Funkcialaj Ekvacioj, 17 (1974), 193-205
On a Bound for Periods of Solutions of
a Certain Nonlinear Differential Epuation (II)
By Tominosuke OTSUKI
(Tokyo Institute of Technology)
Introduction.
As is shown in [6], the following nonlinear differential equation:
§0.
(E)
$nx(1-x^{2})¥frac{d^{2}x}{dt^{2}}+(¥frac{dx}{dt})^{2}+(1-x^{2})(nx^{2}-1)=0$
where
$n¥geqq 2$
, is the equation for the support function
2-dimensional Riemannian manifold
$o_{n}^{2}$
$x(t)$
,
of a geodesic in the
with the metric:
$ds^{2}=(1-u^{2}-v^{2})^{n-2}¥{(1-v^{2})du^{2}+2uvdudv+(1-u^{2})dv^{2}¥}$
(0. 1)
in the unit disk: $u^{2}+v^{2}<1$ .
can be considered for any real number
and
represented
is
isometrically as a surface of revolution in the 4-dimensional
Lorentzian space which is taken off a singular point from a closed one, when
$o_{n}^{2}$
$n>1$ ,
in [9].
$n$
For the generalized
$o_{n}^{2}$
, (E) has the same meaning as mentioned
above.
Any non constant solution
$x(t)$
of (E) such that
$x^{2}+(¥frac{dx}{dt})^{2}<1$
is periodic and its period
(0. 2)
$T$
is given by the improper integral:
$T=2¥int_{a¥mathrm{o}}^{a_{1}}¥frac{dx}{¥sqrt 1-x^{2}-C(¥frac{1}{x^{2}}-1)^{a}}$
,
where
(0. 3)
$C=(a^{22}¥mathrm{o})^{¥alpha}(1-a¥mathrm{o})^{1-a}=(a^{2}1)^{a}(1-a^{2}1)^{1-¥alpha}$
$(0<a_{0}<¥sqrt{¥alpha}<a_{1}<1, ¥alpha=1/n)$
is the integral constant of (E) and $0<C<A=¥alpha^{¥alpha}(1-¥alpha)^{1-¥alpha}$ .
The following is known in [4]:
(i)
(ii)
$ T>¥pi$
,
.
and
By means of a numerical analysis and observation about (E) done by M. Urabe,
the following inequality:
$¥mathrm{l}¥mathrm{i}¥mathrm{m}¥mathrm{c}¥rightarrow 0¥mathrm{T}=¥pi$
$¥mathrm{l}¥mathrm{i}¥mathrm{m}c¥rightarrow AT=¥sqrt{2}¥pi$
194
T. OTSUKI
(U)
$ T</¥overline{2}¥pi$
was conjectured in [5] and [10], when
is an integer $>1$ .
The inequality (U) was proved in [2] and [5] when $n=2$ and in [8] when
is any real number
. In the present paper, the author will prove it when
$1<n<3$ , using the notation in [8]. First, he will prove it in the case $2<n<3$
and then the rest case $1<n<2$ , by proving a duality of the equations (E) for
and $n=m$ such that $1/n+1/m=1$ .
$n$
$¥geqq 3$
$n$
$n$
§1.
Preliminaries.
Replacing
and
can be written as
$nx^{2}$
(1. 1)
$nC$
by
and
$x$
$C$
respectively, the period
$T$
given by (0. 2)
,
$T=T_{n}(x_{0})=¥int_{x_{0}}^{x_{1}}¥frac{dx}{¥mathit{1}¥overline{x(n-x)-C¥ddot{x}^{1-¥mathrm{a}}(n-x)^{¥alpha}}}$
where
(1. 2)
$C=x_{0}^{¥alpha}(n-x_{0})^{1-¥alpha}=x_{1}^{¥alpha}(n-x_{1})^{1-a}$
and
$0<x_{0}<1<x_{1}<n$ .
(1. 3)
Using the auxiliary function
(1. 4)
$T$
$X=X_{n}(x)$
for
$0¥leqq x¥leqq 1$
$x(n-x)^{n-1}=X(n-X)^{n-1}$ ,
by
$1¥leqq X¥leqq n$
,
can be written as follows:
(1. 5)
$ T_{n}(x_{0})=¥int_{x_{0}}^{1}¥{¥frac{¥mathit{1}¥overline{x(n-x)}}{1-x}+¥frac{¥mathit{1}¥overline{X_{n}(x^{¥mathfrak{l}})(n-X_{n}(x))}}{X_{n}(x)-1}¥}/¥overline{¥frac{B-¥varphi(x)}{¥varphi(x)}}¥times$
$¥frac{d¥varphi(x)}{J^{¥frac{}{(B-¥varphi(x))(¥varphi(x)-C)}}}$
,
where
(1. 6)
$(0¥leqq x¥leqq n)$
$¥varphi(x):=x^{¥alpha}(n-x)^{1-¥mathrm{a}}$
and
(1. 7)
$B=¥varphi(1)=nA$
By Lemma 1. 3 in [8], the function
$F(x)$
.
defined by
$(0¥leqq x¥leqq n, x¥neq 1)$
(1. 8)
$F(x)=¥int_{1}¥frac{x(n-x)}{(1-x)^{2}}21_{(x=1)}.¥frac{B-¥varphi(x))}{¥varphi(x)}$
is smooth and positive in
$(0, n)$
.
Then, (1. 5) can be written as
On a Bound for Periods of Solutions
(1. 9)
of a Certa
in
Nonlinear Diflerential Equation
$T_{n}(x_{0})=¥int_{x¥mathrm{o}}^{1}¥{¥sqrt{F(x)}+¥mathit{1}¥overline{F(X_{n}(x))}¥}¥frac{d¥varphi(x)}{¥mathit{1}¥overline{(B-¥varphi(x))(¥varphi(x)-C)}}$
Hence, the inequality
inequality:
(1. 10)
$ T</¥overline{2}¥pi$
$(0<x<1)$ .
is the the constant defined by
$¥Lambda(n-¥Lambda)^{n-1}¥{n+(n-2)¥Lambda¥}^{n}=(n-1)^{n-1}¥{n-1+n¥Lambda-¥Lambda^{2}¥}^{n}$
then the function
$n)$
.
will be proved if we can prove the following
$¥sqrt{F(x)}+/¥overline{F(X_{n}(x))}</¥overline{2}$
$¥Lambda(1<¥Lambda<n)$
195
$F(x)$
,
is monotone increasing in (0, ] and decreasing in
$¥Lambda$
$[¥Lambda$
,
.
§2.
Properties of
$F(x)$
.
Lemma 2. 1. $n/2<¥Lambda<n-1$ for $n>2$ .
Proof. By Lemma 2. 2 in [8], this inequality is true for
suppose $2<n<3$ . Since the function
(2. 1)
$[n/2, n]$
$g_{0}(n-1)=¥frac{(n^{2}-2n_{¥mathrm{t}}12)^{n}}{2^{n}(n-1)^{n-1}}<(n-1)^{n-1}$
Let us
by Lemma 2. 1 in
,
which is equivalent to
.
$¥frac{2(n-1)}{n}¥cdot¥log(n-1)>¥log(1-n+¥frac{n^{2}}{2})$
Setting $n=2+u(0<u<1)$ and substituting it in (2. 2) we get
(2. 3)
.
$g_{0}(x):=¥frac{x(n-x)^{n-1}¥{n+(n-2)x¥}^{n}}{(n-1+nx-x^{2})^{n}}$
is monotone increasing in $[0, n/2]$ and decreasing in
[8], it is sufficient to prove that
(2. 2)
$n¥geqq 3$
.
$u¥log(1+u)>(2+u)¥log¥{1+¥frac{u^{2}}{2(1+u)}¥}$
Now, we have easily
$0<¥frac{u^{2}}{2(1+u)}<¥frac{1}{4}$
for
$0<u<1$
and
$¥log(1+t)<t-¥frac{2}{5}t^{2}$
for
$¥mathrm{o}<t<¥frac{1}{4}$
.
Hence (2. 3) is true, if we can prove the following inequality:
T. OTSUKI
196
$u¥log(1+u)>(2+u)¥frac{u^{2}}{2(1+u)}¥{1-¥frac{u^{2}}{5(1+u)}¥}$
$¥mathrm{i}.¥mathrm{e}$
,
.
(2. 4)
$¥log(1+u)>(2+u)u¥{¥frac{1}{2(1+u)}-¥frac{u^{2}}{10(1+u)^{2}}¥}$
.
Since we have
$(¥log(1+u))^{¥prime}-[(2+u)u¥{¥frac{1}{2(1+u)}-¥frac{u^{2}}{10(1+u)^{2}}¥}]^{¥prime}$
$=¥frac{1}{1+u}-2(1+|u)¥{¥frac{1}{2(1+u)}-¥frac{u^{2}}{10(1+u)^{2}}¥}$
$+¥frac{(2+u)u}{2(1+u)^{2}}+¥frac{(2+u)u^{2}}{5(1+u)^{3}}=¥frac{u^{2}(1+u+2u^{2})}{10(1+u)^{3}}>0$
,
(2. 4) is clearly true for $0<u<1$ .
Lemma 2. 2.
Q. E. D.
$F(x)<¥frac{n^{2}}{n^{2}+4n-4}$
$(n>2)$ .
Proof. By Theorem 2. 3, Lemma 2. 4 and Theorem 2. 5 in [8] and Lemma
2. 1, we obtain easily
$F(x)¥leqq F(¥Lambda)=¥frac{¥Lambda(n-¥Lambda)}{(¥Lambda-1)^{2}}¥cdot¥frac{B-¥varphi(¥Lambda)}{¥varphi(¥Lambda)}=¥frac{¥Lambda(n-¥Lambda)}{n-1+n¥Lambda-¥Lambda^{2}}<¥frac{n^{2}}{n^{2}+4n-4}$
.
Q. E. D.
By Lemma 3. 1 in [8], $¥sqrt{F(x)}+/¥overline{F(X_{n}(x))}$ is increasing at $x(0<x<1)$ , if
and only if $f(x)>f(X_{n}(x))$ , where
$(0¥leqq x¥leqq n, x¥neq 1)$
(2. 5)
$f(x):=¥int_{¥backslash }¥frac{/¥overline{x(n-x))}}{(6(1)^{3}/nB}¥frac{--x2)}{n-1}[¥{(x=1)n+(n.-2)x¥}¥{B-¥varphi(x)¥}-B(1-x)^{2}]$
.
and $f(x)<0$ in
By Lemma 3. 2 in [8], $f(x)>0$ in
$f(x)$
decreasing
, if and
at
[8],
$x(0<x<n)$
by
is
3
in
Lemma 3.
Furthermore,
only if
$(¥Lambda, n)$
$(0, ¥Lambda)$
$[P(x)-3(1-x)^{2}¥{n+(n-2)x¥}]B<P(x)¥varphi(x)$ ,
where
(2. 6)
$P(x):=n(n+2)-^{t}¥ulcorner 2(4n^{2}-5n-2)x+(3n^{2}-16n+16)x^{2}$
Since $P(x)-3(1-x)^{2}¥{n+(n-2)x¥}>0$ in
(2. 7)
$[0, n]$
, setting
$g(x):=¥frac{P(x)¥varphi(x)}{P(x)-3(1-x)^{2}¥{n+(n-2)x¥}}$
$(0<x<n)$ ,
.
On a Bound for Periods of Solutions of a Certain Nonlinear Diflerential Equation
in (0, 1) and
in $(1, n)$ by Lemma 3. 5
and decreasing
Hence $g(x)$ is monotone increasing in (0, ] and
$g(1)=
¥
varphi(1)=B$
$n)$
equation:
Hence,
.
the
and [ , . We have
$g^{r}(x)=0(0<x<n)$ has unique roots
in [8].
in
$f$
$¥gamma$
$[1, ¥overline{¥gamma}]$
$¥mathcal{T}$
$[¥mathcal{T}, 1]$
197
$¥gamma-$
$[P(x)-3(1-x)^{2}¥{n+(n-2)x¥}]B=P(x)¥varphi(x)$ ,
$ 0<x<¥nu$ ,
has the two solutions a in $(0, T)$ and
decreasing in
and increasing in
$x¥neq 1$
,
in
and
$(¥overline{¥gamma}, n)$
$¥overline{¥sigma}$
$[0, ¥sigma]$
$[¥sigma,¥overline{¥sigma}]$
for
$f(¥sigma)¥geqq f(x)¥geqq f(¥overline{¥sigma})$
Lemma 2.3. $/¥overline{F(x)}+¥sqrt{F(X_{n}(x)}<J¥overline{2}$
Proof. Since we have
.
$[¥overline{¥sigma}, n]$
Then,
and
$0¥leqq x¥leqq n$
for
$f(x)$
is monotone
.
$¥sigma¥leqq x<1$
.
$(¥sqrt{F(x)}+/¥overline{F(X_{n}(x))})^{¥prime}=¥frac{1-x}{2x(n-x)¥sqrt{¥varphi(x)¥{B-¥varphi(x)¥}}}[f(x)-f(X_{n}(x))]$
for $0<x<1$ by (3. 2) and (3. 4) in [8], we have
for
$(¥sqrt{F(x)}+/¥overline{F(X_{n}(x))})^{¥prime}>0$
when
$X_{n}(x)¥leqq¥overline{¥sigma}$
.
,
On the other hand, we have
$f(¥Lambda)=0¥geqq f(¥overline{¥sigma})$
and $f(x)>0$ in
$¥sigma¥leqq x<1$
$(0, ¥Lambda)$
and $f(x)<0$ in
$(¥Lambda, n)$
$¥Lambda¥leqq¥overline{¥sigma}$
,
, it follows that
.
Thus, we have also
$(¥sqrt{F(x)}+¥sqrt{F(X_{n}(x))})^{¥prime}>0$
when
$X_{n}(x)>¥overline{¥sigma}$
. There-fore, we
for
$¥sigma¥leqq x<1$
,
get
.
$¥sqrt{F(x)}+¥sqrt{F(X_{n}(x))}<J¥overline{F(1)}+¥sqrt{F(X_{n}(1))}=¥sqrt{2}$
Q. E. D.
for $2<n<3$ .
An Estimation of
$T<1/5$
but this does not hold for $2<n<3$ .
proved
for
that
In [8], we
Injfact,
is the root in (0, 1) of the equation of order 2:
§3.
$¥gamma$
$n¥geqq 3$
$¥gamma$
(3. 1)
$¥mathrm{a}¥mathrm{n}¥mathrm{d}_{¥overline{¥overline{¥ddot{4}}}}¥mathrm{t}¥mathrm{h}¥mathrm{e}$
$n^{2}(n+2)-n(9n^{2}-2n+8)x+4(3n^{2}-2n+2)x^{2}=0$
value of the left hand side of (3. 1) at
$x=1/5$
is equal to
$-¥frac{4}{25}¥{5n^{3}-18n^{2}+12n-2¥}_{;}$
which is negative for
when $n=5/2$.
Lemma 3. 1. $T<1/4$
$n¥geqq 3$
and non negative for $2<n<3$ , for instance 51/50
for
$2¥leqq n¥leqq 3$
.
198
T.
$¥mathrm{o}_{¥mathrm{T}¥mathrm{S}¥mathrm{U}¥mathrm{K}¥mathrm{I}}$
The value of the left hand side of (3. 1) at
Proof.
$x=1/4$
multiplied by 4
is equal to
$-5n^{3}+13n^{2}-10n+2=-¥{n(5n-3)(n-2)_{¥mathrm{t}}|2(2n-1)¥}<0$
for
.
Hence we have
Lemma3.2. When
$2¥leqq n¥leqq 3$
$0<¥gamma<1/4$
$2¥leqq n_{-¥leq_{¥_}3}$
.
Q. E. D.
, we have
$/¥overline{F(x)}+¥sqrt{F(X_{n}(x))}<¥frac{1}{3}¥sqrt{4(n-1)(¥frac{4n-1}{n-1})^{1/n}-(4n-1)}+¥frac{n}{¥vee¥overline{n^{2}+4n-4}}$
for
$ 0<x¥leqq¥sigma$
.
Proof. Since we have
, we have
and
$0<¥sigma<T<1/4$ , $F(x)$
is monotone increasing in
$(0, ¥Lambda)$
$ 1<¥Lambda$
$F(x)<F(¥frac{1}{4})=¥frac{1}{9}¥{4(n-1)(¥frac{4n-1}{n-¥mathrm{k}})^{1/n}-(4n-1)¥}$
for
$ 0<x¥leqq¥sigma$
Since
.
This inequality and Lemma 2. 2 imply immediately this lemma.
Q. E. D.
$¥frac{n}{J¥overline{n^{2}+4n-4}}$
, as function of
$¥tau¥iota$
, is increasing for
for
$¥frac{n}{¥sqrt{n^{2}+4n-4}}¥leqq¥frac{3}{/¥overline{17}}$
Hence, when
$2¥leqq n¥leqq 3$
$2¥leqq n¥leqq 3$
$n¥geqq 2$
, we have
.
, in order to prove that
$¥sqrt{F(x)}+¥sqrt{F(X_{n}(x))}</¥overline{2}$
for
$ 0<x¥leqq¥sigma$
,
it is sufficient to prove that
$¥frac{1}{3}¥sqrt{4(n-1)(¥frac{4n-1}{n-1})^{1/n}-(4n-1)}<¥sqrt{2}-¥frac{3}{/¥overline{17}}$
,
which is equivalent to
(3. 2)
$4(n-1)(¥frac{4n-1}{n-1})^{1/n}-(4n-1)<¥frac{9(43-6¥sqrt{34})}{17}$
.
Setting
(3. 3)
$n=2+u$
$(0¥leqq u¥leqq 1)$
,
the above inequality can be written as
(3. 4)
Remark.
have
$(1+u)(¥frac{7+4u}{1+u})^{1/(2+u)}-u<¥frac{253-27¥sqrt{34}}{34}$
.
Regarding the inequality (3. 4), by numerical computation we
On a Bound for Periods of Solutions of a Certain Nonlinear Diflerential Equation
.
$¥frac{253-27¥sqrt{34}}{34}=.$
2. 81071467
199
$¥cdots$
and
$[(1+u)(¥frac{7+4u}{1+u})^{1/(2+u)}-u]_{u=0}=¥sqrt{7}¥mathrm{i}$
.
2. 64575131
.
$[(1+u)(¥frac{7+4u}{1+u})^{1/(2+u)}-u]_{u=1}=(44)^{1/¥theta}-1=.$
$¥cdots$
,
2. 53034833
$¥cdots$
.
Proof of $¥sqrt{F(x)}+¥sqrt{F(X_{n}(x))}<¥sqrt{2}$ , when ¥ ¥
.
In this section, we shall prove the inequality (1. 10), when
. By
Lemma 2. 3 and the argument in §3, it suffices to prove the inequality (3. 4).
Proposition 4. 1.
§4.
$2 leqq n leqq 3$
$2¥leqq n¥leqq 3$
Proof.
$0<u<1$ .
for
$(1+u)(¥frac{7+4u}{1+u})^{1/(2+u)}-u<¥sqrt{7}$
We rewrite the above inequality as
(4. 1)
$(¥frac{7+4u}{1+u})^{1/(2+u)}<¥frac{¥sqrt{7}+u}{1+u}$
,
which is equivalent to
(4. 2)
$¥frac{1}{2+u}¥log¥frac{7+4u}{1+u}<¥log¥frac{¥sqrt¥overline{7}+u}{1+u}$
.
Since we have
$¥log¥frac{¥sqrt¥overline{7}+u}{1+u}=¥frac{1}{2}¥log 7¥mp¥dagger¥log(^{¥mathrm{Y}}1-_{¥frac{7-¥sqrt¥overline{7}}{7}}¥cdot¥frac{u}{1+u})$
$0-<¥frac{7-¥sqrt¥overline{7}}{7}¥cdot¥frac{u}{1+u}<¥frac{7-¥sqrt{7}}{14}$
for
,
$0<u<1$
and
.
$¥frac{7-¥sqrt¥overline{7}}{14}=.$
0. 31101776 ,
$¥cdots$
we provide here the following
Lemma 4. 2.
$¥log(1-t)>-t-¥frac{7-¥sqrt{7}}{6}t^{2}$
Proof.
for
$¥mathrm{o}<t<¥frac{7-J¥overline{7}}{14}$
.
The both sides of the above inequality take the value 0 at
$(¥log(1-t)+t+¥frac{7-¥sqrt{7}}{6}t^{2})^{¥prime}=-¥frac{1}{1-t}+1+¥frac{7-¥sqrt{7}}{3}t$
$t=0$
and
T. OTSUKI
.’00
for
$=¥frac{t}{3(1-t)}¥{4-¥sqrt{7}-(7-¥sqrt{7})t¥}>0$
$¥mathrm{o}<t<¥frac{7-¥sqrt{7}}{14}$
.
Hence we have
$¥log(1-t)>-t-¥frac{7-¥sqrt{7}}{6}t^{2}$
for
$¥mathrm{o}<t<¥frac{7-¥sqrt{7}}{14}$
.
Q. E. D.
By this lemma, we get the following inequality:
$u^{2}$
$¥log¥frac{¥vee¥overline{7}+u}{1+u}>¥frac{1}{2}¥log 7-¥frac{7-¥sqrt{7}}{7}¥cdot¥frac{u}{1+u}-¥frac{7-¥vee¥overline{7}}{6}(7-¥sqrt{7})^{2}$
$.¥overline{7^{2}}.¥overline{(1+u)^{2}}$
$¥mathrm{i}.¥mathrm{e}$
’
.
(4. 3)
$¥log¥frac{¥sqrt{7}+u}{1+u}>¥frac{1}{2}¥log 7-¥frac{7-¥sqrt{7}}{7}¥cdot¥frac{u}{1+u}-¥frac{35-11/¥overline{7}}{21}¥cdot¥frac{u^{2}}{(1+u)^{2}}$
for $0<u<1$ .
Thus, the inequality (4. 2) is true for $0<u<1$ , if we can prove that
(4. 4)
$(2+u)¥{¥frac{1}{2}¥log 7-¥frac{7-J¥overline{7}}{7}¥cdot¥frac{u}{1+u}-¥frac{35-11¥sqrt{7}}{21}¥cdot¥frac{u^{2}}{(1+u)^{2}}¥}>¥log¥frac{7+4u}{1+u}$
for $0<u<1$ .
The both sides of (4. 4) take the value log7 at
the left hand side is equal to
$u=0$
and the derivative of
$¥frac{1}{2}¥log 7-¥frac{7-¥sqrt{7}}{7}¥cdot¥frac{u}{1+u}-¥frac{35-11/¥overline{7}}{21}¥cdot¥frac{u^{2}}{(1+u)^{2}}$
$-(2+u)¥{¥frac{7-/¥overline{7}}{7}¥cdot¥frac{1}{(1+u)^{2}}+¥frac{35-11/¥overline{7}}{21}¥cdot¥frac{2u}{1+u}¥cdot¥frac{1}{(1+u)^{2}}¥}$
$=¥frac{1}{2}1o¥mathrm{g}7-¥frac{7-¥sqrt{7}}{7}+¥frac{7-¥sqrt{7}}{7}¥cdot¥frac{1}{1+u}-¥frac{35-11¥sqrt{7}}{21}+¥frac{70-22¥sqrt{7}}{21}¥cdot¥frac{1}{1+u}$
$-¥frac{35-11¥sqrt{7}}{21}¥cdot¥frac{1}{(1+u)^{2}}-¥frac{7-¥sqrt{7}}{7}¥cdot¥frac{1}{1+u}-¥frac{7-¥sqrt{7}}{7}¥cdot¥frac{1}{(1+u)^{2}}$
$-¥frac{70-22¥sqrt{7}}{21}¥cdot¥frac{1}{1+u}+¥frac{70-22¥sqrt{7}}{21}¥cdot¥frac{1}{(1+u)^{3}}$
$---¥frac{1}{2}¥log 7-¥frac{2(4-¥sqrt{7})}{3}-¥frac{2(4-¥sqrt{7})}{3}¥cdot¥frac{1}{(1+u)^{2}}+¥frac{70-22¥vee¥overline{7}}{21}¥cdot¥frac{1}{(1+u)^{3}}$
Hence (4. 4) is true for $0<u<1$ , if we can prove that
(4. 5)
$¥frac{1}{2}¥log 7-¥frac{2(4-J¥overline{7})}{3}+¥frac{1}{1+u}-¥frac{2(4-¥sqrt{7})}{3}¥cdot¥frac{1}{(1+u)^{2}}$
$+¥frac{70-22¥sqrt{7}}{21}¥cdot¥frac{1}{(1+u)^{3}}>¥frac{4}{7+4u}$
Now, for simplicity, setting
for
$¥mathrm{o}<u<1$
.
.
On a Bound for Periods of Solutions of a Certain Nonlinear Diflerential Equation
$¥frac{1}{1+u}=s$
201
,
and
(4. 6).
$¥lambda(s):=b_{0}+s-b_{2}s^{2}+b_{3}s^{3}$
,
$¥mu(s):=¥frac{4s}{4+3s}$
,
where
.
$b_{0}=¥frac{1}{2}¥log 7-¥frac{2(4-¥sqrt{7})}{3}=.$
$b_{2}=¥acute{3}=.$. 0.
.
$b_{3}=¥frac{70-22¥sqrt{7}}{21}=.$
0. 07012256 ,
90283245
$¥cdots$
$¥cdots$
,
0. 56159386 ,
$¥cdots$
(4. 5) will be true, if we can prove the following
Lemma 4. 3. $¥lambda(s)>¥mu(s)$
for $1/2<s<1$ .
Proof. Since we have
$¥lambda^{¥prime}(s)=1-2b_{2}s+3b_{3}s^{2}$
and
$b_{2}^{2}-3b_{3}<1^{2}-3¥cdot¥frac{1}{2}=-¥frac{1}{2}<0$
,
the polynomial
of order 3 in has no maximal and minimal values.
be the value of
which gives the point of inflexion of the graph of
Then, we have
$¥lambda(s)$
$s$
$s$
$s_{0}$
(4. 7)
$s_{0}=¥frac{b_{2}}{3b_{3}}=_{¥frac{2(4-¥sqrt{7})}{9}}¥cdot¥frac{21}{70-22/¥overline{7}}=¥frac{7(4-¥vee¥overline{7})}{3(35-11/¥overline{7})}$
.
$=¥frac{7+¥sqrt{7^{¥neg}}}{18}=.$
0. 53587507 ,
$¥cdots$
and so
$¥frac{1}{2}<s_{0}<1$
.
Now we have
$¥min_{-¥infty<s<¥infty}¥lambda^{¥prime}(s)=¥lambda^{¥prime}(s_{0})=1-2b_{2}s_{0}+3b_{3}s_{0}^{2}$
$=1-2b_{2}¥cdot¥frac{b_{2}}{3b_{3}}+3b_{3}¥cdot¥frac{b_{2}^{2}}{9b_{3}^{2}}=1-¥frac{b_{2}^{2}}{3b_{3}}$
$=1-¥acute{9(35-11/¥overline{7})}=¥frac{2+¥sqrt{7}}{9}$
,
Let
$¥lambda(s)$
.
T. OISUKI
202
that is
(4. 8)
.
$¥min_{-¥infty<s<¥infty}¥lambda^{¥prime}(s)=¥lambda^{¥prime}(s_{0})=¥frac{2+¥vee¥overline{7}}{9}=.$
0. 51619459
$¥cdots$
.
On the other hand, we have
(4. 9)
,
$¥mu^{¥prime}(s)=¥frac{16}{(4-¥vdash 3s)^{2}}$
which is monotone decreasing in $1/2<s<1$ .
Here, we divide the interval $1/2<s<1$ into the subintervals
$s_{0}¥leqq s<1$
For
$1/2<s_{=}^{/_{¥backslash }}s_{0}$
.
$1/2<s¥leqq s_{0}$
, using (4. 8) and (4. 6) we have
$¥lambda(s)-¥mu(s)=¥int_{1/2}^{s}¥{¥lambda^{¥prime}(s)-¥mu^{¥prime}(s)¥}ds+¥lambda(¥frac{1}{2})-¥mu(¥frac{1}{2})$
$>l_{/2}^{s}¥{¥lambda^{¥prime}(s_{0})-¥mu^{¥prime}(¥frac{1}{2})¥}ds+¥lambda(¥frac{1}{2})-¥mu(¥frac{1}{2})$
$=¥{¥lambda^{¥prime}(s_{0})-¥mu^{¥prime}(¥frac{1}{2})¥}(s-¥frac{1}{2})+¥lambda(¥frac{1}{2})-¥mu(¥frac{1}{2})$
.
Since we have
$¥mu^{¥prime}(¥frac{1}{2})=¥frac{64}{121}$
,
$¥lambda^{¥prime}(s_{0})-¥mu^{¥prime}(¥frac{1}{2})=¥frac{2+¥sqrt{7}}{9}-¥frac{64}{121}=-¥frac{334-121¥vee¥overline{7}}{9¥times 121}$
.
$=.$
?0. 01273102 ,
$¥cdots$
$¥lambda(¥frac{1}{2})=b_{0}+¥frac{1^{¥mathrm{x}}}{2}-¥frac{1}{4}b_{2}+¥frac{1}{8}b_{3}=¥frac{1}{2}¥log 7+¥frac{1}{2}-¥frac{5}{4}b_{2}+¥frac{1}{8}b_{3}$
$=¥frac{1}{2}¥log 7^{1}-¥ulcorner¥frac{1}{2}-¥frac{5}{4}.¥frac{2(4-¥sqrt{7})}{3}+¥frac{1}{8}.-¥frac{2(35-11¥sqrt{7})}{21}$
$=¥frac{1}{2}1¥mathrm{o}g7-¥frac{203-59¥parallel¥overline{7}}{84}=..0.41461368¥cdots$
.
$¥mu(¥frac{1}{2})=¥frac{4}{11}¥mathrm{i}$
0. 36363636
,
$¥cdots$
and
.
$¥lambda(¥frac{1}{2})-¥mu(¥frac{1}{2})=¥frac{1}{2}¥log 7-¥frac{203-59/¥overline{7}}{84}-¥frac{4}{11}¥mathrm{i}$
0. 05097732 ,
$¥cdots$
we get from the above inequality and (4. 6)
$¥lambda(s)-¥mu(s)>¥{¥lambda^{¥prime}(s_{0})-¥mu^{¥prime}(¥frac{1}{2})¥}(s_{0}-¥frac{1}{2})+¥lambda(¥frac{1}{2})-¥mu(¥frac{1}{2})$
$>-0.013¥times 0.04+0.05=0.04948>0$ ,
and
On a Bound for Periods of Solutions of a Certain Nonlinear Diflerential Equation
203
hence
$¥lambda(s)>¥mu(s)$
Next, for
have
$s_{0}¥leqq s<1$
for
$¥frac{1}{2}<s¥leqq s_{0}$
.
, we shall prove the above inequality.
Using (4. 7), we
$¥mu^{¥prime}(s_{0})=¥frac{16}{(4+3s_{0})^{2}}=¥frac{16¥times 36}{(31+¥sqrt{7})^{2}}=¥frac{32¥times(484-31¥sqrt{7})}{159¥times 159}$
.
$=.$
0. 50881747 ,
$¥cdots$
and hence by (4. 8) we obtain
$¥lambda^{¥prime}(s_{0})>¥mu^{¥prime}(s_{0})$
Thus, for
$s_{0}¥leqq s<1$
.
, we have
$¥lambda^{¥prime}(s)-¥mu^{¥prime}(s)¥geqq¥lambda^{¥prime}(s_{0})-¥mu^{¥prime}(s_{0})>0$
.
We have
$¥lambda(s_{0})=b_{0}+s_{0}-b_{2}s_{0}^{2}+b_{3}s_{0}^{3}$
$=¥frac{1}{2}¥log 7+¥frac{7+¥vee¥overline{7}}{18}-¥check{3}¥cdot¥{1+¥frac{(7+¥sqrt{7})^{2}}{18¥times 18}¥}$
$+¥frac{2(35-11¥sqrt{7})}{21}¥cdot¥frac{(7¥dashv-¥sqrt{7})^{3}}{18¥times 18¥times 18}$
$=¥frac{1}{2}¥log 7-¥frac{41-13¥sqrt{7}}{18}$
$+¥{¥frac{(35-11¥sqrt{7})(7+¥sqrt{7})}{21¥times 9}-¥frac{2(4-¥vee¥overline{7})}{3}¥cdot¥frac{56+14¥sqrt{7}}{18¥times 18}$
$=¥frac{1}{2}¥log 7-¥frac{41-13/¥overline{7}}{18}-¥frac{14}{81}$
$¥mathrm{i}.¥mathrm{e}$
,
.
(4. 10)
$¥lambda(s_{0})=¥frac{1}{2}¥log 7+¥frac{13/¥overline{7}}{18}-¥frac{397}{162}=..0.43315813¥cdots$
,
and
(4. 11)
.
$¥mu(s_{0})=¥frac{4s_{0}}{3s_{0}+4}=¥frac{28}{3(35-4/¥overline{7})}=¥frac{28(35+4/¥overline{7})}{3¥times 1113}=.$
0. 38224742 .
$¥cdots$
From (4. 10) and (4. 11), we obtain
$¥lambda(s_{0})>¥mu(s_{0})$
.
Now, using the above equalities and inequalities, we obtain for
$¥lambda(s)-¥mu(s)=¥int_{s_{0}}^{s}¥{¥lambda^{¥prime}(s)-¥mu^{¥prime}(s)¥}ds+¥lambda(s_{0})-¥mu(s_{0})$
$s_{0}¥leqq s<1$
204
T. oTSUKI
$¥geqq¥int_{s_{0}}^{s}¥{¥lambda^{¥prime}(s_{0})-¥mu^{¥prime}(s_{0})¥}ds+¥lambda(s_{0})-¥mu(s_{0})$
$=¥{¥lambda^{¥prime}(s_{0})-¥mu^{¥prime}(s_{0})¥}(s-s_{0})+¥lambda(s_{0})-¥mu(s_{0})>0$
,
hence
$¥lambda(s)>¥mu(s)$
for
$s_{0}¥leqq s<1$
.
Q. E. D.
Thus, by the above argument we have completed the proof of Proposition
4. 1. Accordingly, by virtue of Lemma 2. 3, Proposition 4. 1 and the argument
in §3, we obtain the following.
, we have the following inequality:
Theorem 1. When
$2¥leqq n¥leqq 3$
for
$¥sqrt{F(x)}+¥sqrt{F(X_{n}(x))}<¥sqrt{2}$
$0<x<1$ .
Since we have always
$¥int_{x_{0}}^{1}¥frac{d¥varphi(x)}{¥sqrt{(B-¥varphi(x))(¥varphi(x)-C)}}=¥pi$
regarding (1. 9), we obtain
$ T_{n}(x_{0})<¥sqrt{2}¥pi$
for
and $0<x_{0}<1$ , by means of Theorem 1 when
Theorem in [8] when
. Thus, we have
given by
Theorem 2. The period function
inequa fity:
$n¥geqq 2$
$2^{¥prime}¥cong n¥leqq 3$
and the Main
$3¥leqq n$
(1. 1) satisfies the
$T_{n}(x_{0})$
$¥pi<T_{n}(x_{0})<¥sqrt{2}¥pi$
for
$n¥geqq 2$
and
$0<x_{0}<1$
Proof of
§5.
Lemma 5. 1.
$X_{n}(x_{0})$
.
$ T<¥sqrt{2}¥pi$
when
$1<n¥leqq 2$
$T_{n}(x_{0})=T_{m}(y_{0})$
,
where
.
$m=¥frac{n}{n-1}$
,
$y_{0}=m-(m-1)x_{1}$ ,
$x_{1}=$
.
Proof.
The period function
(5. 1)
$T_{n}(x_{0})$
is given by (1. 1)
$T_{n}(x_{0})=¥int_{x_{0}}^{x_{1}}¥frac{dx}{/¥overline{x(n-x)-Cx^{1-1/n}(n-x)^{1/n}}}$
,
where
$C=x_{0}^{1/n}(n-x_{0})^{1-1/n}=x_{1}^{1/n}(n-x_{1})^{1-1/n}$
and
$0<x_{0}<1<x_{1}<n$ .
Supposing
$n>1$ ,
we have $m=n|/(n-1)>1$ .
Changing the integral parameter
On a Bound for Periods of Solutions of a Certain Nonlinear Diflerential Equation
$x$
in (5. 1) to
$y$
205
by
$y=m-(m-1)x$ ,
then we have
$x=¥frac{m-y}{m-1}$
,
$n-x=¥frac{y}{m-1}$
and
$¥underline{-dy}$
$T_{n}(x_{0})=¥int_{y_{1}}^{y0}¥frac{m-1}{/^{¥frac{}{¥frac((m-y)ym-1)^{2}-C(¥frac{m-y}{m-1})^{1/m}(¥frac{y}{m-1})^{¥perp-1/m}}}}$
,
$=¥int_{y0}^{y_{1}}¥frac{dy}{¥mathit{1}¥overline{y(m-y)-(m-1)Cy^{1-1/m}(m-y)^{1/m}}}=T_{m}(y_{0})$
where
$y_{0}=m-(m-1)x_{1}$ ,
$y_{1}=m-(m-1)x_{0}$ .
We have easily
,
$(m-1)C=y_{0}^{1/m}(m-y_{0})^{1-1/m}=y_{1}^{1/m}(m-y_{1})^{1-1/m}$
$0<y_{0}<1<y_{1}<m$ .
In Lemma 5. 1, suppose
$1<n¥leqq 2$
, then
$m¥geqq 2$
.
Q. E. D.
Hence by Theorem 2 we have
$ T_{n}(x_{0})=T_{m}(y_{0})<¥sqrt{2}¥pi$
Thus, from Theorem 2 and Lemma 5. 1 we obtain finally the following
Main Theorem. For $1<n$ and $0<x_{0}<1$ , we have
$¥pi<¥int_{x¥mathrm{o}}^{x_{1}}¥frac{dx}{/¥overline{x(n-x)-}C¥overline{x^{1-1/n}(n-x)}^{1¥overline{/n}}}</¥overline{2}¥pi$
,
where
$C=x_{0}^{1/n}(n-x_{0})^{1-1/n}=x_{1}^{1/n}(n-x_{1})^{1-1/n}$
,
$1<x_{1}<n$
.
References
[1] S. S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere
with second fundamental foom of constant length, Functional Analysis and
Related Fields, Springer-Verlag, 1970, 60-75.
[2] S. Furuya, On periods of periodic solutions of a certain nonlinear differential
equation, Japan-United States Seminar on Ordinary Differential and Functional
Equations, Springer-Verlag, 1972, 320-323.
[3] Wu-Yi Hsiang and H. B. Lawson, Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geometry, 5 (1970), 1-38.
[4] T. Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curva-
T. OTSUKI
206
J. Math., 92 (1970), 145-173.
T. Otsuki, On integral inequalities related with a certain nonlinear differential
equation, Proc. Japan Acad., 48 (1972), 9-12.
T. Otsuki, On a 2-dimensional Riemannian manifold, Differential Geometry, in
honor of K. Yano Kinokuniya, Tokyo, 1972, 401-414.
T. Otsuki, On a family of Riemannian manifolds defined on an m-disk, Math.
J. Okayama Univ., 16 (1973), 85-97.
T.Otsuki, Onabound for periods of solutions of a certain nonlinear differential
equation (I), J. Math. Soc. Japan, 26 (1974), 206-233.
M.Maeda and T. Otsuki, Models of the Riemannian manifolds 0 n2 in the Lorentzian 4-space, J. Differential Geometry, 9 (1974), 97-108.
M. Urabe, Computations of periods of a certain nonlinear autonomous oscillations, Study of algorithms of numerical computations, Surikaiseki Kenkyusho
Kokyu-roku, 149 (1972), 111-129 (in Japanese).
ture, Amer.
[5]
[6]
[7]
[8]
[9]
[10]
nuna adreso:
Department of Mathematics
Tokyo Institute of Technology
Oh-okayama, Meguro-ku, Tokyo
Japan
(Ricevita la 11-an de Oktobro, 1973)