On the extension of Hölder functions

Costica Mustata
(original), Approximation Theory, paper

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Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

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C. Mustăţa, On the extension of Hölder functions, Seminar of Functional Analysis and Numerical Methods, Preprint no. 1 (1985), 85-92.

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Preprint

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“Babes-Bolyai” University Cluj-Napoca, Romania

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MR # 87h: 46068

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[1] Aronsson, G., Extension of functions satisfying Lipschitz conditions, Arkiv for Mathematik 6 (1967), 551-561.
[2] Czipser, J., Geher, I., Extension of functions satisfying a Lipschitz condition, Acta Math., Sci. Hungar 6 (1955), 213-220.
[3] Danzer, L., Grunbaum, B., Klee, V., Helly’s theorem and its relatives, Proc. Sympos. Pure Mathemm. 7, Amar. Math. Soc., Providence, R.I. (1963), 101-180.
[4] Fiett, T.M., Extension of Lipschitz functions, J. London Math. Soc., 7 (1974), 604-608.
[5] Grunbaum, B., A generalization of Theorems of Kirszbraum and Minty, Proc. Amer. Math. Soc., 13 (1962), 812-814.
[6] Hiriart-Urruti, J.B., Extension of Lipschitz functions, Preprint 1980, (23 pp.).
[7] Mustata, C., Best Approximation and Unique Extension of Lipschitz Functions, Journal of Approx. Theory 19, 3 (1977), 222-230.
[8] Mustata, C., Cobzas, S., Norm Preserving Extension of Convex Lipschitz Functions, Journal of Approx. Theory 24, 3 (1978), 236-244.
[9] Mustata, C., About the determination of extrema of a  Holder funcitons, Seminar of Funct. Anal. Numerical Methods, Babes-Bolyai University, Fac. of Math. Preprint Nr.1 (1983), 107-116.
[10] Mustata, C., Asupra unicității prelungirii funcțiilor Holder reale, Seminarul itinerant de Ecuații Funcționale, aproximare și Convexitate, Cluj-Napoca, 1980.
[11] Minty, G.J., On the extension of Lipschitz, Lipschitz-Holder continuous, and monotone functions, Bull. Amer. Math. Soc. 76 (1970), 334-339.
[12] Schonbeck, S.O., Extension of nonlinear contractions, Bull. Amer. Math. Soc. 72 (1966), 99-101.
[13] Schonbeck, S.O., On the extension of Lipschitz maps, Arkiv for Mathematik 7 (1967-1969), 201-209.
[14] McShane, E.J., Extension of range of functions, Bull. Amer. Math. Soc., 40 (1934),, 837-842.

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1985-Mustata-UBB-Seminar-On-the-extension-of-Holder-functions
ON THE EXTENSION OF HOLDER FUNCTIONS

Costică Mustăţa

  1. Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) and ( Y , ρ ) ( Y , ρ ) (Y,rho)(Y, \rho)(Y,ρ) be two metric spaces. For α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1] a function f : X → Y f : X → Y f:X rarr Yf: X \rightarrow Yf:X→Y is called Hölder of class α α alpha\alphaα if there exists M ⩾ 0 M ⩾ 0 M >= 0M \geqslant 0M⩾0 such that
    (1) ( E ( x ) , P ( y ) ) ≤ U ( d ( x , y ) ) α ( E ( x ) , P ( y ) ) ≤ U ( d ( x , y ) ) α quad(E(x),P(y)) <= U(d(x,y))^(alpha)quad\quad(E(x), P(y)) \leq U(d(x, y))^{\alpha} \quad(E(x),P(y))≤U(d(x,y))α,
    for all x v , y ∈ X x v , y ∈ X x_(v),y in Xx_{v}, y \in Xxv,y∈X.
    Denote by Λ α ( X , Y ) Λ α ( X , Y ) Lambda_(alpha)(X,Y)\Lambda_{\alpha}(X, Y)Λα(X,Y) the set of all Hölder functions of class α α alpha\alphaα from X X XXX to Y Y YYY.
If I I III is a metric linear space then, equiped with the pointvise operations of addition and multiplication by scalars, ∧ a ( X , Y ) ∧ a ( X , Y ) ^^_(a)(X,Y)\wedge_{a}(X, Y)∧a(X,Y) become a linear space. If Y = R Y = R Y=RY=RY=R then ⋀ α ( X , R ) ⋀ α   ( X , R ) ^^^_(alpha)(X,R)\bigwedge_{\alpha}(X, R)⋀α(X,R) is also a lattice (the order is defined pointwisely too).
For f ∈ ∧ α ( X , R ) f ∈ ∧ α ( X , R ) f in^^_(alpha)(X,R)f \in \wedge_{\alpha}(X, R)f∈∧α(X,R) put
(2) ‖ f ‖ α = sup { | f ( x ) − f ( y ) | / ( d ( x , y ) ) α : x , y ∈ X , x ∈ y } ‖ f ‖ α = sup | f ( x ) − f ( y ) | / ( d ( x , y ) ) α : x , y ∈ X , x ∈ y ||f||_(alpha)=s u p{|f(x)-f(y)|//(d(x,y))^(alpha):quad x,y in X,quad x in y}\|f\|_{\alpha}=\sup \left\{|f(x)-f(y)| /(d(x, y))^{\alpha}: \quad x, y \in X, \quad x \in y\right\}‖f‖α=sup{|f(x)−f(y)|/(d(x,y))α:x,y∈X,x∈y}, the smallest number M ⩾ 0 M ⩾ 0 M >= 0M \geqslant 0M⩾0 for which the inequality
(3) | f ( x ) − f ( y ) | ≤ M ( d ( x , y ) ) α | f ( x ) − f ( y ) | ≤ M ( d ( x , y ) ) α quad|f(x)-f(y)| <= M(d(x,y))^(alpha)\quad|f(x)-f(y)| \leq M(d(x, y))^{\alpha}|f(x)−f(y)|≤M(d(x,y))α,
holds for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X.
Obviously ‖ f ‖ α ⩾ 0 ‖ f ‖ α ⩾ 0 ||f||_(alpha) >= 0\|f\|_{\alpha} \geqslant 0‖f‖α⩾0 and ‖ f ‖ α = 0 ‖ f ‖ α = 0 ||f||_(alpha)=0\|f\|_{\alpha}=0‖f‖α=0 if and only if f = f = f=f=f= const. , for all f ∈ Λ α ( X , R ) f ∈ Λ α ( X , R ) f inLambda_(alpha)(X,R)f \in \Lambda_{\alpha}(X, R)f∈Λα(X,R).
Let x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X be fixed and let
.. (4) Λ α ( x 0 , X , R ) = { f ∈ Λ α ( X , R ) , f ( x 0 ) = 0 } Λ α x 0 , X , R = f ∈ Λ α ( X , R ) , f x 0 = 0 quadLambda_(alpha)(x_(0),X,R)={f inLambda_(alpha)(X,R),f(x_(0))=0}\quad \Lambda_{\alpha}\left(x_{0}, X, R\right)=\left\{f \in \Lambda_{\alpha}(X, R), f\left(x_{0}\right)=0\right\}Λα(x0,X,R)={f∈Λα(X,R),f(x0)=0}.
Then ⋀ α ( x 0 , x , R ) ⋀ α   x 0 , x , R ^^^_(alpha)(x_(0),x,R)\bigwedge_{\alpha}\left(x_{0}, x, R\right)⋀α(x0,x,R) is a subspace of Λ α ( X , R ) Λ α ( X , R ) Lambda_(alpha)(X,R){\Lambda_{\alpha}}(X, R)Λα(X,R) and the functional defined by (2) is a norm on this subspace and is called the Hölder norm of f f fff.
We say that two functions f , g ∈ ⋀ α ( X , R ) f , g ∈ ⋀ α   ( X , R ) f,g in^^^_(alpha)(X,R)f, g \in \bigwedge_{\alpha}(X, R)f,g∈⋀α(X,R) are equivalent if f − g = f − g = f-g=f-g=f−g= const. and we shall denote this by frug.
It is immediate that the quatient space of ⋂ α ( X , R ) ⋂ α   ( X , R ) nnn_(alpha)(X,R)\bigcap_{\alpha}(X, R)⋂α(X,R) by this equivalence relation is isomorphic to ⋀ α ( x 0 , X , R ) ⋀ α   x 0 , X , R ^^^_(alpha)(x_(0),X,R)\bigwedge_{\alpha}\left(x_{0}, X, R\right)⋀α(x0,X,R).
A very important problem in the theory of Hölder functions is the extension problem. Hore exactly, let ( X , d ) , ( Y , ρ ) ( X , d ) , ( Y , ρ ) (X,d),(Y,rho)(X, d),(Y, \rho)(X,d),(Y,ρ) be two metric spaces and let ZCX . The extension problem is the following: for f ∈ Λ α ( Z , Y ) f ∈ Λ α ( Z , Y ) f inLambda_(alpha)(Z,Y)f \in \Lambda_{\alpha}(Z, Y)f∈Λα(Z,Y) find F ∈ Λ α ( X , Y ) F ∈ Λ α ( X , Y ) F inLambda_(alpha)(X,Y)F \in \Lambda_{\alpha}(X, Y)F∈Λα(X,Y) such that
(5) f = η | Z and ‖ f ‖ α = ‖ F ‖ α . (5) f = η Z  and  ‖ f ‖ α = ‖ F ‖ α . {:(5)f= eta|_(Z)quad" and "quad||f||_(alpha)=||F||_(alpha)quad.:}\begin{equation*} f=\left.\eta\right|_{Z} \quad \text { and } \quad\|f\|_{\alpha}=\|F\|_{\alpha} \quad . \tag{5} \end{equation*}(5)f=η|Z and ‖f‖α=‖F‖α.
The function F F FFF is called a norm preserving extension of f f fff.
For α = 1 α = 1 alpha=1\alpha=1α=1 (the case of Lipschitz functions) the problem was extensively studied. The existence of a norm preserving extension for every f ∈ Λ d ( Z , Y ) f ∈ Λ d ( Z , Y ) f inLambda_(d)(Z,Y)f \in \Lambda_{d}(Z, Y)f∈Λd(Z,Y) depends on the properties of the sets Z Z ZZZ and Y Y YYY. A positive solution for the extension problem in the case Y = B Y = B Y=BY=BY=B and for X X XXX arbitrary was given by Mc SHANE [14] and by G G GGG. MLNTY [11] in the case when X X XXX and Y Y YYY are Hilber spaces.
If X X XXX and Y Y YYY are arbitrary metric spaces (ever Banach spaces) the extensions is not always possible as was shown by B. Gi JNBAUIA [5] and s.o. SCHÖENBECK [12], [13] .
T.U. PLETT [4] proved that if X X XXX and Y Y YYY are normed spaces and Z ⊂ X Z ⊂ X ZsubX\mathrm{Z} \subset \mathrm{X}Z⊂X is convex, closed, bounded of diameter δ δ delta\deltaδ and contains a ball of radius r > 0 r > 0 r > 0r>0r>0 then f o r f o r forf o rfor every f ∈ Λ 1 ( Z , X ) f ∈ Λ 1 ( Z , X ) f inLambda_(1)(Z,X)f \in \Lambda_{1}(Z, X)f∈Λ1(Z,X) there exists
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    F ∈ Λ 1 ( X , Y ) F ∈ Λ 1 ( X , Y ) F inLambda_(1)(X,Y)F \in \Lambda_{1}(X, Y)F∈Λ1(X,Y) such that F | 2 = f F 2 = f F|_(2)=f\left.F\right|_{2}=fF|2=f and ‖ F ‖ 1 = δ r ⋅ ‖ f ‖ 1 ‖ F ‖ 1 = δ r ⋅ ‖ f ‖ 1 ||F||_(1)=(delta )/(r)*||f||_(1)\|F\|_{1}=\frac{\delta}{r} \cdot\|f\|_{1}‖F‖1=δr⋅‖f‖1.
    If every function f ∈ Λ α ( Z , Y ) f ∈ Λ α ( Z , Y ) f inLambda_(alpha)(Z,Y)f \in \Lambda_{\alpha}(Z, Y)f∈Λα(Z,Y) has an extension F ∈ Λ α ( X , Y ) F ∈ Λ α ( X , Y ) F inLambda_(alpha)(X,Y)F \in \Lambda_{\alpha}(X, Y)F∈Λα(X,Y)
    it is natural to ask if this extension is unique or not. It was shown that the question of the unicity of the norm preserving extension is closely related to some approximation problems in the space Λ α ( X , Y ) Λ α ( X , Y ) Lambda_(alpha)(X,Y)\Lambda_{\alpha}(X, Y)Λα(X,Y) (see [7], [8], [10] ) .
  1. In the following we shall denote Lip ( X , Y ) = Λ 1 ( X , Y ) Lip ( X , Y ) = Λ 1 ( X , Y ) Lip(X,Y)=Lambda_(1)(X,Y)\operatorname{Lip}(X, Y)=\Lambda_{1}(X, Y)Lip(X,Y)=Λ1(X,Y). If X X XXX is a Banach space and S S SSS is a closed ball of radius r > 0 r > 0 r > 0r>0r>0 in X X XXX then as was shown by T.M. FLETT [4] there exists a function F ∈ Lip ( X , X ) F ∈ Lip ( X , X ) F in Lip(X,X)F \in \operatorname{Lip}(X, X)F∈Lip(X,X) such that
    (6) ‖ F ‖ 1 = 2 ∣ I ‖ 1 ‖ F ‖ 1 = 2 ∣ I ‖ 1 quad||F||_(1)=2∣I||_(1)\quad\|F\|_{1}=2 \mid I \|_{1}‖F‖1=2∣I‖1
    where f = F | S f = F S f=F|_(S)f=\left.F\right|_{S}f=F|S.
    THEOREM 1. Let X X XXX be a Banach space and let f ∈ Ilp ( X , X ) f ∈ Ilp ( X , X ) f in Ilp(X,X)f \in \operatorname{Ilp}(X, X)f∈Ilp(X,X). Suppose that the following conditions hold true :
    a) There exists a convex, closed, bounded set C of diameter ρ ρ rho\rhoρ and containing a ball of radius δ > 0 δ > 0 delta > 0\delta>0δ>0 guch that
‖ r | a ‖ 1 < δ ρ , r a 1 < δ ρ , ||r|_(a)||_(1) < (delta )/(rho),\left\|\left.r\right|_{a}\right\|_{1}<\frac{\delta}{\rho},‖r|a‖1<δρ,
b) Every extension F F FFF of. f f fff yexifies
(7) | f − F | 2 < 1 − | I | G | 2 ρ δ | f − F | 2 < 1 − | I | G 2 ρ δ quad|f-F|_(2) < 1-|I|_(G)|_(2)(rho )/(delta)\quad|f-F|_{2}<1-\left.|I|_{G}\right|_{2} \frac{\rho}{\delta}|f−F|2<1−|I|G|2ρδ.
Then there exists a unique x ′ ∈ X x ′ ∈ X x^(')in Xx^{\prime} \in Xx′∈X auch that f ( x − ) = x ∗ f x − = x ∗ f(x^(-))=x^(**)f\left(x^{-}\right)=x^{*}f(x−)=x∗. (The function f f fff has a unique fix point x ∗ ∈ x x ∗ ∈ x x^(**)in xx^{*} \in xx∗∈x ).
Proof. Let f ∈ Lip ( X , I ) f ∈ Lip ( X , I ) f in Lip(X,I)f \in \operatorname{Lip}(X, I)f∈Lip(X,I) and C ⊂ X C ⊂ X C sub XC \subset XC⊂X such that condition a) is verified. By the above quated result of Mett there exists F ∈ Lip ( x , x ) F ∈ Lip ( x , x ) F in Lip(x,x)F \in \operatorname{Lip}(x, x)F∈Lip(x,x) such that F | c = 4 | c F c = 4 c F|_(c)=4|_(c)\left.F\right|_{c}=\left.4\right|_{c}F|c=4|c and | F | 1 = | f | 0 | 1 f | F | 1 = | f | 0 1 f |F|_(1)=|f|_(0)|_(1)f|F|_{1}=\left.|f|_{0}\right|_{1} f|F|1=|f|0|1f. Then ‖ f ‖ 1 = ‖ f − F + F I 2 ≤ ‖ f − F ‖ 2 + | I 1 < 1 − | F | C | I ρ δ + ‖ P | C ‖ I ρ σ = 1 ‖ f ‖ 1 = f − F + F I 2 ≤ f − F 2 + I 1 < 1 − | F | C I ρ δ + P C ‖ I ρ σ = 1 ||f||_(1)=||f-F+FI_(2) <= ||f-F||_(2)+|I_(1) < 1-|F|_(C)|_(I)(rho )/(delta)+||P|_(C)||_(I)(rho )/(sigma)=1\|f\|_{1}=\left\|f-F+F I_{2} \leq\right\| f-\left.F\left\|_{2}+\left|I_{1}<1-|F|_{C}\right|_{I} \frac{\rho}{\delta}+\right\| P\right|_{C} \|_{I} \frac{\rho}{\sigma}=1‖f‖1=‖f−F+FI2≤‖f−F‖2+|I1<1−|F|C|Iρδ+‖P|C‖Iρσ=1.
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Since ‖ f ( x ) − f ( y ) ‖ ≤ ‖ P ‖ 4 ‖ x − y ‖ ‖ f ( x ) − f ( y ) ‖ ≤ ‖ P ‖ 4 ‖ x − y ‖ ||f(x)-f(y)|| <= ||P||_(4)||x-y||\|f(x)-f(y)\| \leq\|P\|_{4}\|x-y\|‖f(x)−f(y)‖≤‖P‖4‖x−y‖ for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X it follows that I I III is a contraction on X X XXX and by Banach contraction principle there exists a unique x ∗ ∈ X x ∗ ∈ X x^(**)in Xx^{*} \in Xx∗∈X such that f ( x ∗ ) = x ∗ f x ∗ = x ∗ f(x^(**))=x^(**)f\left(x^{*}\right)=x^{*}f(x∗)=x∗. Theorem is proved .
COROLLARY 1. Let X X XXX be a Banach space and f ∈ Lip ( X , X ) f ∈ Lip ( X , X ) f in Lip(X,X)f \in \operatorname{Lip}(X, X)f∈Lip(X,X). Suppose that there exists a closed ball s ∈ X s ∈ X s in Xs \in Xs∈X of radius δ > 0 δ > 0 delta > 0\delta>0δ>0 such that every extension F F FFF of f / s f / s f//sf / sf/s verifies the condition :
(8) | f − F | 1 < 1 − 2 | f | s | 1 . (8) | f − F | 1 < 1 − 2 | f | s 1 . {:(8)|f-F|_(1) < 1-2|f|_(s)|_(1).:}\begin{equation*} |f-F|_{1}<1-\left.2|f|_{s}\right|_{1} . \tag{8} \end{equation*}(8)|f−F|1<1−2|f|s|1.
Then f f fff has a unique fix point in X X XXX.
Proof. The diameter of S S SSS is ρ = 2 δ ρ = 2 δ rho=2delta\rho=2 \deltaρ=2δ and by (8) ‖ f | S ‖ 1 < 1 2 f S 1 < 1 2 ||f|_(S)||_(1) < (1)/(2)\left\|\left.f\right|_{S}\right\|_{1}<\frac{1}{2}‖f|S‖1<12 so that the condition a) and b) from Theorem 1 are verified.
Bemark 1. Let c c ccc be as in Theorem 1 and f ∈ Lip ( X , X ) f ∈ Lip ( X , X ) f in Lip(X,X)f \in \operatorname{Lip}(X, X)f∈Lip(X,X). If ‖ P | C ‖ 1 = 0 P C 1 = 0 ||P|_(C)||_(1)=0\left\|\left.P\right|_{C}\right\|_{1}=0‖P|C‖1=0 then ‖ P ( x ) − f ( y ) ‖ = 0 ‖ P ( x ) − f ( y ) ‖ = 0 ||P(x)-f(y)||=0\|P(x)-f(y)\|=0‖P(x)−f(y)‖=0 for all x , y ∈ C x , y ∈ C x,y in Cx, y \in Cx,y∈C and f ( x ) = f ( y ) = z ∈ X f ( x ) = f ( y ) = z ∈ X f(x)=f(y)=z in Xf(x)=f(y)=z \in Xf(x)=f(y)=z∈X for all x , y ∈ C x , y ∈ C x,y in Cx, y \in Cx,y∈C. Since F | C = f F C = f F|_(C)=f\left.F\right|_{C}=fF|C=f and ‖ F ‖ 1 == 0 ρ f = 0 ‖ F ‖ 1 == 0 ρ f = 0 ||F||_(1)==0(rho )/(f)=0\|F\|_{1}= =0 \frac{\rho}{f}=0‖F‖1==0ρf=0 it follows that F ( x ) = z F ( x ) = z F(x)=zF(x)=zF(x)=z for all x ∈ X x ∈ X x in Xx \in Xx∈X. Therefore the condition (7) from Theorem 1 becomes
( 7 ∘ ) 7 ∘ {:7^(@))\left.7^{\circ}\right)7∘)
‖ x ‖ 1 < 1 . ‖ x ‖ 1 < 1 . ||x||_(1) < 1.\|x\|_{1}<1 .‖x‖1<1.
1.e. f f fff is a contraction on X X XXX.
If ‖ P | G ‖ 1 = 0 P G 1 = 0 ||P|_(G)||_(1)=0\left\|\left.P\right|_{G}\right\|_{1}=0‖P|G‖1=0 then P = P = P=P=P= const. on C C CCC and the extension I ∈ I 4 p ( s ¯ , X ) I ∈ I 4 p ( s ¯ , X ) I in I4p( bar(s),X)I \in I 4 p({ } \bar{s}, X)I∈I4p(s¯,X) is uniqus.
3. We consider the following problem t t ttt for a metric space X X XXX, a subset M M MMM of X X XXX and a function f ∈ Λ α ( M , R ) f ∈ Λ α ( M , R ) f inLambda_(alpha)(M,R)f \in \Lambda_{\alpha}(M, R)f∈Λα(M,R) find
(9) min { f ( y ) : y ∈ M } (9) min { f ( y ) : y ∈ M } {:(9)min{f(y):y in M}:}\begin{equation*} \min \{f(y): y \in M\} \tag{9} \end{equation*}(9)min{f(y):y∈M}
In conorete problems the set 1 I is usually determined by some restriotions and the function f f fff is replaced by the function f ¯ f ¯ bar(f)\bar{f}f¯
defined by
P ( x ) = { f ( x ) , x ∈ M + ∞ , x ∈ X M P ( x ) = f ( x )      ,      x ∈ M + ∞      ,      x ∈ X M P(x)={[f(x),",",x in M],[+oo,",",x in XM]:}P(x)=\left\{\begin{array}{lll} f(x) & , & x \in M \\ +\infty & , & x \in X M \end{array}\right.P(x)={f(x),x∈M+∞,x∈XM
Obviously min { f ( y ) : y ∈ M } = min { I ¯ ( x ) : x ∈ X } min { f ( y ) : y ∈ M } = min { I ¯ ( x ) : x ∈ X } min{f(y):y inM}=min{ bar(I)(x):x in X}\min \{f(y): y \in \mathbb{M}\}=\min \{\bar{I}(x): x \in X\}min{f(y):y∈M}=min{I¯(x):x∈X}.
HIRIART-URRUTY [6] proved that if X X XXX is a Banach space M ⊂ X M ⊂ X M sub XM \subset XM⊂X is closed and f ∈ Lip ( M , R ) f ∈ Lip ( M , R ) f in Lip(M,R)f \in \operatorname{Lip}(M, R)f∈Lip(M,R), then the problem
min { f ( y ) : y ∈ M } min { f ( y ) : y ∈ M } min{f(y):y in M}\min \{f(y): y \in M\}min{f(y):y∈M}
can be replaced by the problem :
min { F 1 ( x ) : x ∈ X } , min F 1 ( x ) : x ∈ X , min{F_(1)(x):x in X},\min \left\{F_{1}(x): x \in X\right\},min{F1(x):x∈X},
where F 1 ( x ) = inf y ∈ M [ f ( y ) + ‖ f ‖ 1 ⋅ ‖ x − y ‖ ] , x ∈ X F 1 ( x ) = inf y ∈ M   f ( y ) + ‖ f ‖ 1 ⋅ ‖ x − y ‖ , x ∈ X F_(1)(x)=i n f_(y in M)[f(y)+||f||_(1)*||x-y||],x in XF_{1}(x)=\inf _{y \in M}\left[f(y)+\|f\|_{1} \cdot\|x-y\|\right], x \in XF1(x)=infy∈M[f(y)+‖f‖1⋅‖x−y‖],x∈X.
In this note we shall give some similar results in the case of a metric space X X XXX and for a function f ∈ Λ a ( M , R ) , 0 < π ≤ 1 f ∈ Λ a ( M , R ) , 0 < π ≤ 1 f inLambda_(a)(M,R),0 < pi <= 1f \in \Lambda_{a}(M, R), 0<\pi \leq 1f∈Λa(M,R),0<π≤1.
Let X X XXX be a metric space, let M M MMM be a closed subset of X X XXX and let f ∈ Λ α ( M , R ) f ∈ Λ α ( M , R ) f inLambda_(alpha)(M,R)f \in \Lambda_{\alpha}(M, R)f∈Λα(M,R). By a result in [9] the function F 1 F 1 F_(1)F_{1}F1 defined by
(10) F 1 ( x ) = inf y ∈ M [ P ( y ) + ‖ I h α ( a ( x , y ) ) α ] , x ∈ X F 1 ( x ) = inf y ∈ M   P ( y ) + ‖ I h α ( a ( x , y ) ) α , x ∈ X quadF_(1)(x)=i n f_(y in M)[P(y)+||Ih_(alpha)(a(x,y))^(alpha)],x in X\quad F_{1}(x)=\inf _{y \in M}\left[P(y)+\| I h_{\alpha}(a(x, y))^{\alpha}\right], x \in XF1(x)=infy∈M[P(y)+‖Ihα(a(x,y))α],x∈X
is in Λ α ( X , R ) Λ α ( X , R ) Lambda_(alpha)(X,R)\Lambda_{\alpha}(X, R)Λα(X,R) and
r 1 | M = f , ‖ r 1 ‖ α = ‖ f ‖ α . r 1 M = f , r 1 α = ‖ f ‖ α . r_(1)|_(M)=f,quad||r_(1)||_(alpha)=||f||_(alpha).\left.r_{1}\right|_{M}=f, \quad\left\|r_{1}\right\|_{\alpha}=\|f\|_{\alpha} .r1|M=f,‖r1‖α=‖f‖α.
A point y 0 ∈ M y 0 ∈ M y_(0)in My_{0} \in My0∈M is called a minimum (maximum) for f f fff if
f ( y 0 ) ⩽ f ( y ) ( f ( y 0 ) ⩾ f ( y ) ) f y 0 ⩽ f ( y ) f y 0 ⩾ f ( y ) f(y_(0)) <= f(y)quad(f(y_(0)) >= f(y))f\left(y_{0}\right) \leqslant f(y) \quad\left(f\left(y_{0}\right) \geqslant f(y)\right)f(y0)⩽f(y)(f(y0)⩾f(y))
for all y ∈ M y ∈ M y inMy \in \mathbb{M}y∈M.
THEORIM 2. Let X X XXX be a metric space, M M MMM a closed subset of X X XXX and f ∈ ∧ α ( H , R ) f ∈ ∧ α ( H , R ) f in^^_(alpha)(H,R)f \in \wedge_{\alpha}(H, R)f∈∧α(H,R). Then y 0 ∈ M y 0 ∈ M y_(0)in My_{0} \in My0∈M is a minimum point for f f fff on M M MMM if and only if J 0 J 0 J_(0)J_{0}J0 is a minimum point for F 1 F 1 F_(1)F_{1}F1 on X X XXX.
Proof. Let y 0 y 0 y_(0)y_{0}y0 be a minimum point for f f fff on U U UUU and let E 1 E 1 E_(1)E_{1}E1 be defined by (10). For every x ∈ M x ∈ M x in Mx \in Mx∈M we have
F 1 ( x ) = f ( x ) ≥ f ( y 0 ) = I 1 ( y 0 ) F 1 ( x ) = f ( x ) ≥ f y 0 = I 1 y 0 F_(1)(x)=f(x) >= f(y_(0))=I_(1)(y_(0))F_{1}(x)=f(x) \geq f\left(y_{0}\right)=I_{1}\left(y_{0}\right)F1(x)=f(x)≥f(y0)=I1(y0)
If x ∉ I x ∉ I x!in Ix \notin Ix∉I, the set I I III being closed, there exists δ > 0 δ > 0 delta > 0\delta>0δ>0 such that a ( x , y ) ≥ 6 > 0 a ( x , y ) ≥ 6 > 0 a(x,y) >= 6 > 0a(x, y) \geq 6>0a(x,y)≥6>0 for all y ∈ M y ∈ M y in My \in My∈M. Therefore
I I ( x ) = inf j ∈ f [ f ( y ) + ‖ f ‖ α ( d ( x , y ) ) α d ] ≤ ⩾ inf j ∈ M [ f ( y ) + ‖ f ‖ α δ ] = ‖ f ‖ α δ α + f ( y 0 ) > f ( y 0 ) I I ( x ) = inf j ∈ f   f ( y ) + ‖ f ‖ α ( d ( x , y ) ) α d ≤ ⩾ inf j ∈ M   f ( y ) + ‖ f ‖ α δ = ‖ f ‖ α δ α + f y 0 > f y 0 {:[I_(I)(x)=i n f_(j in f)[f(y)+||f||_(alpha)(d(x,y))^(alpha d)] <= ],[ >= i n f_(j in M)[f(y)+||f||_(alpha)delta]=||f||_(alpha)delta^(alpha)+f(y_(0)) > f(y_(0))]:}\begin{aligned} I_{I}(x) & =\inf _{j \in f}\left[f(y)+\|f\|_{\alpha}(d(x, y))^{\alpha d}\right] \leq \\ & \geqslant \inf _{j \in M}\left[f(y)+\|f\|_{\alpha} \delta\right]=\|f\|_{\alpha} \delta^{\alpha}+f\left(y_{0}\right)>f\left(y_{0}\right) \end{aligned}II(x)=infj∈f[f(y)+‖f‖α(d(x,y))αd]≤⩾infj∈M[f(y)+‖f‖αδ]=‖f‖αδα+f(y0)>f(y0)
so that y 0 y 0 y_(0)y_{0}y0 is a minimum point for F 1 F 1 F_(1)F_{1}F1 on X X XXX.
Conversely, suppose that Y 0 Y 0 Y_(0)Y_{0}Y0 is a minimum point for P 1 P 1 P_(1)P_{1}P1 on X X XXX. If we would aher that y 0 ∈ M y 0 ∈ M y_(0)in My_{0} \in My0∈M then, as I 1 | M = f I 1 M = f I_(1)|_(M)=f\left.I_{1}\right|_{M}=fI1|M=f, it would follow that y 0 y 0 y_(0)y_{0}y0 is a minimum point for f f fff on M M MMM.
Suppose, on the cuntrary, that y 0 ∉ M y 0 ∉ M y_(0)!in My_{0} \notin My0∉M. Then, since M M MMM is clossd,
d ( J 0 , M ) = inf { d ( J 0 , J ) , y ∈ M } = q > 0 . d J 0 , M = inf d J 0 , J , y ∈ M = q > 0 . d(J_(0),M)=i n f{d(J_(0),J),y inM}=q > 0.d\left(J_{0}, M\right)=\inf \left\{d\left(J_{0}, J\right), y \in \mathbb{M}\right\}=q>0 .d(J0,M)=inf{d(J0,J),y∈M}=q>0.
By the definition of F 1 F 1 F_(1)F_{1}F1 we have
F 1 ( y 0 ) = inf g ∈ M [ f ( y ) + ‖ f ‖ α ( α ( y 0 , y ) ) α ] , F 1 y 0 = inf g ∈ M   f ( y ) + ‖ f ‖ α α y 0 , y α , F_(1)(y_(0))=i n f_(g in M)[f(y)+||f||_(alpha)(alpha(y_(0),y))^(alpha)],F_{1}\left(y_{0}\right)=\inf _{g \in M}\left[f(y)+\|f\|_{\alpha}\left(\alpha\left(y_{0}, y\right)\right)^{\alpha}\right],F1(y0)=infg∈M[f(y)+‖f‖α(α(y0,y))α],
so that, for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0, there exists y ε ∈ I y ε ∈ I y_(epsi)inIy_{\varepsilon} \in \mathbb{I}yε∈I such that
T 1 ( y 0 ) + ε > f ( y E ) + ‖ f ‖ α ( d ( y 0 , J E ) ) α . T 1 y 0 + ε > f y E + ‖ f ‖ α d y 0 , J E α . T_(1)(y_(0))+epsi > f(y_(E))+||f||_(alpha)(d(y_(0),J_(E)))^(alpha).T_{1}\left(y_{0}\right)+\varepsilon>f\left(y_{E}\right)+\|f\|_{\alpha}\left(d\left(y_{0}, J_{E}\right)\right)^{\alpha} .T1(y0)+ε>f(yE)+‖f‖α(d(y0,JE))α.
For ε n = I f 1 q n ε n = I f 1 q n epsi_(n)=(If_(1)q)/(n)\varepsilon_{n}=\frac{I f_{1} q}{n}εn=If1qn, denoting y n = y E n y n = y E n y_(n)=y_(E_(n))y_{n}=y_{E_{n}}yn=yEn, one obtains
L ( y ε n ) = F 1 ( y n ) ⩾ F 1 ( y 0 ) > f ( y n ) + ‖ f ‖ α ( a ( y 0 , y n ) ) α − ‖ f ‖ α q n L y ε n = F 1 y n ⩾ F 1 y 0 > f y n + ‖ f ‖ α a y 0 , y n α − ‖ f ‖ α q n L(y_(epsi_(n)))=F_(1)(y_(n)) >= F_(1)(y_(0)) > f(y_(n))+||f||_(alpha)(a(y_(0),y_(n)))^(alpha)-(||f||_(alpha)q)/(n)\mathcal{L}\left(y_{\varepsilon_{n}}\right)=F_{1}\left(y_{n}\right) \geqslant F_{1}\left(y_{0}\right)>f\left(y_{n}\right)+\|f\|_{\alpha}\left(a\left(y_{0}, y_{n}\right)\right)^{\alpha}-\frac{\|f\|_{\alpha} q}{n}L(yεn)=F1(yn)⩾F1(y0)>f(yn)+‖f‖α(a(y0,yn))α−‖f‖αqn
which implies
‖ r ‖ α ⋅ ( ( α ( y 0 , y n ) ) α − q n ) ≤ 0 . ‖ r ‖ α ⋅ α y 0 , y n α − q n ≤ 0 . ||r||_(alpha)*((alpha(y_(0),y_(n)))^(alpha)-(q)/(n)) <= 0.\|r\|_{\alpha} \cdot\left(\left(\alpha\left(y_{0}, y_{n}\right)\right)^{\alpha}-\frac{q}{n}\right) \leq 0 .‖r‖α⋅((α(y0,yn))α−qn)≤0.
If ‖ f ‖ α = 0 ‖ f ‖ α = 0 ||f||_(alpha)=0\|f\|_{\alpha}=0‖f‖α=0 then f = f = f=f=f= const on M M MMM and J 0 ( J 0 ( J_(0)quad(J_{0} \quad(J0( as every other point in M M MMM ). will be a minimum point for f f fff on M M MMM.
If ‖ f ‖ α > 0 ‖ f ‖ α > 0 ||f||_(alpha) > 0\|f\|_{\alpha}>0‖f‖α>0 then
( α ( y 0 , y n ) ) α − q n ≤ 0 α y 0 , y n α − q n ≤ 0 (alpha(y_(0),y_(n)))^(alpha)-(q)/(n) <= 0\left(\alpha\left(y_{0}, y_{n}\right)\right)^{\alpha}-\frac{q}{n} \leq 0(α(y0,yn))α−qn≤0
so that
0 < q ⩽ d ( y 0 , y n ) ⩽ ( q n ) 1 d . 0 < q ⩽ d y 0 , y n ⩽ q n 1 d .  0 < q <= d(y_(0),y_(n)) <= ((q)/(n))^((1)/(d))". "0<q \leqslant d\left(y_{0}, y_{n}\right) \leqslant\left(\frac{q}{n}\right)^{\frac{1}{d}} \text {. }0<q⩽d(y0,yn)⩽(qn)1d. 
Letting n → ∞ n → ∞ n rarr oon \rightarrow \inftyn→∞ in the inequality 0 < q ⩽ ( q n ) 1 4 0 < q ⩽ q n 1 4 0 < q <= ((q)/(n))^((1)/(4))0<q \leqslant\left(\frac{q}{n}\right)^{\frac{1}{4}}0<q⩽(qn)14 one obtains a contradiction. Theorem 2 is proved.
Let f ∈ ∧ a ( M , R ) f ∈ ∧ a ( M , R ) f in^^_(a)(M,R)f \in \wedge_{a}(M, R)f∈∧a(M,R) and let
F 2 ( x ) = sup y ∈ M [ f ( y ) ′ − ‖ f ‖ d ( d ( x , y ) ) d ] , x ∈ X F 2 ( x ) = sup y ∈ M   f ( y ) ′ − ‖ f ‖ d ( d ( x , y ) ) d , x ∈ X F_(2)(x)=s u p_(y in M)[f(y)^(')-||f||_(d)(d(x,y))^(d)],x in XF_{2}(x)=\sup _{y \in M}\left[f(y)^{\prime}-\|f\|_{d}(d(x, y))^{d}\right], x \in XF2(x)=supy∈M[f(y)′−‖f‖d(d(x,y))d],x∈X
The function B 2 B 2 B_(2)B_{2}B2 has the properties :
F 2 | M = P and ‖ F 2 ‖ α = ‖ P R α , F 2 M = P  and  F 2 α = ‖ P R α , F_(2)|_(M)=P quad" and "quad||F_(2)||_(alpha)=||PR_(alpha),\left.F_{2}\right|_{M}=P \quad \text { and } \quad\left\|F_{2}\right\|_{\alpha}=\| P R_{\alpha},F2|M=P and ‖F2‖α=‖PRα,
(see [9] ) .
THEOREM 3. Let X X XXX be a metric space, M a closed subset of X X XXX and f ∈ Λ α ( M , R ) f ∈ Λ α ( M , R ) f inLambda_(alpha)(M,R)f \in \Lambda_{\alpha}(M, R)f∈Λα(M,R). Then y 0 ∈ M y 0 ∈ M y_(0)in My_{0} \in My0∈M is a maximum point for f f fff on M M MMM if and anly if y 0 y 0 y_(0)y_{0}y0 is a maximum point for F 2 F 2 F_(2)F_{2}F2 on X X XXX.
The proof of this Theorem is simillar to the proof of Theorem 2.
Remark 2. If X X XXX is a metric linear space, M M MMM a closed convex subset of X X XXX and f ∈ ∧ α ( M , R ) f ∈ ∧ α ( M , R ) f in^^_(alpha)(M,R)f \in \wedge_{\alpha}(M, R)f∈∧α(M,R) is convex, then f f fff has minimum point on M M MMM. The function F 1 F 1 F_(1)F_{1}F1, defined by (10), has the same minimum on X X XXX as f f fff on M M MMM. Furthermore the function F F FFF is conver too (see [8]).
If f f fff is a concave function on M M MMM, then the function F 2 F 2 F_(2)F_{2}F2 is concave too on X X XXX and the maximum of F 2 F 2 F_(2)F_{2}F2 on X X XXX equals the maximum of f f fff on M M M\mathbb{M}M.

REFFERENCES

  1. ARONSSON, G. Brtension of runctions satisfying Lipschitz conditions, Arictiv for Mathematik 6 (1967), 551 - 561.
  2. CZIPBKR, J., GEHER, Io, Extension of functions satisfying a Lipgchite condition , Acta Math. Sci. Hungar 6 (1955), 213 − 220 213 − 220 213-220213-220213−220.
  3. DANZER, L., GRUNBAMM, B., KLKE, Ψ 0 Ψ 0 Psi_(0)\Psi_{0}Ψ0, Helly's theorem and its relatives , Proc. Sympos. Pure Mathem. 7, Amer. Math. Soc., Providence, R.I. (1963), 101-180.
  4. FIEIT, T.M., Extension of Iipschits functions, J. London Math. Soc. 7 (1974), 604-608.
  5. GRUNBAUM, B., A generalization of Theorems of Kirszbraum and Mintri . Proc. Amer. Math. Soc, 13 (1962) , 812 − 814 812 − 814 812-814812-814812−814.
  6. HIRIART-URRUTY, J.B., Extension of Lipschitz functions, Preprint 1980 ( 23 pp. ).
  7. MUSTATA, C., Best Approximation and Unique Extension of Ilpschitz Functions, Journal of Approx. Theory 19, 3 (1977), 222-230.
  8. MUSTATTA, C., COBZAS, B., Norm Preserving Extension of Convex Lipschitz Functions, Journal of Approx. Theory 24, 3 (1978), 236-244.
  9. MUSTATA, C., About the determination of extrema of a
Hölder functions, Seminar of Funct.Anal. Nunerical Methods, Babes-Bolyai University, Fac. of Math. Preprint Nr. 1 (1983), 107-116.
10. MUSTÁTA, C., Asupra unicitătil prelungiril functilior Hölder reale , Seminarul itinerant de Bcuatii Functionale, Aproximars si Convexitate, ClujNapoca, 1930 .
11. UINIY, G.J., on the extension of Lipschitz, LipschitzHölder contruous, and monotone functions, Bull. Amer. Math. Soc. 76 (1970), 334-339 .
12. SCHONBACK, S.O., Extension of nonlinear contractions, Bull. Amer. Math . Soc. 72 (1966), 99-101 .
13. SCHONBECK, S.O., On the extension of Lipschitz maps. Arkiv för Mathematik 7 (1967-1969), 201-209.
14. MCSHANE, E.J., Extension of range of functions, Bull. Amer. Math. Soc., 40 (1934), 837 - 842 .
15. VALENTINL, F.A., A Lipschitz condition preserving extension for a vector function, Amer. J. Math. 67 (1945) , 83-93.
1985

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