数理解析研究所講究録 1445 巻 2005 年 1-11

1

A solution of the equation $f’(x)=\lambda^{2}f(\lambda x), \lambda>1$ .
米田 剛 (Tsuyoshi Yoneda)

大阪教育大学

Department of Mathematics
Osaka Kyoiku University

1. INTRODUCTION
The purpose of this paper is to give solutions for the functional-differential equa-tion of advanced type$f’(x)=\lambda^{2}f(\lambda x)$

(1.1)

$\{$

,

$x\in \mathbb{R}=(-\infty, +\infty)$,$f(0)=0$ ,where A is a constant, A $>1$ . Our solutions are infinitely differentiable on R.
Moreover, if$\lambda\geq 2$, then the solutions are bounded and have arbitrarily large zeros.

Our methods give numerical data readily.

Frederickson $[1, 2]$ (1971) investigated functional-differential equations of advanced type
(1.2)

$f’(x)=af$ (Ar)$+\lambda f(x\rangle$,

here A $>1$ , and proved several properties of solutions. Later, Kato and McLeod [5](1971) and Kato [4] (1972) studied the asymptotic behaviour of solutions of (1.2).
Frederickson [1] provided a global existence theorem for equations$f’(x)=F(f(2x))$ ,

where

$F$

$x\in \mathbb{R}$,is an odd, continuous function with $F(s)>0$ for $s>0$ , by applicationof the Schauder fixed point theorem. He showed that the absolute value of thesolution $|f(x)|$ is periodic for $x\geq 0$ . Frederickson [2] also provided a constructive
method for solutions for equations$f’(z)=af(\lambda z)+bf(z)$ ,2where ,
$a$$b\in \mathbb{C}$and A $>1$ . He further gave solutions in the form of a Diriclet series$\varphi(z, \beta)=\sum_{n\in \mathbb{Z}}c_{n}e^{\beta\lambda^{n}z}$where$\beta$,$\Re(\beta z)\leq 0$,

is allowed to vary as a parameter. In the case of $b=0$ andsolution is analytic in the upper half plane lsz $>0$ , continuous online$s^{\alpha}z=0$$\beta=\mathrm{i}$$s^{\infty}z\geq 0$, the, and theis a natural boundary, From his result it follows that our solutions of(1.1) cannot be real analytic,
Ivanov, Kitamura, Kusano and Shevelo [3] (1982) investigated the higher orderfunctional-differential equations of the form
(1.3)

$f^{\{n)}(x)=p(x)F(f(g(x)))$ ,where ,
$p$

$F$

and

$g$

satisfy appropriate conditions. Kusano [6] (1984) also investigated

the functional differential equation(1.4)$f^{(n)}(x)=p(x)f(g(x))$

where

$n$

is even,$p$:$[0, \infty)arrow \mathbb{R}$is nondecreasing and

and

$g$

$\lim_{t\prec\infty}g(t)=\infty$:$[0, \infty)arrow \mathbb{R}$are continuous, $p(t)>0$ , $g(t)$. They $[3, 6]$ gave sufficient conditions thatthe solutions are oscillatory.
If
(1.5)

$f$

is a solution of (1.1), then$f$is also a solution of the equations$f”(x)=\lambda^{4}\lambda f(\lambda^{2}x)$,$x\in \mathbb{R}\}$

and

(1.6)

$f’(x)=\lambda^{6}\lambda^{3}f(\lambda^{3}x)$,$x\in$

R.

However, (1.5) and (1.6) don’t satisfy the sufficient conditions in $[3, 6]$ .
Recently, the author [9] constructed solutions of (1.1) with$\lambda=2$

by using a little

different method from this paper.
We state the main theorem (Theorem 2.3) and application in next section. We caneasly apply the solution for the case$\lambda=2$

to Friedrichs’ molifier theorem and we canrewrite de ferential operator. For the proof of main theorem, see [10]. In the third
section, we give graphs of solutions for the case of A $=4,3,2$ , 31/16, 15/8, 7/4, 2/3, 5/4.

3

In the last section, we will give Mathematica programs.

2. MAIN RESULTSFirst, we state two lemmas. Let$\hat{f}(\xi)=\mathcal{F}[f](\xi)=\int_{\mathbb{R}}f(x)e^{-ix\xi}dx$,$\mathcal{F}^{-1}[f](\xi)=\frac{1}{2\pi}\int_{\mathbb{R}}f(x)e^{ix\xi}dx$

andgive $\xi=\{$,$\xi=0$ .

,$\sin(\pi\xi)/(\pi\xi)$$\xi\neq 0$

1,

Lemma 2.1. The product$\prod_{k=1}^{\infty}$

is converges pointwise and ingive$L^{1}(\mathbb{R})$$( \frac{\xi}{2\lambda^{k}\pi})$,$\xi\in \mathbb{R}$.

Lemma 2.2, Let
(2.2)

Then

$u=\mathcal{F}^{-1}[U]$$u$,$U( \xi)=\exp(-\frac{\mathrm{i}\xi}{2(\lambda-1)})\prod_{k=1}^{\infty}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}$ $( \frac{\xi}{2\lambda^{k}\pi})$

has the following properties:$u\in C^{\infty}(\mathbb{R})$$u(x)>0$ for $x\in(0,$$\frac{1}{\lambda-1})$,, $u(x)$ $=0$ ffoorr $x\not\in(0,$$\frac{1}{\lambda-1})$$u(x)=u(1/(\lambda-1)-x)$ ,$\int_{\mathbb{R}}u(x)dx=1$

,

and
(2.2)

for $x\in[0,$$u’(x)=\lambda^{2}u(\lambda x)$

Let we define the operator(2.3)
Then the function
(2.4)

.

$T$

:

$L^{1}arrow L^{1}$$\min(\frac{1}{\lambda}$.

as follows.
,$Tf(x)=\lambda(\chi_{[0_{2}1]}*f)(\lambda x)$$u$$\frac{1}{\lambda(\lambda-1)})\ovalbox{\tt\small REJECT}$}$f\in L^{1}$.

in Lemma 2.2 is given by the following equation.
$u=$$\lim_{k}$$T^{k}\chi_{[0,1]}$

.

,

,

4

Secondly, we define sequences$n_{2}=1$$n_{1}=0_{7}$

(2.5)

$\{$

and

$\{n_{k}\}_{k=1}^{\infty}$$\{y_{k}\}_{k=1}^{\infty}$

as follows:$n_{k}=1$$(k\geq 2)$,, $n_{2k}=0$ ,$n_{2k-1}=0$ , $n_{2k}=1$ ,ifif$n_{2k-1}=1$,$(k\geq 2)$ ,$n_{k}=0$

and
(2.6)

$y_{k}= \sum_{l=1}^{\infty}C_{k,l}\lambda^{l-1}$$k=1$ , 2,,,

$3_{7}\cdots$

where $C_{k,l}\in\{0, 1\}$ $(l=1,2, 3, \cdots)$ are coefficients of the binary system such that$k-1= \sum_{l=1}^{\infty}C_{k,l}2^{l-1}$,$k=1,2_{\dagger}3,$ $\cdots$

Then we have the following relations.
$(-1)^{n_{2k-1}}=(-1)^{n_{k}}$

(2.7)

$\{$

,

$k=1,2$ ,(-1) . $(-1)^{n_{2k}}=(-1)^{n_{k}}$ ,$y_{2k-1}/\lambda=y_{k}$

(2.8)

$\{$

$y_{2k}/$A,$=y_{k}+1/\lambda$$k=1,2,3$ ,,,$3_{7}\cdots$,$\cdots$

and
(2.9)

Hence

$y_{k}\geq\lambda^{j}$if. If A$\geq 2$$\lim_{karrow\infty}y_{k}=\infty$$k-1\geq 2^{j}$, then$y_{k}$,$j=0,1,2$ ,$\cdots$

is strictly increasing. For example,$\{n_{k}\}_{k=1}^{\infty}=\{0, 1, 1, 0, 1_{7}0_{2}0, 1, 1, 0, 0, 1, 0, 1, 1, 0 \cdots\}$$\{y_{k}\}_{k=1}^{\infty}=\{0,1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, \cdots\}$for$\{y_{k}\}_{k=1}^{\infty}=\{0, 1, 4_{7}5,16,17, 20, 21, 64, 65, 68, 69, 80, \cdots\}$$\{y_{k}\}_{k=1}^{\infty}=\{0,1$, , , ,$\frac{3}{2}$$\frac{5}{2}$$\frac{9}{4}$$\frac{13}{4}$,$\frac{15}{4}$,$\frac{19}{4}$,$\frac{27}{8}$,$\frac{35}{8}$,$\frac{39}{8}$,$\frac{47}{8}$,$\frac{45}{8}$,,$\lambda=2$

for$\cdots\}$,$\lambda=4$,

for $\lambda=3/2$ .

Our main result is the following:
Theorem 2.3. Let$\lambda>1$. Then a solution f of (1.1) can be found as$f(x)= \sum_{k=1}^{\infty}(-1)^{n_{k}}u(x-y_{k})$where ,$u$solution$\{n_{k}\}_{k=1}^{\infty}$

$f$

is in

and

$C^{\infty}(\mathbb{R})$,are as in (2.1), (2.5) and (2.6), respectively. The
and $f(x)=0$ for $x\leq 0$ . If A
, then is bounded$\{y_{k}\}_{k=1}^{\infty}$$\geq 2$$f$

5

Let we define a function space$L^{1,\nu}(\mathbb{R})$.

$L^{1,\nu}=\{f\in L_{loc}^{1};||f||_{L^{1,\nu}}<\infty\}$$||f||_{L^{1,\nu}}= \sup_{r>0}\frac{1}{r^{\nu}}\int_{-r}^{+r}|f(x)|dx$

Theorem 2.4. The solution f of (1.1) is in$C^{\infty}\cap L^{1,1/\log_{2}\lambda}$.

Remark 2.1. The solution of (1.1) is tempered distribution.
Remark 2.2. A constant times$f$is also a solution.

Theorem 2.5. Let f be the solution in Theorem 2.3 for$\lambda=2$$G_{k,\epsilon}(x)=(2^{k(k-1\rangle/2}\epsilon^{k+1})^{-1}(f\chi_{[0,2^{h}]})(x/\epsilon)$

If$v\in C^{k}(\mathbb{R})$or$v\in L:(\mathbb{R})$(A$\geq 0$

uniformly on each compact subset in$G_{k,\epsilon}$

is inFriedrichs’ molifie$C^{\infty}(\mathbb{R})$.

, $1\leq p<\infty$ ), then$\frac{d^{k}v}{dx^{k}}=\lim_{\epsilonarrow 0}v*G_{k,\epsilon}$

Remark 2.3.

and$\mathbb{R}$

or in

,

$L^{p}(\mathbb{R})$, respectively.

with compact support. To prove the theorem we use$\frac{d^{k}v}{dx^{k}}*u_{\delta}=v$$\mathrm{r}$$* \frac{d^{k}u_{\delta}}{dx^{k}}$, where$u_{\delta}=u(x/\delta)/\delta$, $\delta>0$ , and$u$is thefunction in Lemma 2.2 (A $=2$ ).

3. EXAMPLESIn this section we give graphs for A $=4$ , 3, 2, 31/16, 15/8, 7/4, 3/2, 5/4.

If A $=2$ , then $\{x>0 :f(x)=0\}=\{1,2, 3, \cdots\}$ .

$0 \}=\bigcup_{k=1}^{\infty}[y_{k}+1/(\lambda-1), y_{k+1}]$$y_{k+1}-y_{k}$, $k=1_{7}2,3$ ,$\cdot$...

If A $>2$ , then { $x>0$ : $f(x)=$and its measure is infinity, since 1/ $(\lambda ・1)<1<$

6

FIGURE 1. u (A $=4$ , 2, 3/2)

FIGURE 2. $f’(x)=4^{2}f(4x)$FIGURE 3. $f’(x)=3^{2}f(3x)$4.

$\mathrm{p}_{\mathrm{R}\mathrm{O}\mathrm{G}\mathrm{R}\mathrm{A}\mathrm{M}}$OF$\mathrm{u}(\mathrm{x})$Mathematica program (Part 1) : The function u [ by using$\mathrm{u}_{-}\{\mathrm{n}+1\}=\mathrm{T}(\mathrm{u}_{-}\{\mathrm{n}\})$]

7

FIGURE 4. $f’(x)=2^{2}f$ (2$)

FIGURE 5. $f’(x)=(31/16)^{2}f(31x/16)$

FIGURE 6. $f’(x)=(15/8)^{2}f(15x/8)$

FIGURE 7. $f’(x)=(7/4)^{2}f(7x/4)$$*$

Setting lambda$(1<\mathrm{l}\mathrm{a}\mathrm{m}<9)$

In [1] :lam $=1.75$ ;$*$Calculation of the data ;In [2] :$\mathrm{u}_{-}\{0\}$

,

...

,

$\mathrm{u}_{-}\{50\}$

data $[01=$Table [If [0$-10000=<$ 1000, lam$<\mathrm{i}$In [3] :Timing [ Do

data

Sum

data

lam

$*$

{ ,Round [lam$\mathrm{i}$$[\mathrm{k}]$$=\mathrm{T}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$$[\mathrm{k} =1]$$*$($\mathrm{j}$–[[[If [1$\mathrm{i}+$–1, 0],$\{\mathrm{i}$, 1, 20000}];

1000,$=<\mathrm{j}-10000=<$

10000 1 ,10000)]- Round [(Dam

–

$1$

)

$*$

10000+ 1,

8

FJGURE 8. f’ $) $=(3/2)^{2}f(3x/2)$

FIGURE 9. f’ $) $=(5/4)^{2}f(5x/4)$
Round [lam($*$$\{\mathrm{j}.

10000)]–$\mathrm{j}$1,20000\}]$,$\}]*0.\mathrm{O}\mathrm{O}\mathrm{l}/(\mathrm{l}\mathrm{a}\mathrm{m} - 1)$$[\mathrm{k}, 1, 50\}]]$Out [3] : {121. 14 Second, Null}$*$

Graph of$\mathrm{u}_{-}\{50\}$

In [4] : ulist$[\mathrm{k}_{-}]$$:=\mathrm{T}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$Part [udata$[\mathrm{k}]$,[{ $(\mathrm{i}-10000)*0.001/$ (lam -1),$\mathrm{i}$

]

$\}$

,

$\{\mathrm{i}, 10000, 11000\}]$In [5] :ListPlot [ulist [50] , PlotJoined $->$ True,$\mathrm{P}\mathrm{l}\mathrm{o}\mathrm{t}\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}arrow\{0,1 .1 *\mathrm{l}\mathrm{a}\mathrm{m}\}]$$*$

Save the data

, 0],

a
In [6] :udata$[50]>>\mathrm{c}:/\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}/\mathrm{u}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}7\mathrm{o}\mathrm{v}4-50$

In [7]: ulist$[50]>>\mathrm{c}:/\mathrm{m}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}/\mathrm{u}\mathrm{l}\mathrm{i}\mathrm{s}\mathrm{t}7\mathrm{o}\mathrm{v}4-50$

In [8] : Export [ ”. eps ,”$\mathrm{c}:/\mathrm{m}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}/\mathrm{u}7\mathrm{o}\mathrm{v}4$.

ListPlot [ulist [50]PlotJoined $->$ True,$\mathrm{P}\mathrm{l}\mathrm{o}\mathrm{t}\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}arrow\{0,1 .1 *\mathrm{l}\mathrm{a}\mathrm{m}\}]]$

5. PROGRAM OF$\mathrm{F}(\mathrm{X})$Mathematica program (Part 2): The solution$*$

Setting lam da$(1<\mathrm{l}\mathrm{a}\mathrm{m}<9)$$\mathrm{f}$on the interval$[\mathrm{O},\mathrm{t}\mathrm{a}\mathrm{u}]$and tau

In [1]: lam $=1$ .75; tau $=30j$
In [2] : kk

$=$

Round

$[{\rm Log} [\mathrm{l}\mathrm{a}\mathrm{m}, \mathrm{t}\mathrm{a}\mathrm{u}]$$+0.5]$Out [2] :7$*$

Load the data

In [3] :udat aIn [4] ; ud$*$$=\mathrm{T}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$Sequences

In [51 :$=<<\mathrm{c}:/\mathrm{m}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}/\mathrm{u}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}7\mathrm{o}\mathrm{v}4-50$$\mathrm{m}_{-}\{\mathrm{k}\}$(Part [udat ,and$\mathrm{m}$

$=$

],

$\{\mathrm{i}, 10000, 11000\}]$;If [Mod$[\mathrm{k},$$2]==0$ ,Mod$[\mathrm{m}$$[\mathrm{k}/2]+1$$[\mathrm{k}, 3,2^{\wedge}\mathrm{k}\mathrm{k}+1\}]$

In [6] : Do$[\mathrm{b}[\mathrm{k}, 1]$$\mathrm{b}$[$\mathrm{k},$$=\mathrm{k}-1$$1+11$$=$;$\mathrm{y}_{-}\{\mathrm{k}\}$$\mathrm{m}[1]=0;\mathrm{m}[2]$ $=1$

Do [ [kl$\mathrm{i}$$\mathrm{a}$;(; Do [$\mathrm{b}[\mathrm{k},$$\mathrm{c}[\mathrm{k}.

Mod1]$ $=$$1]-\mathrm{c}[\mathrm{k},$$1]$$[\mathrm{b} [\mathrm{k}, 1]$

)/2,

, 2],

.

$\mathrm{m}[(\mathrm{k}+1)/2]$

],

2] ;

{1. 1, kk +1}],$\{\mathrm{k}, 1, 2^{\wedge}\mathrm{k}\mathrm{k}+1\}]$

In [7] : Do$[\mathrm{y}[\mathrm{k}]$

Sum

$*$

$=$

$[\mathrm{c} [\mathrm{k}, 1]*1\mathrm{a}\mathrm{m}^{\wedge}(1$–1 , {1, 1, kk $+1$ } , {
$]$

$)$

$\mathrm{k}$.

1,Calculation of the solution
as the sum of$(-1)^{\wedge}\{\mathrm{m}_{-}\{\mathrm{k}\}\}\mathrm{u}(\mathrm{x}-\mathrm{y}_{-}\{\mathrm{k}\})$,$\mathrm{k}=1$, 2,\ldots,$2^{\wedge}\mathrm{k}\mathrm{k}$.

$2^{\wedge}$

kk $+1$ }

$]$

10

In [8] : Do [yy [klIn [91 :$\mathrm{z}\mathrm{z}[1]$$=$$[0, \{\mathrm{i}, 1, \mathrm{y}\mathrm{y}[\mathrm{k}] \}]$Do [zz$\mathrm{n}$$[10]$:$\mathrm{u}\mathrm{d}\mathrm{y}[1]$Join
I [1] :$\mathrm{n}$

$*$

; Do

$=\mathrm{T}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}[0, \{\mathrm{i}, 1, \mathrm{y}\mathrm{y} [2^{\wedge}\mathrm{k}\mathrm{k}]\}]$$\mathrm{T}\mathrm{h}\mathrm{e}$I

Round $[\mathrm{y}[\mathrm{k}]*1000*$ (lam -1)],$\mathrm{f}\mathrm{d}=$$[\mathrm{k}]=$$=\mathrm{J}\mathrm{o}\mathrm{i}\mathrm{n}$$\mathrm{T}\mathrm{h}\mathrm{e}$[$\mathrm{u}\mathrm{d}$$[0,$$\{\mathrm{i}$

,

, 1,

, zz [1]] ; Do [udy$[\mathrm{k}]$$[\mathrm{u}\mathrm{d}\mathrm{y}[\mathrm{k}], \{\mathrm{k} , 1, 2^{\wedge}\mathrm{k}\mathrm{k}\}]$$2^{\wedge}\mathrm{k}\mathrm{k}+1\}$

]

$=$

;

$\mathrm{y}\mathrm{y}[2^{\wedge}\mathrm{k}\mathrm{k}]-\mathrm{y}\mathrm{y}[\mathrm{k}]$

$\}]$

,

$\{\mathrm{k}$, 2,$2^{\wedge}\mathrm{k}\mathrm{k}\}$

] ;

$=$

$[\mathrm{z}[\mathrm{k}], \mathrm{u}\mathrm{d}*(-1)^{\wedge}\mathrm{m}[\mathrm{k}], \mathrm{z}\mathrm{z}[\mathrm{k}]]$Sum$[\mathrm{z}[\mathrm{k}]$$\{\mathrm{k}, 2,2^{\wedge}\mathrm{k}\mathrm{k}\}]$, 1,$\{\mathrm{k}$,$\{\mathrm{k}, 2,2^{\wedge}\mathrm{k}\mathrm{k}\}]$;

Save the graph of the solution

In [121 :$\mathrm{i}\mathrm{i}=\mathrm{t}\mathrm{a}\mathrm{u}*(\mathrm{l}\mathrm{a}\mathrm{m}-1)*1000$

In [13]: flistPart;$=\mathrm{T}\mathrm{h}\mathrm{e}[\{\mathrm{i}*0.001/$$[\mathrm{f}\mathrm{d}, \mathrm{i}]\}$

In [14] :Export [”,$\{\mathrm{i}.

$(\mathrm{l}\mathrm{a}\mathrm{m} -1)$

1, \mathrm{i}\mathrm{i}\}]$$\mathrm{c}:/\mathrm{m}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}/\mathrm{f}7\mathrm{o}\mathrm{v}4$,$\mathrm{j}$.eps” ,

ListPlot flist, PlotJoined $->$ True, AspectEatio$arrow$

Automat$\mathrm{i}\mathrm{c}$6. ACKNOWLEDGEMENT
The author would like to thank Professors Eiichi Nakai and Hiroyuki Usami for
their useful comments and the author also would like to thank Professor Eiichi
Nakai for their useful Mathematica proglam.

REFERENCES[1] P. O. Frederickson, Global solutions to certain nonlinear functional differential
equations. J. Math. Anal . AppL 33 (1971), 355-358.
[2] P. O. Frederickson, Dirichlet series solutions for certain functional differential

equations. Japan-United States Seminar on Ordinary Differential and Functional

Equations (Kyoto, 1971), 249-254. Lecture Notes in Math., Vol. 243, Springer,
Berlin, 1971

]] ;

11

[3] A. F. Ivanov, Y. Kitamura, T. Kusano and V. N. Shevelo, Oscillatory solutionsof functional differential equations Generated by deviation of arguments of mixed
type,Hiroshima Math. J,12 (1982), 645-655[4] T. Kato Asymptotic behavior of solutions of the functional differential equation$y’(x)=ay(\lambda x)+by(x)$ ,Delay and functional differential equations and theirapplications (Proc, Conf., Park City, Utah, 1972), Academic Press, New York,1972, 197-217.
[5] T. Kato and J. B. McLeod, The functional-differential equation $y’(x)=ay(\lambda x)+$by(x), Bull Amer. Math. Soc. 77 (1971), 891-937.
[6] T. Kusano, Oscillation of even order Linear Functional Differential Equations

with Deviating Arguments of Mixed Type,3. Math. Anal. Appl. 98 (1984), 341-

347
[7] Zhi-cheng Wang, Ionannis P. Stavroulakis, Xiang-zheng Qian

A Survey On the Oscillation Of Solutions Of First Order Linear Differential
Equations With Deviating Arguments, P Applied Mathematics -Notes, $2(2002)$ ,$\mathrm{E}$171-191[8] Hiroshi Onose

Oscillatory Properties of the First Order Differential In equarities with Deviating
Argumen ?,Funkcialaj Ekvacioj,26(1983), 189-195[9] T. Yoneda, On the functional-differential equation of advanced type $f’(x)=$a $f(2x)$ with $f(0)=0$ , preprint.
[10] T. Yoneda, On the functional-differential equation of advanced type $f’(x)=$of (Ax), A $>1$ with $f(0)=0$ , preprint.
[i1] T. Yoneda, On the functional-differential equation of advanced type $f’(x)=$$\sum_{j}o_{j}f(\lambda x-b_{j})$, in preparation.

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