On the Extensions Preserving the Shape of a Semi-Hölder Function

Costica Mustata
(original), Approximation Theory, ISI/JCR, paper

Abstract

We present some results concerning the extension of a semi-Hölder real-valued function defined on a subset of a quasi-metric space, preserving some shape properties: the smallest semi-Hölder constant, the radiantness and the global minimum (maximum) of the extended function.

Authors

Costică Mustăța
”Tiberiu Popoviciu” Institute of Numerical Analysis Cluj-Napoca, Romanian Academy

Keywords

Quasi-metric space; semi-Holder function; radiant function

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Mustăţa, C., On the extensions preserving the shape of a semi-Hölder function, Results. Math. 63 (2013), 425–433.
doi: 10.1007/s00025-011-0206-x

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Results in Mathematics

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Springer

DOI

10.1007/s00025-011-0206-x

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1422-6383

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2013-Mustata-On the Extensions Preserving-Result Math

On the Extensions Preserving the Shape of a Semi-Hölder Function

Costică Mustăţa

Abstract

We present some results concerning the extension of a semiHölder real-valued function defined on a subset of a quasi-metric space, preserving some shape properties: the smallest semi-Hölder constant, the radiantness and the global minimum (maximum) of the extended function.

Mathematics Subject Classification (2000). 46A22, 54E25.
Keywords. Quasi-metric space, semi-Hölder function, radiant function.

1. Introduction

Let X X XXX be a nonvoid set. A mapping d : X × X → [ 0 , ∞ ) d : X × X → [ 0 , ∞ ) d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty)d:X×X→[0,∞) satisfying the following conditions:
( QM 1 ) d ( x , y ) = d ( y , x ) = 0 QM 1 d ( x , y ) = d ( y , x ) = 0 (QM_(1))d(x,y)=d(y,x)=0quad\left(\mathrm{QM}_{1}\right) d(x, y)=d(y, x)=0 \quad(QM1)d(x,y)=d(y,x)=0 iff x = y x = y x=yx=yx=y,
( QM 2 ) d ( x , y ) ≤ d ( x , z ) + d ( z , y ) QM 2 d ( x , y ) ≤ d ( x , z ) + d ( z , y ) (QM_(2))d(x,y) <= d(x,z)+d(z,y)\left(\mathrm{QM}_{2}\right) d(x, y) \leq d(x, z)+d(z, y)(QM2)d(x,y)≤d(x,z)+d(z,y),
for all x , y , z ∈ X x , y , z ∈ X x,y,z in Xx, y, z \in Xx,y,z∈X is called a quasi-metric (asymmetric metric) on X X XXX, and the pair ( X , d X , d X,dX, dX,d ) is called a quasi-metric space [ 17,18 ].
Because, in general, d ( x , y ) ≠ d ( y , x ) , x , y ∈ X d ( x , y ) ≠ d ( y , x ) , x , y ∈ X d(x,y)!=d(y,x),x,y in Xd(x, y) \neq d(y, x), x, y \in Xd(x,y)≠d(y,x),x,y∈X, one defines the conjugate d ¯ d ¯ bar(d)\bar{d}d¯ of quasi-metric d d ddd as the quasi-metric d ¯ ( x , y ) = d ( y , x ) , x , y ∈ X d ¯ ( x , y ) = d ( y , x ) , x , y ∈ X bar(d)(x,y)=d(y,x),x,y in X\bar{d}(x, y)=d(y, x), x, y \in Xd¯(x,y)=d(y,x),x,y∈X.
For example, an asymmetric norm ‖ ∣ ‖ ∣ ||∣\| \mid‖∣ on a linear space X X XXX (see [6], Ch. IX, § 5) or [2], where a functional analysis in asymmetric normed space is presented) defines a quasi-metric d ‖ ∣ d ‖ ∣ d_(||∣)d_{\| \mid}d‖∣through the formula:
d ‖ | ( x , y ) = ‖ y − x | , x , y ∈ X . d ‖ | ( x , y ) = ‖ y − x | , x , y ∈ X . d_(||)|(x,y)=||y-x|,x,y in X.d_{\|}|(x, y)=\| y-x|, x, y \in X .d‖|(x,y)=‖y−x|,x,y∈X.
Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space. A sequence ( x k ) k ≥ 1 x k k ≥ 1 (x_(k))_(k >= 1)\left(x_{k}\right)_{k \geq 1}(xk)k≥1 is d d ddd-convergent to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X (or forward convergent to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X ) if
lim k → ∞ d ( x 0 , x k ) = 0 lim k → ∞   d x 0 , x k = 0 lim_(k rarr oo)d(x_(0),x_(k))=0\lim _{k \rightarrow \infty} d\left(x_{0}, x_{k}\right)=0limk→∞d(x0,xk)=0
and d ¯ d ¯ bar(d)\bar{d}d¯-convergent to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X (or backward convergent to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X ) if
lim k → ∞ d ( x k , x 0 ) = lim k → ∞ d ¯ ( x 0 , x k ) = 0 lim k → ∞   d x k , x 0 = lim k → ∞   d ¯ x 0 , x k = 0 lim_(k rarr oo)d(x_(k),x_(0))=lim_(k rarr oo) bar(d)(x_(0),x_(k))=0\lim _{k \rightarrow \infty} d\left(x_{k}, x_{0}\right)=\lim _{k \rightarrow \infty} \bar{d}\left(x_{0}, x_{k}\right)=0limk→∞d(xk,x0)=limk→∞d¯(x0,xk)=0
We say that the set Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X is d d ddd-closed ( d ¯ d ¯ bar(d)\bar{d}d¯-closed) if every d d ddd-convergent ( d ¯ d ¯ bar(d)\bar{d}d¯-convergent) sequence ( y n ) n ≥ 1 ⊂ Y y n n ≥ 1 ⊂ Y (y_(n))_(n >= 1)sub Y\left(y_{n}\right)_{n \geq 1} \subset Y(yn)n≥1⊂Y has limit in Y Y YYY.
We say that a set Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X is d d ddd-sequentially compact (forward sequentially compact) if every sequence in Y Y YYY has a d d ddd-convergent (forward convergent) subsequence with limit in Y Y YYY (Definition 4.1 in [3]). Finally, the set Y Y YYY in ( X , d ) ( X , d ) (X,d)(X, d)(X,d) is called ( d , d ¯ d , d ¯ d, bar(d)d, \bar{d}d,d¯ )-sequentially compact if every sequence ( y n ) n ≥ 1 y n n ≥ 1 (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1}(yn)n≥1 in Y Y YYY has a subsequence ( y n k ) k ≥ 1 d y n k k ≥ 1 d (y_(n_(k)))_(k >= 1)d\left(y_{n_{k}}\right)_{k \geq 1} d(ynk)k≥1d-convergent to u ∈ Y u ∈ Y u in Yu \in Yu∈Y and d ¯ d ¯ bar(d)\bar{d}d¯-convergent to v ∈ Y v ∈ Y v in Yv \in Yv∈Y. For other properties and results in asymmetric metric spaces, see also [3,5,7,11-18].

2. Extension of Semi-Hölder Functions

Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X be a nonvoid subset of X X XXX and α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1] a given number.
Definition 1. A function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R is called d d ddd-semi-Hölder (of exponent α α alpha\alphaα ) if there exists a constant K Y ( f ) ≥ 0 K Y ( f ) ≥ 0 K_(Y)(f) >= 0K_{Y}(f) \geq 0KY(f)≥0 such that
(1) f ( x ) − f ( y ) ≤ K Y ( f ) d α ( x , y ) (1) f ( x ) − f ( y ) ≤ K Y ( f ) d α ( x , y ) {:(1)f(x)-f(y) <= K_(Y)(f)d^(alpha)(x","y):}\begin{equation*} f(x)-f(y) \leq K_{Y}(f) d^{\alpha}(x, y) \tag{1} \end{equation*}(1)f(x)−f(y)≤KY(f)dα(x,y)
for all x , y ∈ Y x , y ∈ Y x,y in Yx, y \in Yx,y∈Y.
For f f fff a function d d ddd-semi-Hölder on Y Y YYY denote by
(2) ‖ f | Y , d α := sup { ( f ( x ) − f ( y ) ) ∨ 0 d α ( x , y ) : d ( x , y ) > 0 , x , y ∈ Y } (2) ‖ f Y , d α := sup ( f ( x ) − f ( y ) ) ∨ 0 d α ( x , y ) : d ( x , y ) > 0 , x , y ∈ Y {:(2)||f|_(Y,d)^(alpha):=s u p{((f(x)-f(y))vv0)/(d^(alpha)(x,y)):d(x,y) > 0,x,y in Y}:}\begin{equation*} \|\left. f\right|_{Y, d} ^{\alpha}:=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d^{\alpha}(x, y)}: d(x, y)>0, x, y \in Y\right\} \tag{2} \end{equation*}(2)‖f|Y,dα:=sup{(f(x)−f(y))∨0dα(x,y):d(x,y)>0,x,y∈Y}
the smallest constant K Y ( f ) K Y ( f ) K_(Y)(f)K_{Y}(f)KY(f), satisfying the inequality (1).
A function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R is called ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing if f ( x ) ≤ f ( y ) f ( x ) ≤ f ( y ) f(x) <= f(y)f(x) \leq f(y)f(x)≤f(y) whenever d ( x , y ) = 0 , x , y ∈ Y d ( x , y ) = 0 , x , y ∈ Y d(x,y)=0,x,y in Yd(x, y)=0, x, y \in Yd(x,y)=0,x,y∈Y.
The set R ≤ d Y R ≤ d Y R_( <= _(d))^(Y)\mathbb{R}_{\leq_{d}}^{Y}R≤dY of all ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing functions on Y Y YYY is a cone in the linear space R Y R Y R^(Y)\mathbb{R}^{Y}RY of all real-valued functions defined on Y Y YYY.
One denotes by
(3) Λ α ( Y , d ) := { f ∈ R ≤ d Y ∣ f is d -semi-Hölder } (3) Λ α ( Y , d ) := f ∈ R ≤ d Y ∣ f  is  d -semi-Hölder  {:(3)Lambda_(alpha)(Y","d):={f inR_( <= _(d))^(Y)∣f" is "d"-semi-Hölder "}:}\begin{equation*} \Lambda_{\alpha}(Y, d):=\left\{f \in \mathbb{R}_{\leq_{d}}^{Y} \mid f \text { is } d \text {-semi-Hölder }\right\} \tag{3} \end{equation*}(3)Λα(Y,d):={f∈R≤dY∣f is d-semi-Hölder }
the set of all d d ddd-semi-Hölder functions on Y Y YYY. This set is a subcone of the cone R ≤ d Y R ≤ d Y R_( <= _(d))^(Y)\mathbb{R}_{\leq_{d}}^{Y}R≤dY.
For y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y fixed, let
(4) Λ α , 0 ( Y , d ) := { f ∈ Λ α ( Y , d ) , f ( y 0 ) = 0 } . (4) Λ α , 0 ( Y , d ) := f ∈ Λ α ( Y , d ) , f y 0 = 0 . {:(4)Lambda_(alpha,0)(Y","d):={f inLambda_(alpha)(Y,d),f(y_(0))=0}.:}\begin{equation*} \Lambda_{\alpha, 0}(Y, d):=\left\{f \in \Lambda_{\alpha}(Y, d), f\left(y_{0}\right)=0\right\} . \tag{4} \end{equation*}(4)Λα,0(Y,d):={f∈Λα(Y,d),f(y0)=0}.
The functional ‖ | Y , d α : Λ α , 0 ( Y , d ) → [ 0 , ∞ ) ‖ Y , d α : Λ α , 0 ( Y , d ) → [ 0 , ∞ ) |||_(Y,d)^(alpha):Lambda_(alpha,0)(Y,d)rarr[0,oo)\|\left.\right|_{Y, d} ^{\alpha}: \Lambda_{\alpha, 0}(Y, d) \rightarrow[0, \infty)‖|Y,dα:Λα,0(Y,d)→[0,∞) defined by (2) satisfies the axioms of an asymmetric norm, and ( Λ α , 0 ( Y , d ) , ‖ | Y , d α Λ α , 0 ( Y , d ) , ‖ Y , d α Lambda_(alpha,0)(Y,d),|||_(Y,d)^(alpha)\Lambda_{\alpha, 0}(Y, d), \|\left.\right|_{Y, d} ^{\alpha}Λα,0(Y,d),‖|Y,dα ) is an asymmetric normed cone (compare with [18]).
Observe that f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d) if and only if − f ∈ Λ α ( Y , d ¯ ) − f ∈ Λ α ( Y , d ¯ ) -f inLambda_(alpha)(Y, bar(d))-f \in \Lambda_{\alpha}(Y, \bar{d})−f∈Λα(Y,d¯); moreover ‖ f | Y , d α = ‖ − f | Y , d ¯ α f Y , d α = − f Y , d ¯ α ||f|_(Y,d)^(alpha)=||-f|_(Y, bar(d))^(alpha)\left\|\left.f\right|_{Y, d} ^{\alpha}=\right\|-\left.f\right|_{Y, \bar{d}} ^{\alpha}‖f|Y,dα=‖−f|Y,d¯α.
Example 1. Let Y Y YYY be a set in a quasi-metric space ( X , d X , d X,dX, dX,d ) and let y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y be fixed. For a number α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1] one considers the function f : Y → R , f ( y ) = d α ( y , y 0 ) f : Y → R , f ( y ) = d α y , y 0 f:Y rarrR,f(y)=d^(alpha)(y,y_(0))f: Y \rightarrow \mathbb{R}, f(y)= d^{\alpha}\left(y, y_{0}\right)f:Y→R,f(y)=dα(y,y0). Then f ∈ Λ α , 0 ( Y , d ) f ∈ Λ α , 0 ( Y , d ) f inLambda_(alpha,0)(Y,d)f \in \Lambda_{\alpha, 0}(Y, d)f∈Λα,0(Y,d). Indeed, for all y 1 , y 2 ∈ Y y 1 , y 2 ∈ Y y_(1),y_(2)in Yy_{1}, y_{2} \in Yy1,y2∈Y,
f ( y 1 ) − f ( y 2 ) = d α ( y 1 , y 0 ) − d α ( y 2 , y 0 ) ≤ d α ( y 1 , y 2 ) . f y 1 − f y 2 = d α y 1 , y 0 − d α y 2 , y 0 ≤ d α y 1 , y 2 . f(y_(1))-f(y_(2))=d^(alpha)(y_(1),y_(0))-d^(alpha)(y_(2),y_(0)) <= d^(alpha)(y_(1),y_(2)).f\left(y_{1}\right)-f\left(y_{2}\right)=d^{\alpha}\left(y_{1}, y_{0}\right)-d^{\alpha}\left(y_{2}, y_{0}\right) \leq d^{\alpha}\left(y_{1}, y_{2}\right) .f(y1)−f(y2)=dα(y1,y0)−dα(y2,y0)≤dα(y1,y2).
The last inequality follows by the following simple lemma:
Lemma 1. Let a , b , c a , b , c a,b,ca, b, ca,b,c be real nonnegative numbers such that a ≤ b + c a ≤ b + c a <= b+ca \leq b+ca≤b+c. Then for α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1] it follows a α ≤ b α + c α a α ≤ b α + c α a^(alpha) <= b^(alpha)+c^(alpha)a^{\alpha} \leq b^{\alpha}+c^{\alpha}aα≤bα+cα.
Since d ( y 1 , y 0 ) ≤ d ( y 1 , y 2 ) + d ( y 2 , y 0 ) d y 1 , y 0 ≤ d y 1 , y 2 + d y 2 , y 0 d(y_(1),y_(0)) <= d(y_(1),y_(2))+d(y_(2),y_(0))d\left(y_{1}, y_{0}\right) \leq d\left(y_{1}, y_{2}\right)+d\left(y_{2}, y_{0}\right)d(y1,y0)≤d(y1,y2)+d(y2,y0) and α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1], Lemma 1 yields d α ( y 1 , y 0 ) ≤ d α ( y 1 , y 2 ) + d α ( y 2 , y 0 ) d α y 1 , y 0 ≤ d α y 1 , y 2 + d α y 2 , y 0 d^(alpha)(y_(1),y_(0)) <= d^(alpha)(y_(1),y_(2))+d^(alpha)(y_(2),y_(0))d^{\alpha}\left(y_{1}, y_{0}\right) \leq d^{\alpha}\left(y_{1}, y_{2}\right)+d^{\alpha}\left(y_{2}, y_{0}\right)dα(y1,y0)≤dα(y1,y2)+dα(y2,y0), i.e., d α ( y 1 , y 0 ) − d α ( y 2 , y 0 ) ≤ d α ( y 1 , y 2 ) d α y 1 , y 0 − d α y 2 , y 0 ≤ d α y 1 , y 2 d^(alpha)(y_(1),y_(0))-d^(alpha)(y_(2),y_(0)) <= d^(alpha)(y_(1),y_(2))d^{\alpha}\left(y_{1}, y_{0}\right)-d^{\alpha}\left(y_{2}, y_{0}\right) \leq d^{\alpha}\left(y_{1}, y_{2}\right)dα(y1,y0)−dα(y2,y0)≤dα(y1,y2).
Example 2. Let ( X , ‖ ∣ X , ‖ ∣ X,||∣X, \| \midX,‖∣ ) be an asymmetric normed space. For a fixed y 0 ∈ X y 0 ∈ X y_(0)in Xy_{0} \in Xy0∈X and α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1] the function h ( x ) = ‖ x − y 0 | α h ( x ) = ‖ x − y 0 α h(x)=||x-y_(0)|^(alpha)h(x)=\| x-\left.y_{0}\right|^{\alpha}h(x)=‖x−y0|α is d ‖ d ‖ d_(||)d_{\|}d‖-semi-Hölder, where d ‖ ∣ y 0 , x ) = ‖ x − y 0 ∣ , x ∈ X d ‖ ∣ y 0 , x = ‖ x − y 0 ∣ , x ∈ X {:d_(||)∣y_(0),x)=||x-y_(0)∣,x in X\left.d_{\|} \mid y_{0}, x\right)=\| x-y_{0} \mid, x \in Xd‖∣y0,x)=‖x−y0∣,x∈X. Using Lemma 1 it follows h ( x 1 ) − h ( x 2 ) ≤ ‖ x 1 − x 2 | α , x 1 , x 2 ∈ X h x 1 − h x 2 ≤ ‖ x 1 − x 2 α , x 1 , x 2 ∈ X h(x_(1))-h(x_(2)) <= ||x_(1)-x_(2)|^(alpha),x_(1),x_(2)in Xh\left(x_{1}\right)-h\left(x_{2}\right) \leq \| x_{1}-\left.x_{2}\right|^{\alpha}, x_{1}, x_{2} \in Xh(x1)−h(x2)≤‖x1−x2|α,x1,x2∈X, and h ( y 0 ) = 0 h y 0 = 0 h(y_(0))=0h\left(y_{0}\right)=0h(y0)=0. This means that h ∈ Λ α , 0 ( X , d ‖ ) h ∈ Λ α , 0 X , d ‖ h inLambda_(alpha,0)(X,d_(||))h \in \Lambda_{\alpha, 0}\left(X, d_{\|}\right)h∈Λα,0(X,d‖).
Remark 1. By Lemma 1 it follows that if d d ddd is a quasi-metric on X X XXX, then d α ( α ∈ ( 0 , 1 ] ) d α ( α ∈ ( 0 , 1 ] ) d^(alpha)(alpha in(0,1])d^{\alpha}(\alpha \in(0,1])dα(α∈(0,1]) is also a quasi-metric on X X XXX. In fact a d d ddd-semi-Hölder function f f fff (of exponent α ∈ ( 0 , 1 ] ) α ∈ ( 0 , 1 ] ) alpha in(0,1])\alpha \in(0,1])α∈(0,1]) on Y Y YYY is a d α d α d^(alpha)d^{\alpha}dα-semi-Lipschitz function on ( Y , d α Y , d α Y,d^(alpha)Y, d^{\alpha}Y,dα ) (see [17], for the definition of semi-Lipschitz functions).
The following theorem holds.
Theorem 1. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, let Y Y YYY be a nonvoid subset of X X XXX, let α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1], and let f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d). Further, let E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) be defined by
(5) E d ( f ) := { H ∈ Λ α ( X , d ) : H | Y = f , ‖ H | X , d α = ‖ f | Y , d α } . (5) E d ( f ) := H ∈ Λ α ( X , d ) : H Y = f , H X , d α = f Y , d α . {:(5)E_(d)(f):={H inLambda_(alpha)(X,d):H|_(Y)=f,||H|_(X,d)^(alpha)=||f|_(Y,d)^(alpha)}.:}\begin{equation*} \mathcal{E}_{d}(f):=\left\{H \in \Lambda_{\alpha}(X, d):\left.H\right|_{Y}=f,\left.\left\|\left.H\right|_{X, d} ^{\alpha}=\right\| f\right|_{Y, d} ^{\alpha}\right\} . \tag{5} \end{equation*}(5)Ed(f):={H∈Λα(X,d):H|Y=f,‖H|X,dα=‖f|Y,dα}.
Then the following statements hold:
1 0 1 0 1^(0)1^{0}10 The function F d ( f ) : X → R F d ( f ) : X → R F_(d)(f):X rarrRF_{d}(f): X \rightarrow \mathbb{R}Fd(f):X→R, defined by
(6) F d ( f ) ( x ) := inf y ∈ Y { f ( y ) + ‖ f | Y , d α d α ( x , y ) } , (6) F d ( f ) ( x ) := inf y ∈ Y   f ( y ) + ‖ f Y , d α d α ( x , y ) , {:(6)F_(d)(f)(x):=i n f_(y in Y){f(y)+||f|_(Y,d)^(alpha)d^(alpha)(x,y)}",":}\begin{equation*} F_{d}(f)(x):=\inf _{y \in Y}\left\{f(y)+\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(x, y)\right\}, \tag{6} \end{equation*}(6)Fd(f)(x):=infy∈Y{f(y)+‖f|Y,dαdα(x,y)},
belongs to E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f).
2 0 2 0 2^(0)2^{0}20 The function G d ( f ) : X → R G d ( f ) : X → R G_(d)(f):X rarrRG_{d}(f): X \rightarrow \mathbb{R}Gd(f):X→R, defined by
(7) G d ( f ) ( x ) := sup y ∈ Y { f ( y ) − ‖ f | Y , d α ⋅ d α ( y , x ) } , (7) G d ( f ) ( x ) := sup y ∈ Y   f ( y ) − ‖ f Y , d α ⋅ d α ( y , x ) , {:(7)G_(d)(f)(x):=s u p_(y in Y){f(y)-||f|_(Y,d)^(alpha)*d^(alpha)(y,x)}",":}\begin{equation*} G_{d}(f)(x):=\sup _{y \in Y}\left\{f(y)-\|\left. f\right|_{Y, d} ^{\alpha} \cdot d^{\alpha}(y, x)\right\}, \tag{7} \end{equation*}(7)Gd(f)(x):=supy∈Y{f(y)−‖f|Y,dα⋅dα(y,x)},
belongs to E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f).
3 0 3 0 3^(0)3^{0}30 Each H ∈ E d ( f ) H ∈ E d ( f ) H inE_(d)(f)H \in \mathcal{E}_{d}(f)H∈Ed(f) satisfies
(8) G d ( f ) ( x ) ≤ H ( x ) ≤ F d ( f ) ( x ) , (8) G d ( f ) ( x ) ≤ H ( x ) ≤ F d ( f ) ( x ) , {:(8)G_(d)(f)(x) <= H(x) <= F_(d)(f)(x)",":}\begin{equation*} G_{d}(f)(x) \leq H(x) \leq F_{d}(f)(x), \tag{8} \end{equation*}(8)Gd(f)(x)≤H(x)≤Fd(f)(x),
whenever x ∈ X x ∈ X x in Xx \in Xx∈X.
Proof. If f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d) then f f fff is d α d α d^(alpha)d^{\alpha}dα-semi-Lipschitz on ( Y , d α ) Y , d α (Y,d^(alpha))\left(Y, d^{\alpha}\right)(Y,dα). By ([12], Theorem 2) it follows that the functions F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f) defined by (6), and G d ( f ) G d ( f ) G_(d)(f)G_{d}(f)Gd(f) defined by (7) satisfy
(9) F d ( f ) | Y = G d ( f ) | Y = f , ‖ F d ( f ) ‖ X , d α = ‖ G d ( f ) ‖ X , d α = ‖ f | Y , d α (9) F d ( f ) Y = G d ( f ) Y = f , F d ( f ) X , d α = G d ( f ) X , d α = ‖ f Y , d α {:(9)F_(d)(f)|_(Y)=G_(d)(f)|_(Y)=f","quad||F_(d)(f)||_(X,d)^(alpha)=||G_(d)(f)||_(X,d)^(alpha)=||f|_(Y,d)^(alpha):}\begin{equation*} \left.F_{d}(f)\right|_{Y}=\left.G_{d}(f)\right|_{Y}=f, \quad\left\|F_{d}(f)\right\|_{X, d}^{\alpha}=\left\|G_{d}(f)\right\|_{X, d}^{\alpha}=\|\left. f\right|_{Y, d} ^{\alpha} \tag{9} \end{equation*}(9)Fd(f)|Y=Gd(f)|Y=f,‖Fd(f)‖X,dα=‖Gd(f)‖X,dα=‖f|Y,dα
Consequently, the statements 1 0 1 0 1^(0)1^{0}10 and 2 0 2 0 2^(0)2^{0}20 are proved.
The inequalities (8) are proved in [12] for semi-Lipschitz functions (see also ([15], Remark 3), so that the statement 3 0 3 0 3^(0)3^{0}30 holds too.
The set E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) defined in (5) is called the set of extensions of f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d) (preserving the smallest constant ‖ f | Y , d α ‖ f Y , d α ||f|_(Y,d)^(alpha)\|\left. f\right|_{Y, d} ^{\alpha}‖f|Y,dα ). The functions F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f), respectively G d ( f ) G d ( f ) G_(d)(f)G_{d}(f)Gd(f) are called the maximal extension, respectively the minimal extension of f f fff [see (8)].
Remark 2. In [15] one gives a direct proof of Theorem 1, by considering the function G d ( f ) G d ( f ) G_(d)(f)G_{d}(f)Gd(f) defined by (7) and proving that G d ( f ) G d ( f ) G_(d)(f)G_{d}(f)Gd(f) is well defined, G d ( f ) | Y = f G d ( f ) Y = f G_(d)(f)|_(Y)=f\left.G_{d}(f)\right|_{Y}=fGd(f)|Y=f and ‖ G d ( f ) | X , d α = ‖ f | Y , d α G d ( f ) X , d α = f Y , d α ||G_(d)(f)|_(X,d)^(alpha)=||f|_(Y,d)^(alpha)\left.\left\|\left.G_{d}(f)\right|_{X, d} ^{\alpha}=\right\| f\right|_{Y, d} ^{\alpha}‖Gd(f)|X,dα=‖f|Y,dα (see also [10,12]).
A natural problem is the following: If f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d) has some supplementtary properties, does there exist H ∈ E d ( f ) H ∈ E d ( f ) H inE_(d)(f)H \in \mathcal{E}_{d}(f)H∈Ed(f) preserving these properties? Such a problem is considered in [9] for Lipschitz functions.
We shall consider two problems of such kind.
For the first one, in the sequel ( X , d X , d X,dX, dX,d ) is a quasi-metric linear space and Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X is a subset of X X XXX.
The set Y Y YYY is said to be radiant if it has the following properties:
(i) Y Y YYY is nonvoid;
(ii) λ y ∈ Y λ y ∈ Y lambda y in Y\lambda y \in Yλy∈Y for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y and all λ ∈ [ 0 , 1 ] λ ∈ [ 0 , 1 ] lambda in[0,1]\lambda \in[0,1]λ∈[0,1].
Let Y Y YYY be a radiant set in X X XXX, and let f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R, and let α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1]. The function f f fff is said to be α α alpha\alphaα-radiant if
(10) f ( λ y ) ≤ λ α f ( y ) , (10) f ( λ y ) ≤ λ α f ( y ) , {:(10)f(lambda y) <= lambda^(alpha)f(y)",":}\begin{equation*} f(\lambda y) \leq \lambda^{\alpha} f(y), \tag{10} \end{equation*}(10)f(λy)≤λαf(y),
for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y and all λ ∈ ( 0 , 1 ] λ ∈ ( 0 , 1 ] lambda in(0,1]\lambda \in(0,1]λ∈(0,1].
The 1 -radiant functions are called, simply, radiant.
Observe that all radiant sets in a linear space X X XXX contain the null element θ θ theta\thetaθ of X X XXX, and every α α alpha\alphaα-radiant function satisfies f ( θ ) ≤ 0 f ( θ ) ≤ 0 f(theta) <= 0f(\theta) \leq 0f(θ)≤0. We consider only functions satisfying f ( θ ) = 0 f ( θ ) = 0 f(theta)=0f(\theta)=0f(θ)=0.
The function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R is said to be α α alpha\alphaα-co-radiant ( α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1] ) if
(11) f ( λ y ) ≥ λ α f ( y ) (11) f ( λ y ) ≥ λ α f ( y ) {:(11)f(lambda y) >= lambda^(alpha)f(y):}\begin{equation*} f(\lambda y) \geq \lambda^{\alpha} f(y) \tag{11} \end{equation*}(11)f(λy)≥λαf(y)
for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y and λ ∈ [ 0 , 1 ] λ ∈ [ 0 , 1 ] lambda in[0,1]\lambda \in[0,1]λ∈[0,1]. The 1-co-radiant functions are called co-radiant [ 4 , 8 ] [ 4 , 8 ] [4,8][4,8][4,8].
The function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R is called α α alpha\alphaα-inverse co-radiant ( α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1] is fixed) if
(12) f ( λ y ) ≤ 1 λ α f ( y ) , (12) f ( λ y ) ≤ 1 λ α f ( y ) , {:(12)f(lambda y) <= (1)/(lambda^(alpha))f(y)",":}\begin{equation*} f(\lambda y) \leq \frac{1}{\lambda^{\alpha}} f(y), \tag{12} \end{equation*}(12)f(λy)≤1λαf(y),
for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y and λ ∈ ( 0 , 1 ] λ ∈ ( 0 , 1 ] lambda in(0,1]\lambda \in(0,1]λ∈(0,1]. The 1 -inverse co-radiant functions are called inverse co-radiant.
Obviously, every nonnegative α α alpha\alphaα-co-radiant function is co-radiant, and every inverse co-radiant function is α α alpha\alphaα-inverse co-radiant.
If Y Y YYY is a convex set in X X XXX, a function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R is called α α alpha\alphaα-convex ( α ∈ [ 0 , 1 ] ) ( α ∈ [ 0 , 1 ] ) (alpha in[0,1])(\alpha \in[0,1])(α∈[0,1]) if
(13) f ( λ x + ( 1 − λ ) y ) ≤ λ α f ( x ) + ( 1 − λ ) α f ( y ) , (13) f ( λ x + ( 1 − λ ) y ) ≤ λ α f ( x ) + ( 1 − λ ) α f ( y ) , {:(13)f(lambda x+(1-lambda)y) <= lambda^(alpha)f(x)+(1-lambda)^(alpha)f(y)",":}\begin{equation*} f(\lambda x+(1-\lambda) y) \leq \lambda^{\alpha} f(x)+(1-\lambda)^{\alpha} f(y), \tag{13} \end{equation*}(13)f(λx+(1−λ)y)≤λαf(x)+(1−λ)αf(y),
for all x , y ∈ Y x , y ∈ Y x,y in Yx, y \in Yx,y∈Y and λ ∈ [ 0 , 1 ] λ ∈ [ 0 , 1 ] lambda in[0,1]\lambda \in[0,1]λ∈[0,1] (see [1]).
If θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y and Y Y YYY is convex, then every convex function ( α = 1 α = 1 alpha=1\alpha=1α=1 ) on Y Y YYY with f ( θ ) = 0 f ( θ ) = 0 f(theta)=0f(\theta)=0f(θ)=0 is radiant, and every α α alpha\alphaα-convex function on Y Y YYY such that f ( θ ) = 0 f ( θ ) = 0 f(theta)=0f(\theta)=0f(θ)=0 is α α alpha\alphaα-radiant, because
f ( λ x ) = f ( λ x + ( 1 − λ ) θ ) ≤ λ α f ( x ) + ( 1 − λ ) α f ( θ ) = λ α f ( x ) , f ( λ x ) = f ( λ x + ( 1 − λ ) θ ) ≤ λ α f ( x ) + ( 1 − λ ) α f ( θ ) = λ α f ( x ) , f(lambda x)=f(lambda x+(1-lambda)theta) <= lambda^(alpha)f(x)+(1-lambda)^(alpha)f(theta)=lambda^(alpha)f(x),f(\lambda x)=f(\lambda x+(1-\lambda) \theta) \leq \lambda^{\alpha} f(x)+(1-\lambda)^{\alpha} f(\theta)=\lambda^{\alpha} f(x),f(λx)=f(λx+(1−λ)θ)≤λαf(x)+(1−λ)αf(θ)=λαf(x),
for all x ∈ Y x ∈ Y x in Yx \in Yx∈Y and λ ∈ [ 0 , 1 ] λ ∈ [ 0 , 1 ] lambda in[0,1]\lambda \in[0,1]λ∈[0,1].
A quasi-metric d d ddd on a quasi-metric linear space X X XXX is called positively homogeneous if
(14) d ( λ x , λ y ) = λ d ( x , y ) (14) d ( λ x , λ y ) = λ d ( x , y ) {:(14)d(lambda x","lambda y)=lambda d(x","y):}\begin{equation*} d(\lambda x, \lambda y)=\lambda d(x, y) \tag{14} \end{equation*}(14)d(λx,λy)=λd(x,y)
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X and λ ≥ 0 λ ≥ 0 lambda >= 0\lambda \geq 0λ≥0. Such a quasi-metric is for example d ‖ ∣ d ‖ ∣ d_(||)∣d_{\|} \midd‖∣, generated by an asymmetric norm ‖ ∣ ‖ ∣ ||∣\| \mid‖∣.
The following result holds.
Theorem 2. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric linear space with d d ddd positively homogeneous, let Y Y YYY be a radiant subset of X X XXX, let α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1] and let f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d). Then the following statements hold:
1 0 1 0 1^(0)1^{0}10 If f f fff is α α alpha\alphaα-radiant, then F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f) is α α alpha\alphaα-radiant.
2 0 2 0 2^(0)2^{0}20 If f f fff is α α alpha\alphaα-co-radiant, then G d ( f ) G d ( f ) G_(d)(f)G_{d}(f)Gd(f) is α α alpha\alphaα-co-radiant.
3 0 3 0 3^(0)3^{0}30 If f f fff is inverse co-radiant, then F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f) is inverse co-radiant.
Proof. Let f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R be radiant, and f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d). Let us consider the maximal extension F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f). Then for all λ ∈ [ 0 , 1 ] λ ∈ [ 0 , 1 ] lambda in[0,1]\lambda \in[0,1]λ∈[0,1] and y ∈ Y y ∈ Y y in Yy \in Yy∈Y,
F d ( f ) ( λ x ) ≤ f ( λ y ) + ‖ f | Y , d α d α ( λ x , λ y ) ≤ λ α f ( y ) + λ α ‖ f | Y , d α d α ( x , y ) = λ α [ f ( y ) + ‖ f | Y , d α d α ( x , y ) ] . F d ( f ) ( λ x ) ≤ f ( λ y ) + ‖ f Y , d α d α ( λ x , λ y ) ≤ λ α f ( y ) + λ α ‖ f Y , d α d α ( x , y ) = λ α f ( y ) + ‖ f Y , d α d α ( x , y ) . {:[F_(d)(f)(lambda x) <= f(lambda y)+||f|_(Y,d)^(alpha)d^(alpha)(lambda x","lambda y)],[ <= lambda^(alpha)f(y)+lambda^(alpha)||f|_(Y,d)^(alpha)d^(alpha)(x","y)],[=lambda^(alpha)[f(y)+||f|_(Y,d)^(alpha)d^(alpha)(x,y)].]:}\begin{aligned} F_{d}(f)(\lambda x) & \leq f(\lambda y)+\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(\lambda x, \lambda y) \\ & \leq \lambda^{\alpha} f(y)+\lambda^{\alpha} \|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(x, y) \\ & =\lambda^{\alpha}\left[f(y)+\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(x, y)\right] . \end{aligned}Fd(f)(λx)≤f(λy)+‖f|Y,dαdα(λx,λy)≤λαf(y)+λα‖f|Y,dαdα(x,y)=λα[f(y)+‖f|Y,dαdα(x,y)].
Taking the infimum with respect to y ∈ Y y ∈ Y y in Yy \in Yy∈Y one gets
F d ( f ) ( λ x ) ≤ λ α F d ( f ) ( x ) , F d ( f ) ( λ x ) ≤ λ α F d ( f ) ( x ) , F_(d)(f)(lambda x) <= lambda^(alpha)F_(d)(f)(x),F_{d}(f)(\lambda x) \leq \lambda^{\alpha} F_{d}(f)(x),Fd(f)(λx)≤λαFd(f)(x),
for every x ∈ X x ∈ X x in Xx \in Xx∈X, showing that F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f) is α α alpha\alphaα-radiant.
Now, let x ∈ X , λ ∈ [ 0 , 1 ] x ∈ X , λ ∈ [ 0 , 1 ] x in X,lambda in[0,1]x \in X, \lambda \in[0,1]x∈X,λ∈[0,1], and f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d) be α α alpha\alphaα-co-radiant. Then, for every y ∈ Y y ∈ Y y in Yy \in Yy∈Y, by considering the minimal extension G d ( f ) ∈ E d ( f ) G d ( f ) ∈ E d ( f ) G_(d)(f)inE_(d)(f)G_{d}(f) \in \mathcal{E}_{d}(f)Gd(f)∈Ed(f) one gets:
G d ( f ) ( λ x ) ≥ f ( λ y ) − ‖ f | Y , d α d α ( λ y , λ x ) ≥ λ α f ( y ) − λ α ‖ f | Y , d α d α ( y , x ) = λ α [ f ( y ) − ‖ f | Y , d α d α ( y , x ) ] G d ( f ) ( λ x ) ≥ f ( λ y ) − ‖ f Y , d α d α ( λ y , λ x ) ≥ λ α f ( y ) − λ α ‖ f Y , d α d α ( y , x ) = λ α f ( y ) − ‖ f Y , d α d α ( y , x ) {:[G_(d)(f)(lambda x) >= f(lambda y)-||f|_(Y,d)^(alpha)d^(alpha)(lambda y","lambda x)],[ >= lambda^(alpha)f(y)-lambda^(alpha)||f|_(Y,d)^(alpha)d^(alpha)(y","x)],[=lambda^(alpha)[f(y)-||f|_(Y,d)^(alpha)d^(alpha)(y,x)]]:}\begin{aligned} G_{d}(f)(\lambda x) & \geq f(\lambda y)-\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(\lambda y, \lambda x) \\ & \geq \lambda^{\alpha} f(y)-\lambda^{\alpha} \|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(y, x) \\ & =\lambda^{\alpha}\left[f(y)-\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(y, x)\right] \end{aligned}Gd(f)(λx)≥f(λy)−‖f|Y,dαdα(λy,λx)≥λαf(y)−λα‖f|Y,dαdα(y,x)=λα[f(y)−‖f|Y,dαdα(y,x)]
Taking the supremum with respect to y ∈ Y y ∈ Y y in Yy \in Yy∈Y, one obtains
G d ( f ) ( λ x ) ≥ λ α G d ( f ) ( x ) , x ∈ X G d ( f ) ( λ x ) ≥ λ α G d ( f ) ( x ) , x ∈ X G_(d)(f)(lambda x) >= lambda^(alpha)G_(d)(f)(x),x in XG_{d}(f)(\lambda x) \geq \lambda^{\alpha} G_{d}(f)(x), x \in XGd(f)(λx)≥λαGd(f)(x),x∈X
and the statement 2 0 2 0 2^(0)2^{0}20 is proved.
Finally, if f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d) is inverse co-radiant on the radiant set Y Y YYY, then for every y ∈ Y y ∈ Y y in Yy \in Yy∈Y and λ ∈ ( 0 , 1 ] λ ∈ ( 0 , 1 ] lambda in(0,1]\lambda \in(0,1]λ∈(0,1], and for the maximal extension F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f) one obtains:
F d ( f ) ( x ) ≤ f ( λ y ) + ‖ f | Y , d α d α ( λ x , λ y ) ≤ 1 λ f ( y ) + λ α ‖ f | Y , d α ⋅ d α ( x , y ) = 1 λ [ f ( y ) + λ α + 1 ‖ f | Y , d α d α ( x , y ) ] ≤ 1 λ [ f ( y ) + ‖ f | Y , d α d α ( x , y ) ] . F d ( f ) ( x ) ≤ f ( λ y ) + ‖ f Y , d α d α ( λ x , λ y ) ≤ 1 λ f ( y ) + λ α ‖ f Y , d α ⋅ d α ( x , y ) = 1 λ f ( y ) + λ α + 1 ‖ f Y , d α d α ( x , y ) ≤ 1 λ f ( y ) + ‖ f Y , d α d α ( x , y ) . {:[F_(d)(f)(x) <= f(lambda y)+||f|_(Y,d)^(alpha)d^(alpha)(lambda x","lambda y)],[ <= (1)/(lambda)f(y)+lambda^(alpha)||f|_(Y,d)^(alpha)*d^(alpha)(x","y)],[=(1)/(lambda)[f(y)+lambda^(alpha+1)||f|_(Y,d)^(alpha)d^(alpha)(x,y)]],[ <= (1)/(lambda)[f(y)+||f|_(Y,d)^(alpha)d^(alpha)(x,y)].]:}\begin{aligned} F_{d}(f)(x) & \leq f(\lambda y)+\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(\lambda x, \lambda y) \\ & \leq \frac{1}{\lambda} f(y)+\lambda^{\alpha} \|\left. f\right|_{Y, d} ^{\alpha} \cdot d^{\alpha}(x, y) \\ & =\frac{1}{\lambda}\left[f(y)+\lambda^{\alpha+1} \|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(x, y)\right] \\ & \leq \frac{1}{\lambda}\left[f(y)+\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(x, y)\right] . \end{aligned}Fd(f)(x)≤f(λy)+‖f|Y,dαdα(λx,λy)≤1λf(y)+λα‖f|Y,dα⋅dα(x,y)=1λ[f(y)+λα+1‖f|Y,dαdα(x,y)]≤1λ[f(y)+‖f|Y,dαdα(x,y)].
Taking the infimum with respect to y ∈ Y y ∈ Y y in Yy \in Yy∈Y it follows
F d ( f ) ( λ x ) ≤ 1 λ F d ( f ) ( x ) , x ∈ X , F d ( f ) ( λ x ) ≤ 1 λ F d ( f ) ( x ) , x ∈ X , F_(d)(f)(lambda x) <= (1)/(lambda)F_(d)(f)(x),x in X,F_{d}(f)(\lambda x) \leq \frac{1}{\lambda} F_{d}(f)(x), x \in X,Fd(f)(λx)≤1λFd(f)(x),x∈X,
and the statement 3 0 3 0 3^(0)3^{0}30 holds.
Another property preserved by extensions is the global minimum (maximum) of a function f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d).
Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, and let Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X be a nonempty subset of X X XXX. An element y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y is called a global minimum (maximum) point of f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d) 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) \leq f(y) \quad\left(f\left(y_{0}\right) \geq f(y)\right)f(y0)≤f(y)(f(y0)≥f(y))
for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y.
Theorem 3. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, let Y Y YYY be a nonvoid subset of X X XXX, let y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y, let α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1], and let f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d). Then the following statements hold:
1 0 1 0 1^(0)1^{0}10 If Y Y YYY is d-closed, then y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y is a global minimum point for f f fff in Y Y YYY if and only if y 0 y 0 y_(0)y_{0}y0 is a global minimum point of F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f) in X X XXX.
2 0 2 0 2^(0)2^{0}20 If Y Y YYY is d ¯ d ¯ bar(d)\bar{d}d¯-closed, then y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y is a global maximum point for f f fff in Y Y YYY if and only if y 0 y 0 y_(0)y_{0}y0 is a global maximum point of G d ( f ) G d ( f ) G_(d)(f)G_{d}(f)Gd(f) in X X XXX.
Proof. 1 0 1 0 quad1^(0)\quad 1^{0}10 Let y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y be a global minimum point of f ∈ Λ α ( Y , d ) f ∈ Λ α ( Y , d ) f inLambda_(alpha)(Y,d)f \in \Lambda_{\alpha}(Y, d)f∈Λα(Y,d). For every y ∈ Y y ∈ Y y in Yy \in Yy∈Y we have
F d ( f ) ( y ) = f ( y ) ≥ f ( y 0 ) = F d ( f ) ( y 0 ) . F d ( f ) ( y ) = f ( y ) ≥ f y 0 = F d ( f ) y 0 . F_(d)(f)(y)=f(y) >= f(y_(0))=F_(d)(f)(y_(0)).F_{d}(f)(y)=f(y) \geq f\left(y_{0}\right)=F_{d}(f)\left(y_{0}\right) .Fd(f)(y)=f(y)≥f(y0)=Fd(f)(y0).
If x ∉ Y , Y x ∉ Y , Y x!in Y,Yx \notin Y, Yx∉Y,Y being d d ddd-closed, there exists δ > 0 δ > 0 delta > 0\delta>0δ>0 such that d ( x , y ) > δ d ( x , y ) > δ d(x,y) > deltad(x, y)>\deltad(x,y)>δ for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y. Consequently,
F d ( f ) ( x ) = inf y ∈ Y { f ( y ) + ‖ f | Y , d α d α ( x , y ) } ≥ inf y ∈ Y { f ( y ) + ‖ f | Y , d α δ α } = f ( y 0 ) + ‖ f | Y , d α δ α ≥ f ( y 0 ) , F d ( f ) ( x ) = inf y ∈ Y   f ( y ) + ‖ f Y , d α d α ( x , y ) ≥ inf y ∈ Y   f ( y ) + ‖ f Y , d α δ α = f y 0 + ‖ f Y , d α δ α ≥ f y 0 , {:[F_(d)(f)(x)=i n f_(y in Y){f(y)+||f|_(Y,d)^(alpha)d^(alpha)(x,y)}],[ >= i n f_(y in Y){f(y)+||f|_(Y,d)^(alpha)delta^(alpha)}],[=f(y_(0))+||f|_(Y,d)^(alpha)delta^(alpha) >= f(y_(0))","]:}\begin{aligned} F_{d}(f)(x) & =\inf _{y \in Y}\left\{f(y)+\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(x, y)\right\} \\ & \geq \inf _{y \in Y}\left\{f(y)+\|\left. f\right|_{Y, d} ^{\alpha} \delta^{\alpha}\right\} \\ & =f\left(y_{0}\right)+\|\left. f\right|_{Y, d} ^{\alpha} \delta^{\alpha} \geq f\left(y_{0}\right), \end{aligned}Fd(f)(x)=infy∈Y{f(y)+‖f|Y,dαdα(x,y)}≥infy∈Y{f(y)+‖f|Y,dαδα}=f(y0)+‖f|Y,dαδα≥f(y0),
so that for every x ∈ X , F d ( f ) ( x ) ≥ f ( y 0 ) = F d ( f ) ( y 0 ) x ∈ X , F d ( f ) ( x ) ≥ f y 0 = F d ( f ) y 0 x in X,F_(d)(f)(x) >= f(y_(0))=F_(d)(f)(y_(0))x \in X, F_{d}(f)(x) \geq f\left(y_{0}\right)=F_{d}(f)\left(y_{0}\right)x∈X,Fd(f)(x)≥f(y0)=Fd(f)(y0).
Conversely, suppose that y 0 ∈ X y 0 ∈ X y_(0)in Xy_{0} \in Xy0∈X is a global minimum point for F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f) in X X XXX. If we would show that y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y, then, as F d ( f ) | Y = f F d ( f ) Y = f F_(d)(f)|_(Y)=f\left.F_{d}(f)\right|_{Y}=fFd(f)|Y=f, it would follow that y 0 y 0 y_(0)y_{0}y0 is a global minimum point for f f fff in Y Y YYY.
Case I: ‖ f | Y , d α = 0 ‖ f Y , d α = 0 ||f|_(Y,d)^(alpha)=0\|\left. f\right|_{Y, d} ^{\alpha}=0‖f|Y,dα=0.
In this case there exists c ∈ R c ∈ R c inRc \in \mathbb{R}c∈R such that f ( y ) = c f ( y ) = c f(y)=cf(y)=cf(y)=c for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y. It follows ‖ F d ( f ) | X , d α = 0 ‖ F d ( f ) X , d α = 0 ||F_(d)(f)|_(X,d)^(alpha)=0\|\left. F_{d}(f)\right|_{X, d} ^{\alpha}=0‖Fd(f)|X,dα=0, so that F d ( f ) = F d ( f ) = F_(d)(f)=F_{d}(f)=Fd(f)= const on X X XXX. Since F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f we must have F d ( f ) ( x ) = F d ( f ) ( x ) = F_(d)(f)(x)=F_{d}(f)(x)=Fd(f)(x)= const , for all x ∈ X x ∈ X x in Xx \in Xx∈X.
Case II: ‖ f | Y , d α > 0 ‖ f Y , d α > 0 ||f|_(Y,d)^(alpha) > 0\|\left. f\right|_{Y, d} ^{\alpha}>0‖f|Y,dα>0.
Let y 0 ∈ X y 0 ∈ X y_(0)in Xy_{0} \in Xy0∈X such that F d ( f ) ( y 0 ) ≤ F d ( f ) ( x ) F d ( f ) y 0 ≤ F d ( f ) ( x ) F_(d)(f)(y_(0)) <= F_(d)(f)(x)F_{d}(f)\left(y_{0}\right) \leq F_{d}(f)(x)Fd(f)(y0)≤Fd(f)(x), for all x ∈ X x ∈ X x in Xx \in Xx∈X. Since F d ( f ) ( y 0 ) = inf [ f ( y ) + ‖ f | Y , d α d α ( y 0 , y ) ] F d ( f ) y 0 = inf f ( y ) + ‖ f Y , d α d α y 0 , y F_(d)(f)(y_(0))=i n f[f(y)+||f|_(Y,d)^(alpha)d^(alpha)(y_(0),y)]F_{d}(f)\left(y_{0}\right)=\inf \left[f(y)+\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}\left(y_{0}, y\right)\right]Fd(f)(y0)=inf[f(y)+‖f|Y,dαdα(y0,y)], for every n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N there exist y n ∈ Y y n ∈ Y y_(n)in Yy_{n} \in Yyn∈Y such that
f ( y n ) + d α ( y 0 , y n ) ‖ f | Y , d α < F d ( f ) ( y 0 ) + ‖ f | Y , d α n α f y n + d α y 0 , y n ‖ f Y , d α < F d ( f ) y 0 + ‖ f Y , d α n α f(y_(n))+d^(alpha)(y_(0),y_(n))||f|_(Y,d)^(alpha) < F_(d)(f)(y_(0))+(||f|_(Y,d)^(alpha))/(n^(alpha))f\left(y_{n}\right)+d^{\alpha}\left(y_{0}, y_{n}\right) \|\left. f\right|_{Y, d} ^{\alpha}<F_{d}(f)\left(y_{0}\right)+\frac{\|\left. f\right|_{Y, d} ^{\alpha}}{n^{\alpha}}f(yn)+dα(y0,yn)‖f|Y,dα<Fd(f)(y0)+‖f|Y,dαnα
The inequalities F d ( f ) ( y 0 ) ≤ F d ( f ) ( y n ) = f ( y n ) F d ( f ) y 0 ≤ F d ( f ) y n = f y n F_(d)(f)(y_(0)) <= F_(d)(f)(y_(n))=f(y_(n))F_{d}(f)\left(y_{0}\right) \leq F_{d}(f)\left(y_{n}\right)=f\left(y_{n}\right)Fd(f)(y0)≤Fd(f)(yn)=f(yn), imply f ( y n ) + ‖ f | Y , d α d α ( y 0 , y n ) < f ( y n ) + ‖ f | Y , d α n − α f y n + f Y , d α d α y 0 , y n < f y n + f Y , d α n − α f(y_(n))+||f|_(Y,d)^(alpha)d^(alpha)(y_(0),y_(n)) < f(y_(n))+||f|_(Y,d)^(alpha)n^(-alpha)f\left(y_{n}\right)+ \left.\left\|\left.f\right|_{Y, d} ^{\alpha} d^{\alpha}\left(y_{0}, y_{n}\right)<f\left(y_{n}\right)+\right\| f\right|_{Y, d} ^{\alpha} n^{-\alpha}f(yn)+‖f|Y,dαdα(y0,yn)<f(yn)+‖f|Y,dαn−α, so that d ( y 0 , y n ) < 1 n , n ∈ N d y 0 , y n < 1 n , n ∈ N d(y_(0),y_(n)) < (1)/(n),n inNd\left(y_{0}, y_{n}\right)<\frac{1}{n}, n \in \mathbb{N}d(y0,yn)<1n,n∈N, i.e., d ( y 0 , y n ) → 0 d y 0 , y n → 0 d(y_(0),y_(n))rarr0d\left(y_{0}, y_{n}\right) \rightarrow 0d(y0,yn)→0. The sequence ( y n ) n ≥ 1 y n n ≥ 1 (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1}(yn)n≥1 in Y Y YYY is d d ddd-convergent to y 0 y 0 y_(0)y_{0}y0, and since Y Y YYY is d d ddd-closed, y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y.
2 0 2 0 2^(0)2^{0}20 Let y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y be a global maximum point of f ∈ Λ α ( Y ) f ∈ Λ α ( Y ) f inLambda_(alpha)(Y)f \in \Lambda_{\alpha}(Y)f∈Λα(Y).
Then for every y ∈ Y y ∈ Y y in Yy \in Yy∈Y,
G d ( f ) ( y ) = f ( y ) ≤ f ( y 0 ) = G d ( f ) ( y 0 ) . G d ( f ) ( y ) = f ( y ) ≤ f y 0 = G d ( f ) y 0 . G_(d)(f)(y)=f(y) <= f(y_(0))=G_(d)(f)(y_(0)).G_{d}(f)(y)=f(y) \leq f\left(y_{0}\right)=G_{d}(f)\left(y_{0}\right) .Gd(f)(y)=f(y)≤f(y0)=Gd(f)(y0).
If x ∉ Y , Y x ∉ Y , Y x!in Y,Yx \notin Y, Yx∉Y,Y being d ¯ d ¯ bar(d)\bar{d}d¯-closed, there exists η > 0 η > 0 eta > 0\eta>0η>0 such that d ¯ ( x , y ) > η d ¯ ( x , y ) > η bar(d)(x,y) > eta\bar{d}(x, y)>\etad¯(x,y)>η (i.e. d ( y , x ) > η d ( y , x ) > η d(y,x) > etad(y, x)>\etad(y,x)>η ) for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y. Therefore
G d ( f ) ( x ) = sup y ∈ Y { f ( y ) − ‖ f | Y , d α d ¯ α ( x , y ) } = sup y ∈ Y { f ( y ) − ‖ f | Y , d α d α ( y , x ) } ≤ sup y ∈ Y { f ( y ) − ‖ f | Y , d α η α } ≤ f ( y 0 ) − ‖ f | Y , d α η α ≤ f ( y 0 ) G d ( f ) ( x ) = sup y ∈ Y   f ( y ) − ‖ f Y , d α d ¯ α ( x , y ) = sup y ∈ Y   f ( y ) − ‖ f Y , d α d α ( y , x ) ≤ sup y ∈ Y   f ( y ) − ‖ f Y , d α η α ≤ f y 0 − ‖ f Y , d α η α ≤ f y 0 {:[G_(d)(f)(x)=s u p_(y in Y){f(y)-||f|_(Y,d)^(alpha) bar(d)^(alpha)(x,y)}],[=s u p_(y in Y){f(y)-||f|_(Y,d)^(alpha)d^(alpha)(y,x)}],[ <= s u p_(y in Y){f(y)-||f|_(Y,d)^(alpha)eta^(alpha)}],[ <= f(y_(0))-||f|_(Y,d)^(alpha)eta^(alpha) <= f(y_(0))]:}\begin{aligned} G_{d}(f)(x) & =\sup _{y \in Y}\left\{f(y)-\|\left. f\right|_{Y, d} ^{\alpha} \bar{d}^{\alpha}(x, y)\right\} \\ & =\sup _{y \in Y}\left\{f(y)-\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}(y, x)\right\} \\ & \leq \sup _{y \in Y}\left\{f(y)-\|\left. f\right|_{Y, d} ^{\alpha} \eta^{\alpha}\right\} \\ & \leq f\left(y_{0}\right)-\|\left. f\right|_{Y, d} ^{\alpha} \eta^{\alpha} \leq f\left(y_{0}\right) \end{aligned}Gd(f)(x)=supy∈Y{f(y)−‖f|Y,dαd¯α(x,y)}=supy∈Y{f(y)−‖f|Y,dαdα(y,x)}≤supy∈Y{f(y)−‖f|Y,dαηα}≤f(y0)−‖f|Y,dαηα≤f(y0)
It follows that G d ( f ) ( x ) ≤ f ( y 0 ) G d ( f ) ( x ) ≤ f y 0 G_(d)(f)(x) <= f(y_(0))G_{d}(f)(x) \leq f\left(y_{0}\right)Gd(f)(x)≤f(y0), for all x ∈ X x ∈ X x in Xx \in Xx∈X.
Conversely, suppose that y 0 ∈ X y 0 ∈ X y_(0)in Xy_{0} \in Xy0∈X is a global maximum point for G d ( f ) G d ( f ) G_(d)(f)G_{d}(f)Gd(f) in X X XXX.
Case I ‖ f | Y , d α = 0 ‖ f Y , d α = 0 ||f|_(Y,d)^(alpha)=0\|\left. f\right|_{Y, d} ^{\alpha}=0‖f|Y,dα=0.
In this case, because f = G d ( f ) | Y f = G d ( f ) Y f=G_(d)(f)|_(Y)f=\left.G_{d}(f)\right|_{Y}f=Gd(f)|Y and ‖ G d ( f ) | X , d α = ‖ f | Y , d α = 0 G d ( f ) X , d α = f Y , d α = 0 ||G_(d)(f)|_(X,d)^(alpha)=||f|_(Y,d)^(alpha)=0\left.\left\|\left.G_{d}(f)\right|_{X, d} ^{\alpha}=\right\| f\right|_{Y, d} ^{\alpha}=0‖Gd(f)|X,dα=‖f|Y,dα=0 it follows that f f fff and G d ( f ) G d ( f ) G_(d)(f)G_{d}(f)Gd(f) are equal with the same constant.
Case II ‖ f | Y , d α > 0 ‖ f Y , d α > 0 ||f|_(Y,d)^(alpha) > 0\|\left. f\right|_{Y, d} ^{\alpha}>0‖f|Y,dα>0.
Let y 0 ∈ X y 0 ∈ X y_(0)in Xy_{0} \in Xy0∈X such that G d ( f ) ( y 0 ) > G d ( f ) ( x ) , x ∈ X G d ( f ) y 0 > G d ( f ) ( x ) , x ∈ X G_(d)(f)(y_(0)) > G_(d)(f)(x),x in XG_{d}(f)\left(y_{0}\right)>G_{d}(f)(x), x \in XGd(f)(y0)>Gd(f)(x),x∈X.
Since
G d ( f ) ( y 0 ) = sup y ∈ Y [ f ( y ) − ‖ f | Y , d α d α ( y , y 0 ) ] G d ( f ) y 0 = sup y ∈ Y   f ( y ) − ‖ f Y , d α d α y , y 0 G_(d)(f)(y_(0))=s u p_(y in Y)[f(y)-||f|_(Y,d)^(alpha)d^(alpha)(y,y_(0))]G_{d}(f)\left(y_{0}\right)=\sup _{y \in Y}\left[f(y)-\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}\left(y, y_{0}\right)\right]Gd(f)(y0)=supy∈Y[f(y)−‖f|Y,dαdα(y,y0)]
for every n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N, there exists y n ∈ Y y n ∈ Y y_(n)in Yy_{n} \in Yyn∈Y such that
f ( y n ) − ‖ f | Y , d α d α ( y n , y 0 ) > G d ( f ) ( y 0 ) − ‖ f | Y , d α n α f y n − ‖ f Y , d α d α y n , y 0 > G d ( f ) y 0 − ‖ f Y , d α n α f(y_(n))-||f|_(Y,d)^(alpha)d^(alpha)(y_(n),y_(0)) > G_(d)(f)(y_(0))-(||f|_(Y,d)^(alpha))/(n^(alpha))f\left(y_{n}\right)-\|\left. f\right|_{Y, d} ^{\alpha} d^{\alpha}\left(y_{n}, y_{0}\right)>G_{d}(f)\left(y_{0}\right)-\frac{\|\left. f\right|_{Y, d} ^{\alpha}}{n^{\alpha}}f(yn)−‖f|Y,dαdα(yn,y0)>Gd(f)(y0)−‖f|Y,dαnα
The inequalities G d ( f ) ( y 0 ) ≥ G d ( f ) ( y n ) = f ( y n ) G d ( f ) y 0 ≥ G d ( f ) y n = f y n G_(d)(f)(y_(0)) >= G_(d)(f)(y_(n))=f(y_(n))G_{d}(f)\left(y_{0}\right) \geq G_{d}(f)\left(y_{n}\right)=f\left(y_{n}\right)Gd(f)(y0)≥Gd(f)(yn)=f(yn), imply f ( y n ) − ‖ f | Y , d α d α ( y n , y 0 ) > f ( y n ) − 1 n α ‖ f | Y , d α f y n − f Y , d α d α y n , y 0 > f y n − 1 n α f Y , d α f(y_(n))-||f|_(Y,d)^(alpha)d^(alpha)(y_(n),y_(0)) > f(y_(n))-(1)/(n^(alpha))||f|_(Y,d)^(alpha)f\left(y_{n}\right)-\left.\left\|\left.f\right|_{Y, d} ^{\alpha} d^{\alpha}\left(y_{n}, y_{0}\right)>f\left(y_{n}\right)-\frac{1}{n^{\alpha}}\right\| f\right|_{Y, d} ^{\alpha}f(yn)−‖f|Y,dαdα(yn,y0)>f(yn)−1nα‖f|Y,dα, so that d ( y n , y 0 ) < 1 n , n ∈ N d y n , y 0 < 1 n , n ∈ N d(y_(n),y_(0)) < (1)/(n),n inNd\left(y_{n}, y_{0}\right) <\frac{1}{n}, n \in \mathbb{N}d(yn,y0)<1n,n∈N, i.e., d ( y n , y 0 ) → 0 d y n , y 0 → 0 d(y_(n),y_(0))rarr0d\left(y_{n}, y_{0}\right) \rightarrow 0d(yn,y0)→0. This means that the sequence ( y n ) n ≥ 1 y n n ≥ 1 (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1}(yn)n≥1 is d ¯ d ¯ bar(d)\bar{d}d¯-convergent to y 0 y 0 y_(0)y_{0}y0. Since Y Y YYY is d ¯ d ¯ bar(d)\bar{d}d¯-closed, y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y.
Remark 3. By Theorem 2.6 in [14], it follows that every f ∈ Λ α , 0 ( Y , d ) f ∈ Λ α , 0 ( Y , d ) f inLambda_(alpha,0)(Y,d)f \in \Lambda_{\alpha, 0}(Y, d)f∈Λα,0(Y,d) is lower semicontinuous on ( Y , d α Y , d α Y,d^(alpha)Y, d^{\alpha}Y,dα ) and attains its minimum on Y Y YYY, provided that Y Y YYY is d α d α d^(alpha)d^{\alpha}dα-sequentially compact. Also, every f ∈ Λ α , 0 ( Y , d ) f ∈ Λ α , 0 ( Y , d ) f inLambda_(alpha,0)(Y,d)f \in \Lambda_{\alpha, 0}(Y, d)f∈Λα,0(Y,d) is upper semicontinuous on ( Y , d α ― Y , d α ¯ Y, bar(d^(alpha))Y, \overline{d^{\alpha}}Y,dα― ) and attains its maximum value on Y Y YYY whenever Y Y YYY is d α ― d α ¯ bar(d^(alpha))\overline{d^{\alpha}}dα―-sequentially compact.
If Y Y YYY is ( d α , d α ― ) d α , d α ¯ (d^(alpha), bar(d^(alpha)))\left(d^{\alpha}, \overline{d^{\alpha}}\right)(dα,dα―)-sequentially compact and ( X , d α X , d α X,d^(alpha)X, d^{\alpha}X,dα ) is a T 1 T 1 T_(1)T_{1}T1-topological space, every f ∈ Λ α , 0 ( Y , d ) f ∈ Λ α , 0 ( Y , d ) f inLambda_(alpha,0)(Y,d)f \in \Lambda_{\alpha, 0}(Y, d)f∈Λα,0(Y,d) attains both the global minimum and the global maximum on Y Y YYY. Moreover, the sequential method for the calculation of the global extremum (maximum and/or minimum) of f f fff, ([14], Th. 3.1) is applicable.

Acknowledgements

I would like to thank the referees for the valuable suggestions in improving the structure of the paper.

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C. Mustăţa
"T. Popoviciu" Institute of Numerical Analysis
P.O. Box 68-1
Cluj-Napoca, Romania
e-mail: cmustata2001@yahoo.com
cmustata@ictp.acad.ro
Received: May 2, 2011.
Accepted: September 12, 2011.
2013

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