Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations
The purpose of this paper is to present new oscillation theorems and nonoscillation theorems for the nonlinear Euler differential equation t²x''′+g(x)=0. Here we assume that xg(x)>0 if x≠0, but we do not necessarily require that g(x) be monotone increasing. The obtained results are best...
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irk-123456789-1655462020-02-15T01:27:23Z Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations Yamaoka, N. Sugie, J. Статті The purpose of this paper is to present new oscillation theorems and nonoscillation theorems for the nonlinear Euler differential equation t²x''′+g(x)=0. Here we assume that xg(x)>0 if x≠0, but we do not necessarily require that g(x) be monotone increasing. The obtained results are best possible in a certain sense. To establish our results, we use Sturm’s comparison theorem for linear Euler differential equations and phase plane analysis for a nonlinear system of Liénard type. Наведено нові осцнляційні та неосцнляційні теореми для нелінійного диференціального рівняння Ейлера t²x''′+g(x)=0, де припускається, що xg(x)>0 при x≠0, але вимога про монотонне зростання g(x) не є обов'язковою. Одержані результати є найкращими у певному сенсі. Для їх встановлення використано порівняльну теорему Штурма для лінійних диференціальних рівнянь Ейлера та фазовий площинний аналіз для нелінійної системи типу Льєнарда. 2006 Article Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations / N. Yamaoka, J. Sugie // Український математичний журнал. — 2006. — Т. 58, № 12. — С. 1704–1714. — Бібліогр.: 15 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165546 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Yamaoka, N. Sugie, J. Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations Український математичний журнал |
| description |
The purpose of this paper is to present new oscillation theorems and nonoscillation theorems for the nonlinear Euler differential equation t²x''′+g(x)=0. Here we assume that xg(x)>0 if x≠0, but we do not necessarily require that g(x) be monotone increasing. The obtained results are best possible in a certain sense. To establish our results, we use Sturm’s comparison theorem for linear Euler differential equations and phase plane analysis for a nonlinear system of Liénard type. |
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| author |
Yamaoka, N. Sugie, J. |
| author_facet |
Yamaoka, N. Sugie, J. |
| author_sort |
Yamaoka, N. |
| title |
Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations |
| title_short |
Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations |
| title_full |
Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations |
| title_fullStr |
Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations |
| title_full_unstemmed |
Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations |
| title_sort |
multilayer structures of second-order linear differential equations of euler type and their application to nonlinear oscillations |
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Інститут математики НАН України |
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2006 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/165546 |
| citation_txt |
Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations / N. Yamaoka, J. Sugie // Український математичний журнал. — 2006. — Т. 58, № 12. — С. 1704–1714. — Бібліогр.: 15 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT yamaokan multilayerstructuresofsecondorderlineardifferentialequationsofeulertypeandtheirapplicationtononlinearoscillations AT sugiej multilayerstructuresofsecondorderlineardifferentialequationsofeulertypeandtheirapplicationtononlinearoscillations |
| first_indexed |
2025-07-14T18:53:09Z |
| last_indexed |
2025-07-14T18:53:09Z |
| _version_ |
1837649563849064448 |
| fulltext |
UDC 517.9
N. Yamaoka (Sophia Univ., Tokyo, Japan), J. Sugie* (Shimane Univ., Matsue, Japan)
MULTILAYER STRUCTURES OF SECOND-ORDER
LINEAR DIFFERENTIAL EQUATIONS OF EULER TYPE
AND THEIR APPLICATION TO NONLINEAR OSCILLATIONS
BAHATOÍAROVI STRUKTURY LINIJNYX
DYFERENCIAL|NYX RIVNQN| DRUHOHO PORQDKU
TYPU EJLERA TA }X ZASTOSUVANNQ
DO NELINIJNYX KOLYVAN|
The purpose of this paper is to present new oscillation theorems and nonoscillation theorems for the
nonlinear Euler differential equation t2
x′′ + g ( x ) = 0 . Here we assume that x g ( x ) > 0 if x ≠ 0, but we
do not necessarily require that g ( x ) be monotone increasing. The obtained results are best possible in a
certain sense. To establish our results, we use Sturm’s comparison theorem for linear Euler differential
equations and phase plane analysis for a nonlinear system of Liénard type.
Navedeno novi oscylqcijni ta neoscylqcijni teoremy dlq nelinijnoho dyferencial\noho rivnqn-
nq Ejlera t2 x′′ + g ( x ) = 0 , de prypuska[t\sq, wo x g ( x ) > 0 pry x ≠ 0, ale vymoha pro mono-
tonne zrostannq g ( x ) ne [ obov’qzkovog. OderΩani rezul\taty [ najkrawymy u pevnomu sensi.
Dlq ]x vstanovlennq vykorystano porivnql\nu teoremu Íturma dlq linijnyx dyferencial\nyx
rivnqn\ Ejlera ta fazovyj plowynnyj analiz dlq nelinijno] systemy typu L\[narda.
1. Introduction and motivation. Let f ( t ) be a continuous function defined on
[ T, ∞ ) for some T > 0. The function f ( t ) is said to be oscillatory if there exists a
sequence { tn } tending to ∞ such that f ( tn ) = 0. Otherwise, f ( t ) said to be
nonoscillatory.
A class of linear differential equations of Euler type has a multilayer structure. To
explain this fact, we first consider the equation
′′ + +
( )
y
t t
y
1 1
42 2
λ
log
= 0, (1.1)
where ′ = d / dt and λ is a positive parameter. Eq. (1.1) is called the Riemann –
Weber version of the Euler differential equation (refer to [1 – 4]). All nontrivial
solutions of Eq. (1.1) are oscillatory if and only if λ > 1 / 4, because Eq. (1.1) has the
general solution
y ( t ) =
t K t K t
t t K K t
z z{ }
{ }
( ) + ( ) ≠
+ ( ) =
−
1 2
1
3 4
1 4
1 4
log log if / ,
log log log if / ,
λ
λ
(1.2)
where Ki , i = 1, 2, 3, 4, are arbitrary constants and z is the root of
z2 – z + λ = 0. (1.3)
Hence, for Eq. (1.1) the critical value of λ is 1 / 4. Such a number is generally called
the oscillation constant.
Letting s = log t and u ( s ) = y ( t ) / t , we can reduce Eq. (1.1) to the basic Euler
differential equation
˙̇u
s
u+ λ
2 = 0, (1.4)
where ⋅ = d / ds. It is well-known that the condition λ > 1 / 4 is necessary and
sufficient for all nontrivial solutions of Eq. (1.4) to be oscillatory (for example, see [5 –
* Supported in part by Grant-in-Aid for Scientific Research 16540152.
© N. YAMAOKA, J. SUGIE, 2006
1704 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
MULTILAYER STRUCTURES OF SECOND-ORDER … 1705
7]). In other words, the oscillation constant for Eq. (1.4) is also 1 / 4. Let us add the
perturbation λ / ( s log s )
2
u to the critical case of Eq. (1.4), namely, ˙̇u + u / ( 4s2
) = 0.
Then we get
˙̇
log
u
s s
u+ +
( )
1 1
42 2
λ = 0, (1.5)
which has the same form of Eq. (1.1). Taking account of this relation between
Eqs. (1.1) and (1.4), we may regard Eqs. (1.1) and (1.4) as the first and the second
stages of linear differential equations of Euler type, respectively. Then what is the
third stage? By putting t = es and y ( t ) = e u ss ( ), Eq. (1.5) is transformed into the
equation
′′ + +
( )
+
( ) ( )
( )
y
t t t t
y1 1
4
1
42 2 2 2log log log log
λ = 0. (1.6)
It is safe to say that Eq. (1.6) is the third stage of Euler’s differential equations.
Repeating the same transformation, we can derive the n th stage of linear differential
equations of Euler type (for details, see [8 – 10]). From the reason above, we see that
linear differential equations of Euler type have a multilayer structure.
The authors [9, 10], have compared the solutions of Eq. (1.6) or the n th stage of
Euler’s differential equations with those of the nonlinear equation
′′ + ( )x
t
g x
1
2 = 0, (1.7)
where g ( x ) satisfies a suitable smoothness condition for the uniqueness of solutions of
the initial value problem and the assumption
x g ( x ) > 0 if x ≠ 0, (1.8)
and established some oscillation theorems and nonoscillation theorems for Eq. (1.7).
For example, we can state the following results which are complementary to each
other.
Theorem A. Assume (1.8) and suppose that there exists a λ > 1 / 4 such that
g x
x x x x
( ) ≥ +
( )
+
( ) ( )( )
1
4
1
4 2 2 2 2 2 2log log log log
λ
for | x | sufficiently large. Then all nontrivial solutions of Eq. (1.7) are oscillatory.
Theorem B. Assume (1.8) and suppose that
g x
x x x x
( ) ≤ +
( )
+
( ) ( )( )
1
4
1
4
1
42 2 2 2 2 2log log log log
for x > 0 or x < 0, | x | sufficiently large. Then all nontrivial solutions of Eq. (1.7)
are nonoscillatory.
Remark 1.1. We can prove that all solutions of Eq. (1.7) exist in the future under
the assumption (1.8) (for the proof, see [11]). Hence, it is worth while to discuss
whether all nontrivial solutions of Eq. (1.7) are oscillatory or nonoscillatory.
As mentioned above, Euler’s differential equations have the multilayer structure
which is built up of stages such as Eqs. (1.4), (1.1) and (1.6). A natural question now
arises. Is the multilayer structure unique? The answer is a “no”. For some a1 > 0, let
t = es
/ a1 and y ( t ) = e u ss ( ). Then Eq. (1.4) is transferred to the equation
′′ + +
( )
y
t a t
y1 1
42
1
2
λ
log
= 0. (1.9)
Rewrite t and y in Eq. (1.9) as s and u, respectively. Then the transformation t =
= es
/ a2 with a2 > 0 yields the equation
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1706 N. YAMAOKA, J. SUGIE
′′ + +
( )
+
( ) ( )
( )
y
t a t a t a a t
y1 1
4
1
42
2
2
2
2
1 2
2log log log log
λ = 0, (1.10)
where y ( t ) = e u ss ( ). Using the same process infinitely many times, we can make
another multilayer structure of linear differential equations of Euler type. For further
details, see the final section.
It is easy to check that Eq. (1.10) has the general solution
y ( t ) =
t a t K a a t K a a t
t a t a a t K K a a t
z zlog log log log log if / ,
log log log log log log if / ,
2 1 1 2 2 1 2
1
2 1 2 3 4 1 2
1 4
1 4
{ ( ) ( ) }
{ ( )}
( ) + ( ) ≠
( ) + ( ) =
− λ
λ
where Ki , i = 1, 2, 3, 4, are arbitrary constants and z is the root of Eq. (1.3). In case
λ > 1 / 4, Eq. (1.3) has conjugate roots z = 1 / 2 ± i α, where α = λ − 1 4/ . Hence,
by Euler’s formula, the real solution can be represented as
y ( t ) = t a t a a t k a a tlog log log cos log log log2 1 2 1 1 2( ) ( ){ ( ( ))α +
+ k a a t2 1 2sin log log log( ( ))}( )α
for some k1 ∈ R and k2 ∈ R. If ( k1 , k2 ) = ( 0, 0 ), then y ( t ) is the trivial solution.
On the other hand, if ( k1 , k2 ) ≠ ( 0, 0 ), then
y ( t ) = k t a t a a t a a t3 2 1 2 1 2log log log sin log log log( ) ( ) +( ( ) )α β . (1.11)
where k3 = k k1
2
2
2+ , sin β = k1 / k3 and cos β = k2 / k3
.
Comparing the solutions of Eq. (1.7) with those of Eq. (1.10), we have the following
pair of an oscillation theorem and a nonoscillation theorem.
Theorem 1.1. Assume (1.8) and suppose that there exists a λ > 1 / 4 such that
g x
x a x a x a a x
( ) ≥ +
( )
+
( ) ( )( )
1
4
1
4 2
2 2
2
2 2
1 2
2 2log log log log
λ
for | x | sufficiently large, where a1 a n d a 2 are arbitrary positive numbers.
Then all nontrivial solutions of Eq. (1.7) are oscillatory.
Theorem 1.2. Assume (1.8) and suppose that
g x
x a x a x a a x
( ) ≤ +
( )
+
( ) ( )( )
1
4
1
4
1
42
2 2
2
2 2
1 2
2 2log log log log
for x > 0 or x < 0, | x | sufficiently large, where a1 and a2 are arbitrary
positive numbers. Then all nontrivial solutions of Eq. (1.7) are nonoscillatory.
Remark 1.2. Let us take 0 < ai < 1, i = 1, 2. Then we find that Theorem 1.2 is
superior to Theorem B.
2. Oscillation theorem. In this section, we will prove Theorem 1.1. To this end,
we need the following results (refer to [10, 12 – 15]).
Lemma 2.1. Let f ( t ) be a positive C 2 -function defined on [ T, ∞ ) for some
T > 0. If the second derivative of f t( ) is negative for t ≥ T, then f t( ) is
nondecreasing on the interval.
Lemma 2.2. Assume (1.8) and suppose that Eq. (1.7) has an eventually positive
solution. Then the positive solution tends to infinity as t → ∞.
Proof of Theorem 1.1. By way of contradiction, we suppose that Eq. (1.7) has a
nonoscillatory solution x0 ( t ). Then the solution is positive or negative eventually. We
consider only the former, because the latter is carried out in the same way. By
assumption, we can find a λ > 1 / 4 and an M > 0 such that
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
MULTILAYER STRUCTURES OF SECOND-ORDER … 1707
g x
x a x a x a a x
( ) ≥ +
( )
+
( ) ( )( )
1
4
1
4 2
2 2
2
2 2
1 2
2 2log log log log
λ (2.1)
for x > M. By virtue of Lemma 2.2, there exists a T > 0 such that
x0 ( t ) > M for t ≥ T.
Changing variable s = log t, we can transform Eq. (1.7) into the equation
˙̇ ˙u u g u− + ( ) = 0, (2.2)
where u ( s ) = x ( t ). Let u0 ( s ) be the solution of Eq. (2.2) corresponding to x0 ( t ).
Then we have
u0 ( log T ) = x0 ( T ) > M.
We next move u0 ( s ) along the s -axis. Let σ0 be a number with 0 < σ0 < 2 log M
and put
u1 ( s ) = u0 ( s – σ0 + log T )
for s ≥ σ0
. Needless to say, u1 ( s ) is also a solution of Eq. (2.2) and it is greater than
the number M for s ≥ σ0
.
We will estimate the growth rate of u1 ( s ) in details. For this purpose, we define
ξ ( s ) = u1 ( s ) e–
s
/
2.
Then, using (2.1), we obtain
˙̇ ˙̇ ˙ / /ξ( ) = ( ) − ( ) + ( )
= − ( ) + ( )
− −( )s u s u s u s e g u s u s es s
1 1 1
2
1 1
21
4
1
4
≤
≤ −
( )
−
( ) ( ( ))
( )
( ) ( ) ( )
−1
4 2 1
2 2
2 1
2 2
1 2 1
2 2 1
2
log log log log
/
a u s a u s a a u s
u s e sλ < 0
for s ≥ σ0
. Hence, it follows from Lemma 2.1 that ξ ( s ) is nondecreasing for s ≥ σ0
,
and therefore, we have
ξ ( s ) ≥ ξ ( σ0 ) = u1 ( σ0 ) e–
σ0
/
2 = u0 ( log T ) e–
σ0
/
2 > M e–
σ0
/
2 > 1
for s ≥ σ0
. From this inequality, we get the lower estimation
u1 ( s ) > es
/
2 for s ≥ σ0
. (2.3)
Let us form an upper estimation of u1 ( s ). By the assumption (1.8), we have
˙̇ ˙u s u s1 1( ) − ( ) = – g ( u1 ( s ) ) < 0
for s ≥ σ0
. This implies that
˙ ˙u s u es
1 1 0
0( ) ≤ ( ) −σ σ for s ≥ σ0
.
Integrate both sides of this inequality to obtain
u1 ( s ) ≤ ˙ ˙u e u us
1 0 1 0 1 0
0( ) − ( ) + ( )−σ σ σσ .
Hence, there exists a σ1 > σ0 such that
a u s2 1( ) < e2s for s ≥ σ1
. (2.4)
To get a sharper estimation than (2.4), we define a function η ( s ) by
s η ( s ) = u1 ( s ) e–
s
/
2.
Differentiating both sides of the equality twice, we have
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1708 N. YAMAOKA, J. SUGIE
s s s u s u s u s e s˙̇ ˙ ˙̇ ˙ /η η( ) + ( ) = ( ) − ( ) + ( )
−2 1
41 1 1
2.
Hence, together with (2.1), (2.3) and (2.4), we obtain
d
ds
s s u s u s u s se g u s u s ses s( ) ( )( ) = ( ) − ( ) + ( )
= − ( ) + ( )
− −2
1 1 1
2
1 1
21
4
1
4
˙ ˙̇ ˙ / /η ≤
≤ −
( )
−
( ) ( ( ))
( )
( ) ( ) ( )
−1
4 2 1
2 2
2 1
2 2
1 2 1
2 2 1
2
log log log log
/
a u s a u s a a u s
u s se sλ ≤
≤ −
( )
< −1
4
1
644 2loge
s
ss
for s ≥ σ1
. From this inequality, we get
s s
s2
1
1
2
1
1
64
˙ log ˙η
σ
σ η σ( ) ≤ − + ( ) for s ≥ σ1
,
and therefore, there exists a σ2 > σ1 such that η̇( )s < 0 for s ≥ σ2
. Hence, we
obtain the upper estimation
u1 ( s ) ≤
u
e
ses1 2
2
2
2
2
( )σ
σ σ /
/ for s ≥ σ2
. (2.5)
We now consider the function
y ( t ) = t a t a a t a a tlog log log sin / log log log2 1 2 1 21 4( ) − ( )( ( ))λ .
Then, as shown in Section 1, the function is an oscillatory solution of Eq. (1.10)
because λ > 1 / 4 (we may take k3 = 1 and β = 0 in the representation (1.11). It is
clear that y ( t ) has infinitely many zeros
em =
1 1
1 42 1a a
m
exp exp exp
/
π
−
λ
for m ∈ N. Let sm = log em
. Then sm tends to infinity as m → ∞ . We can easily
check that
e
s
s
m
m / 2
1+
→ ∞ as m → ∞.
Hence, we can choose an m0 ∈ N so that
σ2 < sm0
and
u e
s
s
m
m
1 2
2
2
1
0
0
( ) <
+
σ
σ
/
. (2.6)
For the sake of simplicity, let σ3 = sm0
and σ4 = sm0 + 1
. Note that σ2 < σ3 < σ4 and
points eσ3 and eσ4 are two successive zeros of y ( t ).
We translate the positive solution u1 ( s ) of Eq. (2.2). Let
u2 ( s ) = u1 ( s – σ3 + σ2 )
for s ≥ σ3 – σ2 + σ0
. From (2.5) and (2.6), we see that
u2 ( s ) <
u
e
s e
u
s es s1 2
2
2 3 2
2 1 2
2
3 2
2
2
3 2 3
( ) ( − + ) = ( ) ( − + )( − + ) ( − )σ
σ
σ σ σ
σ
σ σσ
σ σ σ
/
/ / <
<
e
s e
s
es s
σ
σ
σ
σ σ σ σ
σ
3
3
2
4
3 2
2 3 2
4
2
/
/ /( − + ) = − +( − )
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
MULTILAYER STRUCTURES OF SECOND-ORDER … 1709
for s ≥ σ3
. Hence, we have
u2 ( s ) < es
/
2 for σ3 ≤ s ≤ σ4
. (2.7)
Let x ( t ) be the solution of Eq. (1.7) corresponding to u2 ( s ). Since x ( t ) is made
by a parallel translation of x0 ( t ), it follows that
x ( t ) > M for t ≥ eσ3. (2.8)
From (2.7), it also turns out that
x ( t ) < t for eσ3 ≤ t ≤ eσ4.
Hence, by (2.1) again, we have
g x t
x t a x t a x t a a x t
( )
( ) ( ) ( )
( )
( )
≥ +
( )
+
( ) ( ( ))
1
4
1
4 2
2 2
2
2 2
1 2
2 2log log log log
λ >
> 1
4
1
4 2
2
2
2
1 2
2+
( )
+
( ) ( )( )log log log loga t a t a a t
λ (2.9)
for eσ3 ≤ t ≤ eσ4. We may regard x ( t ) as an eventually positive solution of the linear
differential equation
′′ + ( )
( )
( )
x
t
g x t
x t
x1
2 = 0.
Remember that y ( t ) is an oscillatory solution of Eq. (1.10), whose successive zeros
are eσ3 and eσ4. Hence, by (2.9) and Sturm’s comparison theorem, x ( t ) has at least
one zero between eσ3 and eσ4. This is a contradiction to (2.8). The proof of Theorem
1.1 is now complete.
3. Nonoscillation theorem. As has been mentioned in the proof of Theorem 1.1,
the change of variable s = log t transfers Eq. (1.7) to Eq. (2.2), which is equivalent to
the system
u̇ = v + u,
(3.1)
v̇ = – g ( u ).
System (3.1) is of Liénard type. Phase plane analysis is frequently made for the
purpose of examining the asymptotic behavior of solutions of system (3.1). We call the
projection of a positive semitrajectory of system (3.1) onto the phase plane a positive
orbit.
Suppose that there exists a nontrivial oscillatory solution x ( t ) of Eq. (1.7). Let t0
be the initial time of x ( t ) and let { tn } be the sequence of zeros of x ( t ). Take s0 =
= log t0 and σ n = log tn . Let ( u ( s ), v ( s ) ) be the solution of system (3.1)
corresponding to x ( t ). Then, it is clear that u ( σn ) = 0. Taking account of the vector
field of system (3.1), we also see that there exists another sequence { τn } with σn <
< τn < σn + 1 such that v ( τn ) = 0. To be precise, the positive orbit of system (3.1)
starting at ( u ( s0 ), v ( s0 ) ) rotates around the origin ( 0, 0 ) clockwise. Since g ( x ) is
smooth enough to guarantee the uniqueness of solutions to the initial value problem
and system (3.1) is autonomous, any positive orbit of system (3.1) fails to cross itself
and all other positive orbits of system (3.1). To sum up, we have the following result.
Lemma 3.1. Under the assumption (1.8), if Eq. (1.7) has a nontrivial oscillatory
solution, then all nontrivial positive orbits of system (3.1) rotate in a clockwise
direction about the origin.
By means of Lemma 3.1 and phase plane analysis for system (3.1), we give the
proof of Theorem 1.2.
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1710 N. YAMAOKA, J. SUGIE
Proof of Theorem 1.2. We prove only the case that
g x
x a x a x a a x
( ) ≤ +
( )
+
( ) ( )( )
1
4
1
4
1
42
2 2
2
2 2
1 2
2 2log log log log
(3.2)
for x > 0 sufficiently large, because the proof of the other case is essentially the same.
The proof is by contradiction. Suppose that there exists a nontrivial oscillatory solution
of Eq. (1.7). Then, from Lemma 3.1 we see that all positive orbits go around the origin
in clockwise order except the trivial positive orbit, namely, the origin.
We choose a number s0 so large that (3.2) holds for x > e as0
2/ and let
P = e
a s s a s
e
a
s s0 0
2 0 0 1 0 2
1
2
1
2
1
2
,
log
− + +
.
Note that the point P belongs to the region R =df
{ ( u, v ) : – u / 2 ≤ v < 0 }. We
consider the solution ( u ( s ), v ( s ) ) of system (3.1) satisfying the initial condition
( u ( s0 ), v ( s0 ) ) = P. (3.3)
Since the positive orbit of system (3.1) corresponding to ( u ( s ), v ( s ) ) also rotates
about the origin, it meets the straight line v = – u / 2 infinitely many times. Let s1 > s0
be the first intersecting time of the positive orbit with the line, and let Q = ( u ( s1 ),
v ( s1 ) ). Then, taking the vector field of system (3.1) into consideration, we see that the
arc P Q of the positive orbit is in the region R. In other words, – u ( s ) / 2 ≤ v ( s ) < 0
for s0 ≤ s ≤ s1
. Hence, we have
u̇ s
u s( ) ≥ ( )
2
for s0 ≤ s ≤ s1
,
and therefore,
u ( s ) ≥ u s e e
a
s s
s
( ) =( − )
0
2
2
0 / for s0 ≤ s ≤ s1
. (3.4)
We consider the function
ξ ( s ) =
v( )
( )
s
u s
.
Then, the function is defined on an open interval containing [ s0
, s1 ]. Taking notice of
(3.3), we see that
ξ ( s0 ) = − + +1
2
1
2
1
20 0 1 0s s a slog
.
Since the point Q is on the line v = – u / 2, we also see that
ξ ( s1 ) = − 1
2
. (3.5)
Differentiating ξ ( s ) and using (3.2) and (3.4), we have
ξ̇ ξ ξ( ) = − ( ) − ( ) − ( )
( )
( )
s s s
g u s
u s
2 ≥
≥ − ( ) +
−
( )
−
( ) ( ( ))( ) ( ) ( )
ξ s
a u s a u s a a u s
1
2
1
4
1
4
2
2
2 2
2
2 2
1 2
2 2log log log log
≥
≥ − ( ) +
− −
( )
ξ s
s s a s
1
2
1
4
1
4
2
2 2
1
2log
(3.6)
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MULTILAYER STRUCTURES OF SECOND-ORDER … 1711
for s0 ≤ s ≤ s1
. To estimate ξ ( s ), we define
η ( s ) = − + +1
2
1
2
1
2 1s s a slog
for s ≥ s0
. It is clear that η ( s0 ) = ξ ( s0 ). Also, it turns out that η ( s ) satisfies
˙
log
η η( ) = − ( ) +
− −
( )
s s
s s a s
1
2
1
4
1
4
2
2 2
1
2 .
Hence, together with (3.6), we obtain
ξ ( s ) ≥ η ( s ) > − 1
2
for s0 ≤ s ≤ s1
.
This is a contradiction to (3.5) at s = s1
. Thus, all nontrivial solutions of Eq. (1.7) are
nonoscillatory. We have completed the proof of Theorem 1.2.
4. Extension to the n-th stage. Having given the proofs of Theorems 1.1 and 1.2,
we may now proceed to natural generalizations. We first make a new multilayer
structure of linear differential equations of Euler type. Let { aj } be any sequence with
aj > 0, j ∈ N. For n ∈ N fixed, we define
log0
n w = w, log log logk
n
n k k
nw a w= ( )− −1 , k = 1, 2, … , n – 1.
We should notice that logk
n w depends on n as well as k. Using the terms, we
describe two sequence of functions as follows:
L wn
1( ) = 1, L w L w wk
n
k
n
k
n
+ ( ) = ( )1 log , k = 1, 2, … , n – 1,
S1 ( w ) = 0, Sn ( w ) = 1
2
1
1
( )( )=
−
∑
L wk
n
k
n
, n ≥ 2.
The sequences are well-defined for w > 0 sufficiently large. Note that
Sn + 1 ( w ) � S w
L w
n
n
n( ) +
( )( )
1
2
unless a1 = a2 = … = an
. To be specific,
L w1
1( ) = 1, L w a w2
2
1( ) = ( )log , L w a w a a w3
3
2 1 2( ) = ( ) ( )( )log log log ,
L w a w a a w a a a w4
4
3 2 3 1 2 3( ) = ( ) ( ) ( ( ))( ) ( )log log log log log log ;
S1 ( w ) = 0, S2 ( w ) = 1, S3 ( w ) = 1 1
2
2+
( )( )log a w
,
S4 ( w ) = 1 1 1
3
2
3
2
2 3
2+
( )
+
( ) ( ( ))( ) ( ) ( )log log log loga w a w a a w
,
and so on.
Consider the linear equation
′′ + ( ) +
( )
( )
y
t
S t
L t
yn
n
n
1 1
42 2
λ = 0, (4.1)
which coincides with Eqs. (1.9) and (1.10) when n = 2 and n = 3, respectively. Then
we have the following result.
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1712 N. YAMAOKA, J. SUGIE
Lemma 4.1. Let n ≥ 2. Then Eq. (4.1) has the general solution
y ( t ) =
tL t K t K t if
tL t K K t if
n
n
n
n z
n
n z
n
n
n
n
− − −
−
−
( ) ( ) + ( ) ≠
( ) + ( ) =
{ }
{ }
1 1 1 2 1
1
3 4 1
1 4
1 4
log log / ,
log log / ,
λ
λ
where Ki , i = 1, 2, 3, 4, are arbitrary constants and z is the root of Eq. (1.3).
From Lemma 4.1, we see that all nontrivial solutions of Eq. (4.1) are oscillatory if
and only if λ > 1 / 4. In case λ > 1 / 4, y ( t ) given in Lemma 4.1 is a complex
solution. By using Euler’s formula, the real solution of Eq. (4.1) can be written in the
form
y ( t ) = tL t k t k tn
n
n
n
n
n( ) ( ) + ( ){ ( ) ( )}− −1 1 2 1cos log log sin log logα α , (4.2)
where ki , i = 1, 2, are arbitrary constants and α = λ − 1 4/ > 0.
Theorems 4.1 and 4.2 below are proven in the same manner as Theorems 1.1 and
1.2, respectively. We give a very brief outline of their proofs.
Theorem 4.1. Let { aj } be any sequence with aj > 0, j ∈ N. Under the
assumption (1.8), if there exists a λ > 1 / 4 such that
g x
x
S x
L x
n
n
n
( ) ≥ ( ) +
( )( )
1
4
2
2 2
λ
for | x | sufficiently large, then all nontrivial solutions of Eq. (1.7) are oscillatory.
Theorem 4.2. Let { aj } be any sequence with aj > 0, j ∈ N. Under the
assumption (1.8), if
g x
x
S x
L x
n
n
n
( ) ≤ ( ) +
( )( )
1
4
1
4
2
2 2
for x > 0 or x < 0, | x | sufficiently large, then all nontrivial solutions of Eq. (1.7)
are nonoscillatory.
Outline of the proof of Theorem 4.1. By contradiction, we suppose that Eq. (1.7)
has an eventually positive solution x0 ( t ). Let M be a number so large that
g x
x
S x
L x
n
n
n
( ) ≥ ( ) +
( )( )
1
4
2
2 2
λ (4.3)
for x > M. From Lemma 2.2, we can choose a T > 0 such that
x0 ( t ) > M for t ≥ T.
Let u0 ( s ) be the solution of
˙̇ ˙u u g u− + ( ) = 0
corresponding to x0 ( t ) and put
u1 ( s ) = u0 ( s – σ0 + log T )
for s ≥ σ0
, where σ0 is a number with 0 < σ0 < 2 log M. As in the proof of Theorem
1.1, we can estimate that
u1 ( s ) ≤
u
e
ses1 2
2
2
2
2
( )σ
σ σ /
/ for s ≥ σ2
,
where σ2 is a number with σ2 > σ0
.
Since λ > 1 / 4, Eq. (4.1) has oscillatory solutions of the form (4.2). We select
y ( t ) = tL t tn
n
n
n( ) − ( )( )−sin / log logλ 1 4 1
from among them. Let em be the zeros of y ( t ). Then we see that
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MULTILAYER STRUCTURES OF SECOND-ORDER … 1713
em = exp exp
/n
n m
−
π
−
1 1 4λ
for m ∈ N, where { }−expn
n t1 is a sequence of function as follows:
exp0
n t = t, exp exp expk
n
k
k
nt
a
t= ( )−
1
1 , k = 1, 2, … , n – 1.
Let sm = log em. Then we obtain
e
s
s
m
m / 2
1+
→ ∞ as m → ∞.
Hence, there exists an m0 ∈ N such that
σ2 < sm0
and
u e
s
s
m
m
1 2
2
2
1
0
0
( ) <
+
σ
σ
/
.
Put σ3 = sm0
and σ4 = sm0 + 1
. We define
u2 ( s ) = u1 ( s – σ3 + σ2 )
for s ≥ σ3 – σ2 + σ0
. Then, we get the estimation
M < u2 ( s ) < es
/
2 for σ3 ≤ s ≤ σ4
.
Let x ( t ) be the solution of Eq. (1.7) corresponding to u2 ( s ). Then, we can rewrite the
above estimation as
M < x ( t ) < t for eσ3 ≤ t ≤ eσ4. (4.4)
Hence, by (4.3) we have
g x t
x t
S t
L t
n
n
n
( )
( )
( )
( )
> ( ) +
( )
1
4 2
λ
for eσ3 ≤ t ≤ eσ4. From this inequality and Sturm’s comparison theorem, we see that
x ( t ) has at least one zero between eσ3 and eσ4, which contradicts (4.4). Thus,
Theorem 4.1 is now proved.
Outline of the proof of Theorem 4.2. We give only the proof of the case that
g x
x
S x
L x
n
n
n
( ) ≤ ( ) +
( )( )
1
4
1
4
2
2 2 (4.5)
for x > 0 sufficiently large. The proof is by contradiction. Suppose that there exists a
nontrivial oscillatory solution of Eq. (1.7). Let
P = e
a L e a
e
a
s
n k
n s
nk
n s
n
0
0
0
1 12 1
1
2
1
2
1
− −= −
− +
( )
∑,
/
,
where s0 is a number so large that (4.5) holds for e as
n
0
1/ − . Let ( u ( s ), v ( s ) ) be
the solution of system (3.1) satisfying the initial condition
( u ( s0 ), v ( s0 ) ) = P.
We consider the positive orbit of system (3.1) corresponding to ( u ( s ), v ( s ) ). Then,
from Lemma 3.1 we see that the positive orbit rotates about the origin, and therefore, it
meets the straight line v = – u / 2 infinitely many times. Let s1 > s0 be the first
intersecting time of the positive orbit with the line. Then we have
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1714 N. YAMAOKA, J. SUGIE
u ( s ) ≥ u s e e
a
s s
s
n
( ) =( − )
−
0
2
1
0 / for s0 ≤ s ≤ s1
. (4.6)
As in the proof of Theorem 1.2, we compare the function
ξ ( s ) =
v( )
( )
s
u s
with a solution
η ( s ) = − +
( )−=
∑1
2
1
2
1
12 L e ak
n s
nk
n
/
of the equation
˙
/
η η( ) = − ( ) +
−
( )( )−=
∑s s
L e ak
n s
nk
n
1
2
1
4
12
1
2
2
.
It follows from (4.5) and (4.6) that
˙
/
ξ ξ( ) ≥ − ( ) +
−
( )( )−=
∑s s
L e ak
n s
nk
n
1
2
1
4
12
1
2
2
for s0 ≤ s ≤ s1
. Since η ( s0 ) = ξ ( s0 ), we conclude that
ξ ( s ) ≥ η ( s ) > − 1
2
for s0 ≤ s ≤ s1
.
This contradicts the fact that ξ ( s1 ) = – 1 / 2. Thus, all nontrivial solutions of Eq. (1.7)
are nonoscillatory, thereby completing the proof of Theorem 4.2.
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Received 10.10.2005
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