New results in $G$-best approximation in $G$-metric spaces
The purpose of this paper is to introduce and discuss the concepts of G-best approximation and $a_0$ -orthogonality in the theory of $G$-metric spaces. We consider the relationship between these concepts and the dual $X$ and obtain some results on subsets of $G$-metric spaces similar to normed space...
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| author | Dehghan, Nezhad A. Mazaheri, H. Дегхан, Нежад А. Мазахері, Х. |
| author_facet | Dehghan, Nezhad A. Mazaheri, H. Дегхан, Нежад А. Мазахері, Х. |
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| description | The purpose of this paper is to introduce and discuss the concepts of G-best approximation and $a_0$ -orthogonality in the theory of $G$-metric spaces. We consider the relationship between these concepts and the dual $X$ and obtain some results on subsets of $G$-metric spaces similar to normed spaces. |
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K O R O T K I P O V I D O M L E N N Q
UDC 517.5
A. Dehghan Nezhad, H. Mazaheri (Yazd Univ., Iran)
NEW RESULTS IN G-BEST APPROXIMATION
IN G-METRIC SPACES
NOVI REZUL|TATY WODO G-NAJKRAWOHO
NABLYÛENNQ V G-METRYÇNYX PROSTORAX
The purpose of this paper is to introduce and discuss the concepts of G-best approximation and a0 -or-
thogonality in G-metric spaces theory. We consider the relation between these concepts and the dual
X, and obtain some results on the subsets of G-metric spaces similar to normed spaces.
Meta roboty — vvesty ta obhovoryty ponqttq G-najkrawoho nablyΩennq ta a0-ortohonal\-
nosti v teori] G-metryçnyx prostoriv. Rozhlqnuto spivvidnoßennq miΩ cymy ponqttqmy ta du-
al\nym X , otrymano deqki rezul\taty wodo pidmnoΩyn G-metryçnyx prostoriv, wo podibni do
normovanyx prostoriv.
1. Introduction. In 2005, Zead Mustafa and Brailey Sims introduced a new structure
of generalized metric spaces (see [1]), which are called G-metric spaces as gene-
ralization of metric space ( , )X d to develop and introduce a new fixed point theory
for a various mappings in this new structure. In this section, we give a brief introducti-
on of G-metric, G-continuous and G-contraction. First, we recall some basic notati-
ons of G-metric theory.
Definition 1 [1]. Let X be a nonempty set and G X X X: [ , )× × → ∞0 a
function satisfying the following properties:
( )G1 G x y z( , , ) = 0 if x = y = z,
( )G2 0 < G x x y( , , ) for all x, y ∈ X with x ≠ y,
( )G3 G x x y( , , ) ≤ G x y z( , , ) for all x, y, z ∈ X with z ≠ y,
( )G4 G x y z( , , ) = G x z y( , , ) = G y z x( , , ) = … (symmetry in all three vari-
ables) and
( )G5 G x y z( , , ) ≤ G x a a G a y z( , , ) ( , , )+ for all x , y , z, a ∈ X (rectangle
inequality).
Then the function G is called a generalized metric or more specifically, a G -met-
ric on X and the pair ( , )X G is called a G-metric space.
Clearly these properties are satisfied when G x y z( , , ) is the perimeter of the tri-
angle with vertices at x, y and z in R2
; moreover, taking a in the interior of the
triangle shows that ( )G5 is the best possible.
If ( , )X d is an ordinary metric space, then ( , )X d can define G-metrics on X by
© A. DEHGHAN NEZHAD, H. MAZAHERI, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4 567
568 A. DEHGHAN NEZHAD, H. MAZAHERI
( )Gs G d x y zs ( ) ( , , ) = d x y d y z d x z( , ) ( , ) ( , )+ + and
( )Gm G d x y zm ( ) ( , , ) = max ( , ), ( , ), ( , )d x y d y z d x z{ } .
Proposition 1 [1]. Let ( , )X G be a G -metric space. Then for any x, y, z
and a X∈ it follows that
(1) if G x y z( , , ) = 0, then x = y = z,
(2) G x y z( , , ) ≤ G x x y G x x z( , , ) ( , , )+ ,
(3) G x y y( , , ) ≤ 2G y x x( , , ) ,
(4) G x y z( , , ) ≤ G x a z G a y z( , , ) ( , , )+ ,
(5) G x y z( , , ) ≤ ( / ) ( , , ) ( , , ) ( , , )2 3 G x y a G x a z G a y z+ +( ) ,
(6) G x y z( , , ) ≤ G x a a G y a a G z a a( , , ) ( , , ) ( , , )+ +( ) .
Proposition 2 [1]. Every G -metric space ( , )X G will define a metric space
( , )X dG by
d x yG ( , ) = G x y y G y x x( , , ) ( , , )+ for all x, y ∈ X .
Definition 2 [1]. Let ( , )X G be a G-metric space. Then for x X0 ∈ and
r > 0 the G -ball with center x0 and radius r is B x rG ( , )0 = y X∈{ :
G x y y r( , , )0 < } .
Proposition 3 [1]. Let ( , )X G be a G -metric space. Then for any x X0 ∈
and r > 0 we have
(i) if G x x y( , , )0 < r, then x, y ∈ B x rG ( , )0 ,
(ii) if y ∈ B x rG ( , )0 , then there exists a δ > 0 such that B yG ( , )δ ⊆
⊆ B x rG ( , )0 .
Definition 3 [1]. Let ( , )X G1 1 and ( , )X G2 2 be two G -metric spaces and
f : ( , ) ( , )X G X G1 1 2 2→ a function, then f is said to be G -continuous at a point
a X∈ 1 if and only if for given ε > 0 , there exists δ > 0 such that x, y ∈ X1 and
G a x y1( , , ) < δ implies G f a f x f y2 ( ( ), ( ), ( )) < ε .
A function f is G-continuous at X1 if and only if it is G-continuous at all
a X∈ 1.
Definition 4 [1]. Let ( , )X G be a G-metric space, ( )xn a sequence of points
of X . A point x X∈ is said to be the limit of the sequence ( )xn i f
lim ( , , )
,n m
n mG x x x
→ ∞
= 0. In this case we say that the sequence ( )xn is G -con-
vergent to x .
Thus if x xn → in a G -metric space ( , )X G , then for any ε > 0 , there exists
l N∈ such that G x x xn m( , , ) < ε for all n, m > l.
Definition 5 [1]. A G -metric space ( , )X G is called symmetric G -metric
space if
G x y y( , , ) = G x x y( , , ) for every x, y ∈ X .
Definition 6. Let ( , )X G1 1 and ( , )X G2 2 be two G-metric spaces. A functi-
on f : X X1 2→ is called G-isometric when f preserves distances, i.e.,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
NEW RESULTS IN G -BEST APPROXIMATION IN G -METRIC SPACES 569
G x y z1( , , ) = G f x f y f z2( ( ), ( ), ( )) for every x, y, z ∈ X .
We can show that onto G-isometric maps are G-homeomorphisms (i.e., G-conti-
nuous inverse).
Definition 7. Let ( , )X G1 1 and ( , )X G2 2 be two G -metric spaces. A
function f : X X1 2→ is called G-contraction when there is a constant 0 ≤ k <
< 1 such that
G f x f y f z2( ( ), ( ), ( )) = kG x y z1( , , ) for every x, y, z ∈ X .
It follows that f is G-continuous, because
G x y z1( , , ) < δ : = ε / k ⇒ G f x f y f z2( ( ), ( ), ( )) ≤ ε .
2. New results. The field of approximation theory has become so vast that it inter-
sects with every other branch of analysis and plays an important role in applications in
the applied sciences and engineering. Fixed point theorems have been used in many
instances in approximation theory. In the subject of approximation theory one often
wishes to know whether some useful properties of the function being approximated are
inherited by the approximating function. In this section we will prove several theo-
rems.
Theorem 1. Let ( , )X G be a G -complete space and f : X X→ a G -
contraction map. Then f has a unique fixed point x = f x( ) .
Proof. We consider the interaction xn+1 = f xn( ) with x0 = a any point in X.
Note that
G x x xn n n( ), ,+ −1 1 = G f x f x f xn n n( )( ), ( ), ( )+ −1 1 ≤ kG x x xn n n( ), ,+ −1 1 .
Hence by induction
G x x xn n n( ), ,+ −1 1 ≤ kG x x x( ), ,2 1 0 .
For all n, m N∈ , n m< , by rectangle inequality that
G x x xn m m( ), , ≤ G x x x G x x x G x xn n n n n n n n( ) ( ) (, , , , ,+ + + + + ++ +1 1 1 2 2 2 ++ +3 3, )xn + …
… + G x x xm m m( ), ,−1 ≤ ( ) ( ), ,k k k G x x xn n m+ + … ++ −1 1
2 1 0 ≤
≤
k
k
G x x x
n
1 2 1 0−
( ), , . (1)
Then G x x xn m m( ), , → 0 as m, n → ∞ and thus ( )xn is a G -Cauchy sequence.
Due to the completeness of ( , )X G , there exists a x X∈ such that x xn → 0 and by
G-continuity of f
x = lim
n
nx
→ ∞ +1 = lim ( )
n
nf x
→ ∞ +1 = f x
n
nlim
→ ∞
= f x( ) .
Moreover, the rate of convergence is given by (1).
Suppose there are two fixed point x = f x( ) and y = f y( ) , then
G x y y( ), , = G f x f y f y( )( ), ( ), ( ) ≤ kG x y y( ), ,
so that G x y y( ), , = 0 since k < 1, therefore, x = y.
The theorem is proved.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
570 A. DEHGHAN NEZHAD, H. MAZAHERI
3. G -best approximations.
Definition 8. Let Y be a subspace of a metric space ( , )X d and x X∈ . The
point x0 is a best approximation to x from X if
(i) x Y0 ∈ and
(ii) d x x( ),0 ≤ d x y( , ) for every y Y∈ .
For a normed linear space X and x y X, ∈ a point x is said to be Birkhoff-ortho-
gonal to y and it denoted by x y⊥ if and only if x x y≤ + α for all scalar α
(see [2 – 4]).
In general if X is a symmetric G -metric space and x, y, a X0 ∈ we call x a0 -
orthogonal to y and denoted by x yG
a⊥ 0 if and only if d x aG ( , )0 ≤ d x yG ( , ) or
G x a a( ), ,0 0 ≤ G x y y( ), , .
Let Y1 and Y2 be subsets of X. Then Y YG
a
1 2
0⊥ if and only if for all y Y1 1∈ ,
y Y2 2∈ , y yG
a
1 2
0⊥ . If x yG
a⊥ 0 then it is not necessary y xg
a⊥ 0 .
Let X be a symmetric G-metric space and Y a subset of X. A point y Y0 ∈ call
a G-best approximation for x X∈ if x YG
y⊥ 0 or d x yG ( , )0 ≤ d x yG ( , ) for every
y Y∈ . The set of all G-best approximations of x in Y is shown by P xG
Y ( ) .
If Y is a subset of X it is clearly that
P xG
Y ( ) = y Y x y y YG
y
0
0∈ ⊥ ∀ ∈{ }: .
In continue we obtain some results on G-best approximation in symmetric G-metric
spaces.
Theorem 2. Let ( , )X G be a G-metric symmetric space. For x, y, z, a X0 ∈
(i) x xG
a⊥ 0 if and only if x a= 0 ,
(ii) if x yG
a⊥ 0 and y zG
x⊥ , then x zG
a⊥ 0 ,
(iii) x aG
a⊥ 0
0 and a xG
a
0
0⊥ .
Proof. (i) If x xG
a⊥ 0 then G x a a( ), ,0 0 ≤ G x x x( ), , = 0. Therefore,
G x a a( ), ,0 0 = 0, i.e., x = a0 .
(ii) Since x yG
a⊥ 0 then G x a a( ), ,0 0 ≤ G x y y( ), , also y zG
x⊥ then
G y x x( ), , ≤ G y z z( ), , and G is symmetric. Hence G x a a( ), ,0 0 ≤ G y z z( ), ,
that is x zG
a⊥ 0 .
(iii) It is clear.
Theorem 3. Let ( , )X G be a symmetric G-metric space and A a subset of
X.
(i) If x X∈ , then x P xG
A∈ ( ) .
(ii) If x A∈ , then P x xG
A( ) { }= .
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
NEW RESULTS IN G -BEST APPROXIMATION IN G -METRIC SPACES 571
Proof. (i) Since G x x x( ), , = 0, therefore, x P xG
A∈ ( ) .
(ii) If x A∈ and y P xG
A
0 ∈ ( ) , then G x y y( ), ,0 0 ≤ 0 = G y x x( ), , . Therefore,
x = y0 .
Theorem 4. Let ( , )X G be a symmetric G-metric space x , a X0 ∈ and A
a subset of X. Then the following statements are equivalent:
(i) x AG
a⊥ 0 ,
(ii) there is a function f X: [ , )→ ∞0 such that d x a f y d x yG G( , ) ( ) ( , )0 ≤ ≤
for all y A∈ .
Proof. (i) ⇒ (ii) we define f X: [ , )→ ∞0 by f z d x zG( ) ( , )= . Suppose
y A∈ since x yG
a⊥ 0
d x aG ( , )0 ≤ d x yG ( , ) = f y( ) .
(ii) ⇒ (i) By definition x yG
a⊥ 0 for every y A∈ .
Corollary 1. Let ( , )X G be a symmetric G-metric space x , a X0 ∈ , Y a
subset of X and y Y0 ∈ . Then the following statements are equivalent:
(i) y P xG
Y
0 ∈ ( ) ,
(ii) there is a function f X: [ , )→ ∞0 such that d x y f y d x yG G( , ) ( ) ( , )0 ≤ ≤
for all y Y∈ .
Corollary 2. Let ( , )X G be a symmetric G-metric space, x , a X0 ∈ , Y a
subset of X and E ⊆ Y . If there is a function f X: [ , )→ ∞0 such that
d x y f y d x yG G( , ) ( ) ( , )0 ≤ ≤ for all y E0 ∈ and for all y Y∈ , then E P xG
Y⊆ ( ) .
Example. We can define a symmetric G-metric on X a b c= { , , } such that
G a b b( ), , = G a a b( ), , = G a c c( ), , = G c a a( ), , = G b c c( ), , = G c b b( ), , = 1,
G a a a( ), , = G b b b( ), , = G c c c( ), , = 0
and
G a b c( ), , = G b c a( ), , = G c a b( ), , = 2.
It is clear that G is a symmetric G-metric. Then a cG
b⊥ , b cG
a⊥ and c bG
a⊥ .
1. Mustafa Z., Sims B. A new approach to generalized metric spaces // J. Nonlinear and Convex Anal.
– 2006. – 7, # 2. – P. 289 – 297.
2. Mazaheri F., Maalek Ghaini F. M. Quasi-orthogonality of the best approximant sets // Nonlinear
Anal. – 2006. – 65. – P. 534 – 537.
3. Mazaheri H., Moshtaghouion S. M. The orthogonality in the vector spaces // Bull. Iran. Math. Soc.
– 2009. – 35, # 1. – P. 119 – 127.
4. Mazaheri H., Vaezpour S. M. Orthogonality and [-orthogonality in Banach spaces // Aust. J.
Math. Anal. and Appl. – 2005. – 2, # 1. – P. 1 – 5.
Received 28.10.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 4
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| spelling | umjimathkievua-article-28882020-03-18T19:39:35Z New results in $G$-best approximation in $G$-metric spaces Нові результати щодо $G$-найкращого наближення в $G$-метричних просторах Dehghan, Nezhad A. Mazaheri, H. Дегхан, Нежад А. Мазахері, Х. The purpose of this paper is to introduce and discuss the concepts of G-best approximation and $a_0$ -orthogonality in the theory of $G$-metric spaces. We consider the relationship between these concepts and the dual $X$ and obtain some results on subsets of $G$-metric spaces similar to normed spaces. Мета роботи — ввести та обговорити поняття $G$-найкращого наближення та $a_0$-ортогональ-пості в теорії $G$-метричпих просторів. Розглянуто співвідношення між цими поняттями та дуальним $X$, отримано деякі результати щодо підмножин $G$-метричиих просторів, що подібні до нормованих просторів. Institute of Mathematics, NAS of Ukraine 2010-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2888 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 4 (2010); 567–571 Український математичний журнал; Том 62 № 4 (2010); 567–571 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2888/2527 https://umj.imath.kiev.ua/index.php/umj/article/view/2888/2528 Copyright (c) 2010 Dehghan Nezhad A.; Mazaheri H. |
| spellingShingle | Dehghan, Nezhad A. Mazaheri, H. Дегхан, Нежад А. Мазахері, Х. New results in $G$-best approximation in $G$-metric spaces |
| title | New results in $G$-best approximation in $G$-metric spaces |
| title_alt | Нові результати щодо $G$-найкращого наближення в $G$-метричних просторах |
| title_full | New results in $G$-best approximation in $G$-metric spaces |
| title_fullStr | New results in $G$-best approximation in $G$-metric spaces |
| title_full_unstemmed | New results in $G$-best approximation in $G$-metric spaces |
| title_short | New results in $G$-best approximation in $G$-metric spaces |
| title_sort | new results in $g$-best approximation in $g$-metric spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2888 |
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