I λ -convergence in intuitionistic fuzzy n-normed linear space. Nabanita Konwar, Pradip Debnath

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1 Annals of Fuzzy Mathematics Informatics Volume 3, No., (January 207), pp ISSN: (print version) ISSN: (electronic version) c Kyung Moon Sa Co. I -convergence in intuitionistic fuzzy n-normed linear space Nabanita Konwar, Pradip Debnath Received 28 May 206; Revised 3 July 206; Accepted 23 July 206 Abstract. The notion of lacunary ideal convergence in intuitionistic fuzzy normed linear space (IFNLS) was introduced by the present corresponding author [P. Debnath, Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces, Comput. Math. Appl., 63 (202), ] an open problem in that paper was whether every lacunary I-convergent sequence is lacunary I-Cauchy. Further, a new concept of convergence of sequences in an intuitionistic fuzzy n-normed linear space (IFnNLS) was given in [M. Sen, P. Debnath, Lacunary statistical convergence in intuitionistic fuzzy n-normed linear spaces, Math. Comput. Modelling, 54 (20), ]. With the help of this new definition of convergence, the main aim of this paper is to introduce the concept of I -convergence in an IFnNLS, where I is an ideal of a family of subsets of positive integers N. We also define I -limit points I -cluster points establish relations between them. Finally we introduce the notion of I -Cauchy sequence in IFnNLS. We improve extend some existing results give a positive answer to the open problem mentioned above in the setting of an IFnNLS. 200 AMS Classification: 46H25, 46H25, 46A99, 60H0 Keywords: Intuitionistic fuzzy n-normed linear space, I -convergence, -convergence, I -Cauchy sequence, I -limit points, I -cluster points. Corresponding Author: P. Debnath (debnath.pradip@yahoo.com). Introduction Many problems that we study in analysis are concerned with large classes of objects most of which turn out to be vector spaces or linear spaces. Since limit process is indispensable in such problems, a metric or topology may be induced in those classes. If the induced metric satisfies the translation invariance property, a norm can be defined in that linear space we get a structure of the space which is compatible with that metric or topology. The resulting structure is a normed linear space. There are situations where crisp norm can not measure the length of a vector

2 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 accurately in such cases the notion of fuzzy norm happens to be useful. There has been a systematic development of fuzzy normed linear spaces (FNLSs) one of the important development over FNLS is the notion of intuitionistic fuzzy normed linear space (IFNLS). The study of analytic propertis of IFNLSs, their topological structure generalizations, therefore, remain well motivated areas of research. The idea of a fuzzy norm on a linear space was introduced by Katsaras [23] in 984. In 992, Felbin [6] introduced the idea of a fuzzy norm whose associated metric is of Kaleva Seikkala [20] type. In 994, Cheng Mordeson [6] introduced another notion of fuzzy norm on a linear space whose associated metric is Kramosil Michalek [26] type. Again in 2003, following Cheng Mordeson, one more notion of fuzzy normed linear space was given by Bag Samanta [3]. The notion of intuitionistic fuzzy set (IFS) introduced by Atanassov [2] has triggered some debate (for details, see [4, 3, 7]) regarding the use of the terminology intuitionistic the term is considered to be a misnomer on the following account: The algebraic structure of IFSs is not intuitionistic, since negation is involutive in IFS theory. Intuitionistic logic obeys the law of contradiction, IFSs do not. Also IFSs are considered to be equivalent to interval-valued fuzzy sets they are particular cases of L-fuzzy sets. In response to this debate, Atanassov justified the terminology in []. Apart from the terminological issues, research in intuitionistic fuzzy setting remains well motivated as IFSs give us a very natural tool for modeling imprecision in real life situations which can not be hled with fuzzy set theory alone also IFS found its application in various areas of science engineering. In 2006, with the help of arbitrary continuous t-norm continuous t-conorm, Saadati Park [3] introduced the concept of IFNLS. There has been further development over IFNLS, e.g., the topological structure of an intuitionistic fuzzy 2-normed space has been studied by Mursaleen Lohani in [29]. Further, generalizing the idea of Saadati Park, an intuitionistic fuzzy n-normed linear space (IFnNLS) has been defined by Vijayabalaji et al. [38] in Some more recent work in similar context can be found in [5, 7, 9, 0,, 2, 4, 30, 32, 37]. The idea of statistical convergence was first introduced by Steinhaus [36] Fast [5]. Karakus [2] studied statistical convergence on probabilistic normed spaces. Then Karakus et al. [22] generalized it on IFNLSs. The notion of I- convergence was initially introduced by Kostyrko et.al [24] as a generalization of statistical convergence which is based on the structure of the ideal I of subsets of natural numbers N. Further, Kostyrko et.al [25] gave some of basic properties of I-convergence deal with external I-limit points. Later on, Esi Hazarika [4] introduced the concept of -ideal convergence in intuitionistic fuzzy 2-normed space. Recently Hazarika, Kumar Guillén [9] studied another generalized ideal convergence in IFNLS. An unambiguous new definition of convergence of sequences in IFnNLS was given in [34, 35] which is different from [38], the notion of lacunary ideal convergence in IFNS was introduced in [8]. Combining these, we extend our work to introduce study the notion of I -convergence in an IFnNLS. 92

3 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., Preliminaries First we collect some preliminaries to be used in this paper. Throughout the paper N R denote the set of natural numbers real numbers respectively. Definition 2. ([8]). Let n N X be a real linear space of dimension d n (d may be infinite). A real valued function. on X X X = X }{{} n is called an n-norm on X, if it satisfies the following properties: (i) x, x 2,..., x n = 0 if only if x, x 2,..., x n are linearly dependent, (ii) x, x 2,..., x n is invariant under any permutation, (iii) x, x 2,..., αx n = α x, x 2,..., x n for any α R, (iv) x, x 2,..., x n, y + z x, x 2,..., x n, y + x, x 2,..., x n, z, the pair (X,. ) is called an n-normed linear space. Definition 2.2 ([38]). An IFnNLS is the five-tuple (X, µ, ν,, ), where X is a linear space over a field F, is a continuous t-norm, is a continuous t-conorm, µ, ν are fuzzy sets on X n (0, ), µ denotes the degree of membership ν denotes the degree of non-membership of (x, x 2,..., x n, t) X n (0, ) satisfying the following conditions for every (x, x 2,..., x n ) X n s, t > 0: (i) µ(x, x 2,..., x n, t) + ν(x, x 2,..., x n, t), (ii) µ(x, x 2,..., x n, t) > 0, (iii) µ(x, x 2,..., x n, t) = if only if x, x 2,..., x n are linearly dependent, (iv) µ(x, x 2,..., x n, t) is invariant under any permutation of x, x 2,..., x n, (v) µ(x, x 2,..., cx n, t) = µ(x, x 2,..., x n, t c ) if c 0, c F, (vi) µ(x, x 2,..., x n, s) µ(x, x 2,..., x n, t) µ(x, x 2,..., x n + x n, s + t), (vii) µ(x, x 2,..., x n, t) : (0, ) [0, ] is continuous in t, (viii) lim t µ(x, x 2,..., x n, t) = lim t 0 µ(x, x 2,..., x n, t) = 0, (ix) ν(x, x 2,..., x n, t) <, (x) ν(x, x 2,..., x n, t) = 0 if only if x, x 2,..., x n are linearly dependent, (xi) ν(x, x 2,..., x n, t) is invariant under any permutation of x, x 2,..., x n, (xii) ν(x, x 2,..., cx n, t) = ν(x, x 2,..., x n, t c ) if c 0, c F, (xiii) ν(x, x 2,..., x n, s) ν(x, x 2,..., x n, t) ν(x, x 2,..., x n + x n, s + t), (xiv) ν(x, x 2,..., x n, t) : (0, ) [0, ] is continuous in t, (xv) lim t ν(x, x 2,..., x n, t) = 0 lim t 0 ν(x, x 2,..., x n, t) =. Example 2.3 ([34]). Let (X,. ) be an n-normed linear space. Also let a b = ab t a b = min{a + b, } for all a, b [0, ], µ(x, x 2,..., x n, t) = ν(x, x 2,..., x n, t) = x,x2,...,xn t+ x,x 2,...,x n. Then (X, µ, ν,, ) is an IFnNLS. n t+ x,x 2,...,x n Definition 2.4 ([34]). Let (X, µ, ν,, ) be an IFnNLS. We say that a sequence x = {x k } in X is convergent to l X with respect to the intuitionistic fuzzy n-norm (µ, ν) n if, for every ɛ > 0, t > 0 y, y 2,..., y n X, there exists k 0 N such that µ(y, y 2,..., y n, x k L, t) > ɛ ν(y, y 2,..., y n, x k l, t) < ɛ for all k k 0. It is denoted by (µ, ν) n lim x = l or x k (µ,ν) n l as k. 93

4 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 Definition 2.5 ([2]). Let (X, µ, ν,, ) be an IFnNLS. For t > 0, we define an open ball B(x, r, t) with center at x X, radius 0 < r < y, y 2,..., y n X as B(x, r, t) = {y X : µ(y, y 2,..., y n, y x, t) > r ν(y, y 2,..., y n, y x, t) < r}. Definition 2.6 ([34]). Let (X, µ, ν,, ) be an IFnNLS. Then the sequence x = {x k } in X is called a Cauchy sequence with respect to the intuitionistic fuzzy n-norm (µ, ν) n if, for every ɛ > 0, t > 0 y, y 2,..., y n X, there exists k 0 N such that µ(y, y 2,..., y n, x k x m, t) > ɛ ν(y, y 2,..., y n, x k x m, t) < ɛ for all k, m k 0. Definition 2.7 ([24]). Let X be a non-empty set. Then a family of sets I P (X) is called an ideal in X, if (i) I, (ii) A, B I implies A B I, (iii) for each A I B A we have B I, where P (X) is the power set of X. Definition 2.8 ([24]). Let X be a non-empty set. Then a non-empty family of sets F P (X) is called a filter on X, if (i) / F, (ii) A, B F implies A B F, (iii) for each A F B A, we have B F. An ideal I is called non-trivial, if I X / I. A non-trivial ideal I P (X) is called an admissible ideal in X, if it contains all singletons, i.e., it contains {{x} : x X}. Definition 2.9 ([36]). If K is a subset of N, the set of natural numbers, then the natural density of K, denoted by δ(k), is given by δ(k) = lim {k n : k K}, n n whenever the limit exists, where A denotes the cardinality of the set A. Definition 2.0 ([27]). Let = ( ) be a non-decreasing sequence of positive numbers tending to infinity such that + +, =. The generalized la Vallée-Poussin mean is defined by t n (x) = (x k ), where J n = [n +, n]. A sequence x = (x k ) is said to be (v, )-summable to a number l if t n (x) l as n. If = n, then (V, )-summability reduces to (C, )-summability. Definition 2. ([28]). A sequence x = (x k ) is said to be -statistically convergent to the number l if for every ɛ > 0, lim n {k J n : x k l ε} = 0. 94

5 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 Let S denote the set of all -statistically convergent sequences. If = n, then S is the same as S. Definition 2.2 ([33]). Let I 2 N be a non-trivial ideal. A sequence x = (x k ) is said to be I [V, ]-summability to a number l if, for every ε > 0 x k l ε} I. Through out the paper, we denote by I as admissible ideal of subsets of N = ( ) is a sequence as defined in Definition 2.9. Now we discuss our main results. 3. Main Results Definition 3.. Let I 2 N let (X, µ, ν,, ) be an IFnNLS. A sequence x = {x k } in X is said to be I - convergent to l X with respect to the intuitionistic fuzzy n-norm (µ, ν) n if, for every ɛ > 0, t > 0 y, y 2,..., y n X, we have µ(y, y 2,..., y n, x k l, t) ɛ or ν(y, y 2,..., y n, x k l, t) ɛ} I. In this case, l is called the I - limit of the sequence x = (x k ) we write I (µ,ν)n lim x = l. Example 3.2. Let (R, ) denote the space of all real numbers with the usual n-norm let a b = ab a b = min{a + b, } for all a, b [0, ]. For all x, x 2,..., x n R every t > 0 consider µ(x, x 2,..., x n, t) = x,x2,...,xn t+ x,x 2,...,x n t t+ x,x 2,...,x n ν(x, x 2,..., x n, t) =. Then (R, µ, ν,, ) is an IFnNLS. If we take I = {A N : δ(a) = 0}, where δ(a) denote natural density of the set A, then I is non trivial admissible ideal. Define a sequence x = (x k ) as follows: {, if k = i x k = 2, i N 0, otherwise. Then for every ɛ (0, ), for any t > 0 y, y 2,..., y n X, the set K(ɛ, t) = µ(y, y 2,..., y n, x k, t) ɛ or ν(y, y 2,..., y n, x k, t) ɛ} will be a finite set. Thus δ(k(ɛ, t)) = 0. So K(ɛ, t) I, i.e.,i (µ,ν)n lim x = 0. Lemma 3.3. Let (X, µ, ν,, ) be an IFnNLS let x = (x k ) be a sequence in X. Then for every ε (0, ), t > 0 y, y 2,..., y n X the following statements are equivalent: 95

6 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 () I (µ,ν)n lim x = l. (2) µ(y, y 2,..., y n, x k l, t) ε} I ν(y, y 2,..., y n, x k l, t) ε} I. (3) µ(y, y 2,..., y n, x k l, t) > ε ν(y, y 2,..., y n, x k l, t) < ε} F (I). (3) µ(y, y 2,..., y n, x k l, t) > ε} F (I) ν(y, y 2,..., y n, x k l, t) < ε} F (I). (4) I lim µ(y, y 2,..., y n, x k l, t) = I lim ν(y, y 2,..., y n, x k l, t) = 0. Theorem 3.4. Let (X, µ, ν,, ) be an IFnNLS. If a sequence x = (x k ) in X is I -convergent to l X with respect to the intuitionistic fuzzy n-norm (µ, ν), then lim x is unique. I (µ,ν)n Proof. If possible suppose, I (µ,ν)n lim x = l I (µ,ν)n lim x = l 2. For a fixed ε (0, ), choose γ (0, ) such that ( γ) ( γ) > ε γ γ < ε. Then for any t > 0 y, y 2,..., y n X define the following sets: K µ, (γ, t) = µ(y, y 2,..., y n, x k l, t 2 ) γ}, K µ,2 (γ, t) = µ(y, y 2,..., y n, x k l 2, t 2 ) γ}, K ν, (γ, t) = ν(y, y 2,..., y n, x k l, t 2 ) γ}, K ν,2 (γ, t) = ν(y, y 2,..., y n, x k l 2, t 2 ) γ}. Now I (µ,ν)n lim x = l implies that (using Lemma 3.3) K µ, (γ, t) I K ν, (γ, t) I for all t > 0. Also using I (µ,ν)n lim x = l 2, we get K µ,2 (γ, t) I K ν,2 (γ, t) I for all t > 0. Again suppose K µ,ν (γ, t) = (K µ, (γ, t) K µ,2 (γ, t)) (K ν, (γ, t) K ν,2 (γ, t)). Then K µ,ν (γ, t) I. This implies that its complement Kµ,ν(γ, C t) is a non-empty set in F (I). If we take n Kµ,ν(γ, C t), first consider the case n (Kµ,(γ, C t) Kµ,2(γ, C t)). Then we have µ(y, y 2,..., y n, x k l, t 2 ) > γ µ(y, y 2,..., y n, x k l 2, t 2 ) > γ. Clearly, we will get a p N such that µ(y, y 2,..., y n, x p l, t 2 ) > µ(y, y 2,..., y n, x k l, t 2 ) > γ 96

7 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 µ(y, y 2,..., y n, x p l 2, t 2 ) > µ(y, y 2,..., y n, x k l 2, t 2 ) > γ. (e.g., consider max{µ(y, y 2,..., y n, x k l, t 2 ), µ(y, y 2,..., y n, x k l 2, t 2 ) : k J n } choose that k as p for which the maximum occurs). Then we have µ(y, y 2,..., y n, l l 2, t) µ(y, y 2,..., y n, x p l, t 2 ) µ(y, y 2,..., y n, x p l 2, t 2 ) > ( γ) ( γ) > ε. Since ε > 0 is arbitrary, we have µ(y, y 2,..., y n, l l 2, t) = for all t > 0, which implies that l = l 2. Again if n (K C ν,(r, t) K C ν,2(r, t)), then using a similar technique it can be proved that ν(y, y 2,..., y n, l l 2, t) < ε, for all t > 0 arbitrary ε > 0. Thus l = l 2. This proves that I (µ,ν)n lim x is unique. The following result shows that the collection of all ideal convergence sequences in an IFnNLS is closed under addition scalar multiplication. Theorem 3.5. Let X be an IFnNLS. Then () If I (µ,ν)n lim x = l I (µ,ν)n lim y = l 2, then I (µ,ν)n lim(x+y) = l +l 2. (2) If I (µ,ν)n lim x = l α R, then I (µ,ν)n lim αx = αl. (3) If I (µ,ν)n lim x = l I (µ,ν)n lim y = l 2, then I (µ,ν)n lim(x y) = l l 2. Proof. () For a given ε > 0, choose γ > 0 such that ( γ) ( γ) > ε. Now for any t > 0, we define the following sets: K µ, (γ, t) = {k N : µ(y, y 2,..., y n, x k l, t 2 ) > γ} K µ,2 (γ, t) = {k N : µ(y, y 2,..., y n, y k l 2, t 2 ) > γ}. Since I (µ,ν)n lim x = l, clearly K µ, (γ, t) F (I). Similarly, K µ,2 (γ, t) F (I). Let K µ (γ, t) = K µ, (γ, t) K µ,2 (γ, t). Then K µ (γ, t) F (I). Now if k K µ (γ, t), we 97

8 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 have µ(y, y 2,..., y n, x k + y k (l + l 2 ), t) µ(y, y 2,..., y n, x k l, t 2 ) µ(y, y 2,..., y n, y k l 2, t 2 ) > ( γ) ( γ) > ε. This shows that {k I n : µ(y, y 2,..., y n, x k + y k (l + l 2 ), t) ɛ} I. Thus I (µ,ν)n lim(x + y) = l + l 2. (2) Let α = 0. Then for every r > 0, t > 0, y, y 2,..., y n X, there exists n 0 = such that µ(y, y 2,..., y n, 0x k 0l, t) = > r ν(y, y 2,..., y n, 0x k 0l, t) = 0 < r, for all k n 0. Thus (µ, ν) lim 0x = 0l. This implies that I (µ,ν)n lim 0x = 0l. Now let α( 0) R let Since I (µ,ν)n K n (ε, t) = {k N : µ(y, y 2,..., y n, x k l, t α ) > ε}. lim x = l, we have K n (ε, t) F (I). Now if k K n (ε, t), then µ(y, y 2,..., y n, αx k αl, t) = µ(y, y 2,..., y n, x k l, t α ) > ε. Thus {k I n : µ(y, y 2,..., y n, αx k αl, t) ε} I. So I (µ,ν)n lim αx = αl. (3) The proof follows from () (2). Theorem 3.6. Let (X, µ, ν,, ) be an IFnNLS let x = (x k ) be a sequence in X. Let I be a non trivial ideal in N. If there is a I (µ,ν)n -convergent sequence y = (y k ) in X such that y k x k, k J n } I, then x is also I (µ,ν)n -convergent to the same limit. Proof. Suppose that {k N : y k x k } I I (µ,ν)n y = l. Then for every ε (0, ), t > 0 y, y 2,..., y n X, we have µ(y, y 2,..., y n, y k l, t) ε or ν(y, y 2,..., y n, y k l, t) ε} I. For every 0 < ε <, t > 0 y, y 2,..., y n X, we have µ(y, y 2,..., y n, x k l, t) ε or ν(y, y 2,..., y n, x k l, t) ε} y k x k, for some k J n } µ(y, y 2,..., y n, y k l, t) ε 98

9 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 or ν(y, y 2,..., y n, y k l, t) ε}. As both the right-h side member of the above equation are in I, we have µ(y, y 2,..., y n, x k l, t) ε or ν(y, y 2,..., y n, x k l, t) ε} I. Thus x is I (µ,ν)n -convergent. 4. Lambda-convergence in IFnNLS In this section, we introduce the concept of -convergence in IFnNLS establish its relation with I -convergence. Definition 4.. Let (X, µ, ν,, ) be an IFnNLS. A sequence x = {x k } in X is -convergent to l X with respect to the intuitionistic fuzzy n-norm (µ, ν) n, if for every t > 0, ε (0, ) y, y 2,..., y n X, there exists n 0 N such that µ(y, y 2,..., y n, x k l, t) > ε for all n n 0. In this case, we write (µ, ν) n lim x = l. ν(y, y 2,..., y n, x k l, t) < ε Theorem 4.2. Let (X, µ, ν,, ) be an IFnNLS x = {x k } be a sequence in X. If x = {x k } is -convergent with respect to the intuitionistic fuzzy n-norm (µ, ν) n, then (µ, ν) n lim x is unique. Proof. Let (µ, ν) n lim x = l (µ, ν) n lim x = l 2 (l l 2 ). For a fixed ε > 0, choose γ (0, ) such that ( γ) ( γ) > ε γ γ < ε. Now, for every t > 0, y, y 2,..., y n X there exists n N such that µ(y, y 2,..., y n, x k l, t) > ε ν(y, y 2,..., y n, x k l, t) < ε, for all n n. Also there exists n 2 N such that µ(y, y 2,..., y n, x k l 2, t) > ε ν(y, y 2,..., y n, x k l 2, t) < ε, for all n n 2. Consider n 0 = max{n, n 2 }. Then for n n 0, we will get a p N such that µ(y, y 2,..., y n, x p l, t 2 ) > µ(y, y 2,..., y n, x k l, t 2 ) > γ 99

10 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 µ(y, y 2,..., y n, x p l 2, t 2 ) > µ(y, y 2,..., y n, x k l 2, t 2 ) > γ. Thus we have µ(y, y 2,..., y n, l l 2, t) µ(y, y 2,..., y n, x p l, t 2 ) µ(y, y 2,..., y n, x p l 2, t 2 ) > ( γ) ( γ) > ε. Since ε > 0 is arbitrary, we have µ(y, y 2,..., y n, l l 2, t) = for all t > 0, which implies that l = l 2. By using similar technique, it can be proved that ν(y, y 2,..., y n, l l 2, t) < ε, for all t > 0 arbitrary ε > 0. So l = l 2. Hence (µ, ν) n lim x is unique. The following theorem shows that -convergence is stronger than I -convergence. Theorem 4.3. Let (X, µ, ν,, ) be an IFnNLS let x = (x k ) in X. If (µ, ν) n lim x = l, then I (µ,ν)n lim x = l. Proof. Suppose that (µ, ν) n lim x = l. Then for every t > 0, ε (0, ) y, y 2,..., y n X, there exists n 0 N such that µ(y, y 2,..., y n, x k l, t) > ε ν(y, y 2,..., y n, x k l, t) < ε, for all n n 0. Thus, we have A = µ(y, y 2,..., y n, x k l, t) ε or ν(y, y 2,..., y n, x k l, t) ε} {, 2,..., n 0 }. But I being admissible, we have A I. so I (µ,ν)n lim x = l. Theorem 4.4. Let (X, µ, ν,, ) be an IFnNLS x = {x k } a sequence in X. If (µ, ν) n lim x = l, then there exists a subsequence {x m k } of x = (x k ) such that (µ, ν) n lim x mk = l. Proof. Let (µ, ν) n lim x = l. Then for every t > 0 ε (0, ), there exists n 0 N such that µ(y, y 2,..., y n, x k l, t) > ɛ ν(y, y 2,..., y n, x k l, t) < ε, 00

11 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 for all n n 0. Clearly, for each n n 0, we can select an m k J n such that µ(y, y 2,..., y n, x mk l, t) > µ(y, y 2,..., y n, x k l, t) > ε ν(y, y 2,..., y n, x mk l, t) < ν(y, y 2,..., y n, x k l, t) < ε. It follows that (µ, ν) n lim x mk = l. Theorem 4.5. Let X be an IFnNLS I be a nontrivial ideal of N. If a sequence x = {x k } is -convergent in X y = {y k } is a sequence in X such that x k y k for some k J n } I, then y is -convergent to the same limit. Proof. Let ε (0, ), t > 0, y, y 2,..., y n X, x k y k for some k J n } I (µ, ν) n lim x = l. Then µ(y, y 2,..., y n, x k l, t) > ε ν(y, y 2,..., y n, x k l, t) < ε}, for all n n 0. Thus µ(y, y 2,..., y n, x k l, t) > ε ν(y, y 2,..., y n, x k l, t) < ε} / I. On one h, µ(y, y 2,..., y n, x k l, t) ε or ν(y, y 2,..., y n, x k l, t) ε} I. Now, for every 0 < ε <, t > 0 y, y 2,..., y n X, we have µ(y, y 2,..., y n, y k l, t) ε or ν(y, y 2,..., y n, y k l, t) ε} x k y k, for some k J n } µ(y, y 2,..., y n, x k l, t) ε or ν(y, y 2,..., y n, x k l, t) ε}. As both the right-h side member of the above equation are in I, we have µ(y, y 2,..., y n, y k l, t) ε or ν(y, y 2,..., y n, y k l, t) ε} I. So y is (µ, ν) n -convergent. 5. Limit point cluster point in IFnNLS Limit points cluster points are essential concepts in the study of closedness of a set. Here we study the analogous concepts in an IFnNLS. Definition 5.. Let (X, µ, ν,, ) be an IFnNLS. l X is called a limit point of of the sequence x = {x k } with respect to the intuitionistic fuzzy n-norm (µ, ν) n provided that there is a subsequence of x that converges to l with respect to the intuitionistic fuzzy n-norm (µ, ν) n. 0

12 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 Definition 5.2. Let (X, µ, ν,, ) be an IFnNLS let x = (x k ) be a sequence in X. Then (i) An element l X is said to be an I -limit point of x = (x k ), if there is a set M = {m < m 2 <... < m k <...} N such that M = m k J n } / I (µ, ν) n lim x m k = l. (ii) An element l X is said to be an I -cluster point of x = (x k ), if for every t > 0, ε (0, ) y, y 2,..., y n X, we have µ(y, y 2,..., y n, x k l, t) > ɛ or ν(y, y 2,..., y n, x k l, t) < ɛ} / I. Let Λ I (µ,ν) (x) denote the set of all I -limit points Γ I (µ,ν) (x) denote the set of all I -cluster points in X, respectively. Theorem 5.3. Let (X, µ, ν,, ) be an IFnNLS. Then for each sequence x X, Λ I (µ,ν) (x) Γ I (µ,ν) (x). Proof. Let l Λ I (µ,ν) (x). Then there exists a set M N such that M / I, where M M are as in the Definition 5.2, satisfies (µ, ν) n lim x m k = l. Thus for every t > 0, ɛ (0, ) y, y 2,..., y n X, there exists n 0 N such that µ(y, y 2,..., y n, x mk l, t) > ε ν(y, y 2,..., y n, x mk l, t) < ε, for all n n 0. So B = µ(y, y 2,..., y n, x k l, t) > ε ν(y, y 2,..., y n, x k l, t) < ε} M \ {m, m 2,... m k0 }. Now, with I being admissible, we must have M \ {m, m 2,... m k0 } / I as such B / I. Hence l Γ I (µ,ν) (x), i.e., Λ I (µ,ν) (x) Γ I (µ,ν) (x). Theorem 5.4. Let (X, µ, ν,, ) be an IFnNLS. For each sequence x = (x k ) in X, the set Γ I (µ,ν) (x) is closed set in X with respect to the usual topology induced by the intuitionistic fuzzy norm (µ, ν) n. Proof. Let y Γ I (µ,ν) (x). Take t > 0 ε (0, ). Then there exists l 0 Γ I (µ,ν) (x) B(y, ε, t). Choose δ > 0 such that B(l 0, δ, t) B(y, ε, t). We have G = µ(y, y 2,..., y n, x k y, t) > ε ν(y, y 2,..., y n, x k y, t) < ε} µ(y, y 2,..., y n, x k l 0, t) > δ ν(y, y 2,..., y n, x k l 0, t) < δ} = H. Thus H / I. So G / I. Hence y Γ I (µ,ν) (x). Therefore Γ I (µ,ν) (x) is closed set in X. 02

13 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 Theorem 5.5. Let (X, µ, ν,, ) be an IFnNLS let x = (x k ) in X. Then the following statements are equivalent: () l is a I -limit point of x. (2) There exists two sequences y k z k in X such that x = y k +z k (µ, ν) n lim y k = l k J n, z k θ} I, where θ is the zero element of X. Proof. Suppose that () holds. Then there exist sets M M as in Definition 5.2 such that M / I (µ, ν) n lim x m k = l. Define the sequences y z as follows: { xk, if k J y k = n ; n M l, otherwise. { θ, if k Jn ; n M z k = x k l, otherwise. We consider the case k J n such that n N M. Then for each ɛ (0, ), t > 0 y, y 2,..., y n X, we have µ(y, y 2,..., y n, y k l, t) = > ɛ ν(y, y 2,..., y n, y k l, t) = 0 < ɛ. Thus µ(y, y 2,..., y n, y k l, t) = > ε ν(y, y 2,..., y n, y k l, t) = 0 < ε. So (µ, ν) n lim y = l k J n, z k θ} I. Suppose that (2) holds. Let M = k J n, z k = θ}. Then, clearly M F (I) so it is an infinite set. Construct the set M = {m < m 2 <... m k <...} N such that m k J n z mk = θ. Since x mk = y mk (µ, ν) n lim y = l, we obtain (µ, ν) n lim x m k = l. 6. Cauchy sequences in IFnNLS Here we define some variants of Cauchy sequences give a positive answer to the open problem mentioned in [8] that every I -convergent sequence is I -Cauchy in the new setting of IFnNLS. Definition 6.. Let (X, µ, ν,, ) be an IFnNLS. A sequence x = (x k ) in X is said to be -Cauchy sequence with respect to the intuitionistic fuzzy norm (µ, ν) n, if for every t > 0, ε (0, ) y, y 2,..., y n X, there exist n 0, m N satisfying µ(y, y 2,..., y n, x k x m, t) > ε ν(y, y 2,..., y n, x k x m, t) < ε, 03

14 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 for all n n 0. Definition 6.2. Let (X, µ, ν,, ) be an IFnNLS. A sequence x = (x k ) in X is said to be I -Cauchy sequence with respect to the intuitionistic fuzzy norm (µ, ν) n, if for every t > 0, y, y 2,..., y n X ε (0, ) there exist m N satisfying µ(y, y 2,..., y n, x k x m, t) > ε ν(y, y 2,..., y n, x k x m, t) < ε} F (I). Definition 6.3. Let (X, µ, ν,, ) be an IFnNLS. A sequence x = (x k ) in X is said to be I -Cauchy sequence with respect to the intuitionistic fuzzy norm (µ, ν)n, if there exists a set M = {m < m 2 <... < m k <...} N such that the set M = m k J n } F (I) the subsequence (x mk ) of x = (x k ) is a Cauchy sequence with respect to the intuitionistic fuzzy norm (µ, ν) n. Theorem 6.4. Let (X, µ, ν,, ) be an IFnNLS. If a sequence x = (x k ) is I - convergent with respect to (µ, ν) n, then it is I -Cauchy. Proof. Suppose that x = (x k ) be a I -convergent sequence converging to l. For a given ε > 0, choose γ > 0 such that ( γ) ( γ) > ε γ γ < ε. Then for any t > 0 y, y 2,..., y n X, define K µ (γ, t) = µ(y, y 2,..., y n, x k l, t 2 ) > γ} K ν (γ, t) = ν(y, y 2,..., y n, x k l, t 2 ) < γ}. Thus K µ (γ, t) F (I) K ν (γ, t) F (I). Let K(γ, t) = K µ (γ, t) K ν (γ, t). Then K(γ, t) F (I). If n K(γ, t) we choose a fixed m K(γ, t), then µ(y, y 2,..., y n, x k x m, t) µ(y, y 2,..., y n, x k l, t 2 ) µ(y, y 2,..., y n, x m l, t 2 ) > ( γ) ( γ) > ε. This clearly implies that Also, µ(y, y 2,..., y n, x k x m, t) > ε. ν(y, y 2,..., y n, x k x m, t) ν(y, y 2,..., y n, x k l, t 2 ) ν(y, y 2,..., y n, x m l, t 2 ) < γ γ < ε. 04

15 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 So, ν(y, y 2,..., y n, x k x m, t) < ε. Hence µ(y, y 2,..., y n, x k x m, t) > ε n N : ν(y, y 2,..., y n, x k x m, t) < ε} F (I). Therefore x = (x k ) is I -Cauchy. The following is an analogue of Theorem 4.3 in the sense that -Cauchyness is stronger than I -Cauchyness of sequences in IFnNLS whose proof follows likewise. Theorem 6.5. Let (X, µ, ν,, ) be an IFnNLS. If a sequence x = (x k ) in X is -Cauchy with respect to (µ, ν) n, then it is I -Cauchy with respect to same. The proof of the following theorems can be easily established using definitions. Theorem 6.6. Let (X, µ, ν,, ) be an IFnNLS. If a sequence x = (x k ) in X is -Cauchy with respect to (µ, ν) n, then there is a subsequence of x = (x k ) which is ordinary Cauchy sequence with respect to the same norm. Theorem 6.7. Let (X, µ, ν,, ) be an IFnNLS. If a sequence x = (x k ) is I -Cauchy, then it is I -Cauchy as well. 7. Conclusions In this paper, the concept of I -convergence has been introduced in IFnNLS based on this concept some existing results are extended some new results are established. We have also introduced the concept of I -Cauchy sequences established the relation between I -convergence I -Cauchy. In establishing most of the proofs, different approach than their classical analogues has been adopted. The results obtained in this paper are more general than the corresponding results for normed spaces. Acknowledgements. The authors thank the reviewers the Editor-in-Chief for their valuable comments suggestions. References [] K. T. Atanassov, Answer to D. Dubois, S. Gottwald, P. Hajek, J. Kacprzk H. Prade s, paper Terminological difficulities in fuzzy set theory-the case of intuitionistic fuzzy sets, Fuzzy Sets Systems 56 (2005) [2] K. T. Atanassov, Intuitionistic Fuzzy Sets, Fuzzy Sets Systems 20 (986) [3] T. Bag S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. (3) (2003) [4] G. Cattaneo D. Ciucci, Basic intuitionistic principle in fuzzy set theories its extensions (a terminological debate on Atanassov IFS), Fuzzy Sets Systems 57 (2006) [5] T. Beaula L. E. Rani, On closed graph theorem Riesz theorem in intuitionistic 2-fuzzy 2-normed space, Ann. Fuzzy Math. Inform. 9 (5) (205) [6] S. C. Cheng J. N. Mordeson, Fuzzy linear operator fuzzy normed linear spaces, Bull. Cal. Math. Soc. 86 (994) [7] S. G. Dapke C. T. Aage, On intuitionistic fuzzy si-2-normed spaces, Ann. Fuzzy Math. Inform. 0 (3) (205)

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17 N. Konwar, P. Debnath/Ann. Fuzzy Math. Inform. 3 (207), No., 9 07 [37] N. Thillaigovindan, S. A, Shanthi Y. B. Jun, On lacunary statistical convergence in intuitionistic fuzzy n-normed linear spaces, Ann. Fuzzy Math. Inform. (2) (20) 9 3. [38] S. Vijayabalaji, N. Thillaigovindan Y. B. Jun, Intuitionistic fuzzy n-normed linear space, Bull. Korean. Math. Soc. 44 (2007) Pradip Debnath (debnath.pradip@yahoo.com) Department of Mathematics, North Eastern Regional Institute of Science Technology, Nirjuli 7909, Arunachal Pradesh, India Nabanita Konwar (nabnitakonwar@gmail.com) Department of Mathematics, North Eastern Regional Institute of Science Technology, Nirjuli 7909, Arunachal Pradesh, India 07