Best Simultaneous Approximation in 2-Normed Almost Linear Space

Best Simultaneous Approximation in 2-Normed Almost Linear Space

Best Simultaneous Approximation in 2-Normed Almost Linear Space

T. Markandeya Naidu & Dr. D. Bharathi

1. 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

   1. x +y X ,

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

   3. x +y = y + x ,

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

   5. 1 x = x ,

   6. (x + y)= x+ y ,

   7. 0 x =0 ,

   8. ( 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

   1. ||| x |||=0 if and only if x=0, ii) ||| x |||=II ||| x ||| ,

      iii) ||| x-z ||||||x-y|||+|||y-z||| for every x,y X and R.

      The concept of linear 2-normed 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

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

      2. II x , y II = II y , x II,

      3. II x ,y II = II II x , y II, and

      4. 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 2-normed almost linear space and established some of the results of best approximation in 2-nomed 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 linear2-normed space. In this paper we extend some of the results of best simultaneous approximation on linear2- normed space in to 2-normed almost linear space.

2. 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 2-normed almost linear space.

   Definition 2.2. Let X be a 2-normed 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 , a-g ||| for every x X\ 2.1

   and

   (A) = 0 G: ||| x , a-0 ||| =

   (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,a-g||| 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,a-0|||=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 2-normed 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, a-g ||| = ||| 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 2-normed 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, a-g |||,

h X\ , g G, a A is a continuous function on X.

Proof: For any a A and g , G we have

||| h, a-g ||| ||| h, a- ||| +

||| h, g ||| , h X\ . Then

||| h, a-g |||

{ ||| 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, a-h ||| =

||| b, a-h ||| , 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, (a-g) + (1- ) (a-) |||

That is (h,g) is continuous on X.

||| x, a-g ||| +

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, a-g ||| +

(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 2-normed 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,a-g |||

This contradicts eq.3.5.

Hence the proof.

Theorem 3.5 Let G be a closed and convex subset of a uniformly convex

1. 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,a-g |||: 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

   1. 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) 33-50.

      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

   2. 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,a-1 ||| =

      ||| x,a-2 ||| and hence

      1 = 2 .

      functions. (Varna 1984 ) Publ . Bulgarium Academy of sciences ; Sofia (1984) 356-363.

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

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

   5. 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) 309-328.

   6. 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.

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

   8. A.L.Garkavi : The conditional Chebyshev center of a compact set of continuous functions. Mat. Zam . 14(1973) 469-478 (Russian) = Mat. Notes of the USSR (1973), 827-831.

   9. J. Mach : On the existence of best simultaneous approximation.

      J. Approx. Theory 25(1979) 258- 265.

   10. J. Mach : Best simultaneous approximation of bounded functions with values in certain Banach spaces. Math. Ann. 240 (1979) 157-164.

   11. J. Mach: Continuity properties of Chebyshev centers J. appr. Theory 29 (1980) 223-238.

   12. P.D. Milman : On best simultaneous approximation in normed linear spaces. J. Approx. Theory 20 (1977) 223-238.

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

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

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

   16. D. Yost: Best approximation and intersection of balls in Banach Spaces, Bull, Austral. Math. Soc. 20 (1979, 285-300).

17.R.Freese and S.G hler,Remarks on semi 2-normed space,Math,Nachr. 105(1982),151-161.

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.

1. S.Elumalai, R.Vijayaragavan : Characterization of best approximation in linear- 2 normed spaces. General Mathematics Vol.17, No.3 (2009), 141-160.

2. S.S.Dragomir : Some

   characterization of Best approximation in Normed linear spaces. Acta Mathematica Vietnamica Volume 25,No.3(2000)pp.359-366.

3. Y.Dominic : Best approximation in uniformly convex 2-normed spaces. Int.journal of Math. Analysis, Vol.6,( 2012), No.21, 1015-1021.

22.S.Elumalai & R.Vijayaragavan:Best simultaneous approximation in linear 2- normed spaces .Genral Mathematics Vol.16,No. 1

(2008),73-81.

23.T.Markandeya naidu & Dr.D.Bharathi:Best approximation in 2-normed almost linear space. International journal of engineering research and technology(IJERT) Vol.2 Issue 12 (December- 2013),3569-3573.