Abstract
A theorem of Baire concerning the approximation of lower semicontinuous real valued functions defined on a metric space, by increasing sequences of continuous functions is extended to the “nonsymmetric” case, i.e. for quasi-metric spaces.
Authors
Costică Mustăţa
“Tiberiu Popoviciu” Institute of Numerical Analysis , Cluj-Napoca, Romania
Keywords
Quasi-metric space, semi-Lipschitz function, approximation.
Paper coordinates
C. Mustăţa, On a theorem of Baire about lower semi-continuous functions, Rev. Anal. Numer. Theor. Approx. 37 (2008) no. 1, 71-75., https://ictp.acad.ro/jnaat/journal/article/view/2008-vol37-no1-art8
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Revue d’Analyse Numer Theorie Approximation
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Publisher House of the Romanian Academy
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[1] Baire, R.,Lecon sur les fonctions discontinues,Paris, Collection Borel, 1905, pp. 121–123.
[2] Borodin, P.A.,The Banach-Mazur theorem for spaces with asymmetric norm and itsapplications in convex analysis, Mat. Zametki,69, no. 3, pp. 329–337, 2001.
[3] Collins, J.and Zimmer, J.An asymmetric Arzela-Ascoli theorem,Topology Appl.,154, no. 11, pp. 2312–2322, 2007.
[4] K unzi, H.P.A., Nonsymmetric distances and their associated topologies: About theorigin of basic ideas in the area of asymmetric topology, in: Handbook of the Historyof General Topology, edited. by C.E. Aull and R. Lower, vol.3, Kluwer Acad. Publ.,Dordrecht, pp. 853–968, 2001.
[5] McShane, E.T., Extension of range of functions, Bull. Amer. Math. Soc.,40, pp. 837–842, 1934.
[6] Menucci, A., On asymmetric distances,Technical Report, Scuola Normale Superiore,Pisa, 2004.
[7] Mustata, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev.Anal. Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.
[8] Nicolescu, M., Mathematical Analysis, Vol. II, Editura Tehnica, Bucharest, p. 119,1958 (in Romanian).
[9] Precupanu, A., Mathematical Analysis: Measure and Integration, Vol. I., EdituraUniversitatii A.I. Cuza Iasi, 2006 (Romanian).
[10] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best-approximation inquasi-metric spaces, J. Approx. Theory,103, pp. 292–301, 2000.
[11] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitzfunctions. Acta Math. Hungar.,108, no. 1-2, pp. 55–70, 2005.
[2] Borodin, P.A.,The Banach-Mazur theorem for spaces with asymmetric norm and itsapplications in convex analysis, Mat. Zametki,69, no. 3, pp. 329–337, 2001.
[3] Collins, J.and Zimmer, J.An asymmetric Arzela-Ascoli theorem,Topology Appl.,154, no. 11, pp. 2312–2322, 2007.
[4] K unzi, H.P.A., Nonsymmetric distances and their associated topologies: About theorigin of basic ideas in the area of asymmetric topology, in: Handbook of the Historyof General Topology, edited. by C.E. Aull and R. Lower, vol.3, Kluwer Acad. Publ.,Dordrecht, pp. 853–968, 2001.
[5] McShane, E.T., Extension of range of functions, Bull. Amer. Math. Soc.,40, pp. 837–842, 1934.
[6] Menucci, A., On asymmetric distances,Technical Report, Scuola Normale Superiore,Pisa, 2004.
[7] Mustata, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev.Anal. Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.
[8] Nicolescu, M., Mathematical Analysis, Vol. II, Editura Tehnica, Bucharest, p. 119,1958 (in Romanian).
[9] Precupanu, A., Mathematical Analysis: Measure and Integration, Vol. I., EdituraUniversitatii A.I. Cuza Iasi, 2006 (Romanian).
[10] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best-approximation inquasi-metric spaces, J. Approx. Theory,103, pp. 292–301, 2000.
[11] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitzfunctions. Acta Math. Hungar.,108, no. 1-2, pp. 55–70, 2005.
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ON A THEOREM OF BAIRE ABOUT LOWER SEMICONTINUOUS FUNCTIONS*
Abstract
A theorem of Baire concerning the approximation of lower semicontinuous real valued functions defined on a metric space, by increasing sequences of continuous functions is extended to the "nonsymmetric" case, i.e. for quasimetric spaces.
MSC 2000. 41A30, 26A15, 26A16, 26A21, 54E25.
Keywords. Quasi-metric space, semi-Lipschitz function, approximation.
Keywords. Quasi-metric space, semi-Lipschitz function, approximation.
1. PRELIMINARIES
In the last years there have been an increasing interest for the study of quasimetric spaces (spaces with asymmetric metric) motivated by their applications in various branches of mathematics, and especially in computer science. A direction of investigation is to study the possibility to extend to quasi-metric spaces known results in metric spaces (see, for example, [2-[7]).
The following classical result of Baire is well known [8, 9. Every lower semicontinuous real valued function defined on a metric space is the pointwise limit of an increasing sequence of continuous functions. Analyzing the proof of this result (see [9], Th. 2.2-23, p. 84), observe that every element of the increasing sequence is a Lipschitz function. This fact suggest to use the semiLipschitz functions defined in [10, 11, to obtain such a theorem for lower semi-continuous real valued functions defined on a quasi-metric space.
This short Note presents some notions connected with quasi-metric spaces and the result of Baire in this framework.
Let XX be a non-empty set. A function d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty) is called a quasi-metric on XX ([10]) if the following conditions hold:
{:Q_(1))d(x,y)=d(y,x)=0\left.Q_{1}\right) d(x, y)=d(y, x)=0 iff x=y;x=y ;
{:Q_(2))d(x,z) <= d(x,y)+d(y,z)\left.Q_{2}\right) d(x, z) \leq d(x, y)+d(y, z), for all x,y,z in Xx, y, z \in X.
The function bar(d):X xx X rarr[0,oo)\bar{d}: X \times X \rightarrow[0, \infty) defined by bar(d)(x,y)=d(y,x)\bar{d}(x, y)=d(y, x), for all x,y in Xx, y \in X is also a quasi-metric on XX, called the conjugate quasi-metric of dd.
{:Q_(1))d(x,y)=d(y,x)=0\left.Q_{1}\right) d(x, y)=d(y, x)=0 iff x=y;x=y ;
{:Q_(2))d(x,z) <= d(x,y)+d(y,z)\left.Q_{2}\right) d(x, z) \leq d(x, y)+d(y, z), for all x,y,z in Xx, y, z \in X.
The function bar(d):X xx X rarr[0,oo)\bar{d}: X \times X \rightarrow[0, \infty) defined by bar(d)(x,y)=d(y,x)\bar{d}(x, y)=d(y, x), for all x,y in Xx, y \in X is also a quasi-metric on XX, called the conjugate quasi-metric of dd.
The function d^(s)(x,y)=max{d(x,y), bar(d)(x,y)}d^{s}(x, y)=\max \{d(x, y), \bar{d}(x, y)\} is a metric on XX. If dd can take the value +oo+\infty, then it is called a quasi-distance on XX.
Each quasi-metric dd on XX induces a topology tau(d)\tau(d) which has as a basis the family of open balls (called forward open balls in [6]):
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 .
This topology is called the forward topology of XX ([3, [6]) and is denoted also by tau^(+)\tau^{+}. Observe that the topology tau^(+)\tau^{+}is a T_(0)T_{0}-topology [10]. If the conditions Q_(1)Q_{1} is replaced by Q_(1)^('):d(x,y)=0Q_{1}^{\prime}: d(x, y)=0 iff x=yx=y, then tau^(+)\tau^{+}is a T_(1)T_{1}-topology ( 10 , [11]).
The pair ( X,dX, d ) is called a quasi-metric space ( T_(0)T_{0}-separated, respectively T_(1)T_{1}-separated). A sequence (x_(n))_(n >= 1)\left(x_{n}\right)_{n \geq 1} in the quasi-metric space ( X,dX, d ) is called tau^(+)\tau^{+}-convergent to x_(0)in Xx_{0} \in X iff lim_(n rarr oo)d(x_(0),x_(n))=0\lim _{n \rightarrow \infty} d\left(x_{0}, x_{n}\right)=0.
Similarly, the topology tau( bar(d))\tau(\bar{d}) has as a basis the family of open balls:
B^(-)(x,epsi):{y in X:d(y,x) < epsi},x in X,epsi > 0B^{-}(x, \varepsilon):\{y \in X: d(y, x)<\varepsilon\}, x \in X, \varepsilon>0
This topology is denoted also by tau^(-)\tau^{-}.
Definition 1. ([10]). Let ( X,dX, d ) be a quasi-metric space. A function ff : X rarrRX \rightarrow \mathbb{R} is called dd-semi-Lipschitz if there exists a number L >= 0L \geq 0 (named a dd-semi-Lipschitz constant for ff ) such that
Definition 1. ([10]). Let ( X,dX, d ) be a quasi-metric space. A function ff : X rarrRX \rightarrow \mathbb{R} is called dd-semi-Lipschitz if there exists a number L >= 0L \geq 0 (named a dd-semi-Lipschitz constant for ff ) such that
{:(1)f(x)-f(y) <= Ld(x","y):}\begin{equation*}
f(x)-f(y) \leq L d(x, y) \tag{1}
\end{equation*}
for all x,y in Xx, y \in X.
A similar definition can be given for bar(d)\bar{d}-semi-Lipschitz functions.
Definition 2. The function f:X rarrRf: X \rightarrow \mathbb{R} is called <= _(d)\leq_{d}-increasing if f(x) <= f(y)f(x) \leq f(y), whenever d(x,y)=0d(x, y)=0.
A similar definition can be given for bar(d)\bar{d}-semi-Lipschitz functions.
Definition 2. The function f:X rarrRf: X \rightarrow \mathbb{R} is called <= _(d)\leq_{d}-increasing if f(x) <= f(y)f(x) \leq f(y), whenever d(x,y)=0d(x, y)=0.
One denotes by R_( <= _(d))^(X)R_{\leq_{d}}^{X} the set of all <= _(d)\leq_{d}-increasing functions on XX. The set R_( <= _(d))^(X)\mathbb{R}_{\leq_{d}}^{X} is a cone in the linear space R^(X)\mathbb{R}^{X} of real valued functions defined on XX [11].
For a dd-semi-Lipschitz function f:X rarrRf: X \rightarrow \mathbb{R}, put
{:(2)||f|_(d)=s u p{((f(x)-f(y))vv0)/(d(x,y)):d(x,y) > 0,x,y in X}:}\begin{equation*}
\|\left. f\right|_{d}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: d(x, y)>0, x, y \in X\right\} \tag{2}
\end{equation*}
Then ||f|_(d)\|\left. f\right|_{d} is the smallest dd-semi-Lipschitz constant of ff (see [7], [10, 11]).
Denote also
Denote also
{:(3)d"-SlipX ":={f inR_( <= d)^(X):||f|_(d) < oo}:}\begin{equation*}
d \text {-SlipX }:=\left\{f \in \mathbb{R}_{\leq d}^{X}: \|\left. f\right|_{d}<\infty\right\} \tag{3}
\end{equation*}
the subcone of the cone R_( <= _(d))^(X)\mathbb{R}_{\leq_{d}}^{X}, of all dd-semi-Lipschitz real valued functions on (X,d)(X, d). If theta in X\theta \in X is a fixed, but arbitrary element, denote
d-Slip_(0)X:={f in d-SlipX:f(theta)=0}.d-\operatorname{Slip}_{0} \mathrm{X}:=\{f \in d-\operatorname{Slip} \mathrm{X}: f(\theta)=0\} .
Then the functional ||*|_(d):d-\|\left.\cdot\right|_{d}: d- Slip _(0)Xrarr[0,oo)_{0} \mathrm{X} \rightarrow[0, \infty) is an asymmetric norm on dd-Slip _(0)X{ }_{0} \mathrm{X}, i.e. this functional is subadditive, positively homogeneous and ||f|_(d)=\|\left. f\right|_{d}=
0 iff f=0f=0. The pair ( dd-Slip _(0)X,||*|_(d){ }_{0} \mathrm{X}, \|\left.\cdot\right|_{d} ) is called the "normed cone" of dd-semiLipschitz real valued functions (vanishing at theta\theta ). The properties of this normed cone are studied in [10, 11.
Definition 3. Let ( X,dX, d ) be a quasi-metric space and f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}}, where bar(R)=[-oo,+oo]\overline{\mathbb{R}}=[-\infty,+\infty] is equipped with the natural topology. The function ff is called tau^(+)\tau^{+}-lower semicontinuous (respectively tau^(+)\tau^{+}-upper semicontinuous) ( tau^(+)\tau^{+}-l.s.c, respectively tau^(+)\tau^{+}-u.s.c., in short) at the point x_(0)in Xx_{0} \in X, if for every epsi > 0\varepsilon>0 there exists r > 0r>0 such that for all x inB^(+)(x_(0),r),f(x) > f(x_(0))-epsix \in B^{+}\left(x_{0}, r\right), f(x)>f\left(x_{0}\right)-\varepsilon (respectively {:f(x) < f(x_(0))+epsi))\left.\left.f(x)<f\left(x_{0}\right)+\varepsilon\right)\right).
Similar definitions can be given for tau^(-)\tau^{-}-l.s.c. and tau^(-)\tau^{-}-u.s.c real valued functions on (X, bar(d))(X, \bar{d}).
Observe that f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}} is tau^(+)\tau^{+}-l.s.c iff -f-f is tau^(-)\tau^{-}-u.s.c and ff is tau^(+)\tau^{+}-u.s.c iff -f-f is tau^(-)\tau^{-}-l.s.c.
The result of Baire in this framework is:
Theorem 4. (Baire). Let ( X,dX, d ) be a quasi-metric space and f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}} be a tau^(+)\tau^{+}- l.s.c function on XX. Then there exists a sequence (F_(n))_(n >= 1),F_(n)in d\left(F_{n}\right)_{n \geq 1}, F_{n} \in d-SlipX such that (F_(n)(x))_(n >= 1),x in X\left(F_{n}(x)\right)_{n \geq 1}, x \in X, is increasing and lim_(n rarr oo)F_(n)(x)=f(x),x in X\lim _{n \rightarrow \infty} F_{n}(x)=f(x), x \in X.
Theorem 4. (Baire). Let ( X,dX, d ) be a quasi-metric space and f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}} be a tau^(+)\tau^{+}- l.s.c function on XX. Then there exists a sequence (F_(n))_(n >= 1),F_(n)in d\left(F_{n}\right)_{n \geq 1}, F_{n} \in d-SlipX such that (F_(n)(x))_(n >= 1),x in X\left(F_{n}(x)\right)_{n \geq 1}, x \in X, is increasing and lim_(n rarr oo)F_(n)(x)=f(x),x in X\lim _{n \rightarrow \infty} F_{n}(x)=f(x), x \in X.
Proof. a) Suppose firstly that f(x) >= 0f(x) \geq 0, for all x in Xx \in X. For x in Xx \in X and n inNn \in \mathbb{N}, let
{:(4)F_(n)(x)=i n f{f(z)+n*d(x","z):z in X}.:}\begin{equation*}
F_{n}(x)=\inf \{f(z)+n \cdot d(x, z): z \in X\} . \tag{4}
\end{equation*}
Obviously that
{:(5)0 <= F_(n)(x) <= f(x)+nd(x","x)=f(x).:}\begin{equation*}
0 \leq F_{n}(x) \leq f(x)+n d(x, x)=f(x) . \tag{5}
\end{equation*}
If x,y,z in Xx, y, z \in X, then
{:[F_(n)(x) <= f(z)+nd(x","z) <= f(z)+nd(x","y)+nd(y","z)],[=(f(z)+nd(y","z))+nd(x","y).]:}\begin{aligned}
F_{n}(x) & \leq f(z)+n d(x, z) \leq f(z)+n d(x, y)+n d(y, z) \\
& =(f(z)+n d(y, z))+n d(x, y) .
\end{aligned}
Taking the infimum with respect to zz, one obtains
F_(n)(x) <= F_(n)(y)+nd(x,y),F_{n}(x) \leq F_{n}(y)+n d(x, y),
i.e.
F_(n)(x)-F_(n)(y) <= n*d(x,y),F_{n}(x)-F_{n}(y) \leq n \cdot d(x, y),
for all x,y in Xx, y \in X. This means that ||F_(n)|_(d) <= n\|\left. F_{n}\right|_{d} \leq n and F_(n)in dF_{n} \in d-SlipX, for every n=1,2,3,dotsn=1,2,3, \ldots
If n <= mn \leq m, by the definition (4) it follows F_(n)(x) <= F_(m)(x),x in XF_{n}(x) \leq F_{m}(x), x \in X, so that the sequence (F_(n)(x))_(n >= 1)\left(F_{n}(x)\right)_{n \geq 1} is increasing and bounded by f(x),x in Xf(x), x \in X. Consequently there exists the limit lim_(n rarr oo)F_(n)(x)=h(x)\lim _{n \rightarrow \infty} F_{n}(x)=h(x), and h(x) <= f(x),x in Xh(x) \leq f(x), x \in X.
In fact h(x)=f(x)h(x)=f(x), for every x in Xx \in X. Indeed, let n inNn \in \mathbb{N} and x in Xx \in X. By definition (4) of F_(n)(x)F_{n}(x), for every epsi > 0\varepsilon>0, there exists z_(n)in Xz_{n} \in X such that
F_(n)(x)+epsi > f(z_(n))+nd(x,z_(n)) >= nd(x,z_(n)),F_{n}(x)+\varepsilon>f\left(z_{n}\right)+n d\left(x, z_{n}\right) \geq n d\left(x, z_{n}\right),
and because F_(n)(x)+epsi <= f(x)+epsiF_{n}(x)+\varepsilon \leq f(x)+\varepsilon, it follows
f(x)+epsi >= nd(x,z_(n))f(x)+\varepsilon \geq n d\left(x, z_{n}\right)
and then
d(x,z_(n)) <= (1)/(n)(f(x)+epsi)d\left(x, z_{n}\right) \leq \frac{1}{n}(f(x)+\varepsilon)
For n rarr oon \rightarrow \infty it follows that the sequence (z_(n))_(n >= 1)\left(z_{n}\right)_{n \geq 1} is tau^(+)\tau^{+}-convergent to xx, and because ff is supposed tau^(+)\tau^{+}-l.s.c.,
lim_(n rarr oo)i n f f(z_(n)) >= f(x)\lim _{n \rightarrow \infty} \inf f\left(z_{n}\right) \geq f(x)
(see [9], p. 127).
Consequently, for every epsi > 0\varepsilon>0 there exists n_(0)inNn_{0} \in \mathbb{N} such that, for every n >= n_(0)n \geq n_{0},
Consequently, for every epsi > 0\varepsilon>0 there exists n_(0)inNn_{0} \in \mathbb{N} such that, for every n >= n_(0)n \geq n_{0},
{:(6)f(z_(n)) >= f(x)-epsi.:}\begin{equation*}
f\left(z_{n}\right) \geq f(x)-\varepsilon . \tag{6}
\end{equation*}
By (4) and (6) it follows
F_(n)(x) > f(z_(n))-epsi+nd(x,z_(n)) >= f(z_(n))-epsi >= f(x)-2epsiF_{n}(x)>f\left(z_{n}\right)-\varepsilon+n d\left(x, z_{n}\right) \geq f\left(z_{n}\right)-\varepsilon \geq f(x)-2 \varepsilon
for n > n_(0)n>n_{0}. By (5) one obtains
0 <= F_(n)(x)uarr f(x)0 \leq F_{n}(x) \uparrow f(x)
for all x in Xx \in X.
b) Now, let ff be a bounded and tau^(+)\tau^{+}-l.s.c function on XX. Then there exists M > 0M>0 such that |f(x)| <= M|f(x)| \leq M, for all x in Xx \in X. Denoting g(x)=f(x)+M,x in Xg(x)=f(x)+M, x \in X, one obtains g(x) >= 0g(x) \geq 0, and gg is tau^(+)\tau^{+}-l.s.c., on XX.
b) Now, let ff be a bounded and tau^(+)\tau^{+}-l.s.c function on XX. Then there exists M > 0M>0 such that |f(x)| <= M|f(x)| \leq M, for all x in Xx \in X. Denoting g(x)=f(x)+M,x in Xg(x)=f(x)+M, x \in X, one obtains g(x) >= 0g(x) \geq 0, and gg is tau^(+)\tau^{+}-l.s.c., on XX.
From the part a), there exists the sequence (G_(n))_(n >= 1),G_(n)in d\left(G_{n}\right)_{n \geq 1}, G_{n} \in d-SlipX, n=1,2,3,dotsn= 1,2,3, \ldots such that 0 <= G_(n)(x)uarr g(x)=f(x)+M0 \leq G_{n}(x) \uparrow g(x)=f(x)+M. This means that the sequence (F_(n))_(n >= 1),F_(n)=G_(n)-M,n=1,2,3dots\left(F_{n}\right)_{n \geq 1}, F_{n}=G_{n}-M, n=1,2,3 \ldots is increasing and converges in every point xx of XX to f(x)f(x). Moreover |F_(n)(x)| <= M,x in X,n=1,2,3,dots\left|F_{n}(x)\right| \leq M, x \in X, n=1,2,3, \ldots.
c) Now, we consider the general case, i.e., f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}} is an arbitrary tau^(+)\tau^{+}-l.s.c. function.
c) Now, we consider the general case, i.e., f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}} is an arbitrary tau^(+)\tau^{+}-l.s.c. function.
Using the Baire function (see [1]) varphi:[-oo,+oo]rarr[-1,1]\varphi:[-\infty,+\infty] \rightarrow[-1,1],
varphi(x)={[-1","" for "x=-oo],[(x)/(1+|x|)","" for "-oo < x < oo],[1","" for "x=+oo]:}\varphi(x)=\left\{\begin{array}{l}
-1, \text { for } x=-\infty \\
\frac{x}{1+|x|}, \text { for }-\infty<x<\infty \\
1, \text { for } x=+\infty
\end{array}\right.
which is a Lipschitz increasing isomorphism, it follows that varphi@f: bar(R)rarr[-1,1]\varphi \circ f: \overline{\mathbb{R}} \rightarrow[-1,1] is bounded and tau^(+)-\tau^{+}-l.s.c. on XX.
By the previous point b), there exists a sequence (H_(n))_(n >= 1)\left(H_{n}\right)_{n \geq 1} with H_(n)in dH_{n} \in d-SlipX such that
H_(n)(x)uarr(varphi@f)(x),x in X.H_{n}(x) \uparrow(\varphi \circ f)(x), x \in X .
Consequently, the sequence (F_(n))_(n >= 1),F_(n)(x)=(varphi^(-1)@H_(n))(x),x in X\left(F_{n}\right)_{n \geq 1}, F_{n}(x)=\left(\varphi^{-1} \circ H_{n}\right)(x), x \in X, is increasing and lim_(n rarr oo)F_(n)(x)=f(x),x in X\lim _{n \rightarrow \infty} F_{n}(x)=f(x), x \in X.
For tau^(+)\tau^{+}-u.s.c functions on XX, one obtains:
Theorem 5. Let ( X,dX, d ) be a quasi-metric space and f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}} a tau^(+)\tau^{+}-u.s.c. function. Then there exists a sequence (G_(n))_(n >= 1),G_(n)in d\left(G_{n}\right)_{n \geq 1}, G_{n} \in d-SlipX, n=1,2,3,dotsn=1,2,3, \ldots such that (G_(n)(x))_(n >= 1),x in X\left(G_{n}(x)\right)_{n \geq 1}, x \in X, is monotonically decreasing and lim_(n rarr oo)G_(n)(x)=f(x),x in X\lim _{n \rightarrow \infty} G_{n}(x)= f(x), x \in X.
Theorem 5. Let ( X,dX, d ) be a quasi-metric space and f:X rarr bar(R)f: X \rightarrow \overline{\mathbb{R}} a tau^(+)\tau^{+}-u.s.c. function. Then there exists a sequence (G_(n))_(n >= 1),G_(n)in d\left(G_{n}\right)_{n \geq 1}, G_{n} \in d-SlipX, n=1,2,3,dotsn=1,2,3, \ldots such that (G_(n)(x))_(n >= 1),x in X\left(G_{n}(x)\right)_{n \geq 1}, x \in X, is monotonically decreasing and lim_(n rarr oo)G_(n)(x)=f(x),x in X\lim _{n \rightarrow \infty} G_{n}(x)= f(x), x \in X.
Proof. If ff is tau^(+)\tau^{+}-u.s.c., then -f-f is tau^(-)\tau^{-}-l.s.c on XX. By Theorem 1, there exists a sequence (F_(n))_(n >= 1)\left(F_{n}\right)_{n \geq 1} in bar(d)\bar{d}-SlipX, monotonically increasing and pointwise convergent to -f-f. Then the sequence G_(n)=-F_(n),n=1,2,dotsG_{n}=-F_{n}, n=1,2, \ldots, has the following properties: G_(n)=-F_(n)in dG_{n}=-F_{n} \in d-SlipX, and G_(n)(x)darr f(x),lim_(n rarr oo)G_(n)=f(x),x in XG_{n}(x) \downarrow f(x), \lim _{n \rightarrow \infty} G_{n}=f(x), x \in X.
REFERENCES
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[3] Collins, J. and Zimmer, J. An asymmetric Arzèla-Ascoli theorem, Topology Appl., 154, no. 11, pp. 2312-2322, 2007.
[4] KünZI, H.P.A., Nonsymmetric distances and their associated topologies: About the origin of basic ideas in the area of asymmetric topology, in: Handbook of the History of General Topology, edited. by C.E. Aull and R. Lower, vol. 3, Kluwer Acad. Publ., Dordrecht, pp. 853-968, 2001.
[5] McShane, E.T., Extension of range of fucntions, Bull. Amer. Math. Soc., 40, pp. 837842, 1934.
[6] Menucci, A., On asymmetric distances, Technical Report, Scuola Normale Superiore, Pisa, 2004.
[7] Mustăţa, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numér. Théor. Approx., 30, no. 1, pp. 61-67, 2001. ©
[8] Nicolescu, M., Mathematical Analysis, Vol. II, Editura Tehnică, Bucharest, p. 119, 1958 (in Romanian).
[9] Precupanu, A., Mathematical Analysis: Measure and Integration, Vol. I., Editura Universităţii A.I. Cuza Iaşi, 2006 (Romanian).
[10] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best-approximation in quasi-metric spaces, J. Approx. Theory, 103, pp. 292-301, 2000.
[11] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitz functions. Acta Math. Hungar., 108, no. 1-2, pp. 55-70, 2005.
[2] Borodin, P.A., The Banach-Mazur theorem for spaces with asymmetric norm and its applications in convex analysis, Mat. Zametki, 69, no. 3, pp. 329-337, 2001.
[3] Collins, J. and Zimmer, J. An asymmetric Arzèla-Ascoli theorem, Topology Appl., 154, no. 11, pp. 2312-2322, 2007.
[4] KünZI, H.P.A., Nonsymmetric distances and their associated topologies: About the origin of basic ideas in the area of asymmetric topology, in: Handbook of the History of General Topology, edited. by C.E. Aull and R. Lower, vol. 3, Kluwer Acad. Publ., Dordrecht, pp. 853-968, 2001.
[5] McShane, E.T., Extension of range of fucntions, Bull. Amer. Math. Soc., 40, pp. 837842, 1934.
[6] Menucci, A., On asymmetric distances, Technical Report, Scuola Normale Superiore, Pisa, 2004.
[7] Mustăţa, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numér. Théor. Approx., 30, no. 1, pp. 61-67, 2001. ©
[8] Nicolescu, M., Mathematical Analysis, Vol. II, Editura Tehnică, Bucharest, p. 119, 1958 (in Romanian).
[9] Precupanu, A., Mathematical Analysis: Measure and Integration, Vol. I., Editura Universităţii A.I. Cuza Iaşi, 2006 (Romanian).
[10] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best-approximation in quasi-metric spaces, J. Approx. Theory, 103, pp. 292-301, 2000.
[11] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitz functions. Acta Math. Hungar., 108, no. 1-2, pp. 55-70, 2005.
Received by the editors: March 27, 2008.
- *This work was supported by MEdC under grant 2CEEX06-11-96/19.09.2006.
^(†){ }^{\dagger} "T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro, cmustata2001@yahoo.com.