 − a3 6= 0.
Note that X1 may be rearranged in the form
X1 =
(
λ2 −
1
2
a2
)2
+ a4 −
1
2
a2
2
so that X1 6= 0 holds if, for example, a4 >
1
4
a2
2. It is also easy to see that X2 6= 0 will hold if
a1a3 < 0, a4 6= 0,
where the condition on a4 here being necessary in order to ensure that λ 6= 0. It should also be
noted that it was shown in Ezeilo [2] that the existence of the relation a4 >
1
4
a2
2 between the
coefficients a2 and a4 of the equation (1) is a sufficient condition that guarantees that the trivial
solution x = 0 of the equation (1) is unstable. The above preamble is merely intended to give
a background to the assumptions which play a dominant role in our treatment here and to give
some indication of how these assumptions stand with respect to the Routh – Hurwitz criteria.
Now, the objective of this paper is to study the instability of the trivial solution X = 0 of
nonlinear vector differential equations of the form
X(4) + Ψ(Ẍ)
...
X +G
(
X, Ẋ, Ẍ,
...
X
)
Ẍ + Θ(Ẋ) + F (X) = 0, (3)
in which X ∈ R
n; Ψ and G are continuous (n × n)-symmetric matrix-valued functions for the
variables indicated explicitly; Θ : R
n → R
n, F : R
n → R
n and Θ(0) = F (0) = 0, and it is
also supposed that the functions Θ and F are continuous for allX, Ẋ ∈ R
n, respectively. In this
case, the equation (3) can be rewritten as
Ẋ = Y, Ẏ = Z, Ż = W,
(4)
Ẇ = −Ψ(Z)W −G(X,Y,Z,W )Z − Θ(Y ) − F (X),
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 1
122 C. TUNÇ
which was obtained by taking Ẋ = Y, Ẍ = Z,
...
X = W in the equation (3). Now, let J(Ψ(Z)Z|Z),
JΘ(Y ) and JF (X), respectively, denote the linear operators from the matrix Ψ(Z) and vectors
Θ(Y ) and F (X) to the matrices
J(Ψ(Z)Z|Z) =
(
∂
∂zj
n
∑
k=1
ψikzk
)
= Ψ(Z) +
(
n
∑
k=1
∂ψik
∂zj
zk
)
,
JΘ(Y ) =
(
∂θi
∂yj
)
, JF (X) =
(
∂fi
∂xj
)
, i, j = 1, 2, . . . , n,
where (x1, x2, . . . , xn), (y1, y2, . . . , yn), (z1, z2, . . . , zn), (ψik), (θ1, θ2, . . . , θn) and (f1, f2, . . . , fn)
are components ofX, Y, Z,Ψ,Θ and F, respectively. Furthermore, it is also assumed throughout
this paper that F (X) and Θ(Ẋ) are gradient vector fields and the matrices JF (X), JΘ(Y ) and
J(Ψ(Z)Z|Z) exist and are symmetric and continuous.
It is also worth mentioning that although a substantial amount of results have been publi-
shed with regard to stability of certain nonlinear differential equations of fourth order, comparati-
vely only a few results have been published which deal directly with the instability of solutions
of certain differential equations of fourth order. In particular, one can refer to the book of
Reissig et al. [3] as a survey for some papers published on the stability of solutions of certain
differential equations of fourth order and the papers of Ezeilo [2, 4, 5], Lu and Liao [6], Sadek
[7], Skrapek [8], Tiryaki [9], Tunç [10 – 12], C. Tunç and E. Tunç [13] and references cited there
for some investigations performed on the instability of solutions of certain fourth order linear
and nonlinear differential equations. Now, to the best of our knowledge, so far in the relevant
literature, the papers performed on the instability of solutions scalar differential equations of
fourth order can be summarized as follows. First, in 1978, Ezeilo [2] has carried out a work on
the instability of the trivial solution of the fourth-order scalar differential equation
x(4) + a1
...
x + a2ẍ+ a3ẋ+ f(x) = 0.
In 1979, Ezeilo [4] also extended his aforementioned instability result in [2] to a much more
general scalar differential equation given by
x(4) + a1
...
x + h(x, ẋ, ẍ,
...
x) ẍ+ g(x) ẋ+ f(x) = 0.
Later, in 1988, Tiryaki [9] interested in a fourth-order nonlinear scalar differential equation of
the form
x(4) + ψ(ẍ)
...
x + φ(ẋ)ẍ+ θ(ẋ) + f(x) = 0. (5)
Under specified conditions imposed on the functions ψ, φ, θ and f, he established sufficient
conditions that guarantee the instability of the trivial solution of the equation (5). After that,
in 1980, Skrapek [8] considered fourth-order scalar differential equations of the form
x(4) + f(x, ẋ, ẍ,
...
x) = 0.
He studied hypotheses on the function f for which the zero solution of the above fourth-order
differential equation is unstable. Similarly, in 1993 by help of Lyapunov’s second method, Lu
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 1
INSTABILITY OF SOLUTIONS FOR CERTAIN NONLINEAR VECTOR DIFFERENTIAL . . . 123
and Liao [6] gave some sufficient conditions for the instability of solutions of a general fourth-
order linear scalar differential equation with variable coefficient, at least one of the characteri-
stic roots of which has positive real part. Finally, in 2000, Ezeilo [5] discussed a similar problem
for the scalar differential equations
x(4) + ψ(ẍ)
...
x + g(x, ẋ, ẍ,
...
x) ẍ+ θ(ẋ) + f(x) = 0, (6)
x(4) + a1
...
x + g(x, ẋ, ẍ,
...
x) ẍ+ h(x)ẋ+ f(x, ẋ, ẍ,
...
x) = 0, (7)
and
x(4) + p(
...
x, ẍ) + q(x, ẋ, ẍ,
...
x) ẍ+ a3ẋ+ a4x = 0, (8)
respectively, where a1, a3 and a4 are constants.
Now, with respect to our observations in the literature, so far instability of solutions of nonli-
near vector differential equations of the form (3) was not investigated in the relevant literature,
except for its scalar version, differential equation (6). It should also be clarified that nearly
throughout all of the papers just stated above it was taken into consideration Krasovskii’s cri-
terion (see Krasovskii [14]) as a fundamental criterion through the proofs, and the Lyapunov’s
[1] second (or direct) method has been used as a basic tool to prove the results established there.
In this paper, in view of Krasovskii’s criterion [14], we use the same method to verify our main
result. The motivation for the present paper has been inspired basically by the paper of Ezeilo
[5] and that just mentioned above. It is worth mentioning that all the papers mentioned above
do not contain an example on the topic. However, our main result includes an explanatory
example.
Throughout this paper, the symbol 〈X,Y 〉 is used to denote the usual scalar product in R
n
for given any X, Y in R
n, that is, 〈X,Y 〉 =
n
∑
i=1
xiyi, thus ‖X‖2 = 〈X,X〉. It is also well-known
that a real symmetric matrix A = (aij), i, j = 1, 2, . . . , n, is said to be positive definite if and
only if the quadratic form XT AX is positive definite, where X ∈ R
n and XT denotes the
transpose of X (see Bellman [15]).
2. Main result. Actually, we prove the following theorem.
Theorem. Assume that the functionG and the functions Ψ,Θ and F in equation (3), respecti-
vely, are continuous and continuously differentiable, and there exist constants a1 and a2 such that
the following conditions are satisfied:
Ψ, G, JΘ and JF are symmetric such that
∂ψij
∂zk
=
∂ψik
∂zj
, i, j, k = 1, 2, . . . , n;
F (X) 6= 0 if X 6= 0, λi(Ψ(Z)) ≥ a1 > 0, λi(G(X,Y,Z,W )) ≤ a2
and
λi(JF (X)) >
1
4
a2
2, i = 1, 2, . . . , n,
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 1
124 C. TUNÇ
for allX, Y, Z,W ∈ R
n,where λi(JF (X)) and λi(G(X,Y,Z,W )), i = 1, 2 . . . , n, are eigenvalues
of G(X,Y,Z,W ) and JF (X).
Then the trivial solution X = 0 of equation (3) is unstable.
Remark 1. In order to prove the above theorem, it will suffice, see Krasovskii [14], to show
that there exists a continuous Lyapunov function V = V (X,Y,Z,W ) which has the following
Krasovskii properties:
(K1) In every neighborhood of (0, 0, 0, 0) there exists a point (ξ, η, ζ, µ) such that
V (ξ, η, ζ, µ) > 0;
(K2) the time derivative V̇ =
d
dt
V (X,Y,Z,W ) along solution paths of the system (4) is
positive semi-definite;
(K3) the only solution (X,Y,Z,W ) = (X(t), Y (t), Z(t),W (t)) of the system (4) which sati-
sfies V̇ = 0 (t ≥ 0) is the trivial solution (0, 0, 0, 0).
Remark 2. It should be clarified that there is no restriction on the eigenvalues of the matrix
JΘ(Y ) obtained from Θ(Y ).
Remark 3. Our result includes and improves the result of Ezeilo [5], which was established
on the instability of zero solution of equation (6).
Proof. For the proof of the theorem, we define a Lyapunov function V = V (X,Y,Z,W )
given by
V = a2〈Y,Z〉 + 〈W,Z〉 + 〈F (X), Y 〉 +
1
∫
0
〈Θ(σ Y ), Y 〉 dσ +
1
∫
0
〈Ψ(σ Z)Z,Z〉 dσ. (9)
Observe that V (0, 0, 0, 0) = 0. Indeed, in view of the assumptions of the theorem, we also have
that
V (0, 0, ε, ε) = 〈ε, ε〉 +
1
∫
0
〈Ψ(σε)ε, ε〉 dσ ≥ 〈ε, ε〉 +
1
∫
0
〈a1ε, ε〉 dσ =
= 〈ε, ε〉
1
2
a1〈ε, ε〉 = ‖ε‖2 +
1
2
a1 ‖ε‖
2 > 0
for all arbitrary ε 6= 0, ε ∈ R
n, which verifies the property (K1). Next let (X,Y,Z,W ) =
= (X(t), Y (t), Z(t),W (t)) be an arbitrary solution of the system (4). In view of the function (9)
and the system (4), a straightforward calculation gives that
V̇ =
d
dt
V (X,Y,Z,W ) = a2〈Z,Z〉 + 〈W,W 〉 + a2〈Y,W 〉 − 〈G(X,Y,Z,W )Z,Z〉+
+ 〈JF (X)Y, Y 〉 − 〈Ψ(Z)W,Z〉 − 〈Θ(Y ), Z〉+
+
d
dt
1
∫
0
〈Θ(σ Y ), Y 〉 dσ +
d
dt
1
∫
0
〈Ψ(σ Z)Z,Z〉 dσ. (10)
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 1
INSTABILITY OF SOLUTIONS FOR CERTAIN NONLINEAR VECTOR DIFFERENTIAL . . . 125
Now, recall that
d
dt
1
∫
0
〈Θ(σ Y ), Y 〉 dσ =
1
∫
0
σ〈JΘ(σ Y )Z, Y 〉 dσ +
1
∫
0
〈Θ(σ Y ), Z〉 dσ =
=
1
∫
0
σ
∂
∂σ
〈Θ(σ Y ), Z〉 dσ +
1
∫
0
〈Θ(σ Y ), Z〉 dσ =
= σ 〈Θ(σ Y ), Z〉|10 = 〈Θ(Y ), Z〉 (11)
and
d
dt
1
∫
0
〈Ψ(σ Z)Z,Z〉 dσ =
1
∫
0
〈Ψ(σ Z)Z,W 〉 dσ +
1
∫
0
〈σ J(Ψ(σ Z)Z|σ Z)W,Z〉 dσ =
=
1
∫
0
〈Ψ(σ Z)Z,W 〉 dσ +
1
∫
0
σ〈J(Ψ(σ Z)Z|σ Z)Z,W 〉 dσ =
=
1
∫
0
〈Ψ(σ Z)Z,W 〉 dσ +
1
∫
0
σ
∂
∂σ
〈Ψ(σ Z)Z,W 〉 dσ =
= σ〈Ψ(σ Z)Z,W 〉|10 = 〈Ψ(Z)Z,W 〉. (12)
Collecting the estimates (11) and (12) into (10), we get
V̇ = a2〈Z,Z〉 + 〈W,W 〉 + a2〈Y,W 〉 − 〈G(X,Y,Z,W )Z,Z〉 + 〈JF (X)Y, Y 〉. (13)
In view of the assumption
λi(G(X,Y,Z,W )) ≤ a2
of the theorem, it follows for the terms
a2〈Z,Z〉 − 〈G(X,Y,Z,W )Z,Z〉,
which are contained in (13), that
a2〈Z,Z〉 − 〈G(X,Y,Z,W )Z,Z〉 ≥ a2〈Z,Z〉 − a2〈Z,Z〉 = 0. (14)
Hence, by using (14), we have
V̇ ≥ 〈W,W 〉 + 〈Y, a2W 〉 + 〈JF (X)Y, Y 〉 ≥
≥ 〈W,W 〉 − |a2| ‖Y ‖ ‖W‖ + 〈JF (X)Y, Y 〉. (15)
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 1
126 C. TUNÇ
In view of (15), the assumption λi(JF (X)) >
1
4
a2
2 of the theorem, we conclude that
V̇ ≥
∥
∥
∥
∥
W +
1
2
a2Y
∥
∥
∥
∥
2
+ 〈JF (X)Y, Y 〉 −
1
4
a2
2〈Y, Y 〉 ≥
≥ 〈JF (X)Y, Y 〉 −
1
4
a2
2〈Y, Y 〉 ≥ 0.
Now, it is clear that
V̇ (0, 0, 0,W ) = 〈W,W 〉 = ‖W‖2 ≥ 0.
Hence, the above discussion shows that the function V̇ is positive semi-definite. This case veri-
fies the property (K2). Next, let (X(t), Y (t), Z(t),W (t)) be an arbitrary solution of the system
(3). We observe from the previous estimate of V̇ in (13) that V̇ = 0, t ≥ 0, necessarily implies
that Y (t) = 0 for all t ≥ 0 and, therefore, also that X = ξ (a constant vector). In its turn this
implies that
X = ξ, Y = Ẋ, Z = Ẏ = 0, W = Ż = 0 for all t ≥ 0.
The substitution of the estimates X = ξ, Y = Ẋ, Z = Ẏ = 0, W = Ż = 0 for all t ≥ 0 into
the system (4) leads to the result F (ξ) = 0 which, by the assumption of the theorem, implies
(only) that ξ = 0. Thus V̇ = 0, t ≥ 0, implies that
X = Y = Z = W = 0 for all t ≥ 0.
Hence the property of (K3) holds for the function V.
The theorem is thereby established.
Example. As a special case of the system (4), let us choose, for the case n = 4, Ψ, G, Θ and
F that appeared in the system (4) to be the following:
Ψ(Z) =
1 + z2
1 0 0 0
0 1 + z2
2 0 0
0 0 1 + z2
3 0
0 0 0 1 + z2
4
,
G(X,Y,Z,W ) =
1 − x2
1 − y2
1 − z2
1 − w2
1 0 0 0
0 2 0 0
0 0 2 −
2
1 + x2
3 + z4
3
0
0 0 0
1
2
−
3
2 + y2
4 + z2
4 + w4
4
,
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 1
INSTABILITY OF SOLUTIONS FOR CERTAIN NONLINEAR VECTOR DIFFERENTIAL . . . 127
Θ(Y ) =
2y1 + y3
1 + y5
1
y2 +
1
3
y3
2
3y3 +
1
5
y5
3
2y4 +
1
3
y3
4
,
and
F (X) =
3x1 +
1
3
x3
1
2x2 +
1
6
x3
2
4x3 + x3
3
5x4 +
1
3
x3
4
.
Then, clearly, eigenvalues of the matrix Ψ(Z) are
λ1(Ψ(Z)) = 1 + z2
1 , λ2(Ψ(Z)) = 1 + z2
2 , λ3(Ψ(Z)) = 1 + z2
3 , λ4(Ψ(Z)) = 1 + z2
4 .
Trivially,
λi(Ψ(Z)) ≥ 1 = a1 > 0, i = 1, 2, . . . , 4.
Next, observe that
λ1(G(X,Y,Z,W )) = 1 − x2 − y2 − z2 − w2 ≤ 1, λ2(G(X,Y,Z,W )) = 2,
λ3(G(X,Y,Z,W )) = 2 −
2
1 + x2 + z4
≤ 2,
λ4(G(X,Y,Z,W )) =
1
2
−
3
2 + y2 + z2 + w2
≤
1
2
.
Thus, one can choose λi(G(X,Y,Z,W )) ≤ a2 = 2, i = 1, 2, . . . , 4. Similarly, it follows that
JΘ(Y ) is a symmetric matrix, namely,
JΘ(Y ) =
2 + 3y2
1 + 5y4
1 0 0 0
0 1 + y2
2 0 0
0 0 3 + y4
3 0
0 0 0 2 + y2
4
,
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 1
128 C. TUNÇ
hence it is readily verified that JΘ(Y ) is a symmetric matrix. Now, the derivative of the function
F (X) gives that
JF (X) =
3 + x2
1 0 0 0
0 2 +
1
2
x2
2 0 0
0 0 4 + 3x2
3 0
0 0 0 5 + x2
4
.
Hence,
λ1(JF (X)) = 3 + x2
1 ≥ 3 > 0, λ2(JF (X)) = 2 +
1
2
x2
2 ≥ 2 > 0,
λ3(JF (X)) = 4 + 3x2
3 ≥ 4 > 0, λ3(JF (X)) = 5 + 3x2
4 ≥ 5 > 0.
Thus
λi(JF (X)) ≥ 2, i = 1, 2, . . . , 4.
Therefore, it can be concluded from the above discussion that
λi(JF (X)) −
1
4
a2
2 = λi(JF (X)) − 1 > 0, i = 1, 2, . . . , 4.
Thus all the conditions of the theorem are satisfied.
Acknowledgement. I would like to express my sincere gratitude to the anonymous referee
for his/her very sensitive corrections, comments and suggestions on this paper.
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Received 16.04.08,
after revision — 08.02.09
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