Ideal convergence of sequences in neutrosophic normed spaces

1 Introduction and background

Theory of fuzzy sets (FSs) was firstly given by Zadeh [1]. This work affected deeply all the scientific fields. The Theory of FSs has submitted to employ imprecise, vagueness and inexact data [1]. FSs, have been widely implemented in different disciplines and technologies. The Theory of FSs cannot always cope with the lack of knowledge of membership degrees. That’s why Atanassov [2] investigated the theory of IFS which the extension of FSs. Atanassov et al. [3] used this concept in decision-making problems. Kramosil and Michalek [4] investigated fuzzy metric space (FMS) utilizing the concepts fuzzy and probabilistic metric space. The FMS as a distance between two points to be a non-negative fuzzy number was examined by Kaleva and Seikkala [5]. George and Veeramani [6] gave some qualifications of FMS. Some basic features of FMS were given and significant theorems were proved in [7]. Moreover, FMS has used by practical researches as for example decision-making, fixed point theory, medical imaging. Park [8] generalized FMSs and defined IF metric space (IFMS). Park utilized George and Veeramani’s [6] opinion of using t-norm and t-conorm to the FMS meantime describing IFMS and investigating its fundamental properties. Saadati and Park [9] initially examined properties of intuitionistic fuzzy normed space (IFNS). The other generalizations are interval-valued FS [10], interval-valued IFS [11], the sets paraconsistent, dialetheist, paradoxist, and tautological [12], Pythagorean fuzzy sets [13].

Smarandache [14, 15] investigated the concept of ‘Neutrosophic set’ (NS) a generalisation of FS, IFS, interval-valued IFS etc. to overcome the complexity arising from uncertainty in solving many practical problems in our routine life activity more precisely. Many times, decision makers face some hesitations beside going to direct approaches (i.e., yes or no) in a decision making. In addition, we can obtain a tri-component outcomes in some real events like sports, procedure for voting etc. Considering all, Smarandache applied the IFS theory by defining a new component namely indeterminacy membership function. Thus, an element belonging to an NS consists of a triplet namely truth-membership function (T), indeterminacy-membership function (I) and falsity-membership function (F). A Neutrosophic set (NS) is determined as a set where every component of the universe has a degree of T, F and I. These three functions are independent in nature in NS. As a result, the concept neutrosophy implies impartial knowledge of thought and then neutral describes the basic difference between neutral, fuzzy, intuitive fuzzy set and logic.

Uncertainty is based on the belongingness degree in IFSs, whereas the uncertainty in NS is considered independently from T and F values. Since no any limitations among the degree of T, F, I, NSs are actually more general than IFS.

Accordingly, Molodtsov [16] introduced soft set theory which has the parametrization adequacy other than different theories dealing with uncertainty. As a result, a scope of works in different directions have been opened to the researchers. Cheng and Mordeson [17] thought an idea of fuzzy norm on a linear space followed by Katsaras [18] and Felbin [19]. Later, Bag and Samanta [20] modified this definition and gave the notion of continuity and boundedness of a linear operator with regards to fuzzy norm [21]. Vijayabalaji et al. [22] worked the IF n-NS and gave some results. Samanta and Jebril [23] studied finite dimensional IFNS in term of t-norm and t-conorm.

Neutrosophic soft linear spaces (NSLSs) were considered by Bera and Mahapatra [24]. Subsequently, in [25, 26], the concept of neutrosophic soft normed linear (NSNLS) was worked and the features of (NSNLS) were examined.

The statistical convergence initially defined by [28]. Statistical convergence in IF normed spaces was given by Karakuş et al [29]. Notable results on this topic can be found in [30–37].

Kirişçi and Şimşek [38] defined new concept known as neutrosophic metric space (NMS) with continuous t-norms and continuous t-conorms. Some notable features of NMS have been examined.

Neutrosophic normed space (NNS) and statistical convergence in NNS has been investigated by Kirişci and Şimşek [39]. Neutrosophic set and neutrosophic logic have been used in applied sciences and theoretical science such as decision making, robotics, summability theory. Some noteworthy results on this topic have been examined in [40–48].

Firstly, we recall some definitions used throughout the paper.

For K ⊂ ℕ and j ∈ ℕ , δ _j  (K) is named jth partial density of K, if δ j ( K ) = | K ∩ { 1 , 2 , . . . , j } | j . If δ ( K ) = lim n → ∞ 1 n | { k ≤ n : k ∈ K } | , ( i . e . , δ ( K ) = lim j → ∞ δ j ( K ) ) exists, it is named the natural density of K. Ψ = { K ⊂ ℕ : δ ( K ) = 0 } is denoted the zero density set.

A sequence (x _n ) is said to be statistically convergent to ξ if for every ɛ > 0, δ ( { n ∈ ℕ : | x k - ξ | ≥ ɛ } ) = 0 , i.e., { n ∈ ℕ : | x k - ξ | ≥ ɛ } ∈ Ψ . We demonstrate st - lim x _n  = ξ or x n → st ξ , (n → ∞).

In the wake of the study of ideal convergence defined by Kostyrko et al. [49], there has been comprehensive research to discover applications and summability studies of the classical theories.

Let ∅ ≠ S be a set, and then a non empty class I ⊆ P ( S ) is said to be an ideal on S iff (i) ∅ ∈ I , (ii) I is additive under union, (iii) for each A ∈ I and each B ⊆ A we find B ∈ I . An ideal I is called non-trivial if I ≠ ∅ and S ∉ I . A non-empty family of sets F is called filter on S iff (i) ∅ ∉ F , (ii) for each A , B ∈ F we get A ∩ B ∈ F , (iii) for every A ∈ F and each B ⊇ A, we obtain B ∈ F . Relationship between ideal and filter is given as follows: F ( I ) = { K ⊂ S : K c ∈ I } , where K ^c  = S - K.

A non-trivial ideal I is (i) an admissible ideal on S iff it contains all singletons.

An admissible ideal I ⊆ 2 N has the property (AP) if for any sequence {A₁, A₂, . . .} of reciprocally disjoint sets of I , there is sequence {B₁, B₂, . . .} of sets such that each symmetric difference A _i ΔB _i (i = 1, 2, . . .) is finite and ⋃ i = 1 ∞ B i ∈ I .

A sequence (x _n ) is said to be ideal convergent to ξ if for every ɛ > 0, i.e. A ( ɛ ) = { n ∈ ℕ : | x n - ξ | ≥ ɛ } ∈ I . Taking I = I δ = { A ⊆ N : δ ( A ) = 0 } , where δ (A) indicates the asymptotic density of set A. If I δ is a non-trivial admissible ideal then ideal convergence coincides with statistical convergence.

We take I as admissible ideal throughout the paper.

Triangular norms (t-norms) (TN) were given by Menger [51]. TNs are used to generalize with the probability distribution of triangle inequality in metric space terms. Triangular conorms (t-conorms) (TC) known as dual operations of TNs. TNs and TCs are important for fuzzy operations (intersections and unions).

Definition 1.1. ([51]) Let ∘ : [0, 1] × [0, 1] → [0, 1] be an operation. When ∘ satisfies following conditions, it is called continuous TN. Take p, q, r, s ∈ [0, 1],

(a) p ∘ 1 = p,

(b) If p ≤ r and q ≤ s, then p ∘ q ≤ r ∘ s,

(c) ∘ is continuous,

(d) ∘ associative and commutative.

Definition 1.2. ([51]) Let • : [0, 1] × [0, 1] → [0, 1] be an operation. When • satisfies following conditions, it is said to be continuous TC.

(a) p • 0 = p,

(b) If p ≤ r and q ≤ s, then p • q ≤ r • s,

(c) • is continuous,

(d) • associative and commutative.

Definition 1.3. ([39]) Let F be a vector space, N = { 〈 u , G ( u ) , B ( u ) , Y ( u ) 〉 : u ∈ F } be a normed space (NS) such that N : F × ℝ + → [ 0 , 1 ] . While following conditions hold, V = ( F , N , ∘ , • ) is said to be NNS. For all u, v ∈ F and λ, μ > 0 and for each σ ≠ 0,

(a) 0 ≤ G ( u , λ ) ≤ 1 , 0 ≤ B ( u , λ ) ≤ 1 , 0 ≤ Y ( u , λ ) ≤ 1 ∀ λ ∈ ℝ + ,

(b) G ( u , λ ) + B ( u , λ ) + Y ( u , λ ) ≤ 3 (for λ ∈ ℝ + ) ,

(c) G ( u , λ ) = 1 (for λ > 0) iff u = 0,

( d ) G ( σ u , λ ) = G ( u , λ | σ | ) ,

(e) G ( u , μ ) ∘ G ( v , λ ) ≤ G ( u + v , μ + λ ) ,

(f) G ( u , . ) is non-decreasing continuous function,

(g) lim λ → ∞ G ( u , λ ) = 1 ,

(h) B ( u , λ ) = 0 (for λ > 0) iff u = 0,

(i) B ( σ u , λ ) = B ( u , λ | σ | ) ,

(j) B ( u , μ ) • B ( v , λ ) ≥ B ( u + v , μ + λ ) ,

(k) B ( u , . ) is non-decreasing continuous function,

(l) lim λ → ∞ B ( u , λ ) = 0 ,

(m) Y ( u , λ ) = 0 (for λ > 0) iff u = 0,

(n) Y ( σ u , λ ) = Y ( u , λ | σ | ) ,

(o) Y ( u , μ ) • Y ( v , λ ) ≥ Y ( u + v , μ + λ ) ,

(p) Y ( u , . ) is non-decreasing continuous function,

(r) lim λ → ∞ Y ( u , λ ) = 0 ,

(s) If λ ≤ 0, then G ( u , λ ) = 0 , B ( u , λ ) = 1 and Y ( u , λ ) = 1 .

Then N = ( G , B , Y ) is called Neutrosophic norm (NN).

Definition 1.4. ([39]) Let V be an NNS, the sequence (x _k ) in V, ɛ ∈ (0, 1) and λ > 0. Then, the sequence (x _k ) is Cauchy in NNS V if there is a N ∈ N such that G ( x k - x m , λ ) > 1 - ɛ , B ( x k - x m , λ ) < ɛ and Y ( x k - x m , λ ) < ɛ for k, m ≥ N.

Definition 1.5. ([39]) A sequence (x _m ) is said to be statistically convergent to ξ ∈ F with regards to NN (SC-NN), if, for each λ > 0 and ɛ > 0 the set P ɛ : = { m ≤ n : G ( x m - ξ , λ ) ≤ 1 - ɛ and B ( x m - ξ , λ ) ≥ ɛ , Y ( x m - ξ , λ ) ≥ ɛ } or equivalently P ɛ : = { m ≤ n : G ( x m - ξ , λ ) > 1 - ɛ or B ( x m - ξ , λ ) < ɛ , Y ( x m - ξ , λ ) < ɛ } . has ND zero. That is d (P _ɛ ) =0 or lim n → ∞ 1 n | { m ≤ n : G ( x m - ξ , λ ) ≤ 1 - ɛ or B ( x m - ξ , λ ) ≥ ɛ , Y ( x m - ξ , λ ) ≥ ɛ } | = 0 . It is denoted by S N -lim x _m  = ξ or x k → ξ ( S N ) . The set of SC-NN will be denoted by S N .

Definition 1.6. ([39]) The sequence (x _k ) is called statistical Cauchy with regards to NN N (SCa-NN) in NNS V, if there exists N = N (ɛ), for every ɛ > 0 and λ > 0 such that C ɛ : = { m ≤ n : G ( x m - x N , λ ) ≤ 1 - ɛ or B ( x m - x N , λ ) ≥ ɛ , Y ( x m - x N , λ ) ≥ ɛ } has ND zero. That is, d (C _ɛ ) = 0.

Ideal convergence in IFS was defined via membership and non-membership functions. Unlike previous studies, this study also takes into account the indeterminacy function while analyzing ideal convergence. The purpose of this study is to present some recent development in NNS. Ideal convergence is more general than statistical convergence for sequences. This has focused us to study the ideal convergence of sequences in NNS. In this study, we investigate some features of this new type of convergence in NNS. Also, it is demonstrated that ideal convergence in NNS is generally different from the ideal convergence in classical normed space, since for there is not any “ N ” function in classical normed space. However; it is shown that if certain conditions are met; every classical normed space can be a NNS at the same time. For the particular case; when the neutrosophic norm is the additive positive integers, our definitions and theorems yield the results of [49, 50]. We have also indicated through an example that, in general I -convergence does not imply I ∗ -convergence in NNS.

2 Main results

Definition 2.1. Take an NNS V. A sequence (x _m ) is said to be ideal convergent to ξ ∈ F with regards to NN ( I C-NN), if, for each ɛ > 0 and λ > 0 { m ∈ ℕ : G ( x m - ξ , λ ) ≤ 1 - ɛ or B ( x m - ξ , λ ) ≥ ɛ , Y ( x m - ξ , λ ) ≥ ɛ } ∈ I . In this case, we write I N -lim x _m  = ξ or x m → ξ ( I N ) . The set of I C-NN will be showed by I N .

Example 2.1. Let (F, || . ||) be a NS, I be a non-trivial admissible ideal. For all a, b ∈ [0, 1], take the TN a ∘ b = ab and the TC a • b = min {a + b, 1}. For all x ∈ F and every λ > 0, we consider G ( x , λ ) = λ λ + ‖ x ‖ , B ( x , λ ) = ‖ x ‖ λ + ‖ x ‖ and Y ( x , λ ) = ‖ x ‖ λ . Then V is an NNS. We define a sequence (x _m ) by x m = { 1 , m = t 2 ( t ∈ ℕ ) 0 , otherwise . Consider A ( ɛ , λ ) = { m ∈ ℕ : G ( x m - ξ , λ ) ≤ 1 - ɛ or B ( x m - ξ , λ ) ≥ ɛ , Y ( x m - ξ , λ ) ≥ ɛ } for every ɛ ∈ (0, 1) and for any λ > 0. Then we have A ( ɛ , λ ) = { m ∈ ℕ : λ λ + ‖ x m ‖ ≤ 1 - ɛ or ‖ x m ‖ λ + ‖ x m ‖ ≥ ɛ , ‖ x m ‖ λ ≥ ɛ } = { m ∈ ℕ : ‖ x m ‖ ≥ λ ɛ 1 - ɛ , or ‖ x m ‖ ≥ λ ɛ } = { m ∈ ℕ : ‖ x m ‖ = 1 } = { m ∈ ℕ : m = t 2 ( t ∈ ℕ ) } will be a finite set. So, δ (A (ɛ, λ)) = 0, and consequently A ( ɛ , λ ) ∈ I , i.e., I N -lim x _k  = 0.

Lemma 2.1. The following situations are equivalent, for every ɛ > 0 and λ > 0, (a) I N -lim x _m  = ξ, (b) { m ∈ ℕ : G ( x m - ξ , λ ) ≤ 1 - ɛ } ∈ I and { m ∈ ℕ : B ( x m - ξ , λ ) ≥ ɛ } ∈ I , { m ∈ ℕ : Y ( x m - ξ , λ ) ≥ ɛ } ∈ I , (c) { m ∈ ℕ : G ( x m - ξ , λ ) > 1 - ɛ and { % B ( x m - ξ , λ ) < ɛ Y ( x m - ξ , λ ) < ɛ } ∈ F ( I ) ,

(d) { m ∈ ℕ : G ( x m - ξ , λ ) > 1 - ɛ } ∈ F ( I ) and { m ∈ ℕ : B ( x m - ξ , λ ) < ɛ } ∈ F ( I ) , { m ∈ ℕ : Y ( x m - ξ , λ ) < ɛ } ∈ F ( I ) ,

(e) I N - lim G ( x m - ξ , λ ) = 1

and I N - lim B ( x m - ξ , λ ) = 0 ,

I N - lim Y ( x m - ξ , λ ) = 0 .

Theorem 2.1. Take an NNS V. If (x _m ) is I C-NN, then I N -lim x _m  = ξ is unique.

Proof. Suppose that I N -lim x _m  = ξ₁ and I N -lim x _m  = ξ₂. Select ɛ > 0. Then, for a given q > 0, (1 - q) ∘ (1 - q) > 1 - ɛ and q • q < ɛ. For any λ > 0, let’s denote the following sets: K G 1 ( q , λ ) = { m ∈ ℕ : G ( x m - ξ 1 , λ 2 ) ≤ 1 - q } , K G 2 ( q , λ ) = { m ∈ ℕ : G ( x m - ξ 2 , λ 2 ) ≤ 1 - q } K B 1 ( q , λ ) = { m ∈ ℕ : B ( x m - ξ 1 , λ 2 ) ≥ q } K B 2 ( q , λ ) = { m ∈ ℕ : B ( x m - ξ 2 , λ 2 ) ≥ q } K Y 1 ( q , λ ) = { m ∈ ℕ : Y ( x m - ξ 1 , λ 2 ) ≥ q } K Y 2 ( q , λ ) = { m ∈ ℕ : Y ( x m - ξ 2 , λ 2 ) ≥ q } . Since I N -lim x _m  = ξ₁, we have K G 1 ( q , λ ) , K B 1 ( q , λ ) , K Y 1 ( q , λ ) ∈ I . Also, using I N -lim x _m  = ξ₂, we get K G 2 ( q , λ ) , K B 2 ( q , λ ) , K Y 2 ( q , λ ) ∈ I .

Now let K N ( q , λ ) : = { K G 1 ( q , λ ) ∪ K G 2 ( q , λ ) } ∩ { K B 1 ( q , λ ) ∪ K B 2 ( q , λ ) } ∩ { K Y 1 ( q , λ ) ∪ K Y 2 ( q , λ ) } . Then, K N ( q , λ ) ∈ I , which implies that ∅ ≠ K N c ( q , λ ) ∈ F ( I ) . If m ∈ K N c ( q , λ ) , then we have three possible cases. That is, m ∈ K G 1 c ( q , λ ) ∩ K G 2 c ( q , λ ) , m ∈ K B 1 c ( q , λ ) ∩ K B 2 c ( q , λ ) or m ∈ K Y 1 c ( q , λ ) ∩ K Y 2 c ( q , λ ) . First, consider that m ∈ K G 1 c ( q , λ ) ∩ K G 2 c ( q , λ ) . Then, we have G ( ξ 1 - ξ 2 , λ ) ≥ G ( x m - ξ 1 , λ 2 ) ∘ G ( x m - ξ 2 , λ 2 ) > ( 1 - q ) ∘ ( 1 - q ) > 1 - ɛ . For arbitrary ɛ > 0, we get G ( ξ 1 - ξ 2 , λ ) = 1 for all λ > 0, which yields ξ₁ = ξ₂. On the other hand, if we take m ∈ K B 1 c ( q , λ ) ∪ K B 2 c ( q , λ ) , then we can write B ( ξ 1 - ξ 2 , λ ) ≤ B ( x m - ξ 1 , λ 2 ) • B ( x m - ξ 2 , λ 2 ) ≤ q • q < ɛ . Therefore, we can see that B ( ξ 1 - ξ 2 , λ ) < ɛ . For all λ > 0, we obtain B ( ξ 1 - ξ 2 , λ ) = 0 , which implies that ξ₁ = ξ₂. Again, for the situation m ∈ K Y 1 c ( q , λ ) ∩ K Y 2 c ( q , λ ) , then we can write Y ( ξ 1 - ξ 2 , λ ) ≤ Y ( x m - ξ 1 , λ 2 ) • Y ( x m - ξ 2 , λ 2 ) ≤ q • q < ɛ . For all λ > 0, we have Y ( ξ 1 - ξ 2 , λ ) = 0 . Thus ξ₁ = ξ₂. In all cases, we conclude that the I N -limit is unique.□

Theorem 2.2. If N -lim x _k  = ξ for NNS V, then I N -lim x _m  = ξ.

Proof. If N -lim x _m  = ξ, then, for every λ > 0 and ɛ > 0, there exists a number r 0 ∈ ℕ such that G ( x m - ξ , λ ) > 1 - ɛ and B ( x m - ξ , λ ) < ɛ , Y ( x m - ξ , λ ) < ɛ , for all m ≥ r₀. Therefore, A ( ɛ ) = { m ∈ ℕ : G ( x m - ξ , λ ) ≤ 1 - ɛ or B ( x m - ξ , λ ) ≥ ɛ , Y ( x m - ξ , λ ) ≥ ɛ } is contained in {1, 2, . . . , r₀ - 1}. We conclude that A ( ɛ ) ∈ I . Hence I N -lim x _m  = ξ.□

Theorem 2.3. Choose an NNS V. Then

(a) If I N -lim x _m  = ξ₁ and I N -lim y _m  = ξ₂, then I N -lim(x _m  + y _m ) = ξ₁ + ξ₂.

(b) If I N -lim x _m  = ξ, then I N -lim αx _m  = αξ.

Proof. (a) Let I N -lim x _m  = ξ₁ and I N -lim y _m  = ξ₂. Choose q > 0. Then, for a given ɛ > 0 (1 - q) ∘ (1 - q) > 1 - ɛ and q • q < ɛ. Then, for any λ > 0, we define the following sets:

K G 1 ( q , λ ) , K G 2 ( q , λ ) , K B 1 ( q , λ ) , K B 2 ( q , λ ) ,

K Y 1 ( q , λ ) and K Y 2 ( q , λ ) as above. Since I N -lim x _m  = ξ₁, we have K G 1 ( q , λ ) , K B 1 ( q , λ ) and K Y 1 ( q , λ ) ∈ I . Further, using I N -lim x _m  = ξ₂, K G 2 ( q , λ ) , K B 2 ( q , λ ) and K Y 2 ( q , λ ) ∈ I . Now let K N ( q , λ ) = { K G 1 ( q , λ ) ∪ K G 2 ( q , λ ) } ∩ { K B 1 ( q , λ ) ∪ K B 2 ( q , λ ) } ∩ { K Y 1 ( q , λ ) ∪ K Y 2 ( q , λ ) } . Then K N ( q , λ ) ∈ I , which implies that ∅ ≠ K N c ( q , λ ) ∈ F ( I ) . Now we have to show that K N c ( q , λ ) ⊂ { m ∈ ℕ : G ( ( x m + y m ) - ( ξ 1 + ξ 2 ) , λ ) > 1 - ɛ and B ( ( x m + y m ) - ( ξ 1 + ξ 2 ) , λ ) < ɛ , Y ( ( x m + y m ) - ( ξ 1 + ξ 2 ) , λ ) < ɛ } . If m ∈ K N c ( ɛ , λ ) , then we get G ( x m - ξ 1 , λ 2 ) > 1 - q , G ( y m - ξ 2 , λ 2 ) > 1 - q , B ( x m - ξ 1 , λ 2 ) < q , B ( y m - ξ 2 , λ 2 ) < q , Y ( x m - ξ 1 , λ 2 ) < q and Y ( y m - ξ 2 , λ 2 ) < q . Therefore G ( ( x m + y m ) - ( ξ 1 + ξ 2 ) , λ ) ≥ G ( x m - ξ 1 , λ 2 ) ∘ G ( y m - ξ 2 , λ 2 ) > ( 1 - q ) ∘ ( 1 - q ) > 1 - ɛ , and B ( ( x m + y m ) - ( ξ 1 + ξ 2 ) , λ ) ≤ B ( x m - ξ 1 , λ 2 ) • B ( y m - ξ 2 , λ 2 ) ≤ q • q < ɛ , Y ( ( x m + y m ) - ( ξ 1 + ξ 2 ) , λ ) ≤ Y ( x m - ξ 1 , λ 2 ) • Y ( y m - ξ 2 , λ 2 ) ≤ q • q < ɛ . This shows that K N c ( q , λ ) ⊂ { m ∈ ℕ : G ( ( x m + y m ) - ( ξ 1 + ξ 2 ) , λ ) > 1 - ɛ and B ( ( x m + y m ) - ( ξ 1 + ξ 2 ) , λ ) < ɛ , Y ( ( x m + y m ) - ( ξ 1 + ξ 2 ) , λ ) < ɛ } . Since

K N c ( q , λ ) ∈ F ( I ) , I N -lim(x _m  + y _m ) = ξ₁ + ξ₂.

(b) This is obvious for α = 0. Let α ≠ 0. Then, for given λ > 0 and ɛ > 0, B ( ɛ ) = { m ∈ ℕ : G ( x m - ξ , λ ) > 1 - ɛ and B ( x m - ξ , λ ) < ɛ , Y ( x m - ξ , λ ) < ɛ } ∈ F ( I ) .(2.1) It is sufficient to show that, for each λ > 0 and ɛ > 0, B ( ɛ ) ⊂ { m ∈ ℕ : G ( α x m - α ξ , λ ) > 1 - ɛ and B ( α x m - α ξ , λ ) < ɛ , Y ( α x m - α ξ , λ ) < ɛ } . Let m ∈ B (ɛ). Then, we have G ( x m - ξ , λ ) > 1 - ɛ and B ( x m - ξ , λ ) < ɛ , Y ( x m - ξ , λ ) < ɛ . So we get G ( α x m - α ξ , λ ) = G ( x m - ξ , λ | α | ) ≥ G ( x m - ξ , λ ) ∘ G ( 0 , λ | α | - λ ) = G ( x m - ξ , λ ) ∘ 1 = G ( x m - ξ , λ ) ≥ 1 - ɛ . Furthermore, B ( α x m - α ξ , λ ) = B ( x m - ξ , λ | α | ) ≤ B ( x m - ξ , λ ) • B ( 0 , λ | α | - λ ) = B ( x m - ξ , λ ) • 0 = B ( x m - ξ , λ ) < ɛ , and Y ( α x m - α ξ , λ ) = Y ( x m - ξ , λ | α | ) ≤ Y ( x m - ξ , λ ) • Y ( 0 , λ | α | - λ ) = Y ( x m - ξ , λ ) • 0 = Y ( x m - ξ , λ ) < ɛ . It is clear that B ( ɛ ) ⊂ { m ∈ ℕ : G ( α x m - α ξ , λ ) > 1 - ɛ and B ( α x m - α ξ , λ ) < ɛ , Y ( α x m - α ξ , λ ) < ɛ } . and from (2.1) we get I N -lim αx _m  = αξ.□

Now, we introduce the concept of I ∗ -convergence in NNS.

Definition 2.2. A sequence (x _m ) is said to be I N ∗ -convergent to ξ ∈ F with regards to NN, if there exists a subset K = { j 1 < j 2 < . . . } ⊂ ℕ such that K ∈ F ( I ) (i.e., ℕ ∖ K ∈ I ) and N - lim k x j k = ξ . In this case, we denote I N ∗ -lim x = ξ.

Theorem 2.4. Take an NNS V. If I N ∗ -lim x = ξ, then I N -lim x = ξ.

Proof. Assume that I N ∗ -lim x = ξ. Then there exists a subset K = { j 1 < j 2 < . . . } ⊂ ℕ such that K ∈ F ( I ) and N - lim k x j k = ξ . But then for each λ > 0 and ɛ > 0, there exists r₀ > 0 such that G ( x j k - ξ , λ ) > 1 - ɛ and B ( x j k - ξ , λ ) < ɛ , Y ( x j k - ξ , λ ) < ɛ for all k > r₀. Since { j k ∈ K : G ( x j k - ξ , λ ) ≤ 1 - ɛ or B ( x j k - ξ , λ ) ≥ ɛ , Y ( x j k - ξ , λ ) ≥ ɛ } is contained in {j₁ < j₂ < . . . < j_r0-1}. Therefore, we get { j k ∈ K : G ( x j k - ξ , λ ) ≤ 1 - ɛ or B ( x j k - ξ , λ ) ≥ ɛ , Y ( x j k - ξ , λ ) ≥ ɛ } ∈ I . Hence, { m ∈ ℕ : G ( x m - ξ , λ ) ≤ 1 - ɛ or B ( x m - ξ , λ ) ≥ ɛ , Y ( x m - ξ , λ ) ≥ ɛ } ⊂ H ∪ { j 1 < j 2 < . . . < j r 0 - 1 } ∈ I for all ɛ > 0 and λ > 0, i.e., I N - lim x = ξ.□

Remark 2.1. The following example denotes that the converse of Theorem 2.4 need not be true.

Example 2.2. Let ( ℝ , ‖ . ‖ ) be a NS. For all a, b ∈ [0, 1], TN a ∘ b = ab and the TC a • b = min {a + b, 1}. For all x ∈ F and every λ > 0, we consider G ( x , λ ) = λ λ + ‖ x ‖ , B ( x , λ ) = ‖ x ‖ λ + ‖ x ‖ and Y ( x , λ ) = ‖ x ‖ λ . Then ( ℝ , N , ∘ , • ) be NNS. Let ℕ = ∪ i Δ i be a decomposition of ℕ such that, for any m ∈ ℕ , each Δ _i contains infinitely many i’s, where i ≥ m and Δ _i ∩ Δ _m  = ∅ for i ≠ m. Now we define a sequence x m = 1 i if m ∈ Δ _i . Then G ( x m , λ ) = λ λ + ‖ x m ‖ → 1 and B ( x m , λ ) = ‖ x m ‖ λ + ‖ x m ‖ → 0 , Y ( x m , λ ) = ‖ x m ‖ λ → 0 as m→ ∞. Hence I N - lim m x m = 0 . Now assume that I N ∗ - lim m x m = 0 . Then there exists a subset K = { m 1 < m 2 < . . . } ⊂ ℕ such that K ∈ F ( I ) (i.e., ℕ ∖ K ∈ I ) and N - lim j x m j = 0 . Since K ∈ F ( I ) , then there exists a set H ∈ F ( I ) such that K = ℕ ∖ H . Now, from the definition of I , there exists, say, q ∈ ℕ such that H ⊂ ⋃ m = 1 q Δ m . But then Δ _p  ⊂ K, and therefore x m j = 1 ( q + 1 ) 2 > 0 for infinitely many m _j ’s from K. This contradicts that N - lim j x m j = 0 . Hence, the assumption I N ∗ - lim m x m = 0 is wrong.

Remark 2.2. Above example gives that I N ∗ -convergence implies I N -convergence but not conversely. The question arises under which condition the converse may hold. For this, under the property (AP) the converse holds.

Theorem 2.5. Let V be an NNS and the ideal I has property (AP). If I N -lim x = ξ, then I N ∗ -lim x = ξ.

Proof. Assume that I satisfies condition (AP) and I N -lim x = ξ. Then for each ɛ > 0 and λ > 0, { m ∈ ℕ : G ( x m - ξ , λ ) ≤ 1 - ɛ or B ( x m - ξ , λ ) ≥ ɛ , Y ( x m - ξ , λ ) ≥ ɛ } ∈ I . We define the set T _n for n ∈ ℕ and λ > 0 as T n = { m ∈ ℕ : 1 - 1 n ≤ G ( x m - ξ , λ ) < 1 - 1 n + 1 or 1 n + 1 < B ( x m - ξ , λ ) ≤ 1 n , 1 n + 1 < Y ( x m - ξ , λ ) ≤ 1 n } . Obviously {T₁, T₂, . . .} is countable and belongs to I , and T _i ∩ T _j  = ∅ for i ≠ j. From the condition (AP), there is countable family of sets { V 1 , V 2 , . . . } ∈ I such that T _i ΔV _i is a finite set for i ∈ ℕ and V = ⋃ i = 1 ∞ V i ∈ I . Using the definition of the associate filter F ( I ) there is a set M ∈ F ( I ) such that M = ℕ ∖ V . To prove the theorem we have to show that the subsequence (x _m ) is convergent to ξ with regards to NN. if, for each γ > 0 and λ > 0. Choose q ∈ ℕ such that 1 q < γ . Then { m ∈ ℕ : G ( x m - ξ , λ ) ≤ 1 - γ or B ( x m - ξ , λ ) ≥ γ , Y ( x m - ξ , λ ) ≥ γ } ⊂ { m ∈ ℕ : G ( x m - ξ , λ ) ≤ 1 - 1 q or B ( x m - ξ , λ ) ≥ 1 q , Y ( x m - ξ , λ ) ≥ 1 q } ⊂ ⋃ i = 1 q + 1 T i . Since T _i ΔV _i , i = 1, 2, . . . , q + 1 are finite, there exists m 0 ∈ ℕ such that ( ⋃ i = 1 q + 1 V i ) ∩ { m ∈ ℕ : m > m 0 } = ( ⋃ i = 1 q + 1 T i ) ∩ { m ∈ ℕ : m > m 0 } .(2.2) If m > m₀ and m ∈ M, so m ∉ ⋃ i = 1 q + 1 V i and by (2.2) m ∉ ⋃ i = 1 q + 1 T i . Hence, for every m > m₀ and m ∈ ℕ , we have G ( x m - ξ , λ ) > 1 - γ and B ( x m - ξ , λ ) < γ , Y ( x m - ξ , λ ) < γ . Since γ > 0 is arbitrary, we get I N ∗ -lim x = ξ.□

Theorem 2.6. Choose an NNS V. Following conditions are equivalent:

(a) I N ∗ -lim x = ξ.

(b) There exist two sequences y = (y _m ) and z = (z _m ) such that x = y + z, N -lim y _m  = ξ and the set { m : z m ≠ θ } ∈ I , where θ indicates the zero element of N .

Proof. Assume that the condition (a) holds. So, there exists a set K = { j 1 < j 2 < . . . } ⊂ ℕ such that K ∈ F ( I ) and N - lim k x j k = ξ .(2.3) We take the sequences y = (y _m ) and z = (z _m ) as follows: y m = { x m , if m ∈ K ξ , if m ∈ K c ; and z _m  = x _m  - y _m for all m ∈ ℕ . For given λ > 0, ɛ > 0 and m ∈ K ^c , we get G ( x m - ξ , λ ) = 1 > 1 - ɛ and { % B ( x m - ξ , λ ) = 0 < ɛ , Y ( x m - ξ , λ ) = 0 < ɛ . Using (2.3), we have N -lim y _m  = ξ.

Since {m : z _m  ≠ θ} ⊂ K ^c , we have { m : z m ≠ θ } ∈ I .

Let the condition (b) holds and K = {m : z _m  ≠ θ}. Obviously K ∈ F ( I ) is an infinite set. Let K = {j₁ < j₂ < . . .}. Since x _m  = y _m and N -lim y _m  = ξ, N -lim x _m  = ξ. Hence I N ∗ -lim x = ξ.□

Now, we define I N - and I N ∗ -Cauchy sequences in NNS and prove that I N -convergence and I N -Cauchy are equivalent in NNS.

Definition 2.3. A sequence x = (x _m ) is called to be I N -Cauchy with regards to NN N ( I Ca - NN ) in NNS V, if, for every λ > 0 and ɛ > 0, there exists N = N (ɛ) such that, for all m ≥ N, { m ∈ ℕ : G ( x m - x N , λ ) ≤ 1 - ɛ or { % B ( x m - x N , λ ) ≥ ɛ , Y ( x m - x N , λ ) ≥ ɛ } ∈ I .

Definition 2.4. A sequence x = (x _k ) is said to be I N ∗ -Cauchy with regards to NN if there exists a subset K = { j 1 < j 2 < . . . } ⊂ ℕ such that K ∈ F ( I ) and the subsequence (x _{j k} ) is an ordinary Cauchy sequence with regards to NN.

The following results are analogues to our Theorems 2.4 and 2.5, respectively, and can be proved on similar lines.

Theorem 2.7. If a sequence x = (x _m ) I N ∗ -Cauchy with regards to NN, then it is I N -Cauchy with regards to NN.

Theorem 2.8. Let V be an NNS and the ideal I satisfy (AP). If a sequence x = (x _m ) is I N -Cauchy with regards to NN, then it is I N ∗ -Cauchy with regards to NN.

Definition 2.5. An NNS is said to be ideal complete if every I N -Cauchy sequence with regards to NN is ideal convergent with regards to NN.

Theorem 2.9. A sequence x = (x _m ) is I C - NN with regards to NN iff if it is I N -Cauchy with regards to NN.

Proof. Necessity: Let x = (x _m ) be I C - NN to ξ with regards to NN, i.e., I N -lim x _m  = ξ. Choose η > 0 such that (1 - η) ∘ (1 - η) > 1 - ɛ and η • η < ɛ. For any λ > 0, we have A = { m ∈ ℕ : G ( x m - ξ , λ ) ≤ 1 - η or { % B ( x m - ξ , λ ) ≥ η , Y ( x m - ξ , λ ) ≥ η } ∈ I .(2.4) This gives that ∅ ≠ A c ∈ F ( I ) .

Let p ∈ A ^c . Then we have G ( x p - ξ , λ ) > 1 - η and B ( x p - ξ , λ ) < η , Y ( x p - ξ , λ ) < η . Now consider B = { m ∈ ℕ : G ( x m - x p , λ ) ≤ 1 - ɛ or B ( x m - x p , λ ) ≥ ɛ , Y ( x m - x p , λ ) ≥ ɛ } . We need to prove that B ⊂ A. Let m ∈ B. Then, we get G ( x m - x p , λ 2 ) ≤ 1 - ɛ or B ( x m - x p , λ 2 ) ≥ ɛ , Y ( x m - x p , λ 2 ) ≥ ɛ . We have there possible cases. We first consider that G ( x m - x p , λ ) ≤ 1 - ɛ . Then, we get G ( x m - ξ , λ 2 ) ≤ 1 - η ; therefore m ∈ A. Otherwise, if G ( x m - ξ , λ 2 ) > 1 - η then 1 - ɛ ≥ G ( x m - x p , λ ) ≥ G ( x m - ξ , λ 2 ) ∘ G ( x p - ξ , λ 2 ) > 1 - ɛ , which is not possible. Hence B ⊂ A.