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 Authors : T. Markandeya Naidu, Dr. D. Bharathi.
 Paper ID : IJERTV3IS10787
 Volume & Issue : Volume 03, Issue 01 (January 2014)
 Published (First Online): 24012014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Best Simultaneous Approximation in 2Normed Almost Linear Space
Best Simultaneous Approximation in 2Normed Almost Linear Space
T. Markandeya Naidu & Dr. D. Bharathi

Department Of Mathematics, P.V.K.N.Govt.Degree College, Chittoor,Andhra Pradesh, India 2 .Department Of Mathematics, S.V.University, Tirupathi, Andhra Pradesh ,India.
ABSTRACT
In this paper we establish some of the results of best simultaneous approximation in linear 2 normed space in the context of 2 normed almost linear space.
1 .INTRODUCTION
In (1) Gliceria Godini introduced the concept almost linear space which is defined as A non empty set X together with two mappings s: XxX X and m:RxX X Where s(x ,y)=x +y and m(,
x) = x is said to be an almost linear space if it satisfies the following properties.
For every x, y, z X and for every , R

x +y X ,

(x + y) +z = x+(y + z) ,

x +y = y + x ,

There exists an element 0 X such that x+0=x ,

1 x = x ,

(x + y)= x+ y ,

0 x =0 ,

( x )=( )x ,ix) ( + )x= x + x for 0,0.
In (1 & 4) Gliceria Godini also introduced the concept normed almost linear space which is defined as an almost linear space X together with  .  : X R is said to be normed almost linear space if it satisfies the following properties

 x =0 if and only if x=0, ii)  x =II  x  ,
iii)  xz xy+yz for every x,y X and R.
The concept of linear 2normed space has been initially investigated by S. G hler(17) and has been extensively by Y.J.Cho, C.Diminnie, R.Freese and many other, which is defined as
a linear space X over R with dim>1 together with II .II is called Linear 2 normed space if II . II satisfy the following properties

II x, y II>0 and II x , y II=0 if and only if x and y are linearly dependent,

II x , y II = II y , x II,

II x ,y II = II II x , y II, and

II x , y+z II = II x , z II + II y ,z II for every x,y,z X and R.

In (23) T.Markandeya naidu and Dr D.Bharathi introduced a new concept called 2normed almost linear space and established some of the results of best approximation in 2nomed almost linear space.In (22) S.Elumalai & R.Vijayaragavan established some of the results of best simultaneous approximation in linear space in the context of linear2normed space. In this paper we extend some of the results of best simultaneous approximation on linear2 normed space in to 2normed almost linear space.


PRELIMINARIES
Definition 2.1. Let X be an almost linear space of dimension> 1 and
 .: X x X R be a real valued function.
If  .  satisfy the following properties
i)  , =0 if and only if and are linearly dependent,
ii)  ,  =  , ,
iii)  a ,  = IaI  , ,
iv)  , –   , –  +  ,
–  for every , , , X and a R.
then (X,.) is called 2normed almost linear space.
Definition 2.2. Let X be a 2normed almost linear space over the real field R and G a non empty subset of . For a bounded sub set A of X let us define
(A) =  x , ag  for every x X\ 2.1
and
(A) = 0 G:  x , a0  =
(A) for every x X\. 2.2
(A) is called the chebyshev radius of A with respect to G and an element 0 (A) is called a best simultaneous approximation or chebyshev centre of A with respect to G.
Definition 2.3. When A is a singleton say A={a}, a X\ then (A) is the distance of a to G, denoted by dist(a,G) and defined by dist(a,G)=  x,ag for every x
X\
2.3
and (A) is the set of all best approximations of a out of G denoted by
(a) and defined by
(a)={ 0G:x,a0=dist(a,G),
for every x X\ }
2.4
It is well known that for any bounded subset A of X we have
(A)= (0())= ( )
(A)= (0())= ( )
Where 0() stands for the convex hull of A and stands for the closure of A.
Definition 2.4. Let X be a 2normed almost linear spaces and G . We difine
(G) X
in the following way
(G) if for each g
such that the following conditions are hold
i)  x, ag  =  x, g  for each
2.5
ii)  x, a   x, –  for every x X\. 2.6
We have (G). If 1 2 then (2)
(1).
Definition 2.5. Let X be a 2normed almost linear space. The set G is said to be proximinal if (a) is nonempty for each a X\ .
Theorem 3.1 Let ( X,  .  ) be a
2 normed almost linear space. Let G X and A be a bounded subset of X. Then the function (h,g) defined by sup  h, ag ,
h X\ , g G, a A is a continuous function on X.
Proof: For any a A and g , G we have
 h, ag   h, a  +
 h, g  , h X\ . Then
 h, ag 
{  h, a  +  h, g  }
a best simultaneous approximation g G to any given compact subset A of X.
Proof: Since A is compact there exist a constant t such that
 a , b  t for all a A and b X. Now we define the subset H of G as G G(0,2t) then
inf.  b, ah  =
 b, ah  , b X\ t Since h is compact the continuous function
(h,b) attains its minimum over H for some g
G.
which is the best simultaneous approximation to A.
Theorem 3.3 Let ( X,  .  ) be a
2 normed almost linear space and let G be a
convex subset of X and A X . If g, G are two best simultaneous approximations to A by elements of G. Then
= g + (1 ) ,0 1 is also best simultaneous approximation to A.
Proof: For x X\ ,
Now if  h, g  then (h,g) (h, ) +
 x, a

By interchanging g and we obtain
=  x, a g + (1 ) 
(h, ) (h,g) + that implies
 (h,g) – (h, ) <
=
 x, (ag) + (1 ) (a) 
That is (h,g) is continuous on X.
 x, ag  +
Theorem 3.2 Let ( X,  .  ) be a 2 normed almost linear space. Let G be a finite dimensional subspace of X. Then there exist
 x, ag  +
(1)  x, a 
(1)  x, a 
= (, ) + (1) (, )
= (, )
3.1
(, ) =  x, a 
 x, a +  = k 3.4
2
Since A is compact there exist an element 0
such that
2
 x, a + 
2
=  x, 0 – + 
=k 3.5
. 3.3  x, 0  k and  x, 0
 k .
Then by strict convexity we have
 x, 0+0  < 2 k
 x, g  3.2
That is  x, 0
+  < k
–
2
(, ) =  x, a  This proves the result.
Theorem 3.4 Let (X, . ) be a strictly convex 2normed almost linear space. Let G be a finite dimensional ubspace of X. Then there exists one and only one best simultaneous approximation from the element G by any given compact subset A of X.
Proof : The existence of a best simultaneous approximation follows from the Theorem3.2.
Suppose and ( ) are best simultaneous approximations to A then for x X\ ,
 x,ag 
This contradicts eq.3.5.
Hence the proof.
Theorem 3.5 Let G be a closed and convex subset of a uniformly convex

Banach space X.Then for any compact subset A of X there exist unique best approximation to A from the element of G.
Proof:
Let k=  x,ag : x X\ and
be any sequence of elements in G Such that
 x,a  =k.
Also =  x,a  ,m1 and x
X\ .
Then k which implies
=
 x, a 
 x, 1 for a A 3.6
=  x, a 
=k 3.3
Now we consider 1 + =
2
+ +
2 +
Then by theorem (3.3),
Let = + .
,
+
+
2
is also
since G is convex, , G.
the best simultaneous approximation. That is
Hence  x,a, k and 
REFERENCES
x, + a – 1 (
+ )  =  x,a,
2
2

G.Godini : An approach to
 ( + ) k( + ). Since A is compact sub
generalizing Banach spaces. Normed
2
2
almost linear spaces. Proceedings of
set of X there exist an a A such that
 x, +  k( + ).
the 12th winter school on Abstract Analysis (Srni.1984).
Suppl.Rend.Circ.Mat.Palermo II. Ser.5, (1984) 3350.
By eq.3.6 and the uniform convexity of the 2
norm it follows that
for given > 0 there exits an N such that  x,
+  < for

G.Godini : Best approximation in normed almost linear spaces. In constructive theory of functions. Proceedings of the International
conference on constructive theory of
m,n > N and x X\ .
Since k as m we can easily see that the sequence is a Cauchy sequence
hence it converges to some g G as G is
closed subset of X.
This provides that G is a best simultaneous approximation.
Assume that there exist two best simultaneous approximations
1 and 2 .
Then there exist sequences and
such that 1 as n
and 2 as m .
Again  x,a  =k =
 x,a  .
This implies that  x,a1  =
 x,a2  and hence
1 = 2 .
functions. (Varna 1984 ) Publ . Bulgarium Academy of sciences ; Sofia (1984) 356363.

G.Godini : A Frame work for best simultaneous approximation. Normed almost linear spaces. J.Approxi. Theory 43, (1985) 338 358.

G.Godini : On Normed almost linear spaces Preprint series Mann. INCREST , Bucuresti 38 (1985).

G.Godini : Operators in Normed almost linear spaces Proceedings of the 14th winter school on Abstract Analysis (Srni.1986). Suppl.Rend.Circ.Mat.Palermo II. Numero 14(1987) 309328.

A.L.Garkavi : The Chebyshev center of a set in a normed space in Investigations on current Problems in Constructive Theory of Functions. Moscow (1961) PP. 328 331.

A.L.Garkavi : On the chebyshev center and the Convex hull of a set. Uspehi Mat. Nank 19 (1964) pp 139 45.

A.L.Garkavi : The conditional Chebyshev center of a compact set of continuous functions. Mat. Zam . 14(1973) 469478 (Russian) = Mat. Notes of the USSR (1973), 827831.

J. Mach : On the existence of best simultaneous approximation.
J. Approx. Theory 25(1979) 258 265.

J. Mach : Best simultaneous approximation of bounded functions with values in certain Banach spaces. Math. Ann. 240 (1979) 157164.

J. Mach: Continuity properties of Chebyshev centers J. appr. Theory 29 (1980) 223238.

P.D. Milman : On best simultaneous approximation in normed linear spaces. J. Approx. Theory 20 (1977) 223238.

B.N. Sahney and S.P. Singh : On best simultaneous prroximation in Approx. Theory New York/London 1980.

I. Singer : Best approximation in normed linear spaces by elements of linear subspaces. Publ. House Acad. Soc. Rep. Romania Bucharest and Springer Verlag, Berlin / Heidelberb / New York (1970).

I. Singer : The theory of best approximation and functional analysis. Regional Conference Series in Applied Mathematics No:13, SIAM; Philadelplhia (1974).

D. Yost: Best approximation and intersection of balls in Banach Spaces, Bull, Austral. Math. Soc. 20 (1979, 285300).

17.R.Freese and S.G hler,Remarks on semi 2normed space,Math,Nachr. 105(1982),151161.
18.S. Elumalai, Y.J.Cho and S.S.Kim : Best approximation sets in linear 2 Normed spaces comm.. Korean
Math. Soc.12 (1997), No.3, PP.619 629.

S.Elumalai, R.Vijayaragavan : Characterization of best approximation in linear 2 normed spaces. General Mathematics Vol.17, No.3 (2009), 141160.

S.S.Dragomir : Some
characterization of Best approximation in Normed linear spaces. Acta Mathematica Vietnamica Volume 25,No.3(2000)pp.359366.

Y.Dominic : Best approximation in uniformly convex 2normed spaces. Int.journal of Math. Analysis, Vol.6,( 2012), No.21, 10151021.
22.S.Elumalai & R.Vijayaragavan:Best simultaneous approximation in linear 2 normed spaces .Genral Mathematics Vol.16,No. 1
(2008),7381.
23.T.Markandeya naidu & Dr.D.Bharathi:Best approximation in 2normed almost linear space. International journal of engineering research and technology(IJERT) Vol.2 Issue 12 (December 2013),35693573.