[1] Aronsson, G., A note on the extension of Lipschitz and Holder functions (preprint, 5 p.)
[2] Aronsson, G., Extension of functions satisfying Lipschitz conditions, Arkiv for Mathematik 6, 28 (1967), 551-561.
[3] Gutier, S., Isac, G., Penot, J.P., Surjectivity of multifunctions under generalized differentiability assumptions, Bull. Austral. Math. Soc., 28 (1983), 13-21.
[4] Mustata, C., Best Approximation and Unique Exctension of Lipschitz Functions, Journal of Approx. Theory 19, 3 (1977), 222-230.
[5] Mustata, C., Extension of bounded Lipschitz functions “Babes-Bolyai” University, Faculty of Mathematics, Research Seminaries, Seminar of Funcitonal Analysis and Numerical Methods, Preprint nr.1, 1980, 1-20.
[6] Mc Shane, E.J., Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842.
[7] Singer, I., Cea mai bună aproximare în spații vectoriale normate prin elemente din subspații vectoriale, Editura Academiei R.S. România, București, 1067.
It was proved in［3］that if XX and YY are two Banach spaces and F：X rarr YX \rightarrow Y is a multifunction such that F(X)F(X) is a closed sub－ set of YY ，then the multifunction FF is surjective if and only if for overy b in P(X)b \in P(X) the tangent cone in bb to the set F(X)F(X) is YY ．
In this note we shall give necesarry and sufficient conditions for the surjectivity of a multifunction F:X rarr XF: X \rightarrow X ，when XX and YY are arbitrary motric spaces．
For a metric space ZZ and a fixed point z_(0)in Zz_{0} \in Z denote by Ifp (Z,R)(\mathrm{Z}, \mathrm{R}) the aet of all real－valued Lipschitz functions on Z which vanish at g_(0),1.0g_{0}, 1.0 ．
（1） Lip_(0)(z_(0),z)={f8,z rarr R,f:}\operatorname{Lip}_{0}\left(z_{0}, z\right)=\left\{f 8, z \rightarrow R, f\right. is Lipschitz and {:f(z_(0))=0}\left.f\left(z_{0}\right)=0\right\} ．
Recall that a function is Z rarr RZ \rightarrow R is called Lipschitz if there ucists 1 >= 01 \geqslant 0 suobs that
（2）quad|x(z_(1))=f(z_(2))| <= Ma(z_(1),z_(2))\quad\left|x\left(z_{1}\right)=f\left(z_{2}\right)\right| \leq M a\left(z_{1}, z_{2}\right)
for all z_(1),z_(2)in zz_{1}, z_{2} \in z ，where dd denotes the metric of ZZ ．The smal－ lest number 1 for which（2）holds is called the Lipschitz norm of 1 and 18 denoted by |f||f| ．The norm is given by ：
（3）｜2｜｜＝⿱⿴囗十力 {∣2(g_(1))-1(g_(2)∣//A(z_(1),z_(2)):z_(1),z_(2)in z,z_(1)!=z_(2)}:}\left\{\mid 2\left(g_{1}\right)-1\left(g_{2} \mid / A\left(z_{1}, z_{2}\right): z_{1}, z_{2} \in z, z_{1} \neq z_{2}\right\}\right. ．
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The space Lip_(0)(Z,R)\operatorname{Lip}_{0}(Z, R) equipped with the norm (3) is a Banach space :
If Z_(1)sub ZZ_{1} \subset Z is such that Z_(0)inZ_(1)Z_{0} \in Z_{1} and f inLip_(0)(Z_(1),R)f \in \operatorname{Lip}_{0}\left(Z_{1}, R\right) then a function bar(f):Z rarr R\bar{f}: Z \rightarrow R, bar(f)inLip_(0)(Z,R)\bar{f} \in \operatorname{Lip}_{0}(Z, R) is called an extension of ff if
(4) quad bar(I)|_(Z_(1))=I\left.\quad \overline{\mathrm{I}}\right|_{\mathrm{Z}_{1}}=\mathrm{I} and || bar(I)||=||I||\|\overline{\mathrm{I}}\|=\|\mathrm{I}\|.
In general, every f in Iip_(0)(Z_(1),R)f \in I i p_{0}\left(Z_{1}, R\right) has a least one extension bar(f)inLip_(o)(Z,R)\bar{f} \in \operatorname{Lip}_{o}(Z, R). Concerning the properties of the set of all extensions of ff see [4], [5].
A subset AA of the metric space (z,d)(z, d) is called uniformly discrete if there exists delta > 0\delta>0 such that a(x,y) >= deltaa(x, y) \geqslant \delta for all x,y in A quad,x!in yx, y \in A \quad, x \notin y.
For Z_(1)sub ZZ_{1} \subset Z denote by Z_(1)^(_|_)Z_{1}^{\perp} the subspace of Lip_(0)(Z_(,)R)\operatorname{Lip}_{0}\left(Z_{,} R\right) defined by
(5) quadz_(1)^(_|_)={e inIip_(0)(z,B),quad:}\quad z_{1}^{\perp}=\left\{e \in \operatorname{Iip}_{0}(z, B), \quad\right. I {:z_(1)=0}\left.z_{1}=0\right\}.
Obviously, Z_(1)^(_|_)Z_{1}^{\perp} is a closed subspace of the Banach space Lip_(0)(Z,R)\operatorname{Lip}_{0}(Z, R).
A subset YY of a normed space XX is called proximinal if every x in Xx \in X has a best approximation element in YY, 1.a. Ior every., x in Xx \in X there exists y_(0)in Yy_{0} \in Y such that
{:(6)||x-y_(0)||=i n f{||x-y||:y in Y}:}\begin{equation*}
\left\|x-y_{0}\right\|=\inf \{\|x-y\|: y \in Y\} \tag{6}
\end{equation*}
If every x in Xx \in X, has exactly one element of best approximation in YY, then the set YY is called a Cebrsextan subset of XX.
Let now XX and YY be aa metric space and I&X rarr YI \& X \rightarrow Y a, multifunction. It was proved in [4] that F(X)^(_|_)F(X)^{\perp} is always a proximinal subset of YY and that F(X)^(_|_)F(X)^{\perp} is Cebyserian if and only if
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I(I)^(_|_)={0}I(I)^{\perp}=\{0\} (Ieaman 1 and Cocollary II in [4] )
The matr wesult of this Jote is the following theorem: InTrorrer 1. Ial X,YX, Y be two matric spaces ; let FsX rarr YF s X \rightarrow Y be e millitunction guch that Gamma(X)\Gamma(X) is closed and non-uniformly discrete and let y_(0)in P(x)y_{0} \in P(x) be fixed . Then the following conditroves area equilaring:
(a) I Is aughestive (( ioe. D(X)=I)D(X)=I);
(b) Evaxy f inIrg_(0)(H(X),F)f \in \operatorname{Irg}_{0}(H(X), F) has exactly one extension bar(I)in bar(It)p_(0)(I,E)\bar{I} \in \overline{I t} p_{0}(I, E);
(c) The gubspace I(X)^(_|_)I(X)^{\perp} is Cebysevian in Lip_(0)(Y_(1)R)\operatorname{Lip}_{0}\left(Y_{1} R\right).
Proof. II I Is dirfective then P(X)=YP(\mathrm{X})=Y so that Itp_(0)(I(X),I)=Itg_(0)(Y_(0),I)\operatorname{Itp}_{0}(I(X), I)=\operatorname{Itg}_{0}\left(Y_{0}, I\right) and I(X)^(_|_)=I^(_|_)={0}I(X)^{\perp}=I^{\perp}=\{0\} which show that the Implicationa (a)=>(b)(a) \Rightarrow(b) and (a)=>(c)(a) \Rightarrow(c) hold true.
suppose now that condition (b) is verified then, by [4, Th.2] It follows that bar(F(X))=Y\overline{F(X)}=Y and since F(X)F(X) is closed in YY it follows that I(X)=I,i_(0).e_(0)(b)=>(a)I(X)=I, i_{0} . e_{0}(b) \Rightarrow(a).
The equivalence (b)<=>(c)(b) \Leftrightarrow(c) follows by Lemma 1 and Corollary 1 from [4]. Theorem 1 is completely proved.
If XX and YY are Banach spaces then condition (b) is equivalent also to the condition :
(d) For every y in F(X)y \in F(X) the tangent cone T_(y)F(X)T_{y} F(X) to F(X)F(X) in yy is I,([3]I,([3], Theorem 3)).
COBOTTIARY 1. Tet X,YX, Y be two Banach spaces and let F:X rarr YF: X \rightarrow Y be a multifunction such that F(X)F(X) is closed. Then
(a)<=>(d)quad;quad(b)<=>(c)" and "(d)=>(b)"; "(a) \Leftrightarrow(d) \quad ; \quad(b) \Leftrightarrow(c) \text { and }(d) \Rightarrow(b) \text {; }
If. furthermore F(X)F(X) is not uniformly discrete then (b)=>(d)(b) \Rightarrow(d) too.
Example. The hypothesis that F(X)F(X) is not uniformly alscrete is essential in Theorem 1.
Let X=Y=[0,1]X=Y=[0,1] and F:X rarr YF: X \rightarrow Y defined by
F(x)={[0",",x in[0","1//2]],[I",",x in(1//2","1]]:}F(x)= \begin{cases}0, & x \in[0,1 / 2] \\ I, & x \in(1 / 2,1]\end{cases}
Then F(X)={0,1}F(X)=\{0,1\} is closed and uniformly discrete. Let Lip_(0)([0,1],R)={f:[0,1]rarr R,f\operatorname{Lip}_{0}([0,1], R)=\{f:[0,1] \rightarrow R, f is Iipschitz and f(0)=0}f(0)=0\} Every function f in Lip({Q,I},R)f \in \operatorname{Lip}(\{Q, I\}, R) has a unique extension bar(f)\bar{f} defined by bar(f)(x)=f(I)x,x in[0,x]\overline{\mathrm{f}}(\mathrm{x})=\mathrm{f}(\mathrm{I}) \mathrm{x}, x \in[0, x]. Indeed
( bar(I))|_({0,1})=1" and "|| bar(I)||=||P||=|I(1)|\left.\bar{I}\right|_{\{0,1\}}=1 \text { and }\|\bar{I}\|=\|P\|=|I(1)|
Therefore (b) does not imply (a).
Also
F({0,1})^(_|_)={f in Iip_(0)([0,I],R),f(1)=0}F(\{0,1\})^{\perp}=\left\{f \in \operatorname{Ii} p_{0}([0, I], R), f(1)=0\right\}
and for every f inLip_(0)([0,1],R)f \in \operatorname{Lip}_{0}([0,1], R) the element g_(0)g_{0} of best approximation for f^(')f^{\prime} in P({0,1})^(_|_)P(\{0,1\})^{\perp} is
g_(0)(x)=f(x)-f(1)x quad,quad x in[0,1].g_{0}(x)=f(x)-f(1) x \quad, \quad x \in[0,1] .
This element is unique since g_(0)(x)=f(x)-f(x)g_{0}(x)=f(x)-f(x) and bar(f)\bar{f} is the only extension of ff (see [4], Lemma 1 ). Therefore (c) does not imply (a) .
REIPERINNCES
I. ARONSSON , G. , A note on the extension of Lipschitz and Hölder functions (preprint, 5 p.)
2. ARONSSON , G., Extension of functions satisfying Lipschitz conditions , Arkiv för Mathematik 6, 28 (1967), 551-561.
3. GATIXI, S., IS40, G, PMOT J. P, Gurfectivity of maltifunctiona under genaralized differentiability assumptionell. Bull. Mastral. Math. Soc., 28 (1983), 13-21.
4. Musrimp, Go. Best Approximation and Untque Bxtension of Higchity Tunctions, Journal of Approx. Theory 19, 3 (1977), 222-230.
5. GUSTITA, C_(n)C_{n}, Batension of bounded Iipschitz functions, "Babeq-Bolyay" University. Faculty of Mathematics, Research Seminaries, Seminar of Functional Analysis and Numerlcal Methods, Preprint Nr.1, 1980, 1-20.
6. Me SHANE, E.J., Extansion of range of functions, Bull. Amer. Hath. Soc. 40 (1934), 837-842.
7. STIVGER, Io, Gea, mai tună aproximare in spatii vectoriale normate prin elemente din subspatii vectoriale, Faltura Academiei R.S. România, Bucureşti 1967.
