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Construction of solutions $f'(x) = af(\lambda x),\lambda >$ 1.(Banach spaces, function spaces, inequalities and their applications)

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(1)

Construction of

solutions

$f’(x)=af(\lambda x),$ $\lambda>1$

.

東大数理

米田剛

(Tsuyoshi

Yoneda)

Graduate School of Mathematical

Sciences

The University

of

Tokyo

1. MAIN

RESULT

We denote

by

$C^{\infty}(\mathbb{R})$

the

set of all infinitely differentiable functions

on

$\mathbb{R}$

.

Let

$C_{0}^{\infty}(\mathbb{R})=\{f\in C^{\infty}(\mathbb{R})$

:

$\frac{d^{p}f}{dx^{p}}(x)arrow 0$

as

$|x|arrow\infty,$ $P=0,1,2,$ $\cdots\}$

,

$C_{comp}^{\infty}(\mathbb{R})=$

{

$f\in C^{\infty}(\mathbb{R})$

:

$f$

has

a

compact

support}.

We denote by

$x_{[r,\ell)}$

the

characteristic

function of

the

interval

$[r, s$).

Let

$X$

be the set of

all step

functions

(1.1)

$p= \sum_{j=1}^{m}c_{j}\chi_{[r_{j-1},r_{j})}$

,

such that

$m=1,2,$$\cdots$

and

$c_{j}\in \mathbb{R}$ $(j=1,2, \cdots m)$

,

$c_{1}=1$

,

$r_{0}=0$

,

$0<r_{1}<r_{2}<\cdots<r_{m}<\infty$

,

$\sum_{j=1}^{m}c_{j}(r_{j}-r_{j-1})=1$

.

We note that

$\int_{\mathbb{R}}p(x)dx=1$

for

$p\in X$

.

For

$\lambda>1$

and

$p\in X$

,

we

define

an

operator

$T=T_{\lambda,p}$

:

$L^{1}(\mathbb{R})arrow$

$L^{1}(\mathbb{R})$

(by

[17]

(p.

149, Remark

2)

and

[2],

we

can

say

that

$T_{\lambda,p}$

:

$B_{1,\infty}^{8}(\mathbb{R})arrow B_{1,\infty}^{\epsilon+1}(\mathbb{R})$

for

$s\in \mathbb{R}$

,

since

$\lambda p(\lambda\cdot)\in B_{1,\infty}^{1}(\mathbb{R}))$

as

follows:

(1.2)

$Tf(x)=T_{\lambda,p}f(x)=\lambda(p*f)(\lambda x)$

.

We

note that

$Tf\in L^{1}(\mathbb{R})\cap C(\mathbb{R})$

, since

$f\in L^{1}(\mathbb{R})$

and

$p\in L_{\infty mp}^{\infty}(\mathbb{R})$

.

If

$f\in B_{1,\infty}^{s}(\mathbb{R})$

we

have

$\hat{f}(1+|\cdot|^{2})^{s/2}\in L^{\infty}(\mathbb{R})$

(see

[15]).

Therefore

(2)

Theorem 1.1.

For

$\lambda>1$

and

$p\in X$,

let

$T$

be

the

operator

defined

by

(1.2).

Then there exists

a

function

$u=u_{\lambda,p}\in C_{comp}^{\infty}(\mathbb{R})$

such that,

for

all

$f\in L^{1}(\mathbb{R})$

(or

for

all

$f \in\bigcup_{s\in \mathbb{R}}B_{1,\infty}^{s}(\mathbb{R})_{f}$

which

function

set

is

strt

ctly

bigger

than

$L^{1}(\mathbb{R}))$

,

$\lim_{\ellarrow\infty}T^{\ell}f(x)=c_{f}u(x)$

uniformly with

resPect

to

$x\in \mathbb{R}$

,

where

$c_{f}=\hat{f}(0)$

.

Mooeover,

(1)

$Tu=u$

;

(2)

$\int_{R}u(x)dx=1$

;

(3)

$\frac{d}{dx}pu(0)=0k$

for

all

$k=0,1,2,$$\cdots$

;

(4)

if

$p\geq 0_{f}$

then

$u\geq 0$;

and,

(5)

if

supp

$p\subset[0, r]$

, then supp

$u\subset[0, r/(\lambda-1)]$

.

In the

above

statement supp

$f$

denotes

the support

of

$f$

.

Remark 1.1. If

$p_{1}\neq p_{2}$

,

then

$u_{\lambda,p_{1}}\neq u_{\lambda,p_{2}}$

.

2.

APPLICATION

To construct solutions for

the

$foUowing$

equation,

(2.1)

$\{\begin{array}{ll}f’(x)=\lambda^{2}f(\lambda x), x\in \mathbb{R},f(0)=0, \end{array}$

we

use

$u=u_{\lambda,p}$

in

Theorem

1.1 with

$\lambda>1$

and

$p= \sum_{j=1}^{m}c_{j}\chi[rr$ ) $\in$

$X$

, and

we

define

a

sequence

of functions

$\{v_{\ell}\}$

inductively

as

follows:

$\{\begin{array}{ll}v_{0}(x)=u(x)+\sum_{j=1}^{m}\overline{c}_{j}u(x-r_{j}), v_{\ell+1}(x)=v_{\ell}(x)+\sum_{j=1}^{m}\tilde{c}_{j}v_{\ell}(x-\lambda^{l+1}r_{j}), \ell=0,1,2, \cdots,\end{array}$

where

$\tilde{c}_{j}=-(c_{j}-c_{j+1}),$ $j=1,2,$ $\cdots m-1,\tilde{c}_{m}=-c_{m}$

.

Then

$v_{\ell}\in C_{\infty mp}^{\infty}(\mathbb{R}),$ $\ell=0,1,2,$ $\cdots$

,

and

$vp(x)=v_{\ell+1}(x)$

,

$x\in(-\infty, \lambda^{\ell+1}r_{1}$

],

$\ell=0,1,2,$ $\cdots$

.

Therefore

we

get

a

function

$f=f_{\lambda p}\in C^{\infty}(\mathbb{R})$

such

that

$f(x)=v_{\ell}(x)$

,

$x\in(-\infty, \lambda^{\ell+1}r_{1}$

],

$\ell=0,1,2,$ $\cdots$

.

(3)

Theorem 2.1.

Let

$\lambda>1$

.

For every

$p\in X$

, the

function

$f=f_{\lambda,p}$

given

by the

above

method

is

a

solution

for

(2.1).

Remark

2.1.

By the

property

(5) in Theorem 1.1 and the

definition

of

$v_{\ell}$

,

supp

$u\subset[0, r_{m}/(\lambda-1)]$, $supp(v_{0}-u)\subset[r_{1}, r_{m}\lambda/(\lambda-1)]$

,

supp

$v_{\ell}\subset[0, r_{m}\lambda^{\ell+1}/(\lambda-1)]$

,

$supp(v_{\ell+1}-v_{\ell})\subset[r_{1}\lambda^{l+1},r_{m}\lambda^{\ell+2}/(\lambda-1)]$

.

Therefore,

if

$r_{m}/(\lambda-1)<r_{1}$

,

then

$f(x)=0$

on

$[r_{m}\lambda^{\ell}/(\lambda-1), r_{1}\lambda^{\ell}]$

,

$P=0,1,2,$ $\ldots$

, and

$f$

is

bounded

(See

the

graph (S12)). Let

$n_{4}$

$p_{i}= \sum_{j=1}c_{i,j}\chi_{[r:_{l-1r_{i.j})}}.,\in X$

,

$i=1,2$

.

If

$p_{1}\neq p_{2}$

and

there

exists

$R>0$

such

taht

$r_{i,m_{1}}/(\lambda-1)<R<r_{i,1}$

,

$i=1,2$

,

then

$f_{\lambda,p_{1}}\neq f_{\lambda,p_{2}}$

by

Remark 1.1, while

$\int_{0}^{R}f_{\lambda,ps}(x)dx=$

$\int_{0}^{R}u_{\lambda,p:}(x)dx=1,$ $i=1,2$

.

Remark 2.2. In the definition

(1.1),

let

$m=\infty$

with

the

condition

$\sum_{j=1}^{\infty}|c_{j+1}-c_{j}|<\infty$

.

Then the derivative of

$p$

in

the

sense

of

dis-tribution

is

a

finite Radon

measure

and

the

Fourier

transform of

the

derivative is

an

almost

periodic

function. With

some

conditions

we

can

ako

construct solutions for

(2.1)

from such functions. For the Fourier

preimage

of the

space

of

all

finite

Radon measures, see, for

example,

$[6, 7]$

.

In the rest of this

section,

we

give graphs

of

solutions for

$f’(x)=$

$4f(2x),$ $f’(x)=(3/2)^{2}f(3x/2)$

and

$f’(x)=9f(3x)$

with

$f(O)=0$

.

Let

$p$

be

as

in

Table

1.

Then, calculating

$T_{\lambda,p}^{\ell}\chi_{[0_{y}1)}$

numerically

by

computer,

we

get

the graphs of

solutions

$(S1)-(S7)$

in Figure 1,

$(S8)-$

(Sll)

in Figure

2 and

$(S12)-(S13)$

in Figure

3.

REFERENCES

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esov

spaces, Proc. SteklovInst. 3(1990),

35-44

(4)

(S1)

(S2)

(S3)

(S4)

(S5)

FIGURE 1.

Solutions

of

$f’(x)=4f(2x)$

.

(5)

TABLE 1.

$\lambda,$ $p$

and the

solution

$f_{p,\lambda}$

.

FIGURE 2.

Solutions

of

$f’(x)=(3/2)^{2}f(3x/2)$

.

(S13)

(6)

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a $cu v\epsilon nt$ collection $sy\epsilon tem$

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(7)

GRADUATESCHOOL OFMATHEMATICALSCIENCES, THEUNIVERSITYOFTOKYO,

3-8-1 KOMABA, MEGURO-KU TOKYO 153-8914, JAPAN E-mail address: yonedadms.u-tokyo.ac.jp

TABLE 1. $\lambda,$ $p$ and the solution $f_{p,\lambda}$ .

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