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“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
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C. Mustăţa, On a surjectivity theorem, Seminar of Functional Analysis and Numerical Methods, Preprint Nr. 1, (1985), 73-84.
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MR # 832505
[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.
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ON A SURJECTIVITY THEOREM
Costică Mustăţa
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. ．
（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}\|.
(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\}.
(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).
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).
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)).
(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
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.
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.