Citation 数理解析研究所講究録 (2007), 1570:

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(2007), 1570: 1-7 Issue Date 2007-10 URL

1 Construction of solutions $f'(x) = Title1(Banach spaces, function spaces, applications) Author(s) Yoneda, Tsuyoshi Citation 数理解析研究所講究録 (2007), 1570: 1-7 Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 Construction of solutions $f (x)=af(\lambda x),$ $\lambda>1$ (Tsuyoshi Yoneda) Graduate School of Mathematical Sciences The University of Tokyo 1 MAIN RESULT on $\mathbb{r}$ $C^{\infty}(\mathbb{R})$ We denote by Let the set of all infinitely differentiable functions $C_{0}^{\infty}(\mathbb{R})=\{f\in C^{\infty}(\mathbb{R})$ : $\frac{d^{p}f}{dx^{p}}(x)arrow 0$ as $C_{comp}^{\infty}(\mathbb{R})=$ { $f\in C^{\infty}(\mathbb{R})$ : $f$ has a compact support} $ x arrow\infty,$ $P=0,1,2,$ $\cdots\}$, We denote by $x_{[r,\ell)}$ the characteristic function of the interval $[r, s$ ) be the set of all step functions Let $X$ (11) $p= \sum_{j=1}^{m}c_{j}\chi_{[r_{j-1},r_{j})}$, 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 $T_{\lambda,p}$ [17] (p 149, Remark 2) and [2], we can say that :, since $s\in \mathbb{r}$ $\lambda p(\lambda\cdot)\in B_{1,\infty}^{1}(\mathbb{R}))$ as follows: $B_{1,\infty}^{8}(\mathbb{R})arrow B_{1,\infty}^{\epsilon+1}(\mathbb{R})$ for (12) $Tf(x)=T_{\lambda,p}f(x)=\lambda(p*f)(\lambda x)$ We note that, since and $Tf\in L^{1}(\mathbb{R})\cap C(\mathbb{R})$ $f\in L^{1}(\mathbb{R})$ $p\in If we have (see [15]) Therefore $f\in B_{1,\infty}^{s}(\mathbb{R})$ $\hat{f}(1+ \cdot ^{2})^{s/2}\in L^{\infty}(\mathbb{R})$ we have for $k>-s-1$ if $T^{k}f\in L^{2}(\mathbb{R})\cap C(\mathbb{R})$ $f\in B_{1,\infty}^{s}(\mathbb{R})$ L_{\infty mp}^{\infty}(\mathbb{r})$

3 2 Theorem 11 For $\lambda>1$ and $p\in X$, let be the operator defined $T$ C_{comp}^{\infty}(\mathbb{R})$ such that, by (12) Then there exists a $u=u_{\lambda,p}\in function for all (or for all $f\in L^{1}(\mathbb{R})$ $f strtctly bigger than $L^{1}(\mathbb{R}))$, Mooeover, \in\bigcup_{s\in \mathbb{r}}b_{1,\infty}^{s}(\mathbb{r})_{f}$ which function set is $\lim_{\ellarrow\infty}t^{\ell}f(x)=c_{f}u(x)$ uniformly with respect to $x\in \mathbb{r}$, where $c_{f}=\hat{f}(0)$ (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 11 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, (21) $\{\begin{array}{ll}f (x)=\lambda^{2}f(\lambda x), x\in \mathbb{r},f(0)=0, \end{array}$ we use $u=u_{\lambda,p}$ in Theorem 11 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 $\tilde{c}_{j}=-(c_{j}-c_{j+1}),$ $j=1,2,$ $\cdots m-1,\tilde{c}_{m}=-c_{m}$ C_{\infty mp}^{\infty}(\mathbb{r}),$ $\ell=0,1,2,$, and $\cdots$ Then $v_{\ell}\in $vp(x)=v_{\ell+1}(x)$, $x\in(-\infty, \lambda^{\ell+1}r_{1}$], $\ell=0,1,2,$ $\cdots$ Therefore we get a function $f=f_{\lambda p}\in C^{\infty}(\mathbb{R})$ such that $x\in(-\infty, $f(x)=v_{\ell}(x)$, \lambda^{\ell+1}r_{1}$], $\ell=0,1,2,$ $\cdots$ Since $f(x)=u(x)$ near $x=0,$ for all $k=0,1,2,$ $\frac{d}{dx}\tau kf(o)=0$ $\cdots$

4 3 Theorem 21 Let $\lambda>1$ For every $p\in X$, the function $f=f_{\lambda,p}$ given by the above method is a solution for (21) Remark 21 By the property (5) in Theorem 11 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 $v_{\ell}\subset[0, r_{m}\lambda^{\ell+1}/(\lambda-1)]$, $supp(v_{\ell+1}-v_{\ell})\subset[r_{1}\lambda^{l+1},r_{m}\lambda^{\ell+2}/(\lambda-1)]$ $[r_{m}\lambda^{\ell}/(\lambda-1), Therefore, if $r_{m}/(\lambda-1)<r_{1}$, then $f(x)=0$ on $P=0,1,2,$ $\ldots$, and is bounded (See the graph (S12)) Let $f$ r_{1}\lambda^{\ell}]$, $n_{4}$ $p_{i}= \sum_{j=1}c_{i,j}\chi_{[r:_{l-1r_{ij})}},\in X$, $i=1,2$ If and there exists $R>0$ such taht $p_{1}\neq p_{2}$ $r_{i,m_{1}}/(\lambda-1)<r<r_{i,1}$, $f_{\lambda,p_{1}}\neq f_{\lambda,p_{2}}$ $i=1,2$, then by Remark 11, while $\int_{0}^{r}f_{\lambda,ps}(x)dx=$ $\int_{0}^{r}u_{\lambda,p:}(x)dx=1,$ $i=1,2$ Remark 22 In the definition (11), let $m=\infty$ with the condition $\sum_{j=1}^{\infty} c_{j+1}-c_{j} <\infty$ Then the derivative of in the sense of distribution is a finite Radon measure and the Fourier transform of the $p$ derivative is an almost periodic function With some conditions we can ako construct solutions for (21) 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 be as in Table 1 Then, calculating $p$ $T_{\lambda,p}^{\ell}\chi_{[0_{y}1)}$ 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 [1] A Augustynowicz, H Leszczy\ oki and W Walter, On some nonlinear $ordina\eta$ differential equations Utth advanced arguments, Nonlinear Anal, 53 (2003), [2] VI Burenkov, On estimates of Fourier transforrns and convolutions in Nikol skij-besov spaces, Proc Steklov Inst 3(1990), 35-44

5 4 (S1) (S2) (S3) (S4) (S5) FIGURE 1 Solutions of $f (x)=4f(2x)$

6 5 $\lambda,$ TABLE 1 $p$ $f_{p,\lambda}$ and the solution FIGURE 2 Solutions of $f (x)=(3/2)^{2}f(3x/2)$ (S13) FIGURE 3 Solutions of $f (x)=9f(3x)$

6 [3] L Fox and D F Mayers, On afunctionai-differential equation, J Iot Maths Applics, 8(1971), 271-307 [4] P O Rederickson, Global solutioo to certain nonlinear functional differential equatioo J

7 6 [3] L Fox and D F Mayers, On afunctionai-differential equation, J Iot Maths Applics, 8(1971), [4] P O Rederickson, Global solutioo to certain nonlinear functional differential equatioo J Math Anal Appl 33 (1971), [5] P O Rederickson, Dirichlet series solutioo for certain functional differential equatioo Japan-United States Seminar on Ordinary Differential and Functional Equations (Kyoto, 1971), Lecture Notes in Math, Vol 243, Springer, Berlin, 1971 [6] Y Giga, K Inui, A Mahalov and S Matsui, Uniform local solvability for the Navier-Stokes equations with Coriolis force, Methods Appl Anal 12 (205), [7] Y Giga, A Mahalov, H Jo and T Yoneda, On time analycity of the Nanier- $Stok\infty$ equations in arotating frame with spatially almoet periodic data, preprint [8] M Heard, A family of solutions of the $IVP$ for the equation $x (t)=ax(\lambda t),$ $\lambda>$ $1,$ $Aequatione8$ Math, 9(1973), [9] A F Ivanov, Y Kitamura, T Kusano and V N Shevelo, $Oscillato\eta$ solu- $tion\epsilon$ of $func\hslash onal$ diffeoential equations Generated by $de\tau\dot{n}ation$ of $a7yuments$ $64\triangleright 655$ of mixed type, Hiroshima Math J 12 (1982), [10] T Kato Asymptotic behavior of solutions of the $diffeoen\hslash al$ fimctional equation $y (x)=ay(\lambda x)+by(x)$, in: Delay and Rnctional Differential Equatioo and Their Applicatioo, Proc Conf, Park City, Utah, 1972, Academic Praes, New York, 1972, [11] T Kato and J B McLeod, The firnctional-differential equation $y (x)=$ $ay(\lambda x)+by(x)$, Btl Amer Math Soc 77 (1971), [12] T Kusano, Oscillation of even order Linear hnctional Differentid $Equation\epsilon$ vrith of Mixed $De\tau\dot{n}a\hslash nga\eta uments$ $\Phi e$, J Math Anal Appl 98 (1984), [13] E Nakai and T Yoneda $Con\epsilon truction$ of solutions for the $func\hslash onal- diffeoent al$ equation $f (x)=af(\lambda x),$, with $\lambda>1$ $f(o)=0$, preprint [14] J R O&endon and A B Tayler, The dynamics of a collection $cu v\epsilon nt$ $sy\epsilon tem$ for an electric $loc\cdot ornotive$, Proc $Roy$ Soc Lond A 322, (1971) [15] Y Sawano and T Yoneda, On the Young theooem for amalgams and Besov $spac\cdot es$, Inteaational Joupal of Pure and Applied Mathematioe, 36, No 2 (2007), [16]Y Sawano and T Yoneda, Quarkonial dxomposition suitable for $intq ution$, to appear in Mathematische Nachriiten [17] H Riebel, $Theo\eta$ of function spaces, Monographs in Mathematioe, 78 Birkhser Verlag, Baael, (1983) [18]W Volk, Properties of subspaces genemted by an infinity oflen diffeoentiable $func\cdot tion$ and its tmnslates, ZAMM Z Angew Math $M\infty h$ $76$ (Suppl 1) (1996) [19] W Volk, $Evalua\hslash on$ of the solution of an integral-fimctional $e\varphi ation,$ JourM of Computational and Appli\ e Mathematioe 174 $(2\mathfrak{X}5) $ [20] T Yoneda, Spline fixnctions and $n- pe\dot{n}odic$ points (Japanese), hansactioo of the Japan Society for Industrial and Appli\ e Mathematics (Japanaee), 15 (2005), [21] T Yoneda, On the functional-diffeoential equation of advanced type $f (x)=$ $af(2x)$ with $f(o)=0$, J Math Anal Appl, 317 (2006), [22]T Yoneda, On the nctional-differential equation of $fu$ $advanc\cdot ed$ type $af(\lambda x),$ $\lambda>1$, with $f(o)=0$, J Math Anal Appl, 332 (2007), $f (x)=$

8 GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, THE UNIVERSITY OF TOKYO, KOMABA, MEGURO-KU TOKYO , JAPAN address: yonedadms u-tokyo ac jp 7

and Function spaces and its applica Citation 数理解析研究所講究録 (2009), 1667:

Convergence of some truncated Riesz generalized Campanato spaces and it Titleuniqueness theorem for nondecaying Stokes equations (The geometrical s and Function spaces and its applica Author(s) Nakai,