BEST CO-APPROXIMATION AND BEST SIMULTANEOUS CO-APPROXIMATION IN INTUITIONISTIC FUZZY NORMED LINEAR SPACES

The main purpose of this paper is to study the t-best co-approximation and t-best simultaneous co-approximation in intuitionistic fuzzy normed spaces. We develop the theory of t-best co-approximation and t-best simultaneous coapproximation in quotient spaces. This new concept is employed by us to improve various characterisations of t-coproximinal and t-co-Chebyshev sets.


INTRODUCTION
The theory of a fuzzy sets was firstly introduced by Zadeh [14] in 1965 and thereafter several authors applied it to different branches of pure and applied mathematics. On the other hand , the notion of fuzzyness has a wide application in many areas of science and engineering .
Katsaras [5]in 1984, first introduced the notion of fuzzy norm on a linear space . The concept of a fuzzy norm on a linear space by assigning a fuzzy real number to each element of the linear spaces introduced by Felbin [4] in 1992.
In 1986 ,Atanassov [2] introduced the concept of intuitionistic fuzzy sets. Park [8] first introduced the concept of intuitionistic fuzzy metric space and Saadati and Park [9] introduced the concept of intuitionistic fuzzy normed space, while the notion of intuitionistic fuzzy n-normed linear space was introduced by S. Vijayabalaji , N. Thillaigovindan and Y. Bae [13].
In 2011 ,Abrishami Moghaddam and Sistani [1], firstly introduced the concept of the set of all t-best co-approximation on fuzzy normed spaces.Surender Reddy [12] in 2012 discussed the concept of the t-Best Co-approximation in fuzzy anti-2-normed linear spaces. J. Kavikumar , N. S. Manian and M.B.K. Moorthy [7] introduced the concept of Best Co-approximation and Best Simultaneous Co-approximation in Fuzzy Normed Spaces .
In this paper we study the set of all t-best co-approximation and t-best simultaneous co-approximation in intuitionistic fuzzy normed linear spaces and we develop the theory of t-best co-approximation and t-best simultaneous co-approximation in quotient spaces. This new concept is employed us to improve various characterizations of t-co-proximinal and t-co-Chebyshevsets. ) is said to be an intuitionistic fuzzy normed linear space (IFNLS) if be a linear space over the field F (R or ) , is a continuous t-norm , is a continuous t-conorm , and , fuzzy sets on ( ) satisfy the following conditions for every :

Definition 3.1: Let (
) be IFNLS and G be a nonempty subset of An element is called an intuitionistic fuzzy -t-best co-approximation to from (IF-t-best co-approximation) if for , ( The set of all IF-t-best co-approximation to from will be denoted by ( )

Remark 3.2: The set
( ) of all IF-t-best co-approximation to from can be written as : similarly , we get ̃ .
Definition 3.5: Let ( ) be a nonempty subset of . If for every has at least one IF-t-best co-approximation in , then is called an intuitionistic fuzzy-t-co-proximinal set (IF-t-co-proximinal set) .
be a nonempty subset of . If for every has exactly one IF-t-best co-approximation in , then is called an intuitionistic fuzzy -t-co-Chebyshev set (IF -t-co-Chebyshev set ) .

Proof : Let
Therefore , for a given , take the natural number such that By assumption and definition 2.4. , we have and for a given ,take the natural number such that ) be an IFNLS and be a subspace of :

( ) is IF-t-co-proximinal set (resp. IF-t-co-Chebyshev set) if and only if
is IF-t-co-proximinal set (resp. IF-t-co-Chebyshev set) for every proof : ( similarly , we get | | is IF-t-co-Chebyshev set .
similarly, we get is IF-t-co-Chebyshev set .    Proof: Let | is an IF-t-co-proximinal with | . Proof: Let | is IF-t-co-Chebyshev with | and has two distinct t-best co-approximation of

t-CO-PROXIMINALITY AND t-CO-CHEBYSHEVITY IN QUOTIENT SPACES
Then is an IF-t-co-Chebyshev with . is IF-t-best co-approximation to from , Then is an IF-t-best co-approximation to from the quotient space .
Proof : Suppose that is IF-t-best co-approximation to from and is not IF-t-best coapproximation to from the quotient space .
this implies that there exists such that is not an IF-t-best co-approximation to from , this contradiction with hypothesis( ) .
Then is an IF-t-best co-approximation to from the quotient space . The set of all IF-t-best simultaneous co-approximation to from , will be denoted by ( ) and define as follows :

Definition 5.2:
Let be a subset of ( ) . It is called IF-t-best simultaneous co-proximinal subset of , if for each IF-bounded set in , there exists at least one IF-t-best simultaneous co-approximation from .

Definition 5.3:
Let be a subset of (