Citation 数理解析研究所講究録 (2008), 1615:

Title EKELAND の $\varepsilon$ 変分不等式あれこれ ( バナッ空間及び関数空間論の最近の進展とその応用 ) Author(s) 鈴木, 智成 Citation 数理解析研究所講究録 (2008), 1615: 117-125 Issue Date 2008-10 URL http://hdlhandlenet/2433/140126 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

$\bullet$ $X$ $\lambda$ 1615 2008 117-125 117 EKELAND $\varepsilon$ (Tomonari SUZUKI) 1, [17, 19] 2 $\varepsilon$ Ekeland ( 1),, Ekeland, [17, 19], [19] [18], 1 (Ekeland [6, 7]) (X, d), $f$ $X$,, $u\in X$ $\lambda>0$ $v\in X$ (i) $f(v)\leq f(u)-\lambda d(u, v)$ (ii) $f(w)>f(v)-\lambda d(v, w)$, $\forall w\in X\backslash \{v\}$, $f$ $(u,$ $f(u))$, (i), $($V, $f(v))$ $\lambda$ (ii), $(v,$ $f(v))$ Ekeland, Ekeland 2 (Kirk [9], Sullivan [11], Weston [21]) $(X, d)$, MSC (2000) $54H25,54E50$ $\tau-$ Palais-Smale, coercivity, Ekeland, distance 804-8550 suzuki-tomns kyut ch ac ip $e$

$\bullet$ $X$ 118 $f,$ $u\in X$ $\lambda>0$, (i), (ii) $v\in X$ 2 Ekeland $\epsilon$,, Ekeland Caristi,, Ekeland Banach 3 (Banach [1]) (X, d), $T$ $X$,, $T$ $r\in[0,1)$ $x,$ $y\in X$ $d(tx, Ty)\leq rd(x, y)$ $X$ $[0_{!}\infty)$ $f$ $f(x)=d(x, Tx)$ Ekeland, $f(w)>f(z)-(1-r)d(z, w)$, $\forall w\in X\backslash \{z\}$ $z\in X$ $z\neq Tz$, $rd(z, Tz)\geq d(\prime 1^{\tau}z, T^{2}z)=f(Tz)$ $>f(z)-(1-r)d(z_{!}tz)=rd(z, Tz)$,, $z$ $T$ Ekeland Caristi $[$4$]$ 4 $($Caristi, Caristi &Kirk $[$5 $])$ $(X, d)$ $f$ 1, $T$ $X$ $x\in X$, $f(tx)+d(x,$ $Tx)\leq f(x)$, $T$ Ekeland, $f(w)>f(z)-d(z, w)$, $\forall w\in X\backslash \{z\}$ $z\in X$ $z\neq Tz$, $f(tz)>f(z)-d(z, Tz)$

119, Caristi, Ekeland, 2,, Takahashi [20],,, Ekeland 3 EKELAND $f$ 1988, Georgiev Ekeland 5 (Georgiev [10]) $(X, d)$ $f$ 1, $u\in X$, $\lambda>0$ $\delta>0$, $v\in X$ (i): $f(v)\leq f(u)-\lambda d(u, v)+\delta$ (ii) $f(w)>f(v)-\lambda d(v, w)_{!}$ (iii) $\lim_{n}(f(x_{n})+\lambda d(v, x_{n}))=f(v)$ $\forall w\in X\backslash \{v\}$ $\{x_{n}\}$ $v$, Ekeland, Ekeland 1 (ii), (iii), $(i)$ $(i)-$(iii),, $X$ $f$, $(i)-$(iii) 2 6([17]) (X, d), $f$ $X$, $u\in X$ $\lambda>0$, $v\in X$ (i) $f(v)\leq f(u)-\lambda d(u, v)$ (ii) $f(w)>f(v)-\lambda d(v, u )$, $\forall w\in X\backslash \{v\}$ (iii) $\lim_{n}(f(x_{n})+\lambda d(v, x_{n}))=f(v)$ {X $v$ 7([17]) $X$, $f$ $X$, $f$ (quasiconvex) $\alpha$,, $\{X :f(x)\leq\alpha\}$, $u\in X$ $\lambda>0$, $(i)-$(iii) $v\in X$, [17], 7

120 8([17]) $X$, $f$ $X$ $*$,, $u\in X$ $\lambda>0$, (i), (ii) $(iii)$ $v\in X$ (iii): $\lim\alpha(f(x_{\alpha})+\lambda d(v, x_{\alpha}))=f(v)$ $\{x_{\alpha}\}$ $v$ $*$ 4 $\tau$-distance EKELAND $\tau$-distance Ekeland, $\tau$-distance 9([12]) $(Xd)$, $p$ $X\cross X$, $p$ $X$ $\tau$-distance $X\cross[0, \infty)$ $[0, \infty)$ $[0, \infty)$ $\eta$ 5 $(\tau 1)p(xz)\leq p(x, y)+p(y, z)$ $x,$ $y,$ $z\in X$ $(\tau 2)\eta(x, 0)=0$ $\eta(x, t)\geq t$ $(x, t)\in X\cross[0, \infty)$, $\eta$ 2 $( \tau 3)\lim_{n}x_{n}=x$ $\lim_{n}\sup\{\eta(z_{n},p(z_{n},x_{m})):m\geq n\}=0$ $p(w, x)\leq$ lim $infnp(w, x_{\eta})$ $w\in X$ $( \tau 4)\lim_{n}\sup\{p(x_{n}, y_{m}) :m\geq n\}=0$ $\lim_{n}\eta(x_{n}, t_{n})=0$ $\lim_{n}\eta(y_{n},t_{n})=0$ $( \tau 5)\lim_{n}\eta(z_{n},p(z_{n}, x_{n}))=0$ $1ini_{n}\eta(z_{n},p(z_{n}, y_{n}))=0$ $d(x_{n},y_{n})=0$ $\lim_{n}$ $d$ 1 $\lceil_{p(x,x)=0\rfloor}$ $\lceil_{p(x}$,, o y) x $=$ $\tau- di_{s}tance$ $\Rightarrow$ $r_{p(x,y)=p(y,x)\rfloor}$ y, $=$, [8, 12-19], $\tau$-distance Ekeland Ekeland 10, [19], 10 ([12]) $(X_{!}d)$ $f$ 1 $p$ $X$ $\tau$-distance, $p(u, u)=0$ $u\in X$ $\lambda>0$, $v\in X$ (i) $f(v)\leq f(u)-\lambda p(u, v)$ (ii) $f(w)>f(v)-\lambda p(v, u)$, $\forall w\in X\backslash \{v\}$

$\bullet$ $X$ 121 11 ([17]) $(X, d)$ $f$ 1 $p$ $X$ $\tau$-distance, $p(u, u)=0$ $u\in X,$ $\lambda>0$ $\delta>0$, $v\in X$ $(i) f(v)\leq f(u)-\lambda p(u, v)+\delta$ (ii) $f(w)>f(v)-\lambda p(v, - w)$, (iii) $\lim_{n}(f(x_{n})+\lambda p(v, x_{n}))=f(v)$ $\forall w\in X\backslash \{v\}$ $p(v, v)=1i_{l}n_{n}p(v, x_{n})=0$ $\{x$ $v$, 5 PS $f$ $X$ $f$ $x\in X$ (G\^ateaux differentiable), $X$ $f (x)$ $y\in X$ $\lim_{tarrow 0}\frac{f(x+ty)-f(x)}{t}=\langle f^{/}(x),$ $y)$ ( ) $f$ (coercive), $\lim_{rarrow\infty}$ $( \inf\{f(x):\vert x\vert\geq r\})=\infty$, $f$ PS (Palais-Smale condition), $\{x_{n}\}$, $\{f(x_{n})\}$ $\lim_{narrow\infty}\vert f^{/}(x_{n})\vert=0$, $\{x_{n}\}$ Ekeland ( 1) 12 ([2, 3] ) $f$ $X$, $f$ PS, $f$ 2, Ekeland $?$, 12, 13 ([19]) $X$ $X=$ { $X$ : $x$ $0$ }

122 $X$ $\ell_{1}$ $\Vert x\vert=\sum_{j_{-}-- 1}^{\infty} x(j) $ $\mathbb{r}$ $(0, \infty)$ $f$ $f(x)= \sum\frac{1}{2^{j}}\exp(\prime 2^{j}x(j))\infty$,,, $f$,, PS, $e_{n}\in X(n\in \mathbb{n})$ $e_{n}(\sim\dot{\uparrow})=\{\begin{array}{ll}0 if j\neq n,1 if j=n\end{array}$ $\mathbb{r}$ $\mathbb{r}$ $\mathbb{n}$, $X$ $\nu$ $f_{j}(j\in \mathbb{n})$ $f_{j}(t)=2^{-j}\exp(2^{j}t)$ $x(j)=0$, $\forall j>\nu(x)$ ( well-defined ) $f$ $x\in X$, $f(x)= \sum_{j=1}\frac{1}{2^{j}}\exp(2^{j}x(j))\infty$ $\nu(x)$ $= \sum\frac{1}{2j}\exp(2^{j}x(j))+$ $\sum^{\infty}$ $j=\nu(x)+1$ $\frac{1}{2j}$ $\nu(x)$ $\leq\sum\frac{1}{2^{j}}\exp(2^{j}x(j))+1$ $<\infty$, $f$ well-defined, ( $f$ ) $X$ $\mathbb{r}$ $x\mapsto f_{j}(x(j))$ $(j\in \mathbb{n})$,, $f$ $f(x)= \sup_{k\in N_{j}}\sum_{=1}f_{j}(x(j))$, $f$ ( ) $f$ $x\in X$, $\epsilon>0$ $\sup\{f(y) : x-y \leq\epsilon\}\geq\sup\{f(x+\epsilon e_{j}) : j>\nu(x)\}$ $=s\iota\iota p\{f(x)+f_{j}(\epsilon)-f_{j}(0):j>\nu(x)\}$ $=\infty$

123, $x\in X$, $f$ ( $f$ ) $x\mapsto f_{j}(x(j))$, $f$ ( $f$ ) $f_{j}(x)>0$ $f(x)>0$ ( ) $f$ $x,$ $y\in X$ $\lim_{tarrow 0}\frac{1}{t}(f(x+ty)-f(x))$ $=t1 iarrow\iota n0\frac{1}{t}\sum(f_{j}(x(j)+ty(j))-f_{j}(x(j)))\nu(y)$ $\nu(y)$ $= \sum f_{j} (x(j))y(j)$ $= \langle\sum_{j=1}\exp(2^{j}x(j))e_{j}\infty,$ $y\rangle$, $f$ $x$, $f (x)= \sum_{j=1}\exp(2^{j}x(j))e_{j}\infty$ ( $f$ PS ) $x\in X$ $\Vert f (x)\vert=\vert\sum_{j=1}\exp\infty(2^{j}x(j))e_{j}\vert=\sup_{j\in \mathbb{n}}(\exp(2^{j}x(j)))\geq 1$, PS $f$ PS ( $f$ ) $X$ $\{u_{n}\}$ $u_{n}(j)=\{\begin{array}{ll}-n if j\leq n,0 if j>n\end{array}$ $f(u_{n})= \sum\frac{1}{2^{j}}\exp(-2^{j}n)+n\sum^{\infty}\frac{1}{2^{j}}\exp(0)$ $j=n+1$ $\leq\sum_{j=1}^{n}\frac{1}{2^{j}}\exp(-n)+\sum_{j=n+1}^{\infty}\frac{1}{2^{j}}$ $\leq\exp(-n)+2^{-n}$

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