by

Let $X$$X$XX$X$ be a normed space, $M$$M$MM$M$ a nonvoid subset of $X$$X$XX$X$ and $x$$x$xx$x$ an element of $X$$X$XX$X$.

The problem of nearest points. Let $d(x,M)=inf\{\Vert x-y\Vert :y\in M\}$$d(x,M)=inf\{\Vert x-y\Vert :y\in M\}$d(x,M)=i n f{||x-y||:y in M}d(x, M)=\inf \{\|x-y\|: y \in M\}$d(x,M)=inf\{\Vert x-y\Vert :y\in M\}$ the distance from $x$$x$xx$x$ to $M$$M$MM$M$, and let $P_M(x)=\{y\in M:\Vert x-y\Vert =d(x,M)\}$$P_M(x)=\{y\in M:\Vert x-y\Vert =d(x,M)\}$P_(M)(x)={y in M:||x-y||=d(x,M)}P_{M}(x)=\{y \in M:\|x-y\|=d(x, M)\}$P_M(x)=\{y\in M:\Vert x-y\Vert =d(x,M)\}$, the set (possibly empty) of nearest points to $x$$x$xx$x$ in $M$$M$MM$M$ (or the set of elements of best approximation of $x$$x$xx$x$ by elements of $M)$$M)$M)M)$M)$. Put $E(M)=\{x\in X$$E(M)=\{x\in X$E(M)={x in XE(M)=\{x \in X$E(M)=\{x\in X$ : $P_M(x)\ne \varnothing \}$$\left.P_M(x)\ne \varnothing \right\}${:P_(M)(x)!=O/}\left.P_{M}(x) \neq \varnothing\right\}$P_M(x)\ne \varnothing \}$ and $Tc(M)=\{x\in X:\mathrm{card}(P_M(x))=1\}$$Tc(M)=\left\{x\in X:\mathrm{card}\left(P_M(x)\right)=1\right\}$Tc(M)={x in X:card(P_(M)(x))=1}T c(M)=\left\{x \in X: \operatorname{card}\left(P_{M}(x)\right)=1\right\}$Tc(M)=\{x\in X:\mathrm{card}(P_M(x))=1\}$. STECKIN [15] proved that if $X$$X$XX$X$ is a uniformly convex Banach space and $M$$M$MM$M$ is a nonvoid closed subset of $X$$X$XX$X$, tehn $X\mathrm{\setminus }Tc(M)$$X\mathrm{\setminus }Tc(M)$X\\Tc(M)X \backslash T c(M)$X\mathrm{\setminus }Tc(M)$ is of first Baire category (in particular, $Tc(M)$$Tc(M)$Tc(M)T c(M)$Tc(M)$ is dense in $X$$X$XX$X$ ). STECKIN [15] asked if this result remains true in a locally uniformly convex Banach space. In [7] it was shown that the answer is no: there exists an equivalent locally uniformly convex norm $p$$p$pp$p$ on $c_0$$c_0$c_(0)c_{0}$c_0$ (namely, Day's norm, see [8]) such that ( $c_0,p$$c_0,p$c_(0),pc_{0}, p$c_0,p$ ) contains a closed bounded symmetric convex body, such that $E(M)=M$$E(M)=M$E(M)=ME(M)=M$E(M)=M$ (such sets are called antiproximinal). Another solution (fortunately, also negative) was kindly comunicated to the author by Professor $P$$P$PP$P$. Kenderov: if $X$$X$XX$X$ is a separable non reflexive Banach space, then, by a result of TROYANSKI [16] there exists on $X$$X$XX$X$ an equivalent locally uniformly convex norm $p.(X,p)$$p.(X,p)$p.(X,p)p .(X, p)$p.(X,p)$ being nonreflexive, by James' theorem there exists a continuous linear functional $x^{\prime }$$x^{\prime }$x^(')x^{\prime}$x^{\prime }$ on $X$$X$XX$X$ which does not attain its norm on the unit ball of ( $X,p$$X,p$X,pX, p$X,p$ ). The corresponding closed hyperplane $H={\left(x^{\prime }\right)}^{-1}(0)$$H={\left(x^{\prime }\right)}^{-1}(0)$H=(x^('))^(-1)(0)H=\left(x^{\prime}\right)^{-1}(0)$H={\left(x^{\prime }\right)}^{-1}(0)$ is antiproximinal in $(X,p)$$(X,p)$(X,p)(X, p)$(X,p)$, i.e. $E(H)=H$$E(H)=H$E(H)=HE(H)=H$E(H)=H$. Recently Ka-SING LAU [13] proved that Stečkin's result holds in reflexive locally uniformly convex Banach spaces.

The problem of farthest points. Suppose further the set $M$$M$MM$M$ bounded and let $h(x,M)=sup\{\Vert x-y\Vert :y\in M\}$$h(x,M)=sup\{\Vert x-y\Vert :y\in M\}$h(x,M)=s u p{||x-y||:y in M}h(x, M)=\sup \{\|x-y\|: y \in M\}$h(x,M)=sup\{\Vert x-y\Vert :y\in M\}$. Put $Q_M(x)=\{y\in M:\Vert x-y\Vert =$$Q_M(x)=\{y\in M:\Vert x-y\Vert =$Q_(M)(x)={y in M:||x-y||=Q_{M}(x)=\{y \in M:\|x-y\|=$Q_M(x)=\{y\in M:\Vert x-y\Vert =$
$=h(x,M)\}$$=h(x,M)\}$=h(x,M)}=h(x, M)\}$=h(x,M)\}$ - the set (possibly empty) of farthest points to $x$$x$xx$x$ in $M$$M$MM$M$ and $e(M)=\{x\in X:Q_M(x)\ne \mathrm{\varnothing }\}$$e(M)=\left\{x\in X:Q_M(x)\ne \mathrm{\varnothing }\right\}$e(M)={x in X:Q_(M)(x)!=O/}e(M)=\left\{x \in X: Q_{M}(x) \neq \emptyset\right\}$e(M)=\{x\in X:Q_M(x)\ne \mathrm{\varnothing }\}$. EDELSTEIN [9] proved that if $M$$M$MM$M$ is a nonvoid closed bounded subset of uniformly convex Banach space $X$$X$XX$X$, then $e(M)$$e(M)$e(M)e(M)$e(M)$ is dense in $X$$X$XX$X$. asplund [1] extended this result proving that if $M$$M$MM$M$ is a nonvoid closed bounded subset of a reflexive locally uniformly convex Banach space $X$$X$XX$X$, then $e(M)$$e(M)$e(M)e(M)$e(M)$ contains a $G_{\delta }$$G_{\delta }$G_(delta)G_{\delta}$G_{\delta }$ set dense in $X$$X$XX$X$. Finally KA-SING IAU [12] proved a similar result for weakly compact subsets of arbitrary Banach spaces, and derived from this one, Asplund's theorem.

Perturbed problems. Let $J$$J$JJ$J$ be a real functional defined on $M$$M$MM$M$. baranger [2] considered the following extensions of the problems of nearest points and of farthest points : Problem $J$$J$JJ$J$-inf (Problem $J$$J$JJ$J$-sup) : for $x\in \overrightarrow{X}$$x\in \overrightarrow{X}$x in vec(X)x \in \vec{X}$x\in \overrightarrow{X}$ find $y_0\in M$$y_0\in M$y_(0)in My_{0} \in M$y_0\in M$ such that $\Vert x-y_0\Vert +J\left(y_0\right)=inf\{\Vert x-y\Vert +J(y):y\in M\}(==sup\{\Vert x-y\Vert +J(y):y\in M\}$$‖x-y_0‖+J\left(y_0\right)=inf\{\Vert x-y\Vert +J(y):y\in M\}(==sup\{\Vert x-y\Vert +J(y):y\in M\}$||x-y_(0)||+J(y_(0))=i n f{||x-y||+J(y):y in M}(==s u p{||x-y||+J(y):y in M}\left\|x-y_{0}\right\|+J\left(y_{0}\right)=\inf \{\|x-y\|+J(y): y \in M\}(= =\sup \{\|x-y\|+J(y): y \in M\}$\Vert x-y_0\Vert +J\left(y_0\right)=inf\{\Vert x-y\Vert +J(y):y\in M\}(==sup\{\Vert x-y\Vert +J(y):y\in M\}$ respectively), and proved that if $X$$X$XX$X$ is a uniformly convex Banach space, $M$$M$MM$M$ a closed nonvoid subset of $X$$X$XX$X$ and $J:M\to R$$J:M\to R$J:M rarr RJ: M \rightarrow R$J:M\to R$ is lower semicontinuous and bounded from below, then the set of all $x\in X$$x\in X$x in Xx \in X$x\in X$ for which the problem $J$$J$JJ$J$-inf has a solution is dense in $X$$X$XX$X$. If $X$$X$XX$X$ is a reflexive locally uniformly convex Banach space, $M$$M$MM$M$ a nonvoid closed bounded subset of $X$$X$XX$X$ and $J:M\to R$$J:M\to R$J:M rarr RJ: M \rightarrow R$J:M\to R$ is upper semicontinuous and bounded from above, then the set of all $x\in X$$x\in X$x in Xx \in X$x\in X$ for which the problem $J$$J$JJ$J$-sup has a solution, contains a $G_{\delta }$$G_{\delta }$G_(delta)G_{\delta}$G_{\delta }$ set dense in $X$$X$XX$X$. Other results along this line were obtained by baranger-temam [3], bidaut [5], ekelland lebourg [11].

The aim of this Note is to extend Ka-Sing Lau's result on farthest points of weakly compact sets to perturbed problems (Problem $J$$J$JJ$J$-sup, with an apropriate $J)$$J)$J)J)$J)$. In the third section some applications to optimal control problems of systems governed by partial differential equations, are given.

In this section we prove the following theorem:
2.1 THEOREM. If $X$$X$XX$X$ is a Banach space, $M$$M$MM$M$ a nonvoid weakly compact subset of $X$$X$XX$X$ and $J:M\to R$$J:M\to R$J:M rarr RJ: M \rightarrow R$J:M\to R$ is an upper semicontinuous and bounded from above functional, then the set of all $x\in X$$x\in X$x in Xx \in X$x\in X$ for which the problem $J$$J$JJ$J$-sup has a solution contains a $G_{\delta }$$G_{\delta }$G_(delta)G_{\delta}$G_{\delta }$ set dense in $X$$X$XX$X$.

In this theorem, and in what follows, by „weak" we mean $\sigma (X,X^{\prime })$$\sigma \left(X,X^{\prime }\right)$sigma(X,X^('))\sigma\left(X, X^{\prime}\right)$\sigma (X,X^{\prime })$, $X^{\prime }$$X^{\prime }$X^(')X^{\prime}$X^{\prime }$ the dual of $X$$X$XX$X$. Our proof follows closely ka-sing lau's proof in [12].

Recall that if $f:X\to R\cup \{\mathrm{\infty }\}$$f:X\to R\cup \{\mathrm{\infty }\}$f:X rarr R uu{oo}f: X \rightarrow R \cup\{\infty\}$f:X\to R\cup \{\mathrm{\infty }\}$ is a function on $X$$X$XX$X$, a subgradient of $f$$f$ff$f$ at a point $x\in X$$x\in X$x in Xx \in X$x\in X$ (such that $f(x)<\mathrm{\infty }$$f(x)<\mathrm{\infty }$f(x) < oof(x)<\infty$f(x)<\mathrm{\infty }$ ) is a continuous linear functional $x^{\prime }$$x^{\prime }$x^(')x^{\prime}$x^{\prime }$ such that

for all $y\in X$$y\in X$y in Xy \in X$y\in X$. The set (possibly empty) of all subgradients of $f$$f$ff$f$ at $x$$x$xx$x$ is denoted by $\partial f(x)$$\partial f(x)$del f(x)\partial f(x)$\partial f(x)$ and is called the subdifferential of $f$$f$ff$f$ at $x$$x$xx$x$. If $f$$f$ff$f$ is contintous, at $x$$x$xx$x$ then, $\partial f(x)$$\partial f(x)$del f(x)\partial f(x)$\partial f(x)$ is a nonempty weakly compact subset of $X^{\prime }$$X^{\prime }$X^(')X^{\prime}$X^{\prime }$ (see [4]).

For $x\in X$$x\in X$x in Xx \in X$x\in X$ put

2.2 lemma. Let $X$$X$XX$X$ a normed space, $M$$M$MM$M$ a nonvoid bounded subset of $X,J:M\to R$$X,J:M\to R$X,J:M rarr RX, J: M \rightarrow R$X,J:M\to R$ a bounded from above functional and let $r:X\to R$$r:X\to R$r:X rarr Rr: X \rightarrow R$r:X\to R$ be defined by (2.2). Then
(i) $r$$r$rr$r$ is convex and Lipschitz, with constant 1, i.e.

(ii) if $x^{\prime }\in \partial r(x)$$x^{\prime }\in \partial r(x)$x^(')in del r(x)x^{\prime} \in \partial r(x)$x^{\prime }\in \partial r(x)$ then $\Vert x^{\prime }\Vert \le 1$$‖x^{\prime }‖\le 1$||x^(')|| <= 1\left\|x^{\prime}\right\| \leq 1$\Vert x^{\prime }\Vert \le 1$, for all $x\in X$$x\in X$x in Xx \in X$x\in X$.

Proof. (i). The functions $r_y:X\to R$$r_y:X\to R$r_(y):X rarr Rr_{y}: X \rightarrow R$r_y:X\to R$, defined by $r_y(x)=\Vert x-y\Vert ++J(y)$$r_y(x)=\Vert x-y\Vert ++J(y)$r_(y)(x)=||x-y||++J(y)r_{y}(x)=\|x-y\|+ +J(y)$r_y(x)=\Vert x-y\Vert ++J(y)$, are convex for all $y\in M$$y\in M$y in My \in M$y\in M$, and so will be their supremum $\gamma$$\gamma$gamma\gamma$\gamma$. Now, for $x,y\in X$$x,y\in X$x,y in Xx, y \in X$x,y\in X$ and $z\in M$$z\in M$z in Mz \in M$z\in M$

so that $r(x)\le \Vert x-y\Vert +r(y)$$r(x)\le \Vert x-y\Vert +r(y)$r(x) <= ||x-y||+r(y)r(x) \leq\|x-y\|+r(y)$r(x)\le \Vert x-y\Vert +r(y)$, or $r(x)-r(y)\le \Vert x-y\Vert$$r(x)-r(y)\le \Vert x-y\Vert$r(x)-r(y) <= ||x-y||r(x)-r(y) \leq\|x-y\|$r(x)-r(y)\le \Vert x-y\Vert$. Interchanging the roles of $x$$x$xx$x$ and $y$$y$yy$y$ one obtains $|r(x)-r(y)|\le \Vert x-y\Vert$$|r(x)-r(y)|\le \Vert x-y\Vert$|r(x)-r(y)| <= ||x-y|||r(x)-r(y)| \leq\|x-y\|$|r(x)-r(y)|\le \Vert x-y\Vert$.
(ii). If $x^{\prime }\in \partial r(x)$$x^{\prime }\in \partial r(x)$x^(')in del r(x)x^{\prime} \in \partial r(x)$x^{\prime }\in \partial r(x)$, then $x^{\prime }(y-x)\le r(y)-r(x)\le \Vert y-x\Vert$$x^{\prime }(y-x)\le r(y)-r(x)\le \Vert y-x\Vert$x^(')(y-x) <= r(y)-r(x) <= ||y-x||x^{\prime}(y-x) \leq r(y)-r(x) \leq\|y-x\|$x^{\prime }(y-x)\le r(y)-r(x)\le \Vert y-x\Vert$, for all $y\in X$$y\in X$y in Xy \in X$y\in X$, which implies $\Vert x^{\prime }\Vert \le 1$$‖x^{\prime }‖\le 1$||x^(')|| <= 1\left\|x^{\prime}\right\| \leq 1$\Vert x^{\prime }\Vert \le 1$. Lemma 2.2 is proved.

If $x^{\prime }\in \partial r(x)$$x^{\prime }\in \partial r(x)$x^(')in del r(x)x^{\prime} \in \partial r(x)$x^{\prime }\in \partial r(x)$. then, by Iemma 2.2 (ii)

so that

for all $x\in X$$x\in X$x in Xx \in X$x\in X$. The following lemma shows that the equality sign holds in (2.3) for all $x\in X$$x\in X$x in Xx \in X$x\in X$, excepting a set of first Baire category.
2.3 LEMMA. Let $X$$X$XX$X$ be a Banach space, $M$$M$MM$M$ a nonvoid closed bounded subset of $X$$X$XX$X$ and let $J:M\to R$$J:M\to R$J:M rarr RJ: M \rightarrow R$J:M\to R$ be bounded from above. If $\nu (x)$$\nu (x)$nu(x)\nu(x)$\nu (x)$ is defined by (2.2), then the set:
$F=\{x\in X:\mathrm{\exists }{x}^{\prime }\in \partial r(x)$$F=\left\{x\in X:\mathrm{\exists }{x}^{\prime }\in \partial r(x)\right.$F={x in X:EEx^(')in del r(x):}F=\left\{x \in X: \exists x^{\prime} \in \partial r(x)\right.$F=\{x\in X:\mathrm{\exists }{x}^{\prime }\in \partial r(x)$ such that $inf\{x^{\prime }(y-x)-J(y):y\in M\}>--r(x)\}$$inf\left\{x^{\prime }(y-x)-J(y):y\in M\right\}>--r(x)\}$i n f{x^(')(y-x)-J(y):y in M} > --r(x)}\inf \left\{x^{\prime}(y-x)-J(y): y \in M\right\}>- -r(x)\}$inf\{x^{\prime }(y-x)-J(y):y\in M\}>--r(x)\}$ is of $F_8$$F_8$F_(8)F_{8}$F_8$ type and of first Baire category.

Proof. For $n\in N$$n\in N$n in Nn \in N$n\in N$, let $F_n=\{x\in X:\mathrm{\exists }{x}^{\prime }\in \partial r(x)$$F_n=\left\{x\in X:\mathrm{\exists }{x}^{\prime }\in \partial r(x)\right.$F_(n)={x in X:EEx^(')in del r(x):}F_{n}=\left\{x \in X: \exists x^{\prime} \in \partial r(x)\right.$F_n=\{x\in X:\mathrm{\exists }{x}^{\prime }\in \partial r(x)$ such that $inf\{x^{\prime }(y-$$inf\left\{x^{\prime }(y-\right.$i n f{x^(')(y-:}\inf \left\{x^{\prime}(y-\right.$inf\{x^{\prime }(y-$
$-x)-J(y):y\in M\}\ge -\gamma (x)+\frac{1}{n}\}$$\left.-x)-J(y):y\in M\}\ge -\gamma (x)+\frac{1}{n}\right\}${:-x)-J(y):y in M} >= -gamma(x)+(1)/(n)}\left.-x)-J(y): y \in M\} \geq-\gamma(x)+\frac{1}{n}\right\}$-x)-J(y):y\in M\}\ge -\gamma (x)+\frac{1}{n}\}$. Obviously $F=\bigcup _{n=1}^{\mathrm{\infty }}{F}_n$$F=\bigcup _{n=1}^{\mathrm{\infty }} F_n$F=uuu_(n=1)^(oo)F_(n)F=\bigcup_{n=1}^{\infty} F_{n}$F=\bigcup _{n=1}^{\mathrm{\infty }}{F}_n$. Therefore, to prove Lemma 2.3, it is sufficient to show that
(a) $F_n$$F_n$F_(n)F_{n}$F_n$ is closed in $X$$X$XX$X$; and
(b) $\mathrm{int}{F}_n=\varnothing$$\mathrm{int}{F}_n=\varnothing$intF_(n)=O/\operatorname{int} F_{n}=\varnothing$\mathrm{int}{F}_n=\varnothing$,
for all $n\in N$$n\in N$n in Nn \in N$n\in N$.
(a). Let $\{x_k:k\in N\}$$\left\{x_k:k\in N\right\}${x_(k):k in N}\left\{x_{k}: k \in N\right\}$\{x_k:k\in N\}$ be a sequence in $F_n$$F_n$F_(n)F_{n}$F_n$ converging to a point $x\in X$$x\in X$x in Xx \in X$x\in X$. For each $k\in N$$k\in N$k in Nk \in N$k\in N$, choose $x_k^{\prime }\in \partial r\left(x_k\right)$$x_k^{\prime }\in \partial r\left(x_k\right)$x_(k)^(')in del r(x_(k))x_{k}^{\prime} \in \partial r\left(x_{k}\right)$x_k^{\prime }\in \partial r\left(x_k\right)$ such that

Since $\Vert x_k^{\prime }\Vert \le 1,k\in N$$‖x_k^{\prime }‖\le 1,k\in N$||x_(k)^(')|| <= 1,k in N\left\|x_{k}^{\prime}\right\| \leq 1, k \in N$\Vert x_k^{\prime }\Vert \le 1,k\in N$, (Lemma 2.2 (ii)), the sequence $\{x_k^{\prime },k\in N\}$$\left\{x_k^{\prime },k\in N\right\}${x_(k)^('),k in N}\left\{x_{k}^{\prime}, k \in N\right\}$\{x_k^{\prime },k\in N\}$ admits a subnet $\{x_i^{\prime },i\in I\}\sigma ((X^{\prime },X)$$\left\{x_i^{\prime },i\in I\right\}\sigma \left(\left(X^{\prime },X\right)\right.${x_(i)^('),i in I}sigma((X^('),X):}\left\{x_{i}^{\prime}, i \in I\right\} \sigma\left(\left(X^{\prime}, X\right)\right.$\{x_i^{\prime },i\in I\}\sigma ((X^{\prime },X)$ - convergent to an element $x^{\prime }$$x^{\prime }$x^(')x^{\prime}$x^{\prime }$ of $X^{\prime }$$X^{\prime }$X^(')X^{\prime}$X^{\prime }$, with $\Vert x^{\prime }\Vert \le 1$$‖x^{\prime }‖\le 1$||x^(')|| <= 1\left\|x^{\prime}\right\| \leq 1$\Vert x^{\prime }\Vert \le 1$. For $z\in X$$z\in X$z in Xz \in X$z\in X$, we have

for all $i\in I$$i\in I$i in Ii \in I$i\in I$, which shows that the net $\left\{x_i^{\prime }(z-x_i)\right\}$$\left\{x_i^{\prime }\left(z-x_i\right)\right\}${x_(i)^(')(z-x_(i))}\left\{x_{i}^{\prime}\left(z-x_{i}\right)\right\}$\left\{x_i^{\prime }(z-x_i)\right\}$ converges to $x^{\prime }(z-x)$$x^{\prime }(z-x)$x^(')(z-x)x^{\prime}(z-x)$x^{\prime }(z-x)$. Since $x_i^{\prime }\in \partial r\left(x_i\right)$$x_i^{\prime }\in \partial r\left(x_i\right)$x_(i)^(')in del r(x_(i))x_{i}^{\prime} \in \partial r\left(x_{i}\right)$x_i^{\prime }\in \partial r\left(x_i\right)$, we have $x^{\prime }(z_i^{\prime }-x_i)+r\left(x_i\right)\le r(z)$$x^{\prime }\left(z_i^{\prime }-x_i\right)+r\left(x_i\right)\le r(z)$x^(')(z_(i)^(')-x_(i))+r(x_(i)) <= r(z)x^{\prime}\left(z_{i}^{\prime}-x_{i}\right)+r\left(x_{i}\right) \leq r(z)$x^{\prime }(z_i^{\prime }-x_i)+r\left(x_i\right)\le r(z)$ (see 2.1)) so that $x^{\prime }(z-x)+r(x)\le r(z)$$x^{\prime }(z-x)+r(x)\le r(z)$x^(')(z-x)+r(x) <= r(z)x^{\prime}(z-x)+r(x) \leq r(z)$x^{\prime }(z-x)+r(x)\le r(z)$, for all $z\in X$$z\in X$z in Xz \in X$z\in X$, which shows that $x^{\prime }\in \partial r(x)$$x^{\prime }\in \partial r(x)$x^(')in del r(x)x^{\prime} \in \partial r(x)$x^{\prime }\in \partial r(x)$. From $x_i^{\prime }(y-x)-J(y)\ge -r(x)+\frac{1}{n}$$x_i^{\prime }(y-x)-J(y)\ge -r(x)+\frac{1}{n}$x_(i)^(')(y-x)-J(y) >= -r(x)+(1)/(n)x_{i}^{\prime}(y-x)-J(y) \geq-r(x)+\frac{1}{n}$x_i^{\prime }(y-x)-J(y)\ge -r(x)+\frac{1}{n}$, follows $x^{\prime }(y-x)-J(y)\ge -r(x)+\frac{1}{n}$$x^{\prime }(y-x)-J(y)\ge -r(x)+\frac{1}{n}$x^(')(y-x)-J(y) >= -r(x)+(1)/(n)x^{\prime}(y-x)-J(y) \geq-r(x)+\frac{1}{n}$x^{\prime }(y-x)-J(y)\ge -r(x)+\frac{1}{n}$ for all $y\in M$$y\in M$y in My \in M$y\in M$. Therefore $x\in F$$x\in F$x in Fx \in F$x\in F$ and the set $F_n$$F_n$F_(n)F_{n}$F_n$ is closed.
(b) int $F_n=\varnothing ,n=1,2,\dots$$F_n=\varnothing ,n=1,2,\dots$F_(n)=O/,n=1,2,dotsF_{n}=\varnothing, n=1,2, \ldots$F_n=\varnothing ,n=1,2,\dots$

Suppose that there exist $k\in N,y_0\in F_n$$k\in N,y_0\in F_n$k in N,y_(0)inF_(n)k \in N, y_{0} \in F_{n}$k\in N,y_0\in F_n$ and a ball $U$$U$UU$U$ of center $y_0$$y_0$y_(0)y_{0}$y_0$ included in $F_k$$F_k$F_(k)F_{k}$F_k$. The set $M$$M$MM$M$ being bounded there exists $\lambda >0$$\lambda >0$lambda > 0\lambda>0$\lambda >0$ such that

for all $z\in M$$z\in M$z in Mz \in M$z\in M$. Let $\epsilon =\lambda [(\lambda +1)k{]}^{-1}$$\epsilon =\lambda [(\lambda +1)k{]}^{-1}$epsi=lambda[(lambda+1)k]^(-1)\varepsilon=\lambda[(\lambda+1) k]^{-1}$\epsilon =\lambda [(\lambda +1)k{]}^{-1}$ and let $z_0\in M$$z_0\in M$z_(0)in Mz_{0} \in M$z_0\in M$ be such that

and let

Since by (2.4), $x_0\in U\subset F_k$$x_0\in U\subset F_k$x_(0)in U subF_(k)x_{0} \in U \subset F_{k}$x_0\in U\subset F_k$, it follows the existence of a $x^{\prime }\in \partial r\left(x_0\right)$$x^{\prime }\in \partial r\left(x_0\right)$x^(')in del r(x_(0))x^{\prime} \in \partial r\left(x_{0}\right)$x^{\prime }\in \partial r\left(x_0\right)$ such that

By (2.5), $r\left(y_0\right)-r\left(x_0\right)\le \Vert y_0-z_0\Vert +J\left(z_0\right)+\epsilon -r\left(x_0\right)$$r\left(y_0\right)-r\left(x_0\right)\le ‖y_0-z_0‖+J\left(z_0\right)+\epsilon -r\left(x_0\right)$r(y_(0))-r(x_(0)) <= ||y_(0)-z_(0)||+J(z_(0))+epsi-r(x_(0))r\left(y_{0}\right)-r\left(x_{0}\right) \leq\left\|y_{0}-z_{0}\right\|+J\left(z_{0}\right)+\varepsilon-r\left(x_{0}\right)$r\left(y_0\right)-r\left(x_0\right)\le \Vert y_0-z_0\Vert +J\left(z_0\right)+\epsilon -r\left(x_0\right)$, and by (2.6), $y_0-z_0=(\lambda +1{)}^{-1}(x_0-z_0)$$y_0-z_0=(\lambda +1{)}^{-1}\left(x_0-z_0\right)$y_(0)-z_(0)=(lambda+1)^(-1)(x_(0)-z_(0))y_{0}-z_{0}=(\lambda+1)^{-1}\left(x_{0}-z_{0}\right)$y_0-z_0=(\lambda +1{)}^{-1}(x_0-z_0)$. Therefore
$r\left(y_0\right)-r\left(x_0\right)\le (\lambda +1{)}^{-1}||x_0-z_0||+J\left(z_0\right)+\epsilon -r\left(x_0\right)\le$$r\left(y_0\right)-r\left(x_0\right)\le (\lambda +1{)}^{-1}||x_0-z_0||+J\left(z_0\right)+\epsilon -r\left(x_0\right)\le$r(y_(0))-r(x_(0)) <= (lambda+1)^(-1)||x_(0)-z_(0)||+J(z_(0))+epsi-r(x_(0)) <=r\left(y_{0}\right)-r\left(x_{0}\right) \leq(\lambda+1)^{-1}| | x_{0}-z_{0}| |+J\left(z_{0}\right)+\varepsilon-r\left(x_{0}\right) \leq$r\left(y_0\right)-r\left(x_0\right)\le (\lambda +1{)}^{-1}||x_0-z_0||+J\left(z_0\right)+\epsilon -r\left(x_0\right)\le$