Matrix Transformations and Statistical Convergence II

Advances in Dynamical Systems and Applications ISSN 0973-532, Volume 6, Number, pp. 7 89 20 http://campus.mst.edu/adsa Matrix Transformations and Statistical Convergence II Bruno de Malafosse LMAH Université du Havre BP 4006 IUT Le Havre 7660 Le Havre, France bdemalaf@wanadoo.fr Vladimir Rakočević University of Niš Department of Mathematics Videgradska 33 8000 Niš, Serbia vrakoc@ptt.rs Abstract In this paper we extend some of our recent results given in [5], so we consider a matrix transformation A = a nk n,k and say that a sequence X = x n n is A-statistically convergent to L V with respect to the intuitionistic fuzzy normed space IFNS V if lim n n {k n : ν [AX] k L, t ε or µ [AX] k L, t ε} = 0 for any ε > 0. The aim of this paper is to give conditions on X to have A-statistical convergence on IFNS V. Then, among other things, we consider the cases when A is either of the matrices Ñq,χ, N q, N p N q, D /τ λ, or C λ. AMS Subject Classifications: 40C05, 40J05, 46A5. Keywords: Matrix transformations, intuitionistic fuzzy normed space, statistical convergence, operator of weighted mean. Introduction Fuzzy set theory was introduced in 965 by Zadeh [22] and used to obtain many results in set theory. Fuzzy logic was used among other things in the study of nonlinear dynamical systems [2], in control of chaos [6], in population dynamics []. The fuzzy topology has many applications in quantum particle physics, see [4]. In this paper we consider the notion of intuitionistic fuzzy normed space briefly IFNS, and we define and deal with A-statistical convergence on IFNS, where A = a nk n,k is an infinite matrix. Then we give applications to the cases when A is either Received October 5, 200; Accepted December 7, 200 Communicated by Mališa R. Žižović

72 Bruno de Malafosse and Vladimir Rakočević of the matrices Ñq,χ, N q, N p N q, D /τ λ, or C λ. These results extend in a certain sense those given in [5]. First recall some notations and definitions used in this paper. In the following we will write I = [0, ]. Definition.. The map T : I I I is said to be of type N if it satisfies the conditions a T is associative, b T is continuous and commutative, c for every a, b, c, d I, the condition a c and b d implies at b ct d. Definition.2 See [20]. The map : I I I is said to be a continuous t-norm if it is of type N with a = a for all a I. Definition.3 See [20]. The map : I I I is said to be a continuous t-conorm if it is of type N with a 0 = a for all a I. There are some examples of maps that are continuous t-norms or continuous t- conorms. Consider the case when V = R, a b = ab, a b = min {a, b}, a b = min {a + b, } and a b = max {a, b} for all a, b I. Definition.4. Let V be a vector space, be a continuous t-norm and a continuous t-conorm and µ, ν be fuzzy sets on V 0, that is, µ, ν : V 0, [0, ]. We say that V, µ, ν,, is an intuitionistic fuzzy normed space IFNS if for every x, y V and s, t > 0 we have a µ x, t + ν x, t, b µ x, t > 0, c µ x, t = if and only if x = 0, d µ αx, t = µ x, t/ α for all α 0, e µ x, t µ y, s µ x + y, t + s, f µ x,. : 0, I is continuous, g lim t µ x, t = and lim t 0 µ x, t = 0, h ν x, t,

Matrix Transformations and Statistical Convergence II 73 i ν x, t = 0 if and only if x = 0, j ν αx, t = ν x, t/ α for all α 0, k ν x, t ν y, s ν x + y, t + s, l ν x,. : 0, I is continuous, m lim t ν x, t = 0 and lim t 0 ν x, t =. In this case µ, ν is called an intuitionistic fuzzy norm briefly IFN. We give a standard example. Example.5. Let V be a vector space with norm and let a b = ab and a b = min {a + b, } for all a, b I. Put µ x, t = t and ν x, t = x t + x It can be easily be shown that V, µ, ν,, is an IFNS. for all x V and t > 0. t + x 2 Convergence with Respect to the IFN µ, ν 2. Convergence on IFNS We write s V for the set of all complex sequences on V, and write s for the set of all complex or real sequences. Definition 2. See [20]. Let V, µ, ν,, be an IFNS. A sequence X = x n n s V is said to be convergent to L V with respect to the IFN µ, ν if for every ε, t > 0 there is k 0 N depending on ε and t such that µ x k L, t > ε and ν x k L, t < ε for all k k 0. Then we write µ, ν lim X = L or x k µ,ν L k. In the same way we will say that µ lim X = L or x k ε, t > 0 there is k 0 N depending on ε and t such that µ L k if for every µ x k L, t > ε for all k k 0. For a given sequence X s V, L V and ε, t > 0, put Γ µ,ν X, ε, t = {k N : µ x k L, t > ε and ν x k L, t < ε}, Γ ν X, ε, t = {k N : ν x k L, t < ε}

74 Bruno de Malafosse and Vladimir Rakočević and Γ µ X, ε, t = {k N : µ x k L, t > ε}. It can easily be seen that x k µ,ν L k if for every ε, t > 0 there is k 0 such that [k 0, + [ Γ µ,ν X, ε, t. µ Similarly, x k L k if for all ε, t > 0 there is k0 with [k 0, + [ Γ µ X, ε, t. To simplify we write Γ µ,ν X, Γ ν X and Γ µ X instead of Γ µ,ν X, ε, t, Γ ν X, ε, t and Γ µ X, ε, t. We state the following elementary lemma. Lemma 2.2. Let V, µ, ν,, be an IFNS. Then for given X s V, L V and ε, t > 0 we have Γ µ,ν X = Γ µ X and Γ µ X Γ ν X. Proof. Trivially Γ µ,ν X Γ µ X. Now take k Γ µ X. Then µ x k L, t > ε and from Definition.4 a we have ν x k L, t µ x k L, t < ε. This shows that k Γ µ,ν X and Γ µ X Γ µ,ν X. We conclude Γ µ,ν X = Γ µ X. Then trivially Γ µ,ν X = Γ µ X Γ ν X. This completes the proof. As a direct consequence of Lemma 2.2, we obtain the next lemma. Lemma 2.3. Let V, µ, ν,, be an IFNS. Then x k µ L k. x k µ,ν L k if and only if µ,ν Proof. If x k L k for every ε, t > 0, then there is k 0 such that [k 0, + [ µ Γ µ,ν X, and since Γ µ,ν X = Γ µ X, we deduce x k L. The converse can be shown similarly. Remark 2.4. The convergence µ, ν lim X = L is equivalent to the convergence µ lim X = L, and it is not necessary to introduce the first convergence. In this paper, we introduce the first convergence, mainly to keep the conventional approach. Furthermore, in the special case, in Example.5, µ, ν-convergence and norm convergence for the sequence x n n are equivalent. Remark 2.5. Let us remark that the notion of intuitionistic fuzzy metric spaces was introduced by Park in [8]. In Gregori, Romaguera, Veeramani [] see also Saadati, Sedghi, Shobe [9], it was shown that Park s definition of intuitionistic fuzzy metric spaces contains extra conditions and can be derived, in an equivalent manner, from the definition of fuzzy metric spaces.

Matrix Transformations and Statistical Convergence II 75 2.2 Statistical Convergence on IFNS 2.2. Statistical Convergence The notion of statistical convergence was first introduced by Steinhaus in 949, see [2], and studied by several authors such as Fast, Fridy [5, 7 0] and Connor. The idea of statistical convergence suggests many other possible lines of investigation. We mention here that recently Di Maio and Kočinac [2] introduced and investigated statistical convergence in topological and uniform spaces and showed how this convergence can be applied to selection principles theory, function spaces and hyperspaces. Di Maio, Djurčić, Kočinac and Žižović [3] considered the set of sequences of positive real numbers in the context of statistical convergence and showed that some of its subclasses have certain nice selection and game-theoretic properties. A sequence X = x n n tends statistically to L if for each ε > 0 we have K X, n, ε = o n, n where K X, n, ε = {k n : x k L ε} and the symbol denotes the number of elements in the enclosed set. In this case we write x k L S. To simplify we put K X, n = K X, n, ε. For the convenience of the reader, we recall the following well-known lemma. Lemma 2.6. Let K and K 2 be two subsets of N with K K 2 then lim n n {k n : k K 2} = 0 lim n n {k n : k K } = 0. Remark 2.7. It is to be noted see [0] that every convergent sequence is statistically convergent with the same limit so that statistical convergence is a natural generalization of the usual convergence of sequences. A sequence which is statistically convergent may neither be convergent nor bounded. This is also demonstrated by the following example. Let us consider the sequence a i i whose terms are i when i = n 2 for all n =, 2, 3, and a i = /i otherwise. Then, the sequence a i i is divergent in the ordinary sense, while 0 is the statistical limit of a i i. Not all properties of convergent sequences are true for statistical convergence. It is well known that a subsequence of a convergent sequence is convergent, however, for statistical convergence this is not true. The sequence b i i whose terms are i, i =, 2, 3, is a subsequence of the statistically convergent sequence a i i, and b i i is statistically divergent. 2.2.2 Statistical Convergence with respect to the IFN Let ε, t > 0, n N and let V, µ, ν,, be an IFNS. For given X s and L V, put K µ X, n, ε, t = {k n : µ x k L, t ε}, K ν X, n, ε, t = {k n : ν x k L, t ε}

76 Bruno de Malafosse and Vladimir Rakočević and K µ,ν X, n, ε, t = {k n : ν x k L, t ε or µ x k L, t ε}. To simplify we write K µ X, n, K ν X, n and K µ,ν X, n instead of K µ X, n, ε, t, K ν X, n, ε, t and K µ,ν X, n, ε, t. Definition 2.8. Let V, µ, ν,, be an IFNS. A sequence X is said to be statistically convergent to L V with respect to the IFN µ, ν provided that for every ε, t > 0 we have n K µ,ν X, n = o n. In this case we write x k L µ,ν S. Definition 2.9. We say that x k L µ S if for each ε, t > 0 we have 2.2.3 Condition ϕ on IFNS n K µ X, n = o n. In the case when V is a set of scalars R or C, we say that an IFN µ, ν satisfies condition ϕ if there is a function ϕ : ]0, [ ]0, [ such that for every x V and t > 0 µ x, t ϕ t x. 2. Consider the classical example where V = R, a b = ab, a b = min {a + b, } and µ x, t = t, ν x, t = µ x, t = x t + x for all x R and t > 0. t + x We have and so ϕ t = /t. ν x, t = x t + x t x Lemma 2.0. Let V, µ, ν,, be an IFNS and let X s, L V, ε, t > 0 and n N. Then i a K µ,ν X, n = K µ X, n and K ν X, n K µ X, n; b x k L µ,ν S if and only if x k L µ S; c x k L µ,ν S implies x k L ν S. ii If V is a set of scalars and µ, ν satisfies condition ϕ, then K µ X, n, εϕ t, t K X, n, ε.

Matrix Transformations and Statistical Convergence II 77 Proof. We first show i. Take k K µ,ν X, n. By Definition.4 a we have µ x k L, t ν x k L, t. 2.2 Then for any given ε, t > 0, ν x k L, t ε implies µ x k L, t ε and K µ,ν X, n K µ X, n. By definition of K µ,ν X, n, it is obvious that K µ X, n K µ,ν X, n. Thus, we conclude K µ,ν X, n = K µ X, n. Now we show K ν X, n K µ X, n. For this we take k K ν X, n. We have ν x k L, t ε, and since 2.2 holds we conclude k K µ,ν X, n = K µ X, n. This proves a, while b and c are direct consequences of a and Lemma 2.6. Next we show ii. For each integer k and t > 0 we have µ x k L, t ϕ t x k L with ϕ : ]0, [ ]0, [. So k K µ X, n, εϕ t, t means that µ x k L, t εϕ t and by 2. that is, k K X, n, ε. x k L [ µ x k L, t] /ϕ t εϕ t /ϕ t = ε, 3 A-Statistical Convergence with Respect to IFN µ, ν In this section we deal with A-statistical convergence with respect to the IFN µ, ν. For this we recall some results on matrix transformations and w 0. Then we give conditions ensuring x k L µ,ν S A. 3. Matrix Transformations For a given infinite matrix A = a nk n,k we define the operators A n for any n N by A n X = a nk x k, 3. k= where X = x n n, the series intervening in the second member being convergent. So we are led to the study of the infinite linear system A n X = b n n N, 3.2 where b = b n n is a one-column matrix and X is the unknown. System 3.2 can be written in the form AX = b, where AX = A n X n. In this paper we also consider A as an operator from a sequence space into another sequence space. We write c 0 for the sets of null sequences. For E, F s we will denote by E, F the set of all matrix transformations A = a nk n,k that map E to F, see [6]. A Banach space E of complex sequences with the norm E is a BK space if each projection P n : X P n X = x n is continuous. A BK space E s is said to have AK

78 Bruno de Malafosse and Vladimir Rakočević if every sequence X = x n n E has a unique representation X = x n e n, where e n is the sequence with in the n-th position, and 0 otherwise. For given F s and a given matrix A, we will put F A = {X s : AX F }. The matrix T is a triangle if [T ] nn 0 for all n and [T ] nk = 0 for k > n. Here we use triangles represented by C λ and λ for a given sequence λ with λ n 0 for all n. We define C λ = c nk n,k by c nk = λ n if k n, 0 otherwise. It was proved in [3] that the matrix λ = c nm n,m with λ n if k = n, c nk = λ n if k = n and n 2, 0 otherwise is the inverse of C λ. Using the notation e =,,..., we write = e and Σ = C e. In the following we use the operators represented by C λ and λ. Let U be the set of all sequences u n n with u n 0 for all n. We define the triangle C λ = c nm n,m for λ = λ n n U by c nk = /λ n for k n. It can be proved that the triangle λ with [ λ] nn = λ n for all n, [ λ] n,n = λ n for n 2, is the inverse of C λ, see [3]. Denote by U + the set of all sequences X = x n n such that x n > 0 for all n. We also use the set Γ of all sequences X U + with lim n x n /x n <. When λ n = n for all n, we get the well known space w 0 or w 0 studied by Maddox see e.g., [], i.e., w 0 = { It is well known that w 0 normed by X = x n n : lim n n X = sup n n } x k = 0. k= x k is a BK space with AK. It can easily be deduced that the class w 0, w 0 is a Banach algebra see [5] normed by AX A = sup. 3.3 X 0 X k= n=

Matrix Transformations and Statistical Convergence II 79 In the following we will apply these results. Obviously, c 0 w 0. Actually, c 0 is a proper subset of w 0. For example, the following sequence x n n is in w 0 but not in c 0 : x n = if n is prime number, and x n = /n otherwise. 3.2 A-Statistical Convergence on IFNS In this subsection we extend some results obtained in [5] where we dealt with A- statistical convergence. First recall some definitions used in [5]. Let L be a scalar and A E, F. For ε > 0, we will use the notation K X, n, A = {k n : [AX] k L ε} where we assume that every series [AX] k = A k X = a km x m for k is convergent. We say that X = x n n s A-statistically convergent to L if for every ε > 0, we have lim K X, n, A = 0. n n We then write x k L S A. Now let X s and let V, µ, ν,, be an IFNS. For given L V and ε > 0, put K µ,ν X, n, A = {k n : ν [AX] k L, t ε or µ [AX] k L, t ε} We say that X = x n n is A-statistically convergent to L with respect to the IFN µ, ν if for every ε, t > 0, we have m= n K µ,ν X, n, A = o n. We then write x k L µ,ν S A. As an immediate consequence of Lemma 2.2, we have K µ,ν X, n, A = K µ X, n, A and K ν X, n, A K µ X, n, A. Now we state a lemma where for a given subvector space Φ s and for L V, we write L +Φ for the set of all sequences of the form X = Le + Y with Y Φ. Let us recall that w 0 A is defined in Subsection 3.. Lemma 3.. Let V be a set of scalars and let V, µ, ν,, be an IFNS and assume µ, ν satisfies condition ϕ. i If x k L S, then x k L µ,ν S. ii Let A be an infinite matrix. If x k L S A, then x k L µ,ν S A.

80 Bruno de Malafosse and Vladimir Rakočević iii Let A be a triangle. If X LA e + w 0 A, then x k L µ,ν S A. Proof. Since K µ X, n K X, n and x k L S, we have x k L µ,ν S. Thus i holds. For ε > 0, the inequalities imply ϕ t [AX] k L µ [AX] k L, t ε K µ,ν X, n, A = K µ X, n, A K X, n, A. Now x k L S A means lim n K X, n, A /n = 0, and by Lemma 2.6, we have lim n K µ,ν X, n, A /n = 0 and x k L µ,ν S A. Thus ii holds. Since A is a triangle, we put l = A e. Putting λ = n n, we obtain [C λ AX Le ] n = n [AX] k L k=. and C λ AX Le = C λ A X L l Thus Then X L l + w 0 A means [C λ AX Le ] n = n lim n n = n [ A k= and we deduce from the preceding that n [ A k= ] X L l [AX] k L k= k KX,n,A ε K X, n, A. n ] X L l lim K X, n, A = 0 n n k k [AX] k L = 0, and x k L S A. By part ii, we conclude x k L µ,ν S A. Thus iii holds.

Matrix Transformations and Statistical Convergence II 8 4 Ñ q,χ -, N q - and N p N q -Statistical Convergence on IFNS In this section we give conditions on the sequence X = x n n guaranteeing x k L µ,ν S Ñq,χ, where Ñq,χ is a matrix band. Similarly we obtain conditions on X to successively have x k L µ,ν S N q and xk L µ,ν S N p N q, where N q is the matrix of weighted means. 4. Ñ q,χ -Statistical Convergence on IFNS To state the next result, we will use an infinite matrix band defined for given χ N and q U + by q Q. q χ 0 Q... Ñ q,χ = q Q. q χ Q 0.. with Q = χ q k. Then we obtain the next result. k= Proposition 4.. Let V, µ, ν,, be an IFNS with ν = µ. If then x k L µ,ν S Proof. We have and since Q = [Ñq,χ X] lim n n lim n n ν x k L, t ν x k+χ L, t = 0, k= Ñq,χ, that is, for every ε, t > 0, { k n : ν [Ñq,χ X] χ q i, we deduce i= k = Q Q χ q i x k+i L, t ε} = 0. i= χ q i x k+i for all X s, i= χ L = q i x k+i QL = k Q Q i= χ q i x k+i L, 4. i=

82 Bruno de Malafosse and Vladimir Rakočević and by Definition.4 j, we get χ χ ν q i x k+i L, t = ν q i x k+i L, Qt. 4.2 Q i= By Definition.4 k, we get χ χ ν q i x k+i L, Qt = ν q i x k+i L, i= and by Definition.4 j, we have i= i= χ tq i i= ν q x k L, q t ν q χ x k+χ L, q χ t, ν q i x k L, q i t = ν x k L, q i t/q i = ν x k L, t, i =, 2,..., χ. Then χ ν q i x k+i L, Qt ν x k L, t ν x k+χ L, t. 4.3 i= Now put k = ν x k L, t ν x k+χ L, t. Since K ν X, n, Ñq,χ [, n], by 4., 4.2 and 4.3, we obtain n k n k= So the condition k k w 0 implies and x k L µ,ν S N q,χ. n ε n ν [Ñq,χ ] L, t k k= ] ν [Ñq,χ k K νx,ñq,χ k K ν X, n, Ñq,χ. Kν X, n, N n q,χ 0 n L, t 4.2 Applications to A-Statistical Convergence on IFNS To study N q - and N p N q -statistical convergence, we need to state some results on the sets W 0 τ C λ and W 0 τ λ.

Matrix Transformations and Statistical Convergence II 83 4.2. The Sets W 0 τ λ and W 0 τ C λ For a given sequence τ = τ n n U +, we define the infinite diagonal matrix D τ = τ n δ nm n,m. For any subset E of s, we will write D τ E for the set of sequences x n n such that x n /τ n n Eand put Wτ 0 = D τ w 0 for τ U +. The space W τ = { X : sup n k= x k τ k < is called the set of sequences that are strongly τ-convergent to zero. Then we will explicitly give the sets Wτ 0 C λ and Wτ 0 λ. Note that we have { } Wτ 0 k C λ = X : lim x i <, n n λ k τ k k= i= { } Wτ 0 λ = X : lim λ k x k λ k x k <. n n τ k k= For a given sequence ρ = ρ n n, we will consider the triangle ρ defined by [ ρ ] nn = for all n, [ ρ ] n,n = ρ n for all n 2. Recall the next result which is a direct consequence of [4, Theorem 5. and Theorem 5.2]. Lemma 4.2. If lim ρ n <, 4.4 n then for any given b w 0, the equation ρ X = b has a unique solution in w 0. We immediately deduce the following result. Lemma 4.3. Let λ, τ U +. Then i If τ Γ, then the operators and Σ are bijective from W 0 τ into itself, and ii } W 0 τ = W 0 τ, W 0 τ Σ = W 0 τ. a If λτ Γ, then W 0 τ C λ = W 0 λτ. b If τ Γ, then W 0 τ λ = W 0 τ/λ. Proof. i By Lemma 2.2, where ρ n = τ n /τ n and λ n = n for all n, we easily see that if τ n lim <, n τ n that is, τ Γ, then D /τ D τ is bijective from w 0 to itself. This means that is bijective from D τ w 0 to itself. Since Σ is also bijective from D τ w 0 to itself, this shows Wτ 0 = Wτ 0 and Wτ 0 Σ = Wτ 0. ii We have X Wτ 0 C λ if and only if ΣX D λτ w 0 = Wλτ. 0 This means that X Wλτ 0 Σ and by i the condition λτ Γ implies Wλτ 0 Σ = W λτ. Then Wτ 0 C λ = Wλτ 0 and C λ is bijective from Wλτ 0 to Wτ 0. Since λ = C λ we conclude λ is bijective from Wτ 0 to Wλτ 0 and Wλτ 0 λ = Wτ 0. We deduce that for τ Γ we have Wτ 0 λ = Wτ/λ. 0

84 Bruno de Malafosse and Vladimir Rakočević 4.2.2 N q - and N p N q -Statistical Convergence on IFNS The operator of weighted means N q is defined for q U + and Q n = q k for all n by k= [ N q ] nk = First we state the following result. { qk Q n for k n, 0 otherwise. Proposition 4.4. Let V be a set of scalars and let µ, ν be an IFN satisfying condition ϕ. If k q i x i L w 0, 4.5 then x k L µ,ν S N q. Proof. First we have Q k i= k X LN q e + w 0 N q. 4.6 Since N q e = e implies that N q e = e, condition 4.6 means that N q X Le w 0 and 4.5 holds. We conclude by Lemma 3. iii that x k L µ,ν S N q. This concludes the proof. Example 4.5. Consider the classical example, where V = R, a b = ab, a b = min {a + b, } and µ x, t = t x, ν x, t = µ x, t = for all x R and t > 0. t + x t + x Assume for every t > 0 we have x i L ν q i, t Q k = q i x i L Q k t + q i x i L Q k and by Proposition 4.4, the condition lim n n implies x k L µ,ν S N q. Q k k= tq k q i x i L for i =, 2,..., k, k q i x i L = 0 i=

Matrix Transformations and Statistical Convergence II 85 Now among other things we consider the case when µ, ν is an IFN satisfying condition ϕ. Theorem 4.6. i Let Q = Q n n Γ. Then x k L S N q for all X L + DQ/q w 0. ii Let V be a set of scalars and let µ, ν be an IFN satisfying condition ϕ. Then a that is, Hence { k n : µ n x k L µ,ν S N q for all X L + DQ/q w 0, Q k lim n n k= q k Q k x k L = 0. k } q i x i L, t ε = 0 n. i= b Let P, P Q/p Γ. If that is, X L + D P Q/pq w 0, then x k L µ,ν S N p N q, Thus, for every ε, t > 0, { k n : µ n = n P k lim n n k p j Q j= j k= p k q k P k Q k x k L = 0. j q i x i L, t ε} i= { k n : µ [ N p N q X ] k L, t ε } = 0 n. Proof. i is a direct consequence of [5, Corollary 3, pp. 38 382]. ii a By Lemma 3. iv, if there is ϕ : ]0, [ ]0, [ such that 2. holds, then x k L S N q implies x k L µ,ν S N q. b We have [ N p N q X Le = N p N q X L ] N p N q e = N p N q X Le since N p N q e = N q if N p e = N q e = e. Then N p N q X Le w 0 if and only X Le w 0 N p N q.

86 Bruno de Malafosse and Vladimir Rakočević We can explicitly give w 0 N p N q. For this note that N q = D /Q ΣD q = D/q Σ D Q and since Σ = we deduce N q = D /q D Q = D /q Q. Similarly we obviously have N p = D /p D P = D /p P. By Lemma 4.3, since P Γ, we immediately get WP 0 = WP 0 and So N p w 0 = D /p WP 0 = D /p WP 0 = WP/p. 0 w 0 N p N q = N q N p Since P Q/p Γ by Lemma 4.3 ii b we conclude w 0 = N WP/p 0 = D /q Q WP/p. 0 Q W 0 P/p = W 0 P/p C Q = W 0 P Q/p, w 0 N p N q = W 0 P Q/pq. Finally X Le w 0 N p N q means that X Le W 0 P Q/pq and x k L S N p N q. Because there is ϕ : ]0, [ ]0, [ such that 2. holds, we conclude x k L µ,ν S N p N q. We also obtain the following result which is a direct consequence of the preceding. Corollary 4.7. Let C, µ, ν,, be an IFNS satisfying condition ϕ. Let < a < b and put k y k = a j j b i x a k b j i. j= i= Then { } ab n k n : µ y k L a b, t ε = 0 n. 4.7 Proof. It is enough to take p = a k and q = b k. So trivially k k P k Q k p k = a k+ a b k+ b a k a b and P Q/p Γ. The calculation gives y k = q ab k+ a b a k = ab k+ a b ab [ N p N q X ] a b. k Then putting L = Lab/ a b, by Theorem 4.6 ii b, we conclude x k L µ,ν S N p N q and 4.7 holds.

Matrix Transformations and Statistical Convergence II 87 5 Other Applications In this section we deal with D /τ λ- and C λ-statistical convergence on IFNS. Theorem 5.. Let V be a set of scalars and let µ, ν be an IFN satisfying condition ϕ. i Let τ Γ. If n k= λ k τ k x k L τ +... + τ k 0 n, 5. λ k then x k L µ,ν S D/τ λ. Thus { } λk x k λ k x k n k n : µ L, t ε = 0 n. ii Let λ Γ. If n k= τ k then x k L µ,ν S C λ. Thus { x +... + x k n k n : µ Proof. i We have x k L µ,ν S D/τ λ if x k L λ k 0 n, 5.2 λ k λ k λ k } L, t ε = 0 n. D /τ λ X Le w 0. 5.3 Further 5.3 holds if D /τ λ X LC λ D τ e w 0, that is, X LC λ D τ e w 0 D /τ λ. But w 0 D /τ λ = W 0 τ λ and since τ Γ, we deduce by Lemma 4.3 ii that w 0 D /τ λ = W 0 τ/λ. Then [C λ D τ e] n = τ +... + τ n λ n and X LC λ D τ e W 0 τ/λ is equivalent to 5.. This concludes the proof of i. ii Here x k L µ,ν S C λ if C λ X Le = C λ X L λ e w 0. 5.4 Since λ Γ and from Lemma 4.3 ii condition 5.4 is equivalent to X L λ e w 0 C λ = W 0 λ. Now [ λ e] n = λ n λ n and X L λ e W 0 λ means that 5.2 holds. This concludes the proof of ii.

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