On the approximation of the global extremum of a semi-Lipschitz function

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

Abstract

In this paper one obtains a sequential procedure for determining the global extremum of a semi-Lipschitz real-valued function defined on a quasi-metric (asymmetric metric) space.

Authors

Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis  Cluj-Napoca, Romanian Academy

Keywords

Spaces with asymmetric metric; semi-Lipschitz functions; extension and approximation

Paper coordinates

C. Mustăța, On the approximation of the global extremum of a semi-Lipschitz function, Mediterr. J. Math. 6 (2009), 169–180,
doi: 10.1007/s00009-009-0003-x

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Mediterranean Journal of Mathematics

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Springer

DOI

10.1007/s00009-009-0003-x

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1660-5446

 

 

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1660-5454

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2009-Mustata-On the approximation of the global extremum-MeditterJ

On the Approximation of the Global Extremum of a Semi-Lipschitz Function

Costică Mustăţa

Abstract

In this paper one obtains a sequential procedure for determining the global extremum of a semi-Lipschitz real-valued function defined on a quasi-metric (asymmetric metric) space.

Mathematics Subject Classification (2000). Primary 68W25; Secondary 46A22.
Keywords. Spaces with asymmetric metric, semi-Lipschitz functions, extension and approximation.

1. Introduction

For a function from a specified class, a method for seeking its extremum deals with the problem of estimating the global maximum or/and minimum values of the function and locating the points where the extremum is attained.
An important class of such methods is the class of sequential methods i.e. in which the choice of each evaluation point, except for the first one, depends on the location and the values of the function at the previous points and, possibly, on the number n n nnn of the evaluations to be performed. In the latter case the method is called an n n nnn-step method. In the following, a sequential method is obtained for evaluating the global maximum and the global minimum of a semi-Lipschitz real-valued function defined on a subset of a quasi-metric space, sometimes called asymmetric metric space (see [7], [27]).
In order to determine the absolute maximum M f M f M_(f)M_{f}Mf of a real semi-Lipschitz function f f fff, the algorithm we propose determines a decreasing sequence of numbers ( M n ) n ≥ 1 M n n ≥ 1 (M_(n))_(n >= 1)\left(M_{n}\right)_{n \geq 1}(Mn)n≥1, having the limit M f M f M_(f)M_{f}Mf. Each number M n ( n = 1 , 2 , … ) M n ( n = 1 , 2 , … ) M_(n)(n=1,2,dots)M_{n}(n=1,2, \ldots)Mn(n=1,2,…) is the absolute maximum of a special semi-Lipschitz function U n ( f ) U n ( f ) U_(n)(f)U_{n}(f)Un(f). This function has a very simple analytical expression compared to the given function f f fff (which is assumed only to be semi-Lipschitz). For determining U n ( f ) ( x ) U n ( f ) ( x ) U_(n)(f)(x)U_{n}(f)(x)Un(f)(x) one requires on one hand
the computation of the value of f f fff at a certain point, and the values of f f fff at the n n nnn point from the previous step, and on the other hand the quasi-distances from the current point x x xxx to the n + 1 n + 1 n+1n+1n+1 points. One can see therefore that the determining of the maximum M n + 1 M n + 1 M_(n+1)M_{n+1}Mn+1 of U n + 1 ( f ) U n + 1 ( f ) U_(n+1)(f)U_{n+1}(f)Un+1(f) requires a small amount of computation. The absolute minimum of f f fff is given by the absolute maximum of − f − f -f-f−f.
We present in the following the framework of the described method.
Let X X XXX be a non-empty set. A function 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,∞) is called a quasimetric on X X XXX [21] (see also [7], [27]) if the following conditions hold
A M 1 ) d ( x , y ) = d ( y , x ) = 0 iff x = y A M 2 ) d ( x , z ) ≤ d ( x , y ) + d ( y , z ) A M 1 )      d ( x , y ) = d ( y , x ) = 0  iff  x = y A M 2 )      d ( x , z ) ≤ d ( x , y ) + d ( y , z ) {:[AM1),d(x","y)=d(y","x)=0" iff "x=y],[AM2),d(x","z) <= d(x","y)+d(y","z)]:}\begin{array}{ll} A M 1) & d(x, y)=d(y, x)=0 \text { iff } x=y \\ A M 2) & d(x, z) \leq d(x, y)+d(y, z) \end{array}AM1)d(x,y)=d(y,x)=0 iff x=yAM2)d(x,z)≤d(x,y)+d(y,z)
for all x , y , z ∈ X x , y , z ∈ X x,y,z in Xx, y, z \in Xx,y,z∈X.
The function d ¯ : X × X → [ 0 , ∞ ) d ¯ : X × X → [ 0 , ∞ ) bar(d):X xx X rarr[0,oo)\bar{d}: X \times X \rightarrow[0, \infty)d¯:X×X→[0,∞) defined by d ¯ ( x , y ) = d ( y , x ) d ¯ ( x , y ) = d ( y , x ) bar(d)(x,y)=d(y,x)\bar{d}(x, y)=d(y, x)d¯(x,y)=d(y,x) for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X is also a quasi-metric on X X XXX, called the conjugate quasi-metric of d d ddd. A pair ( X , d X , d X,dX, dX,d ), where X X XXX is a non-empty set and d d ddd a quasi-metric on X X XXX, is called a quasi-metric space. Obviously, the function d s ( x , y ) = max { d ( x , y ) , d ¯ ( x , y ) } d s ( x , y ) = max { d ( x , y ) , d ¯ ( x , y ) } d^(s)(x,y)=max{d(x,y), bar(d)(x,y)}d^{s}(x, y)=\max \{d(x, y), \bar{d}(x, y)\}ds(x,y)=max{d(x,y),d¯(x,y)} is a metric on X X XXX. Each quasi-metric d d ddd on X X XXX induces a topology τ ( d ) τ ( d ) tau(d)\tau(d)τ(d) on X X XXX which has as a base the family of balls (forward open balls [7]).
B + ( x , ε ) := { y ∈ X : d ( x , y ) < ε } , x ∈ X , ε > 0 . B + ( x , ε ) := { y ∈ X : d ( x , y ) < ε } , x ∈ X , ε > 0 . B^(+)(x,epsi):={y in X:d(x,y) < epsi},x in X,epsi > 0.B^{+}(x, \varepsilon):=\{y \in X: d(x, y)<\varepsilon\}, x \in X, \varepsilon>0 .B+(x,ε):={y∈X:d(x,y)<ε},x∈X,ε>0.
This topology is called the forward topology of X ( [ 7 ] , [ 15 ] ) X ( [ 7 ] , [ 15 ] ) X([7],[15])X([7],[15])X([7],[15]) and is denoted by τ + τ + tau_(+)\tau_{+}τ+. Analogously, the quasi-metric d ¯ d ¯ bar(d)\bar{d}d¯ induces the topology τ ( d ¯ ) τ ( d ¯ ) tau( bar(d))\tau(\bar{d})τ(d¯) on X X XXX which has as a base the family of backward open balls ([7])
B − ( x , ε ) := { y ∈ X : d ( y , x ) < ε } , x ∈ X , ε > 0 . B − ( x , ε ) := { y ∈ X : d ( y , x ) < ε } , x ∈ X , ε > 0 . B^(-)(x,epsi):={y in X:d(y,x) < epsi},x in X,epsi > 0.B^{-}(x, \varepsilon):=\{y \in X: d(y, x)<\varepsilon\}, x \in X, \varepsilon>0 .B−(x,ε):={y∈X:d(y,x)<ε},x∈X,ε>0.
This topology is called the backward topology of X X XXX ([7], [15]) and is denoted by τ − τ − tau_(-)\tau_{-}τ−。
Note that the topology τ + τ + tau_(+)\tau_{+}τ+is a T 0 T 0 T_(0)T_{0}T0-topology. If the condition AM1) is replaced by the condition: AM0) d ( x , y ) = 0 d ( x , y ) = 0 d(x,y)=0d(x, y)=0d(x,y)=0 iff x = y x = y x=yx=yx=y, then τ + τ + tau_(+)\tau_{+}τ+is a T 1 T 1 T_(1)T_{1}T1-topology. The pair ( X , d X , d X,dX, dX,d ) is called a T 0 T 0 T_(0)T_{0}T0 quasi-metric space, respectively a T 1 T 1 T_(1)T_{1}T1 quasi-metric space (see [21] and [22]).
Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) be a quasi-metric space. A sequence ( x k ) k ≥ 1 d x k k ≥ 1 d (x_(k))_(k >= 1)d\left(x_{k}\right)_{k \geq 1} d(xk)k≥1d-converges to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X (respectively d ¯ d ¯ bar(d)\bar{d}d¯-converges to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X ) iff
lim k → ∞ d ( x 0 , x k ) = 0 , respectively lim k → ∞ d ( x k , x 0 ) = lim k → ∞ d ¯ ( x 0 , x k ) = 0 lim k → ∞   d x 0 , x k = 0 ,  respectively  lim k → ∞   d x k , x 0 = lim k → ∞   d ¯ x 0 , x k = 0 lim_(k rarr oo)d(x_(0),x_(k))=0," respectively "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_{0}, x_{k}\right)=0, \text { respectively } \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(x0,xk)=0, respectively limk→∞d(xk,x0)=limk→∞d¯(x0,xk)=0
A set K ⊂ X K ⊂ X K sub XK \subset XK⊂X is called d d ddd-compact if every open cover of K K KKK with respect to the forward topology has a finite subcover. We say that K K KKK is d d ddd-sequentially compact if every sequence in K K KKK has a d d ddd-convergent subsequence with limit in K K KKK (Definition 4.1 in [7]). Finally, the set Y Y YYY in ( X , d X , d X,dX, dX,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 ) d y n k d (y_(n_(k)))d\left(y_{n_{k}}\right) d(ynk)d-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.
Observe that, if ( X , d X , d X,dX, dX,d ) is a quasi-metric space ( d , d ¯ d , d ¯ d, bar(d)d, \bar{d}d,d¯ )-sequentially compact and T 0 T 0 T_(0)T_{0}T0-separated, then it is possible to find sequences with all subsequences both d d ddd-convergent and d ¯ d ¯ bar(d)\bar{d}d¯-convergent, but to different limits. For example, let X = [ 0 , 1 ] X = [ 0 , 1 ] X=[0,1]X=[0,1]X=[0,1]
and d ( x , y ) = ( y − x ) ∨ 0 , x , y ∈ [ 0 , 1 ] d ( x , y ) = ( y − x ) ∨ 0 , x , y ∈ [ 0 , 1 ] d(x,y)=(y-x)vv0,x,y in[0,1]d(x, y)=(y-x) \vee 0, x, y \in[0,1]d(x,y)=(y−x)∨0,x,y∈[0,1]. Then d ¯ ( x , y ) = ( x − y ) ∨ 0 d ¯ ( x , y ) = ( x − y ) ∨ 0 bar(d)(x,y)=(x-y)vv0\bar{d}(x, y)=(x-y) \vee 0d¯(x,y)=(x−y)∨0 and the sequence ( 1 n ) n ≥ 1 1 n n ≥ 1 ((1)/(n))_(n >= 1)\left(\frac{1}{n}\right)_{n \geq 1}(1n)n≥1 satisfies the property that every subsequences d d ddd-converges to 0 and d ¯ d ¯ bar(d)\bar{d}d¯ converges to 1 . But if ( X , d X , d X,dX, dX,d ) is ( d , d ¯ d , d ¯ d, bar(d)d, \bar{d}d,d¯ )-sequentially compact and T 1 T 1 T_(1)T_{1}T1-separated, then by Lemma 3.1 of [7] it follows that if ( x n ) n ≥ 1 ⊂ X x n n ≥ 1 ⊂ X (x_(n))_(n >= 1)sub X\left(x_{n}\right)_{n \geq 1} \subset X(xn)n≥1⊂X is d d ddd-convergent to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X and d ¯ d ¯ bar(d)\bar{d}d¯-convergent to y 0 ∈ X y 0 ∈ X y_(0)in Xy_{0} \in Xy0∈X, then x 0 = y 0 x 0 = y 0 x_(0)=y_(0)x_{0}=y_{0}x0=y0. This fact is essential in the proof of Theorem 3.1 from bellow.
Definition 1.1 ([21]). Let Y Y YYY be a non-empty subset of a quasi-metric space ( X , d X , d X,dX, dX,d ). A function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R is called d d ddd-semi-Lipschitz if there exists L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 (named a d d ddd-semi-Lipschitz constant for f ) f ) f)f)f) such that
(1.1) f ( x ) − f ( y ) ≤ L d ( x , y ) , for all x , y ∈ Y . (1.1) f ( x ) − f ( y ) ≤ L d ( x , y ) ,  for all  x , y ∈ Y . {:(1.1)f(x)-f(y) <= Ld(x","y)","" for all "x","y in Y.:}\begin{equation*} f(x)-f(y) \leq L d(x, y), \text { for all } x, y \in Y . \tag{1.1} \end{equation*}(1.1)f(x)−f(y)≤Ld(x,y), for all x,y∈Y.
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 d ( x , y ) = 0 d(x,y)=0d(x, y)=0d(x,y)=0.
Denote by R ≤ d Y R ≤ d Y R_( <= _(d))^(Y)\mathbb{R}_{\leq_{d}}^{Y}R≤dY the set of all ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing functions on Y Y YYY. This set 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, i.e., for each f , g ∈ R ≤ d Y f , g ∈ R ≤ d Y f,g inR_( <= _(d))^(Y)f, g \in \mathbb{R}_{\leq_{d}}^{Y}f,g∈R≤dY and λ ≥ 0 λ ≥ 0 lambda >= 0\lambda \geq 0λ≥0 it follows that f + g ∈ R ≤ d Y f + g ∈ R ≤ d Y f+g inR_( <= _(d))^(Y)f+g \in \mathbb{R}_{\leq_{d}}^{Y}f+g∈R≤dY and λ f ∈ R ≤ d Y λ f ∈ R ≤ d Y lambda f inR_( <= _(d))^(Y)\lambda f \in \mathbb{R}_{\leq_{d}}^{Y}λf∈R≤dY.
For a d d ddd-semi-Lipschitz function f f fff on Y Y YYY, put [21]
(1.2) | | f | d = sup d ( x , y ) > 0 x , y ∈ Y ( f ( x ) − f ( y ) ) ) ∨ 0 d ( x , y ) . (1.2) | f | d = sup d ( x , y ) > 0 x , y ∈ Y   ( f ( x ) − f ( y ) ) ) ∨ 0 d ( x , y ) . {:(1.2)||f|_(d)=s u p_({:[d(x","y) > 0],[x","y in Y]:})((f(x)-f(y)))vv0)/(d(x,y)).:}:}\begin{equation*} \left||f|_{d}=\sup _{\substack{d(x, y)>0 \\ x, y \in Y}} \frac{(f(x)-f(y))) \vee 0}{d(x, y)} .\right. \tag{1.2} \end{equation*}(1.2)||f|d=supd(x,y)>0x,y∈Y(f(x)−f(y)))∨0d(x,y).
Then | | f | d | f | d ||f|_(d):}\left||f|_{d}\right.||f|d is the smallest d d ddd-semi-Lipschitz constant for f ( [ 18 ] ) f ( [ 18 ] ) f([18])f([18])f([18]).
For a fixed element θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y denote
(1.3) d − S Lip 0 Y := { f ∈ R ≤ d Y : ‖ f | d < ∞ and f ( θ ) = 0 } . (1.3) d − S  Lip  0 Y := f ∈ R ≤ d Y : ‖ f d < ∞  and  f ( θ ) = 0 . {:(1.3)d-S" Lip "_(0)Y:={f inR_( <= _(d))^(Y):||f|_(d) < oo" and "f(theta)=0}.:}\begin{equation*} d-S \text { Lip }_{0} Y:=\left\{f \in \mathbb{R}_{\leq_{d}}^{Y}: \|\left. f\right|_{d}<\infty \text { and } f(\theta)=0\right\} . \tag{1.3} \end{equation*}(1.3)d−S Lip 0Y:={f∈R≤dY:‖f|d<∞ and f(θ)=0}.
If ( X , d X , d X,dX, dX,d ) is a T 1 T 1 T_(1)T_{1}T1 quasi-metric space, then every f ∈ R X f ∈ R X f inR^(X)f \in \mathbb{R}^{X}f∈RX is ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing ([21]).
The set defined by (1.3) is a subcone of the cone R ≤ d Y R ≤ d Y R_( <= _(d))^(Y)\mathbb{R}_{\leq_{d}}^{Y}R≤dY, and the functional ‖ | d : d − ‖ d : d − |||_(d):d-\|\left.\right|_{d}: d-‖|d:d− SLip 0 Y → [ 0 , ∞ ) 0 Y → [ 0 , ∞ ) _(0)Y rarr[0,oo)_{0} Y \rightarrow[0, \infty)0Y→[0,∞) defined by (1.2) is an asymmetric norm, i.e., it is subadditive, positive homogeneous and ‖ f | d = 0 ‖ f d = 0 ||f|_(d)=0\|\left. f\right|_{d}=0‖f|d=0 iff f ≡ 0 f ≡ 0 f-=0f \equiv 0f≡0. The pair ( d d ddd-SLip 0 Y , ‖ | d 0 Y , ‖ d _(0)Y,|||_(d){ }_{0} Y, \|\left.\right|_{d}0Y,‖|d ) is called the normed cone of real semi-Lipschitz functions on Y Y YYY, vanishing at the fixed point θ ∈ Y ( [ 22 ] ) θ ∈ Y ( [ 22 ] ) theta in Y([22])\theta \in Y([22])θ∈Y([22]).
In [22] some properties of the normed cone ( d d ddd-SLip 0 Y , ‖ | d 0 Y , ‖ d _(0)Y,|||_(d){ }_{0} Y, \|\left.\right|_{d}0Y,‖|d ) are presented. Similar properties in the case of semi-Lipschitz functions on a quasi-metric space with values in a quasi-normed linear space (space with asymmetric norm) are discussed in [24]. For more information concerning other properties of quasi-metric spaces and their applications, see [7], [8], [13], [20], [26].

2. Results

Let f ∈ d − S f ∈ d − S f in d-Sf \in d-Sf∈d−S Lip 0 Y 0 Y _(0)Y_{0} Y0Y. A function F F FFF in d − S d − S d-Sd-Sd−S Lip 0 X 0 X _(0)X_{0} X0X satisfying the inequality
F ( u ) − F ( v ) ≤ ‖ f | d d ( u , v ) , F ( u ) − F ( v ) ≤ ‖ f d d ( u , v ) , F(u)-F(v) <= ||f|_(d)d(u,v),F(u)-F(v) \leq \|\left. f\right|_{d} d(u, v),F(u)−F(v)≤‖f|dd(u,v),
for all u , v ∈ X u , v ∈ X u,v in Xu, v \in Xu,v∈X and such that F ( y ) = f ( y ) F ( y ) = f ( y ) F(y)=f(y)F(y)=f(y)F(y)=f(y) for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y is called an extension of f f fff (preserving the asymmetric norm ‖ f | d ‖ f d ||f|_(d)\|\left. f\right|_{d}‖f|d ).
It follows that each extension F ∈ d − S F ∈ d − S F in d-SF \in d-SF∈d−S Lip 0 X 0 X _(0)X_{0} X0X of f ∈ d − S f ∈ d − S f in d-Sf \in d-Sf∈d−S Lip 0 Y 0 Y _(0)Y_{0} Y0Y satisfies
(2.1) F | Y = f and ‖ F | d = ‖ f | d (2.1) F Y = f  and  F d = f d {:(2.1)F|_(Y)=f" and "||F|_(d)=||f|_(d):}\begin{equation*} \left.F\right|_{Y}=f \text { and }\left.\left\|\left.F\right|_{d}=\right\| f\right|_{d} \tag{2.1} \end{equation*}(2.1)F|Y=f and ‖F|d=‖f|d
The existence of such an extension for each f ∈ d − S f ∈ d − S f in d-Sf \in d-Sf∈d−S Lip 0 Y 0 Y _(0)Y_{0} Y0Y follows from the following theorem proved in [18]. For the sake of completeness we include the proof.
Theorem 2.1. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X a fixed element, and Y Y YYY a subset of X X XXX with θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y. Then for every f ∈ d − S f ∈ d − S f in d-Sf \in d-Sf∈d−S Lip 0 Y 0 Y _(0)Y_{0} Y0Y there exists at least a function F ∈ d F ∈ d F in dF \in dF∈d-SLip 0 X 0 X _(0)X{ }_{0} X0X such that F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f and ‖ F | d = ‖ f | d F d = f d ||F|_(d)=||f|_(d)\left.\left\|\left.F\right|_{d}=\right\| f\right|_{d}‖F|d=‖f|d.
Proof. For f ∈ d − S f ∈ d − S f in d-Sf \in d-Sf∈d−S Lip 0 Y 0 Y _(0)Y_{0} Y0Y let
(2.2) F d ( f ) ( x ) = inf y ∈ Y [ f ( y ) + ‖ f | d d ( x , y ) } , x ∈ X (2.2) F d ( f ) ( x ) = inf y ∈ Y   f ( y ) + ‖ f d d ( x , y ) , x ∈ X {:(2.2)F_(d)(f)(x)=i n f_(y in Y)[f(y)+||f|_(d)d(x,y)}","x in X:}\begin{equation*} F_{d}(f)(x)=\inf _{y \in Y}\left[f(y)+\|\left. f\right|_{d} d(x, y)\right\}, x \in X \tag{2.2} \end{equation*}(2.2)Fd(f)(x)=infy∈Y[f(y)+‖f|dd(x,y)},x∈X
First we show that F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f) is well defined.
Let x ∈ X x ∈ X x in Xx \in Xx∈X. For any y ∈ Y y ∈ Y y in Yy \in Yy∈Y we have
f ( y ) + ‖ f | d d ( x , y ) = ‖ f | d d ( x , y ) − ( f ( θ ) − f ( y ) ) ≥ ‖ f | d d ( x , y ) − ‖ f | d d ( θ , y ) = ‖ f | d ( d ( x , y ) − d ( θ , y ) ) ≥ − ‖ f | d d ( θ , x ) f ( y ) + ‖ f d d ( x , y ) = ‖ f d d ( x , y ) − ( f ( θ ) − f ( y ) ) ≥ f d d ( x , y ) − f d d ( θ , y ) = f d ( d ( x , y ) − d ( θ , y ) ) ≥ − f d d ( θ , x ) {:[f(y)+||f|_(d)d(x","y)=||f|_(d)d(x","y)-(f(theta)-f(y))],[ >= ||f|_(d)d(x,y)-||f|_(d)d(theta","y)],[=||f|_(d)(d(x,y)-d(theta,y)) >= -||f|_(d)d(theta","x)]:}\begin{aligned} f(y)+\|\left. f\right|_{d} d(x, y) & =\|\left. f\right|_{d} d(x, y)-(f(\theta)-f(y)) \\ & \geq\left.\left\|\left.f\right|_{d} d(x, y)-\right\| f\right|_{d} d(\theta, y) \\ & =\left.\left\|\left.f\right|_{d}(d(x, y)-d(\theta, y)) \geq-\right\| f\right|_{d} d(\theta, x) \end{aligned}f(y)+‖f|dd(x,y)=‖f|dd(x,y)−(f(θ)−f(y))≥‖f|dd(x,y)−‖f|dd(θ,y)=‖f|d(d(x,y)−d(θ,y))≥−‖f|dd(θ,x)
showing that for every x ∈ X x ∈ X x in Xx \in Xx∈X the set
{ f ( y ) + | | f | d d ( x , y ) : y ∈ Y } f ( y ) + | f | d d ( x , y ) : y ∈ Y {f(y)+||f|_(d)d(x,y):y in Y}:}\left\{f(y)+\left||f|_{d} d(x, y): y \in Y\right\}\right.{f(y)+||f|dd(x,y):y∈Y}
is bounded from below and, consequently, the infimum in (2.2) is finite.
Now we show that F d ( f ) | Y = f , F d ( f ) ∈ d − S F d ( f ) Y = f , F d ( f ) ∈ d − S F_(d)(f)|_(Y)=f,quadF_(d)(f)in d-S\left.F_{d}(f)\right|_{Y}=f, \quad F_{d}(f) \in d-SFd(f)|Y=f,Fd(f)∈d−S Lip 0 X 0 X _(0)X_{0} X0X and ‖ F d ( f ) | d = ‖ f | d F d ( f ) d = f d ||F_(d)(f)|_(d)=||f|_(d)\left.\left\|\left.F_{d}(f)\right|_{d}=\right\| f\right|_{d}‖Fd(f)|d=‖f|d. For every y ∈ Y y ∈ Y y in Yy \in Yy∈Y we have
F d ( f ) ( x ) ≤ f ( y ) + ‖ f | d d ( x , y ) , x ∈ X , F d ( f ) ( x ) ≤ f ( y ) + ‖ f d d ( x , y ) , x ∈ X , F_(d)(f)(x) <= f(y)+||f|_(d)d(x,y),x in X,F_{d}(f)(x) \leq f(y)+\|\left. f\right|_{d} d(x, y), x \in X,Fd(f)(x)≤f(y)+‖f|dd(x,y),x∈X,
which for x = y x = y x=yx=yx=y yields
F d ( f ) ( y ) ≤ f ( y ) . F d ( f ) ( y ) ≤ f ( y ) . F_(d)(f)(y) <= f(y).F_{d}(f)(y) \leq f(y) .Fd(f)(y)≤f(y).
On the other hand, for y ∈ Y y ∈ Y y in Yy \in Yy∈Y and all y ′ ∈ Y y ′ ∈ Y y^(')in Yy^{\prime} \in Yy′∈Y,
f ( y ) − f ( y ′ ) ≤ ‖ f | d d ( y , y ′ ) f ( y ) − f y ′ ≤ ‖ f d d y , y ′ f(y)-f(y^(')) <= ||f|_(d)d(y,y^('))f(y)-f\left(y^{\prime}\right) \leq \|\left. f\right|_{d} d\left(y, y^{\prime}\right)f(y)−f(y′)≤‖f|dd(y,y′)
implies
f ( y ) ≤ f ( y ′ ) + ‖ f | d d ( y , y ′ ) . f ( y ) ≤ f y ′ + ‖ f d d y , y ′ . f(y) <= f(y^('))+||f|_(d)d(y,y^(')).f(y) \leq f\left(y^{\prime}\right)+\|\left. f\right|_{d} d\left(y, y^{\prime}\right) .f(y)≤f(y′)+‖f|dd(y,y′).
Taking the infimum with respect to y ′ ∈ Y y ′ ∈ Y y^(')in Yy^{\prime} \in Yy′∈Y one obtains f ( y ) ≤ F d ( f ) ( y ) f ( y ) ≤ F d ( f ) ( y ) f(y) <= F_(d)(f)(y)f(y) \leq F_{d}(f)(y)f(y)≤Fd(f)(y), so that
F d ( f ) ( y ) = f ( y ) , y ∈ Y . F d ( f ) ( y ) = f ( y ) , y ∈ Y . F_(d)(f)(y)=f(y),y in Y.F_{d}(f)(y)=f(y), y \in Y .Fd(f)(y)=f(y),y∈Y.
Let x 1 , x 2 ∈ X x 1 , x 2 ∈ X x_(1),x_(2)in Xx_{1}, x_{2} \in Xx1,x2∈X and ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. Choosing y ∈ Y y ∈ Y y in Yy \in Yy∈Y such that
F d ( f ) ( x 1 ) ≥ f ( y ) + ‖ f | d d ( x 1 , y ) − ε F d ( f ) x 1 ≥ f ( y ) + ‖ f d d x 1 , y − ε F_(d)(f)(x_(1)) >= f(y)+||f|_(d)d(x_(1),y)-epsiF_{d}(f)\left(x_{1}\right) \geq f(y)+\|\left. f\right|_{d} d\left(x_{1}, y\right)-\varepsilonFd(f)(x1)≥f(y)+‖f|dd(x1,y)−ε
we get
F d ( f ) ( x 2 ) − F d ( f ) ( x 1 ) ≤ f ( y ) + ‖ f | d ⋅ d ( x 2 , y ) − ( f ( y ) + ‖ f | d d ( x 1 , y ) − ε ) = ‖ f | d ( d ( x 2 , y ) − d ( x 1 , y ) ) + ε . F d ( f ) x 2 − F d ( f ) x 1 ≤ f ( y ) + ‖ f d ⋅ d x 2 , y − f ( y ) + ‖ f d d x 1 , y − ε = ‖ f d d x 2 , y − d x 1 , y + ε . {:[F_(d)(f)(x_(2))-F_(d)(f)(x_(1)) <= f(y)+||f|_(d)*d(x_(2),y)-(f(y)+||f|_(d)d(x_(1),y)-epsi)],[=||f|_(d)(d(x_(2),y)-d(x_(1),y))+epsi.]:}\begin{aligned} F_{d}(f)\left(x_{2}\right)-F_{d}(f)\left(x_{1}\right) & \leq f(y)+\|\left. f\right|_{d} \cdot d\left(x_{2}, y\right)-\left(f(y)+\|\left. f\right|_{d} d\left(x_{1}, y\right)-\varepsilon\right) \\ & =\|\left. f\right|_{d}\left(d\left(x_{2}, y\right)-d\left(x_{1}, y\right)\right)+\varepsilon . \end{aligned}Fd(f)(x2)−Fd(f)(x1)≤f(y)+‖f|d⋅d(x2,y)−(f(y)+‖f|dd(x1,y)−ε)=‖f|d(d(x2,y)−d(x1,y))+ε.
Because d ( x 2 , y ) − d ( x 1 , y ) ≤ d ( x 2 , x 1 ) d x 2 , y − d x 1 , y ≤ d x 2 , x 1 d(x_(2),y)-d(x_(1),y) <= d(x_(2),x_(1))d\left(x_{2}, y\right)-d\left(x_{1}, y\right) \leq d\left(x_{2}, x_{1}\right)d(x2,y)−d(x1,y)≤d(x2,x1) it follows that
F d ( f ) ( x 2 ) − F d ( f ) ( x 1 ) ≤ ‖ f | d d ( x 2 , x 1 ) . F d ( f ) x 2 − F d ( f ) x 1 ≤ ‖ f d d x 2 , x 1 . F_(d)(f)(x_(2))-F_(d)(f)(x_(1)) <= ||f|_(d)d(x_(2),x_(1)).F_{d}(f)\left(x_{2}\right)-F_{d}(f)\left(x_{1}\right) \leq \|\left. f\right|_{d} d\left(x_{2}, x_{1}\right) .Fd(f)(x2)−Fd(f)(x1)≤‖f|dd(x2,x1).
This means that F d ( f ) ∈ d − S L i p 0 X F d ( f ) ∈ d − S L i p 0 X F_(d)(f)in d-SLip_(0)XF_{d}(f) \in d-S L i p_{0} XFd(f)∈d−SLip0X, and by the last inequality
‖ F d ( f ) | d ≤ ‖ f | d F d ( f ) d ≤ f d ||F_(d)(f)|_(d) <= ||f|_(d)\left.\left\|\left.F_{d}(f)\right|_{d} \leq\right\| f\right|_{d}‖Fd(f)|d≤‖f|d
By the definitions of an asymmetric norm
‖ F d ( f ) | d ≥ ‖ F d ( f ) | Y | d = ‖ f | d F d ( f ) d ≥ F d ( f ) Y d = ‖ f d ||F_(d)(f)|_(d) >= ||F_(d)(f)|_(Y)|_(d)=||f|_(d)\left.\left.\left\|\left.F_{d}(f)\right|_{d} \geq\right\| F_{d}(f)\right|_{Y}\right|_{d}=\|\left. f\right|_{d}‖Fd(f)|d≥‖Fd(f)|Y|d=‖f|d
so that the equality ‖ F d ( f ) | d = ‖ f | d F d ( f ) d = f d ||F_(d)(f)|_(d)=||f|_(d)\left.\left\|\left.F_{d}(f)\right|_{d}=\right\| f\right|_{d}‖Fd(f)|d=‖f|d holds.
The following Remarks 2.2 and 2.3 are taken from [18] and [19].
Remark 2.2. By Theorem 2.1 it follows that for every f ∈ d f ∈ d f in df \in df∈d-SLip 0 Y 0 Y _(0)Y{ }_{0} Y0Y, the set of all extensions preserving the asymmetric norm ‖ f | d ‖ f d ||f|_(d)\|\left. f\right|_{d}‖f|d, i.e.
(2.3) E d ( f ) = { H ∈ d − S L i p 0 X : H | Y = f and ‖ H | d = | | f | d } (2.3) E d ( f ) = H ∈ d − S L i p 0 X : H Y = f  and  ‖ H d = | f | d {:(2.3)E_(d)(f)={H in d-SLip_(0)X:H|_(Y)=f" and "||H|_(d)=||f|_(d)}:}:}\begin{equation*} \mathcal{E}_{d}(f)=\left\{H \in d-S L i p_{0} X:\left.H\right|_{Y}=f \text { and } \|\left. H\right|_{d}=\left||f|_{d}\right\}\right. \tag{2.3} \end{equation*}(2.3)Ed(f)={H∈d−SLip0X:H|Y=f and ‖H|d=||f|d}
is nonempty, because F d ( f ) ∈ E d ( f ) F d ( f ) ∈ E d ( f ) F_(d)(f)inE_(d)(f)F_{d}(f) \in \mathcal{E}_{d}(f)Fd(f)∈Ed(f) where F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f) is given by (2.2).
Analogously, one proves that the function
(2.4) G d ( f ) = sup y ∈ Y { f ( y ) − ‖ f | d d ¯ ( x , y ) } , x ∈ X (2.4) G d ( f ) = sup y ∈ Y   f ( y ) − ‖ f d d ¯ ( x , y ) , x ∈ X {:(2.4)G_(d)(f)=s u p_(y in Y){f(y)-||f|_(d)( bar(d))(x,y)}","x in X:}\begin{equation*} G_{d}(f)=\sup _{y \in Y}\left\{f(y)-\|\left. f\right|_{d} \bar{d}(x, y)\right\}, x \in X \tag{2.4} \end{equation*}(2.4)Gd(f)=supy∈Y{f(y)−‖f|dd¯(x,y)},x∈X
is in E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f).
Remark 2.3. Obviously, the set E d ( f ) E d ( f ) E_(d)(f)\mathcal{E}_{d}(f)Ed(f) is convex, i.e. for every H 1 , H 2 ∈ E d ( f ) H 1 , H 2 ∈ E d ( f ) H_(1),H_(2)inE_(d)(f)H_{1}, H_{2} \in \mathcal{E}_{d}(f)H1,H2∈Ed(f) and λ ∈ [ 0 , 1 ] λ ∈ [ 0 , 1 ] lambda in[0,1]\lambda \in[0,1]λ∈[0,1] it follows λ H 1 + ( 1 − λ ) H 2 ∈ E d ( f ) λ H 1 + ( 1 − λ ) H 2 ∈ E d ( f ) lambdaH_(1)+(1-lambda)H_(2)inE_(d)(f)\lambda H_{1}+(1-\lambda) H_{2} \in \mathcal{E}_{d}(f)λH1+(1−λ)H2∈Ed(f). Moreover for every H ∈ E d ( f ) H ∈ E d ( f ) H inE_(d)(f)H \in \mathcal{E}_{d}(f)H∈Ed(f) we have:
(2.5) G d ( f ) ( x ) ≤ H ( x ) ≤ F d ( f ) ( x ) , x ∈ X . (2.5) G d ( f ) ( x ) ≤ H ( x ) ≤ F d ( f ) ( x ) , x ∈ X . {:(2.5)G_(d)(f)(x) <= H(x) <= F_(d)(f)(x)","x in X.:}\begin{equation*} G_{d}(f)(x) \leq H(x) \leq F_{d}(f)(x), x \in X . \tag{2.5} \end{equation*}(2.5)Gd(f)(x)≤H(x)≤Fd(f)(x),x∈X.
The function F d ( f ) F d ( f ) F_(d)(f)F_{d}(f)Fd(f) defined by (2.2) is called the maximal extension of f f fff, and G d ( f ) G d ( f ) G_(d)(f)G_{d}(f)Gd(f) defined by (2.4) is called the minimal extension of f f fff.
Remark 2.4. If θ ∈ Y 1 ⊂ Y 2 ⊂ Y θ ∈ Y 1 ⊂ Y 2 ⊂ Y theta inY_(1)subY_(2)sub Y\theta \in Y_{1} \subset Y_{2} \subset Yθ∈Y1⊂Y2⊂Y and f ∈ d f ∈ d f in df \in df∈d-SLip 0 Y 0 Y _(0)Y{ }_{0} Y0Y, then for each u ∈ Y u ∈ Y u in Yu \in Yu∈Y we can easily obtain:
inf y ∈ Y 1 { f ( y ) + ‖ f | d d ( u , y ) } ≥ inf y ∈ Y 2 { f ( y ) + ‖ f | d d ( u , y ) } inf y ∈ Y 1   f ( y ) + ‖ f d d ( u , y ) ≥ inf y ∈ Y 2   f ( y ) + ‖ f d d ( u , y ) i n f_(y inY_(1)){f(y)+||f|_(d)d(u,y)} >= i n f_(y inY_(2)){f(y)+||f|_(d)d(u,y)}\inf _{y \in Y_{1}}\left\{f(y)+\|\left. f\right|_{d} d(u, y)\right\} \geq \inf _{y \in Y_{2}}\left\{f(y)+\|\left. f\right|_{d} d(u, y)\right\}infy∈Y1{f(y)+‖f|dd(u,y)}≥infy∈Y2{f(y)+‖f|dd(u,y)}
and
sup y ∈ Y 1 { f ( y ) − ‖ f | d d ¯ ( u , y ) } ≤ sup y ∈ Y 2 { f ( y ) − ‖ f | d d ¯ ( u , y ) } . sup y ∈ Y 1   f ( y ) − ‖ f d d ¯ ( u , y ) ≤ sup y ∈ Y 2   f ( y ) − ‖ f d d ¯ ( u , y ) . s u p_(y inY_(1)){f(y)-||f|_(d)( bar(d))(u,y)} <= s u p_(y inY_(2)){f(y)-||f|_(d)( bar(d))(u,y)}.\sup _{y \in Y_{1}}\left\{f(y)-\|\left. f\right|_{d} \bar{d}(u, y)\right\} \leq \sup _{y \in Y_{2}}\left\{f(y)-\|\left. f\right|_{d} \bar{d}(u, y)\right\} .supy∈Y1{f(y)−‖f|dd¯(u,y)}≤supy∈Y2{f(y)−‖f|dd¯(u,y)}.
Remark 2.5. Observe that Theorem 2.1 is the "nonsymmetric" analog of McShane's theorem [14] for metric spaces.
Theorem 2.6. Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) be a quasi-metric space and Y ⊆ X Y ⊆ X Y sube XY \subseteq XY⊆X. Then
(a) Every f ∈ d f ∈ d f in df \in df∈d-SLip Y Y YYY is upper semicontinuous on ( Y , d ¯ ) ( Y , d ¯ ) (Y, bar(d))(Y, \bar{d})(Y,d¯);
(b) If Y Y YYY is d ¯ d ¯ bar(d)\bar{d}d¯-sequentially compact, then every f ∈ d f ∈ d f in df \in df∈d-SLip Y Y YYY attains its maximum value on Y Y YYY.
Proof. Let f ∈ d f ∈ d f in df \in df∈d-SLipY. If ‖ f | d = 0 ‖ f d = 0 ||f|_(d)=0\|\left. f\right|_{d}=0‖f|d=0 then f ( y ) = f ( y ) = f(y)=f(y)=f(y)= constant for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y and this function is upper semicontinuous. Let y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y and ‖ f | d > 0 ‖ f d > 0 ||f|_(d) > 0\|\left. f\right|_{d}>0‖f|d>0. Then the inequality
f ( y ) − f ( y 0 ) ≤ ‖ f | d d ( y , y 0 ) f ( y ) − f y 0 ≤ ‖ f d d y , y 0 f(y)-f(y_(0)) <= ||f|_(d)d(y,y_(0))f(y)-f\left(y_{0}\right) \leq \|\left. f\right|_{d} d\left(y, y_{0}\right)f(y)−f(y0)≤‖f|dd(y,y0)
implies
f ( y ) ≤ f ( y 0 ) + ‖ f | d d ( y , y 0 ) . f ( y ) ≤ f y 0 + ‖ f d d y , y 0 . f(y) <= f(y_(0))+||f|_(d)d(y,y_(0)).f(y) \leq f\left(y_{0}\right)+\|\left. f\right|_{d} d\left(y, y_{0}\right) .f(y)≤f(y0)+‖f|dd(y,y0).
For ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 and y ∈ Y y ∈ Y y in Yy \in Yy∈Y such that d ( y , y 0 ) < ε ‖ f | d d y , y 0 < ε ‖ f d d(y,y_(0)) < (epsi)/(||f|_(d))d\left(y, y_{0}\right)<\frac{\varepsilon}{\|\left. f\right|_{d}}d(y,y0)<ε‖f|d it follows
f ( y ) ≤ f ( y 0 ) + ε f ( y ) ≤ f y 0 + ε f(y) <= f(y_(0))+epsif(y) \leq f\left(y_{0}\right)+\varepsilonf(y)≤f(y0)+ε
showing that f f fff is upper semicontinuous on ( Y , d ¯ ) ( Y , d ¯ ) (Y, bar(d))(Y, \bar{d})(Y,d¯).
Let Y Y YYY be d ¯ d ¯ bar(d)\bar{d}d¯-sequentially compact in ( X , d X , d X,dX, dX,d ) and M = sup f ( Y ) M = sup f ( Y ) M=s u p f(Y)M=\sup f(Y)M=supf(Y), where M ∈ R ∪ { + ∞ } M ∈ R ∪ { + ∞ } M inRuu{+oo}M \in \mathbb{R} \cup\{+\infty\}M∈R∪{+∞}. Then there exists a 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 such that lim n → ∞ f ( y n ) = M lim n → ∞   f y n = M lim_(n rarr oo)f(y_(n))=M\lim _{n \rightarrow \infty} f\left(y_{n}\right)= Mlimn→∞f(yn)=M. Because Y Y YYY is d ¯ d ¯ bar(d)\bar{d}d¯-sequentially compact there exists y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y and a subsequence ( y n k ) k ≥ 1 y n k k ≥ 1 (y_(n_(k)))_(k >= 1)\left(y_{n_{k}}\right)_{k \geq 1}(ynk)k≥1 of ( y n ) n ≥ 1 y n n ≥ 1 (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1}(yn)n≥1 such that lim k → ∞ d ( y n k , y 0 ) = 0 lim k → ∞   d y n k , y 0 = 0 lim_(k rarr oo)d(y_(n_(k)),y_(0))=0\lim _{k \rightarrow \infty} d\left(y_{n_{k}}, y_{0}\right)=0limk→∞d(ynk,y0)=0. By the upper semicontinuity of f f fff in y 0 y 0 y_(0)y_{0}y0 it follows:
M = lim k → ∞ f ( y n k ) = lim sup f ( y n k ) ≤ f ( y 0 ) ≤ M M = lim k → ∞   f y n k = lim sup f y n k ≤ f y 0 ≤ M M=lim_(k rarr oo)f(y_(n_(k)))=l i m   s u p f(y_(n_(k))) <= f(y_(0)) <= MM=\lim _{k \rightarrow \infty} f\left(y_{n_{k}}\right)=\limsup f\left(y_{n_{k}}\right) \leq f\left(y_{0}\right) \leq MM=limk→∞f(ynk)=lim supf(ynk)≤f(y0)≤M
implying M < ∞ M < ∞ M < ooM<\inftyM<∞ and f ( y 0 ) = M f y 0 = M f(y_(0))=Mf\left(y_{0}\right)=Mf(y0)=M.
By Theorem 2.6 it follows that for Y d ¯ Y d ¯ Y bar(d)Y \bar{d}Yd¯-sequentially compact, the functional ‖ | ∞ : d − S ‖ ∞ : d − S |||_(oo):d-S\|\left.\right|_{\infty}: d-S‖|∞:d−S Lip 0 Y → [ 0 , ∞ ) 0 Y → [ 0 , ∞ ) _(0)Y rarr[0,oo)_{0} Y \rightarrow[0, \infty)0Y→[0,∞) defined by
‖ f | ∞ = max { f ( y ) : y ∈ Y } ‖ f ∞ = max { f ( y ) : y ∈ Y } ||f|_(oo)=max{f(y):y in Y}\|\left. f\right|_{\infty}=\max \{f(y): y \in Y\}‖f|∞=max{f(y):y∈Y}
is an asymmetric norm on d − S d − S d-Sd-Sd−S Lip 0 Y 0 Y _(0)Y_{0} Y0Y.
Indeed, for every f f fff in d d ddd-SLip 0 Y 0 Y _(0)Y{ }_{0} Y0Y we have ‖ f | ∞ ≥ f ( θ ) = 0 ‖ f ∞ ≥ f ( θ ) = 0 ||f|_(oo) >= f(theta)=0\|\left. f\right|_{\infty} \geq f(\theta)=0‖f|∞≥f(θ)=0. If ‖ f | ∞ > 0 ‖ f ∞ > 0 ||f|_(oo) > 0\|\left. f\right|_{\infty}>0‖f|∞>0 then there exists y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y such that f ( y 0 ) > 0 = f ( θ ) f y 0 > 0 = f ( θ ) f(y_(0)) > 0=f(theta)f\left(y_{0}\right)>0=f(\theta)f(y0)>0=f(θ). Consequently, because f ∈ R ≤ d Y f ∈ R ≤ d Y f inR_( <= _(d))^(Y)f \in \mathbb{R}_{\leq_{d}}^{Y}f∈R≤dY it follows d ( y 0 , θ ) > 0 d y 0 , θ > 0 d(y_(0),theta) > 0d\left(y_{0}, \theta\right)>0d(y0,θ)>0, and
| | f | d ≥ f ( y 0 ) − f ( θ ) d ( y 0 , θ ) > 0 . | f | d ≥ f y 0 − f ( θ ) d y 0 , θ > 0 . ||f|_(d) >= (f(y_(0))-f(theta))/(d(y_(0),theta)) > 0.:}\left||f|_{d} \geq \frac{f\left(y_{0}\right)-f(\theta)}{d\left(y_{0}, \theta\right)}>0 .\right.||f|d≥f(y0)−f(θ)d(y0,θ)>0.
It follows f ≠ 0 f ≠ 0 f!=0f \neq 0f≠0, because ‖ | d ‖ d |||_(d)\|\left.\right|_{d}‖|d is asymmetric norm on d d ddd-SLip 0 Y 0 Y _(0)Y{ }_{0} Y0Y. Obviously, | | f + g | ∞ ≤ | | f | ∞ + | | g | ∞ and | | λ f | ∞ = λ | | f | ∞ | f + g | ∞ ≤ | f | ∞ + | g | ∞  and  λ f ∞ = λ | | f ∞ ||f+g|_(oo) <= ||f|_(oo)+||g|_(oo)" and "||lambda f|_(oo)= lambda||f|_(oo)\left||f+g|_{\infty} \leq\left||f|_{\infty}+\left||g|_{\infty} \text { and }\right|\right| \lambda f\right|_{\infty}=\left.\lambda| | f\right|_{\infty}||f+g|∞≤||f|∞+||g|∞ and ||λf|∞=λ||f|∞ for all f , g ∈ d − S f , g ∈ d − S f,g in d-Sf, g \in d-Sf,g∈d−S Lip 0 Y 0 Y _(0)Y_{0} Y0Y and λ ≥ 0 λ ≥ 0 lambda >= 0\lambda \geq 0λ≥0.

3. The sequential method

Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X a fixed element, and Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X with θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y. Suppose that Y Y YYY is d ¯ d ¯ bar(d)\bar{d}d¯-sequentially compact, and f ∈ d f ∈ d f in df \in df∈d-SLip 0 Y 0 Y _(0)Y{ }_{0} Y0Y. Let
M f = sup { f ( y ) : y ∈ Y } M f = sup { f ( y ) : y ∈ Y } M_(f)=s u p{f(y):y in Y}M_{f}=\sup \{f(y): y \in Y\}Mf=sup{f(y):y∈Y}
and
E f = { y ∈ Y : f ( y ) = M f } . E f = y ∈ Y : f ( y ) = M f . E_(f)={y in Y:f(y)=M_(f)}.E_{f}=\left\{y \in Y: f(y)=M_{f}\right\} .Ef={y∈Y:f(y)=Mf}.
We want to find the maximum value M f M f M_(f)M_{f}Mf of f f fff and a point y 0 ∈ E f y 0 ∈ E f y_(0)inE_(f)y_{0} \in E_{f}y0∈Ef.
For this goal we consider the following sequential method, supposing that q > 0 q > 0 q > 0q>0q>0 is an upper bound for ‖ f | d ‖ f d ||f|_(d)\|\left. f\right|_{d}‖f|d on Y Y YYY, i.e. ‖ f | d ≤ q ‖ f d ≤ q ||f|_(d) <= q\|\left. f\right|_{d} \leq q‖f|d≤q.
Firstly, let Z Z ZZZ be a nonempty subset of Y Y YYY with θ ∈ Z θ ∈ Z theta in Z\theta \in Zθ∈Z. From the proof of Theorem 2.1, the functions
U ( f ) ( y ) = inf { f ( z ) + q d ( y , z ) : z ∈ Z } , y ∈ Y U ( f ) ( y ) = inf { f ( z ) + q d ( y , z ) : z ∈ Z } , y ∈ Y U(f)(y)=i n f{f(z)+qd(y,z):z in Z},y in YU(f)(y)=\inf \{f(z)+q d(y, z): z \in Z\}, y \in YU(f)(y)=inf{f(z)+qd(y,z):z∈Z},y∈Y
and
u ( f ) ( y ) = sup { f ( z ) − q d ( z , y ) : z ∈ Z } , y ∈ Y u ( f ) ( y ) = sup { f ( z ) − q d ( z , y ) : z ∈ Z } , y ∈ Y u(f)(y)=s u p{f(z)-qd(z,y):z in Z},y in Yu(f)(y)=\sup \{f(z)-q d(z, y): z \in Z\}, y \in Yu(f)(y)=sup{f(z)−qd(z,y):z∈Z},y∈Y
satisfy the conditions:
U ( f ) | Z = u ( f ) | Z = f | Z U ( f ) Z = u ( f ) Z = f Z U(f)|_(Z)=u(f)|_(Z)=f|_(Z)\left.U(f)\right|_{Z}=\left.u(f)\right|_{Z}=\left.f\right|_{Z}U(f)|Z=u(f)|Z=f|Z
and
‖ U ( f ) | d = ‖ u ( f ) | d = q ≥ ‖ f | d on Y . U ( f ) d = u ( f ) d = q ≥ ‖ f d  on  Y .  ||U(f)|_(d)=||u(f)|_(d)=q >= ||f|_(d)" on "Y". "\left.\left\|\left.U(f)\right|_{d}=\right\| u(f)\right|_{d}=q \geq \|\left. f\right|_{d} \text { on } Y \text {. }‖U(f)|d=‖u(f)|d=q≥‖f|d on Y. 
Moreover
u ( f ) ( y ) ≤ f ( y ) ≤ U ( f ) ( y ) , y ∈ Y . u ( f ) ( y ) ≤ f ( y ) ≤ U ( f ) ( y ) , y ∈ Y . u(f)(y) <= f(y) <= U(f)(y),y in Y.u(f)(y) \leq f(y) \leq U(f)(y), y \in Y .u(f)(y)≤f(y)≤U(f)(y),y∈Y.
Indeed, for y ∈ Y y ∈ Y y in Yy \in Yy∈Y and each z ∈ Z ⊂ Y z ∈ Z ⊂ Y z in Z sub Yz \in Z \subset Yz∈Z⊂Y we have
f ( y ) − f ( z ) ≤ ‖ f | d d ( y , z ) ≤ q d ( y , z ) f ( y ) − f ( z ) ≤ ‖ f d d ( y , z ) ≤ q d ( y , z ) f(y)-f(z) <= ||f|_(d)d(y,z) <= qd(y,z)f(y)-f(z) \leq \|\left. f\right|_{d} d(y, z) \leq q d(y, z)f(y)−f(z)≤‖f|dd(y,z)≤qd(y,z)
and
f ( y ) ≤ f ( z ) + q d ( y , z ) . f ( y ) ≤ f ( z ) + q d ( y , z ) . f(y) <= f(z)+qd(y,z).f(y) \leq f(z)+q d(y, z) .f(y)≤f(z)+qd(y,z).
Taking the infimum with respect to z ∈ Z z ∈ Z z in Zz \in Zz∈Z it follows
f ( y ) ≤ U ( f ) ( y ) , y ∈ Y . f ( y ) ≤ U ( f ) ( y ) , y ∈ Y . f(y) <= U(f)(y),y in Y.f(y) \leq U(f)(y), y \in Y .f(y)≤U(f)(y),y∈Y.
Analogously,
f ( z ) − f ( y ) ≤ q d ( z , y ) , f ( z ) − f ( y ) ≤ q d ( z , y ) , f(z)-f(y) <= qd(z,y),f(z)-f(y) \leq q d(z, y),f(z)−f(y)≤qd(z,y),
implies
f ( y ) ≥ f ( z ) − q d ( z , y ) . f ( y ) ≥ f ( z ) − q d ( z , y ) . f(y) >= f(z)-qd(z,y).f(y) \geq f(z)-q d(z, y) .f(y)≥f(z)−qd(z,y).
Taking the supremum with respect to z ∈ Z z ∈ Z z in Zz \in Zz∈Z one obtains
u ( f ) ( y ) ≤ f ( y ) , y ∈ Y . u ( f ) ( y ) ≤ f ( y ) , y ∈ Y . u(f)(y) <= f(y),y in Y.u(f)(y) \leq f(y), y \in Y .u(f)(y)≤f(y),y∈Y.
If
M U := max { U ( f ) ( y ) : y ∈ Y } , M U := max { U ( f ) ( y ) : y ∈ Y } , M_(U):=max{U(f)(y):y in Y},M_{U}:=\max \{U(f)(y): y \in Y\},MU:=max{U(f)(y):y∈Y},
then
M f ≤ M U . M f ≤ M U . M_(f) <= M_(U).M_{f} \leq M_{U} .Mf≤MU.
We define now two sequences ( y n ) n ≥ 0 y n n ≥ 0 (y_(n))_(n >= 0)\left(y_{n}\right)_{n \geq 0}(yn)n≥0 in Y Y YYY and ( M n ) n ≥ 0 M n n ≥ 0 (M_(n))_(n >= 0)\left(M_{n}\right)_{n \geq 0}(Mn)n≥0 in R R R\mathbb{R}R in the following way. Let
U 0 ( f ) ( y ) = f ( θ ) + q d ( y , θ ) = q d ( y , θ ) , y ∈ Y , U 0 ( f ) ( y ) = f ( θ ) + q d ( y , θ ) = q d ( y , θ ) , y ∈ Y , U_(0)(f)(y)=f(theta)+qd(y,theta)=qd(y,theta),y in Y,U_{0}(f)(y)=f(\theta)+q d(y, \theta)=q d(y, \theta), y \in Y,U0(f)(y)=f(θ)+qd(y,θ)=qd(y,θ),y∈Y,
i.e. U 0 ( f ) U 0 ( f ) U_(0)(f)U_{0}(f)U0(f) is an extension (the maximal extension) of f | { θ } f { θ } f|_({theta})\left.f\right|_{\{\theta\}}f|{θ} with the semi-Lipschitz constant q q qqq. Then, by the above considerations, it follows
f ( y ) ≤ U 0 ( f ) ( y ) , y ∈ Y , U 0 ( f ) ∈ d − S L i p 0 Y . f ( y ) ≤ U 0 ( f ) ( y ) , y ∈ Y , U 0 ( f ) ∈ d − S L i p 0 Y . {:[f(y) <= U_(0)(f)(y)","y in Y","],[U_(0)(f) in d-SLip_(0)Y.]:}\begin{aligned} f(y) & \leq U_{0}(f)(y), y \in Y, \\ U_{0}(f) & \in d-S L i p_{0} Y . \end{aligned}f(y)≤U0(f)(y),y∈Y,U0(f)∈d−SLip0Y.
If y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y is such that
U 0 ( f ) ( y 0 ) = M 0 := sup U 0 ( f ) ( Y ) , U 0 ( f ) y 0 = M 0 := sup U 0 ( f ) ( Y ) , U_(0)(f)(y_(0))=M_(0):=s u pU_(0)(f)(Y),U_{0}(f)\left(y_{0}\right)=M_{0}:=\sup U_{0}(f)(Y),U0(f)(y0)=M0:=supU0(f)(Y),
then
M f ≤ M 0 . M f ≤ M 0 . M_(f) <= M_(0).M_{f} \leq M_{0} .Mf≤M0.
Let Z 1 = { θ , y 0 } Z 1 = θ , y 0 Z_(1)={theta,y_(0)}Z_{1}=\left\{\theta, y_{0}\right\}Z1={θ,y0} and let
U 1 ( f ) ( y ) = inf z ∈ Z 1 { f ( z ) + q d ( y , z ) } , y ∈ Y , U 1 ( f ) ( y ) = inf z ∈ Z 1   { f ( z ) + q d ( y , z ) } , y ∈ Y , U_(1)(f)(y)=i n f_(z inZ_(1)){f(z)+qd(y,z)},y in Y,U_{1}(f)(y)=\inf _{z \in Z_{1}}\{f(z)+q d(y, z)\}, y \in Y,U1(f)(y)=infz∈Z1{f(z)+qd(y,z)},y∈Y,
be the maximal extension of f | Z 1 f Z 1 f|_(Z_(1))\left.f\right|_{Z_{1}}f|Z1 with semi-Lipschitz constant q q qqq. Then U 1 ∈ d U 1 ∈ d U_(1)in dU_{1} \in dU1∈d S S SSS Lip 0 Y 0 Y _(0)Y_{0} Y0Y and by Remark 2.3, it follows:
f ( y ) ≤ U 1 ( f ) ( y ) ≤ U 0 ( f ) ( y ) , y ∈ Y , f | Z 1 = U 1 ( f ) | Z 1 = U 0 ( f ) | Z 1 . f ( y ) ≤ U 1 ( f ) ( y ) ≤ U 0 ( f ) ( y ) , y ∈ Y , f Z 1 = U 1 ( f ) Z 1 = U 0 ( f ) Z 1 . {:[f(y) <= U_(1)(f)(y) <= U_(0)(f)(y)","y in Y","],[f|_(Z_(1))=U_(1)(f)|_(Z_(1))=U_(0)(f)|_(Z_(1)).]:}\begin{aligned} f(y) & \leq U_{1}(f)(y) \leq U_{0}(f)(y), y \in Y, \\ \left.f\right|_{Z_{1}} & =\left.U_{1}(f)\right|_{Z_{1}}=\left.U_{0}(f)\right|_{Z_{1}} . \end{aligned}f(y)≤U1(f)(y)≤U0(f)(y),y∈Y,f|Z1=U1(f)|Z1=U0(f)|Z1.
If y 1 ∈ Y y 1 ∈ Y y_(1)in Yy_{1} \in Yy1∈Y is such that
U 1 ( f ) ( y 1 ) = M 1 := sup U 1 ( f ) ( Y ) , U 1 ( f ) y 1 = M 1 := sup U 1 ( f ) ( Y ) , U_(1)(f)(y_(1))=M_(1):=s u pU_(1)(f)(Y),U_{1}(f)\left(y_{1}\right)=M_{1}:=\sup U_{1}(f)(Y),U1(f)(y1)=M1:=supU1(f)(Y),
then
M f ≤ M 1 ≤ M 0 . M f ≤ M 1 ≤ M 0 . M_(f) <= M_(1) <= M_(0).M_{f} \leq M_{1} \leq M_{0} .Mf≤M1≤M0.
Let now
Z 2 = { θ , y 0 , y 1 } Z 2 = θ , y 0 , y 1 Z_(2)={theta,y_(0),y_(1)}Z_{2}=\left\{\theta, y_{0}, y_{1}\right\}Z2={θ,y0,y1}
Supposing that, following the described procedure, we have constructed the sets
Z n = { θ , y 0 , y 1 , … , y n − 1 } and { M 0 , M 1 , M 2 , … , M n − 1 } . Z n = θ , y 0 , y 1 , … , y n − 1  and  M 0 , M 1 , M 2 , … , M n − 1 . Z_(n)={theta,y_(0),y_(1),dots,y_(n-1)}" and "{M_(0),M_(1),M_(2),dots,M_(n-1)}.Z_{n}=\left\{\theta, y_{0}, y_{1}, \ldots, y_{n-1}\right\} \text { and }\left\{M_{0}, M_{1}, M_{2}, \ldots, M_{n-1}\right\} .Zn={θ,y0,y1,…,yn−1} and {M0,M1,M2,…,Mn−1}.
Put
U n ( f ) ( y ) = inf z ∈ Z n { f ( z ) + q d ( y , z ) } , y ∈ Y . U n ( f ) ( y ) = inf z ∈ Z n   { f ( z ) + q d ( y , z ) } , y ∈ Y . U_(n)(f)(y)=i n f_(z inZ_(n)){f(z)+qd(y,z)},y in Y.U_{n}(f)(y)=\inf _{z \in Z_{n}}\{f(z)+q d(y, z)\}, y \in Y .Un(f)(y)=infz∈Zn{f(z)+qd(y,z)},y∈Y.
It follows
f ( y ) ≤ U n ( f ) ( y ) ≤ … ≤ U 1 ( f ) ( y ) ≤ U 0 ( f ) ( y ) f ( y ) ≤ U n ( f ) ( y ) ≤ … ≤ U 1 ( f ) ( y ) ≤ U 0 ( f ) ( y ) f(y) <= U_(n)(f)(y) <= dots <= U_(1)(f)(y) <= U_(0)(f)(y)f(y) \leq U_{n}(f)(y) \leq \ldots \leq U_{1}(f)(y) \leq U_{0}(f)(y)f(y)≤Un(f)(y)≤…≤U1(f)(y)≤U0(f)(y)
for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y.
Choose y n ∈ Y y n ∈ Y y_(n)in Yy_{n} \in Yyn∈Y such that
U n ( f ) ( y n ) = M n := sup U n ( Y ) . U n ( f ) y n = M n := sup U n ( Y ) . U_(n)(f)(y_(n))=M_(n):=s u pU_(n)(Y).U_{n}(f)\left(y_{n}\right)=M_{n}:=\sup U_{n}(Y) .Un(f)(yn)=Mn:=supUn(Y).
Continuing in this manner we obtain the sequences
(3.1) { θ , y 0 , y 1 , … , y n , … } ⊂ Y , and { M 0 , M 1 , … , M n , … } ⊂ R . (3.1) θ , y 0 , y 1 , … , y n , … ⊂ Y , and  M 0 , M 1 , … , M n , … ⊂ R .  {:[(3.1){theta,y_(0),y_(1),dots,y_(n),dots} sub Y", and "],[{M_(0),M_(1),dots,M_(n),dots} subR". "]:}\begin{align*} \left\{\theta, y_{0}, y_{1}, \ldots, y_{n}, \ldots\right\} & \subset Y \text {, and } \tag{3.1}\\ \left\{M_{0}, M_{1}, \ldots, M_{n}, \ldots\right\} & \subset \mathbb{R} \text {. } \end{align*}(3.1){θ,y0,y1,…,yn,…}⊂Y, and {M0,M1,…,Mn,…}⊂R. 
The following theorem contains the properties of these two sequences, if Y Y YYY is ( d , d ¯ d , d ¯ d, bar(d)d, \bar{d}d,d¯ )-sequentially compact.
Theorem 3.1. Let ( X , d X , d X,dX, dX,d ) be a T 1 T 1 T_(1)T_{1}T1 quasi-metric space, θ ∈ X θ ∈ X theta in X\theta \in Xθ∈X fixed, and Y Y YYY a ( d , d ¯ ) ( d , d ¯ ) (d, bar(d))(d, \bar{d})(d,d¯) sequentially compact subset of X X XXX with θ ∈ Y θ ∈ Y theta in Y\theta \in Yθ∈Y. Let f ∈ d − S f ∈ d − S f in d-Sf \in d-Sf∈d−S Lip 0 Y , q ≥ ‖ f | d 0 Y , q ≥ ‖ f d _(0)Y,q >= ||f|_(d)_{0} Y, q \geq \|\left. f\right|_{d}0Y,q≥‖f|d and let ( y n y n y_(n)y_{n}yn ) and ( M n M n M_(n)M_{n}Mn ) be the sequences in (3.1). Then
(a) ( M n ) M n (M_(n))\left(M_{n}\right)(Mn) converges to M f M f M_(f)M_{f}Mf;
(b) lim n → ∞ inf { d ( y n , y ) : y ∈ E f } = 0 lim n → ∞   inf d y n , y : y ∈ E f = 0 lim_(n rarr oo)i n f{d(y_(n),y):y inE_(f)}=0\lim _{n \rightarrow \infty} \inf \left\{d\left(y_{n}, y\right): y \in E_{f}\right\}=0limn→∞inf{d(yn,y):y∈Ef}=0.
Proof. (a). Since for every n ≥ 1 n ≥ 1 n >= 1n \geq 1n≥1
U n ( f ) ( y ) ≤ U n − 1 ( f ) ( y ) , y ∈ Y , U n ( f ) ( y ) ≤ U n − 1 ( f ) ( y ) , y ∈ Y , U_(n)(f)(y) <= U_(n-1)(f)(y),y in Y,U_{n}(f)(y) \leq U_{n-1}(f)(y), y \in Y,Un(f)(y)≤Un−1(f)(y),y∈Y,
it follows
M n = sup U n ( f ) ( Y ) ≤ sup U n − 1 ( f ) ( Y ) = M n − 1 M n = sup U n ( f ) ( Y ) ≤ sup U n − 1 ( f ) ( Y ) = M n − 1 M_(n)=s u pU_(n)(f)(Y) <= s u pU_(n-1)(f)(Y)=M_(n-1)M_{n}=\sup U_{n}(f)(Y) \leq \sup U_{n-1}(f)(Y)=M_{n-1}Mn=supUn(f)(Y)≤supUn−1(f)(Y)=Mn−1
Therefore, the sequence ( M n M n M_(n)M_{n}Mn ) is decreasing. Since U n ( f ) ( θ ) = 0 U n ( f ) ( θ ) = 0 U_(n)(f)(theta)=0U_{n}(f)(\theta)=0Un(f)(θ)=0 we have M n ≥ 0 M n ≥ 0 M_(n) >= 0M_{n} \geq 0Mn≥0 for all n ≥ 0 n ≥ 0 n >= 0n \geq 0n≥0. It follows that there exists M ≥ 0 M ≥ 0 M >= 0M \geq 0M≥0 such that
M = lim n → ∞ M n M = lim n → ∞   M n M=lim_(n rarr oo)M_(n)M=\lim _{n \rightarrow \infty} M_{n}M=limn→∞Mn
Since Y Y YYY is ( d , d ¯ ) ( d , d ¯ ) (d, bar(d))(d, \bar{d})(d,d¯)-sequentially compact, the sequence ( y n ) y n (y_(n))\left(y_{n}\right)(yn) contains a subsequence ( y n k ) k ≥ 1 y n k k ≥ 1 (y_(n_(k)))_(k >= 1)\left(y_{n_{k}}\right)_{k \geq 1}(ynk)k≥1 which is d d ddd - and d ¯ d ¯ bar(d)\bar{d}d¯-convergent to an element y ¯ ∈ Y y ¯ ∈ Y bar(y)in Y\bar{y} \in Yy¯∈Y, i.e.,
lim k → ∞ d ( y n k , y ¯ ) = 0 and lim k → ∞ d ( y ¯ , y n k ) = 0 lim k → ∞   d y n k , y ¯ = 0  and  lim k → ∞   d y ¯ , y n k = 0 lim_(k rarr oo)d(y_(n_(k)),( bar(y)))=0" and "lim_(k rarr oo)d(( bar(y)),y_(n_(k)))=0\lim _{k \rightarrow \infty} d\left(y_{n_{k}}, \bar{y}\right)=0 \text { and } \lim _{k \rightarrow \infty} d\left(\bar{y}, y_{n_{k}}\right)=0limk→∞d(ynk,y¯)=0 and limk→∞d(y¯,ynk)=0
Furthermore
lim k → ∞ U n k ( f ) ( y n k ) = lim k → ∞ M n k = M . lim k → ∞   U n k ( f ) y n k = lim k → ∞   M n k = M . lim_(k rarr oo)U_(n_(k))(f)(y_(n_(k)))=lim_(k rarr oo)M_(n_(k))=M.\lim _{k \rightarrow \infty} U_{n_{k}}(f)\left(y_{n_{k}}\right)=\lim _{k \rightarrow \infty} M_{n_{k}}=M .limk→∞Unk(f)(ynk)=limk→∞Mnk=M.
On the other hand, by the upper semicontinuity of the function f f fff, it follows
lim sup k → ∞ f ( y n k ) ≤ f ( y ¯ ) ≤ M f . lim sup k → ∞   f y n k ≤ f ( y ¯ ) ≤ M f . l i m   s u p_(k rarr oo)f(y_(n_(k))) <= f( bar(y)) <= M_(f).\limsup _{k \rightarrow \infty} f\left(y_{n_{k}}\right) \leq f(\bar{y}) \leq M_{f} .lim supk→∞f(ynk)≤f(y¯)≤Mf.
By the definitions of the extensions U n ( f ) ( n ≥ 1 ) U n ( f ) ( n ≥ 1 ) U_(n)(f)(n >= 1)U_{n}(f)(n \geq 1)Un(f)(n≥1) we have
U n k ( f ) ( y n k ) − U n k ( f ) ( y n k − 1 ) ≤ q d ( y n k , y n k − 1 ) ≤ q ( d ( y n k , y ¯ ) + d ( y ¯ , y n k − 1 ) ) → 0 , as k → ∞ . U n k ( f ) y n k − U n k ( f ) y n k − 1 ≤ q d y n k , y n k − 1 ≤ q d y n k , y ¯ + d y ¯ , y n k − 1 → 0 ,  as  k → ∞ . {:[U_(n_(k))(f)(y_(n_(k)))-U_(n_(k))(f)(y_(n_(k)-1)) <= qd(y_(n_(k)),y_(n_(k)-1))],[ <= q(d(y_(n_(k)),( bar(y)))+d(( bar(y)),y_(n_(k)-1)))rarr0","" as "k rarr oo.]:}\begin{aligned} U_{n_{k}}(f)\left(y_{n_{k}}\right)-U_{n_{k}}(f)\left(y_{n_{k}-1}\right) & \leq q d\left(y_{n_{k}}, y_{n_{k}-1}\right) \\ & \leq q\left(d\left(y_{n_{k}}, \bar{y}\right)+d\left(\bar{y}, y_{n_{k}-1}\right)\right) \rightarrow 0, \text { as } k \rightarrow \infty . \end{aligned}Unk(f)(ynk)−Unk(f)(ynk−1)≤qd(ynk,ynk−1)≤q(d(ynk,y¯)+d(y¯,ynk−1))→0, as k→∞.
It follows that for every ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 there exists k 0 ∈ N k 0 ∈ N k_(0)inNk_{0} \in \mathbb{N}k0∈N such that for all k ≥ k 0 k ≥ k 0 k >= k_(0)k \geq k_{0}k≥k0,
U n k ( f ) ( y n k ) − f ( y n k − 1 ) < ε , U n k ( f ) y n k − f y n k − 1 < ε , U_(n_(k))(f)(y_(n_(k)))-f(y_(n_(k)-1)) < epsi,U_{n_{k}}(f)\left(y_{n_{k}}\right)-f\left(y_{n_{k}-1}\right)<\varepsilon,Unk(f)(ynk)−f(ynk−1)<ε,
or equivalently,
U n k ( f ) ( y n k ) < f ( y n k − 1 ) + ε . U n k ( f ) y n k < f y n k − 1 + ε . U_(n_(k))(f)(y_(n_(k))) < f(y_(n_(k)-1))+epsi.U_{n_{k}}(f)\left(y_{n_{k}}\right)<f\left(y_{n_{k}-1}\right)+\varepsilon .Unk(f)(ynk)<f(ynk−1)+ε.
Taking lim sup as k → ∞ k → ∞ k rarr ook \rightarrow \inftyk→∞, we get
M ≤ lim k → ∞ sup f ( y n k − 1 ) + ε ≤ M f + ε . M ≤ lim k → ∞   sup f y n k − 1 + ε ≤ M f + ε . M <= lim_(k rarr oo)s u p f(y_(n_(k)-1))+epsi <= M_(f)+epsi.M \leq \lim _{k \rightarrow \infty} \sup f\left(y_{n_{k}-1}\right)+\varepsilon \leq M_{f}+\varepsilon .M≤limk→∞supf(ynk−1)+ε≤Mf+ε.
As ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 was arbitrarily chosen, we obtain M ≤ M f M ≤ M f M <= M_(f)M \leq M_{f}M≤Mf. Because the inequality M f ≤ M M f ≤ M M_(f) <= MM_{f} \leq MMf≤M is also true, it follows that (a) holds.
(b). For the proof of (b), supposing that the sequence
( inf { d ( y n , y ) : y ∈ E f } ) n ≥ 1 inf d y n , y : y ∈ E f n ≥ 1 (i n f{d(y_(n),y):y inE_(f)})_(n >= 1)\left(\inf \left\{d\left(y_{n}, y\right): y \in E_{f}\right\}\right)_{n \geq 1}(inf{d(yn,y):y∈Ef})n≥1
does not converge to 0 , then there exist ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 and an infinite sequence n 1 < n 2 < … n k < … n 1 < n 2 < … n k < … n_(1) < n_(2) < dotsn_(k) < dotsn_{1}<n_{2}< \ldots n_{k}<\ldotsn1<n2<…nk<… such that
inf { d ( y n k , y ) : y ∈ E f } ≥ ε , ∀ k ∈ N . inf d y n k , y : y ∈ E f ≥ ε , ∀ k ∈ N . i n f{d(y_(n_(k)),y):y inE_(f)} >= epsi,AA k inN.\inf \left\{d\left(y_{n_{k}}, y\right): y \in E_{f}\right\} \geq \varepsilon, \forall k \in \mathbb{N} .inf{d(ynk,y):y∈Ef}≥ε,∀k∈N.
By the ( d , d ¯ d , d ¯ d, bar(d)d, \bar{d}d,d¯ )-sequentially compactness of Y Y YYY, the sequence ( y n k ) k ≥ 1 y n k k ≥ 1 (y_(n_(k)))_(k >= 1)\left(y_{n_{k}}\right)_{k \geq 1}(ynk)k≥1 contains a subsequence ( y n k i ) i ≥ 1 y n k i i ≥ 1 (y_(n_(k_(i))))_(i >= 1)\left(y_{n_{k_{i}}}\right)_{i \geq 1}(ynki)i≥1 that converges to an element y ¯ ∈ Y y ¯ ∈ Y bar(y)in Y\bar{y} \in Yy¯∈Y such that f ( y ¯ ) = M f f ( y ¯ ) = M f f( bar(y))=M_(f)f(\bar{y})=M_{f}f(y¯)=Mf, i.e. y ¯ ∈ E f y ¯ ∈ E f bar(y)inE_(f)\bar{y} \in E_{f}y¯∈Ef, in contradiction to the inequality
inf { d ( y n k , y ) : y ∈ E f } ≥ ε . inf d y n k , y : y ∈ E f ≥ ε . i n f{d(y_(n_(k)),y):y inE_(f)} >= epsi.\inf \left\{d\left(y_{n_{k}}, y\right): y \in E_{f}\right\} \geq \varepsilon .inf{d(ynk,y):y∈Ef}≥ε.
The theorem is proved.
Remark 3.2. Let M ¯ n = max { f ( θ ) , f ( y 0 ) , f ( y 1 ) , … , f ( y n ) } M ¯ n = max f ( θ ) , f y 0 , f y 1 , … , f y n bar(M)_(n)=max{f(theta),f(y_(0)),f(y_(1)),dots,f(y_(n))}\bar{M}_{n}=\max \left\{f(\theta), f\left(y_{0}\right), f\left(y_{1}\right), \ldots, f\left(y_{n}\right)\right\}M¯n=max{f(θ),f(y0),f(y1),…,f(yn)}. Then M ¯ n ≤ M f ≤ M n M ¯ n ≤ M f ≤ M n bar(M)_(n) <= M_(f) <= M_(n)\bar{M}_{n} \leq M_{f} \leq M_{n}M¯n≤Mf≤Mn for every n = 1 , 2 , 3 , … n = 1 , 2 , 3 , … n=1,2,3,dotsn=1,2,3, \ldotsn=1,2,3,…. It follows that
M f − M ¯ n ≤ M n − M ¯ n , n = 1 , 2 , … M f − M ¯ n ≤ M n − M ¯ n , n = 1 , 2 , … M_(f)- bar(M)_(n) <= M_(n)- bar(M)_(n),n=1,2,dotsM_{f}-\bar{M}_{n} \leq M_{n}-\bar{M}_{n}, n=1,2, \ldotsMf−M¯n≤Mn−M¯n,n=1,2,…
The last inequality is a convenient upper bound for the error M f − M ¯ n M f − M ¯ n M_(f)- bar(M)_(n)M_{f}-\bar{M}_{n}Mf−M¯n.
Because
U n ( f ) ( y ) = inf z ∈ { θ , y 0 , y 1 , … , y n − 1 } = Z n . { f ( z ) + q d ( y , z ) } , y ∈ Y U n ( f ) ( y ) = inf z ∈ θ , y 0 , y 1 , … , y n − 1 = Z n .   { f ( z ) + q d ( y , z ) } , y ∈ Y U_(n)(f)(y)=i n f_(z in{theta,y_(0),y_(1),dots,y_(n-1)}=Z_(n).){f(z)+qd(y,z)},y in YU_{n}(f)(y)=\inf _{z \in\left\{\theta, y_{0}, y_{1}, \ldots, y_{n-1}\right\}=Z_{n} .}\{f(z)+q d(y, z)\}, y \in YUn(f)(y)=infz∈{θ,y0,y1,…,yn−1}=Zn.{f(z)+qd(y,z)},y∈Y
has a simple expression depending essentially on d ( y , z ) , z ∈ Z n d ( y , z ) , z ∈ Z n d(y,z),z inZ_(n)d(y, z), z \in Z_{n}d(y,z),z∈Zn and y ∈ Y y ∈ Y y in Yy \in Yy∈Y, it is easy - at least in principle - to compute the number
M n = max U n ( f ) ( Y ) M n = max U n ( f ) ( Y ) M_(n)=maxU_(n)(f)(Y)M_{n}=\max U_{n}(f)(Y)Mn=maxUn(f)(Y)
Also
0 ≤ U n + 1 ( f ) ( y n + 1 ) − U n + 1 ( f ) ( y n ) ≤ q d ( y n + 1 , y n ) 0 ≤ U n + 1 ( f ) y n + 1 − U n + 1 ( f ) y n ≤ q d y n + 1 , y n 0 <= U_(n+1)(f)(y_(n+1))-U_(n+1)(f)(y_(n)) <= qd(y_(n+1),y_(n))0 \leq U_{n+1}(f)\left(y_{n+1}\right)-U_{n+1}(f)\left(y_{n}\right) \leq q d\left(y_{n+1}, y_{n}\right)0≤Un+1(f)(yn+1)−Un+1(f)(yn)≤qd(yn+1,yn)
i.e.
0 ≤ M n + 1 − f ( y n ) ≤ q d ( y n + 1 , y n ) 0 ≤ M n + 1 − f y n ≤ q d y n + 1 , y n 0 <= M_(n+1)-f(y_(n)) <= qd(y_(n+1),y_(n))0 \leq M_{n+1}-f\left(y_{n}\right) \leq q d\left(y_{n+1}, y_{n}\right)0≤Mn+1−f(yn)≤qd(yn+1,yn)
and because M ¯ n + 1 ≥ f ( y n ) M ¯ n + 1 ≥ f y n bar(M)_(n+1) >= f(y_(n))\bar{M}_{n+1} \geq f\left(y_{n}\right)M¯n+1≥f(yn) it follows that
0 ≤ M n + 1 − M ¯ n + 1 ≤ M n + 1 − f ( y n ) ≤ q d ( y n + 1 , y n ) . 0 ≤ M n + 1 − M ¯ n + 1 ≤ M n + 1 − f y n ≤ q d y n + 1 , y n . 0 <= M_(n+1)- bar(M)_(n+1) <= M_(n+1)-f(y_(n)) <= qd(y_(n+1),y_(n)).0 \leq M_{n+1}-\bar{M}_{n+1} \leq M_{n+1}-f\left(y_{n}\right) \leq q d\left(y_{n+1}, y_{n}\right) .0≤Mn+1−M¯n+1≤Mn+1−f(yn)≤qd(yn+1,yn).
This means that
M n + 1 − M ¯ n + 1 = O ( d ( y n + 1 , y n ) ) M n + 1 − M ¯ n + 1 = O d y n + 1 , y n M_(n+1)- bar(M)_(n+1)=O(d(y_(n+1),y_(n)))M_{n+1}-\bar{M}_{n+1}=O\left(d\left(y_{n+1}, y_{n}\right)\right)Mn+1−M¯n+1=O(d(yn+1,yn))
and, consequently,
M f − M n = O ( d ( y n , y n − 1 ) ) M f − M n = O d y n , y n − 1 M_(f)-M_(n)=O(d(y_(n),y_(n-1)))M_{f}-M_{n}=O\left(d\left(y_{n}, y_{n-1}\right)\right)Mf−Mn=O(d(yn,yn−1))
Remark 3.3. A function f f fff belongs to d d ddd-SLip 0 Y 0 Y _(0)Y{ }_{0} Y0Y if and only if − f − f -f-f−f belongs to d ¯ d ¯ bar(d)\bar{d}d¯ S L i p 0 Y S L i p 0 Y SLip_(0)YS L i p_{0} YSLip0Y and for every f ∈ d − S L i p 0 Y , | | f | d = | | − f | d ¯ f ∈ d − S L i p 0 Y , | f | d = | − f | d ¯ f in d-SLip_(0)Y,||f|_(d)=||-f|_( bar(d)):}f \in d-S L i p_{0} Y,\left||f|_{d}=\left||-f|_{\bar{d}}\right.\right.f∈d−SLip0Y,||f|d=||−f|d¯ ([22], Corollary 1, page 59).
It follows that − f − f -f-f−f is upper semicontinuous on ( Y , d Y , d Y,dY, dY,d ) and attains its maximum on Y Y YYY, if Y Y YYY is d d ddd-sequentially compact (see Theorem 2.6).
By Theorem 2.1 and Remark 2.2 it follows that the maximal extension of − f − f -f-f−f in d ¯ − S L i p 0 Y d ¯ − S L i p 0 Y bar(d)-SLip_(0)Y\bar{d}-S L i p_{0} Yd¯−SLip0Y is
(3.2) F d ¯ ( − f ) ( x ) = inf { ( − f ) ( y ) + ‖ f | d d ¯ ( x , y ) } , x ∈ X (3.2) F d ¯ ( − f ) ( x ) = inf ( − f ) ( y ) + ‖ f d d ¯ ( x , y ) , x ∈ X {:(3.2)F_( bar(d))(-f)(x)=i n f{(-f)(y)+||f|_(d)( bar(d))(x,y)}","x in X:}\begin{equation*} F_{\bar{d}}(-f)(x)=\inf \left\{(-f)(y)+\|\left. f\right|_{d} \bar{d}(x, y)\right\}, x \in X \tag{3.2} \end{equation*}(3.2)Fd¯(−f)(x)=inf{(−f)(y)+‖f|dd¯(x,y)},x∈X
i.e.
( − f ) | Y = F d ¯ ( − f ) | Y and ‖ f | d = ‖ − f | d ¯ = ‖ F d ¯ ( − f ) | d ¯ ( − f ) Y = F d ¯ ( − f ) Y  and  f d = − f d ¯ = ‖ F d ¯ ( − f ) d ¯ (-f)|_(Y)=F_( bar(d))(-f)|_(Y)" and "||f|_(d)=||-f|_( bar(d))=||F_( bar(d))(-f)|_( bar(d))\left.(-f)\right|_{Y}=\left.F_{\bar{d}}(-f)\right|_{Y} \text { and }\left\|\left.f\right|_{d}=\right\|-\left.f\right|_{\bar{d}}=\|\left. F_{\bar{d}}(-f)\right|_{\bar{d}}(−f)|Y=Fd¯(−f)|Y and ‖f|d=‖−f|d¯=‖Fd¯(−f)|d¯
The algorithm described above may be applied for searching the global maximum of − f − f -f-f−f, i.e. the global minimum of f f fff, if the set Y Y YYY is ( d , d ¯ ) ( d , d ¯ ) (d, bar(d))(d, \bar{d})(d,d¯)-sequentially compact, and X X XXX is T 1 T 1 T_(1)T_{1}T1-separated.

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Costică Mustăţa
"Tiberiu Popoviciu" Institute of Numerical Analysis
P.O. Box. 68-1
400110 Cluj-Napoca
Romania
e-mail: cmustata2001@yahoo.com
Received: June 19, 2008.
Revised: September 29, 2008.
Accepted: March 5, 2009.
  1. This research has been supported by the Romanian Ministry of Education and Research under Grant 2-CEx-06-11-96/2006.
2009

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