Matrix Transformations and Statistical Convergence II
|
|
- Beatrice Beasley
- 5 years ago
- Views:
Transcription
1 Advances in Dynamical Systems and Applications ISSN , Volume 6, Number, pp 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 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
2 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,
3 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}
4 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.
5 Matrix Transformations and Statistical Convergence II 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 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}
6 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 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.
7 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
8 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=
9 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.
10 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.
11 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=
12 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.
13 Matrix Transformations and Statistical Convergence II 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
14 84 Bruno de Malafosse and Vladimir Rakočević 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=
15 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 ]. 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.
16 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.
17 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 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 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.
18 88 Bruno de Malafosse and Vladimir Rakočević Acknowledgement The authors thank the referee for his/her contribution to improve some results and the presentation of the paper. The second author is supported by Grant No of the Ministry of Science and Technological Development, Republic of Serbia. References [] L. C. Barros, R. C. Bassanezi and P. A. Tonelli, Fuzzy modelling in population dynamics, Ecol. Model , [2] G. Di Maio and Lj. D. R. Kočinac, Statistical convergence in topology, Topology Appl , [3] G. Di Maio, D. Djurčić, Lj. D. R. Kočinac and M. R. Žižović, Statistical convergence, selection principles and asymptotic analysis, Chaos, Solitons and Fractals , [4] M. S. El Naschie, On uncertainty of Cantorian geometry and two-slit experiment, Chaos, Solitons and Fractals 9 998, [5] H. Fast, Sur la convergence statistique, Colloq. Math. 2 95, [6] A. L. Fradkov and R. J. Evans, Control of chaos: Methods and applications in engineering, Chaos, Solitons and Fractals , [7] J. A. Fridy, On statistical convergence, Analysis 5 985, [8] J. A. Fridy, Statistical limit points, Proc. Amer. Math. Soc , [9] J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math , [0] J. A. Fridy and C. Orhan, Statistical core theorems, J. Math. Anal. Appl , [] V. Gregori, S. Romaguera and P. Veeramani, A note on intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals , [2] L. Huang and J. Q. Sun, Bifurcations of fuzzy nonlinear dynamical systems, Commun. Nonlinear Sci. Numer. Simul. 2006, 2. [3] B. de Malafosse, On some BK space, Internat. J. Math. Sci. Math ,
19 Matrix Transformations and Statistical Convergence II 89 [4] B. de Malafosse and E. Malkowsky, The Banach algebra w λ, w λ, Far East J. Math. Sci., in press 200. [5] B. de Malafosse and V. Rakočević, Matrix transformations and statistical convergence, Linear Algebra Appl , [6] I. J. Maddox, Infinite Matrices of Operators, Springer-Verlag, Berlin, Heidelberg and New York, 980. [7] E. Malkowsky and V. Rakočević, An introduction into the theory of sequence spaces and measure of noncompactness, Zbornik radova, Matematički institut SANU , [8] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals , [9] R. Saadati, S. Sedghi and S. N. Shobe, Modified intuitionistic fuzzy metric spaces and some fixed point theorems, Chaos, Solitons and Fractals , [20] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math , [2] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 95, [22] L. A. Zadeh, Fuzzy sets, Inform. Control 8 965,
Czechoslovak Mathematical Journal
Czechoslovak Mathematical Journal Oktay Duman; Cihan Orhan µ-statistically convergent function sequences Czechoslovak Mathematical Journal, Vol. 54 (2004), No. 2, 413 422 Persistent URL: http://dml.cz/dmlcz/127899
More informationFuzzy Statistical Limits
Fuzzy Statistical Limits Mark Burgin a and Oktay Duman b a Department of Mathematics, University of California, Los Angeles, California 90095-1555, USA b TOBB Economics and Technology University, Faculty
More information1. Introduction EKREM SAVAŞ
Matematiqki Bilten ISSN 035-336X Vol.39 (LXV) No.2 205 (9 28) UDC:55.65:[57.52:59.222 Skopje, Makedonija I θ -STATISTICALLY CONVERGENT SEQUENCES IN TOPOLOGICAL GROUPS EKREM SAVAŞ Abstract. Recently, Das,
More informationSOME FIXED POINT RESULTS FOR ADMISSIBLE GERAGHTY CONTRACTION TYPE MAPPINGS IN FUZZY METRIC SPACES
Iranian Journal of Fuzzy Systems Vol. 4, No. 3, 207 pp. 6-77 6 SOME FIXED POINT RESULTS FOR ADMISSIBLE GERAGHTY CONTRACTION TYPE MAPPINGS IN FUZZY METRIC SPACES M. DINARVAND Abstract. In this paper, we
More informationIntuitionistic Fuzzy Metric Groups
454 International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012 Intuitionistic Fuzzy Metric Groups Banu Pazar Varol and Halis Aygün Abstract 1 The aim of this paper is to introduce the structure
More informationOn the statistical type convergence and fundamentality in metric spaces
Caspian Journal of Applied Mathematics, Ecology and Economics V. 2, No, 204, July ISSN 560-4055 On the statistical type convergence and fundamentality in metric spaces B.T. Bilalov, T.Y. Nazarova Abstract.
More informationContraction Maps on Ifqm-spaces with Application to Recurrence Equations of Quicksort
Electronic Notes in Theoretical Computer Science 225 (2009) 269 279 www.elsevier.com/locate/entcs Contraction Maps on Ifqm-spaces with Application to Recurrence Equations of Quicksort S. Romaguera 1,2
More informationBanach Algebras of Matrix Transformations Between Spaces of Strongly Bounded and Summable Sequences
Advances in Dynamical Systems and Applications ISSN 0973-532, Volume 6, Number, pp. 9 09 20 http://campus.mst.edu/adsa Banach Algebras of Matrix Transformations Between Spaces of Strongly Bounded and Summable
More informationLacunary Statistical and Lacunary Strongly Convergence of Generalized Difference Sequences in Intuitionistic Fuzzy Normed Linear Spaces
Bol. Soc. Paran. Mat. (3s.) v. 38 (2020): 7 29. c SPM ISSN-275-88 on line ISSN-0037872 in press SPM: www.spm.uem.br/bspm doi:0.5269/bspm.v38i.3484 Lacunary Statistical and Lacunary Strongly Convergence
More informationExtremal I-Limit Points Of Double Sequences
Applied Mathematics E-Notes, 8(2008), 131-137 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Extremal I-Limit Points Of Double Sequences Mehmet Gürdal, Ahmet Şahiner
More informationSome topological properties of fuzzy cone metric spaces
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 2016, 799 805 Research Article Some topological properties of fuzzy cone metric spaces Tarkan Öner Department of Mathematics, Faculty of Sciences,
More informationOn Ideal Convergent Sequences in 2 Normed Spaces
Thai Journal of Mathematics Volume 4 006 Number 1 : 85 91 On Ideal Convergent Sequences in Normed Spaces M Gürdal Abstract : In this paper, we investigate the relation between I-cluster points and ordinary
More informationOn Statistical Limit Superior, Limit Inferior and Statistical Monotonicity
International Journal of Fuzzy Mathematics and Systems. ISSN 2248-9940 Volume 4, Number 1 (2014), pp. 1-5 Research India Publications http://www.ripublication.com On Statistical Limit Superior, Limit Inferior
More informationLacunary Statistical Convergence on Probabilistic Normed Spaces
Int. J. Open Problems Compt. Math., Vol. 2, No.2, June 2009 Lacunary Statistical Convergence on Probabilistic Normed Spaces Mohamad Rafi Segi Rahmat School of Applied Mathematics, The University of Nottingham
More informationOn Pointwise λ Statistical Convergence of Order α of Sequences of Fuzzy Mappings
Filomat 28:6 (204), 27 279 DOI 0.2298/FIL40627E Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Pointwise Statistical Convergence
More informationApplied Mathematics Letters
Applied Mathematics Letters 25 (2012) 545 549 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On the equivalence of four chaotic
More informationLacunary statistical cluster points of sequences
Mathematical Communications (2006), 39-46 39 Lacunary statistical cluster points of sequences S. Pehlivan,M.Gürdal and B. Fisher Abstract. In this note we introduce the concept of a lacunary statistical
More informationKeywords. 1. Introduction.
Journal of Applied Mathematics and Computation (JAMC), 2018, 2(11), 504-512 http://www.hillpublisher.org/journal/jamc ISSN Online:2576-0645 ISSN Print:2576-0653 Statistical Hypo-Convergence in Sequences
More informationOn the Spaces of Nörlund Almost Null and Nörlund Almost Convergent Sequences
Filomat 30:3 (206), 773 783 DOI 0.2298/FIL603773T Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On the Spaces of Nörlund Almost
More informationLinear Algebra and its Applications
Linear Algebra and its Applications 436 2012 41 52 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Compactness of matrix
More informationBulletin of the. Iranian Mathematical Society
ISSN: 1017-060X Print) ISSN: 1735-8515 Online) Bulletin of the Iranian Mathematical Society Vol. 41 2015), No. 2, pp. 519 527. Title: Application of measures of noncompactness to infinite system of linear
More informationI and I convergent function sequences
Mathematical Communications 10(2005), 71-80 71 I and I convergent function sequences F. Gezer and S. Karakuş Abstract. In this paper, we introduce the concepts of I pointwise convergence, I uniform convergence,
More informationCOMMON FIXED POINT THEOREMS FOR SIX WEAKLY COMPATIBLE SELF-MAPPINGS IN M- FUZZY METRIC SPACES
COMMON FIXED POINT THEOREMS FOR SIX WEAKLY COMPATIBLE SELF-MAPPINGS IN M- FUZZY METRIC SPACES 1 N. Appa Rao, 2 Dr. V. Dharmaiah, 1 Department of Mathematics Dr. B.R. Ambedkar Open University Hyderabad
More informationThe small ball property in Banach spaces (quantitative results)
The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence
More informationON WEAK STATISTICAL CONVERGENCE OF SEQUENCE OF FUNCTIONALS
International Journal of Pure and Applied Mathematics Volume 70 No. 5 2011, 647-653 ON WEAK STATISTICAL CONVERGENCE OF SEQUENCE OF FUNCTIONALS Indu Bala Department of Mathematics Government College Chhachhrauli
More informationIN PROBABILITY. Kragujevac Journal of Mathematics Volume 42(1) (2018), Pages
Kragujevac Journal of Mathematics Volume 4) 08), Pages 5 67. A NOTION OF -STATISTICAL CONVERGENCE OF ORDER γ IN PROBABILITY PRATULANANDA DAS, SUMIT SOM, SANJOY GHOSAL, AND VATAN KARAKAYA 3 Abstract. A
More informationFUZZY H-WEAK CONTRACTIONS AND FIXED POINT THEOREMS IN FUZZY METRIC SPACES
Gulf Journal of Mathematics Vol, Issue 2 203 7-79 FUZZY H-WEAK CONTRACTIONS AND FIXED POINT THEOREMS IN FUZZY METRIC SPACES SATISH SHUKLA Abstract. The purpose of this paper is to introduce the notion
More informationZweier I-Convergent Sequence Spaces
Chapter 2 Zweier I-Convergent Sequence Spaces In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generation builds
More informationI λ -convergence in intuitionistic fuzzy n-normed linear space. Nabanita Konwar, Pradip Debnath
Annals of Fuzzy Mathematics Informatics Volume 3, No., (January 207), pp. 9 07 ISSN: 2093 930 (print version) ISSN: 2287 6235 (electronic version) http://www.afmi.or.kr @FMI c Kyung Moon Sa Co. http://www.kyungmoon.com
More informationOn the statistical and σ-cores
STUDIA MATHEMATICA 154 (1) (2003) On the statistical and σ-cores by Hüsamett in Çoşun (Malatya), Celal Çaan (Malatya) and Mursaleen (Aligarh) Abstract. In [11] and [7], the concets of σ-core and statistical
More informationThe generalized multiplier space and its Köthe-Toeplitz and null duals
MATHEMATICAL COMMUNICATIONS 273 Math. Commun. 222017), 273 285 The generalized multiplier space and its Köthe-Toeplitz and null duals Davoud Foroutannia and Hadi Roopaei Department of Mathematics, Vali-e-Asr
More informationλ-statistical convergence in n-normed spaces
DOI: 0.2478/auom-203-0028 An. Şt. Univ. Ovidius Constanţa Vol. 2(2),203, 4 53 -statistical convergence in n-normed spaces Bipan Hazarika and Ekrem Savaş 2 Abstract In this paper, we introduce the concept
More informationFIBONACCI STATISTICAL CONVERGENCE ON INTUITIONISTIC FUZZY NORMED SPACES arxiv: v1 [math.gm] 3 Dec 2018
FIBONACCI STATISTICAL CONVERGENCE ON INTUITIONISTIC FUZZY NORMED SPACES arxiv:82.024v [math.gm] 3 Dec 208 MURAT KIRIŞCI Abstract. In this paper, we study the concept of Fibonacci statistical convergence
More informationFUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES. Yılmaz Yılmaz
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 33, 2009, 335 353 FUNCTION BASES FOR TOPOLOGICAL VECTOR SPACES Yılmaz Yılmaz Abstract. Our main interest in this
More informationSigned Measures and Complex Measures
Chapter 8 Signed Measures Complex Measures As opposed to the measures we have considered so far, a signed measure is allowed to take on both positive negative values. To be precise, if M is a σ-algebra
More informationI-CONVERGENT TRIPLE SEQUENCE SPACES OVER n-normed SPACE
Asia Pacific Journal of Mathematics, Vol. 5, No. 2 (2018), 233-242 ISSN 2357-2205 I-CONVERGENT TRIPLE SEQUENCE SPACES OVER n-normed SPACE TANWEER JALAL, ISHFAQ AHMAD MALIK Department of Mathematics, National
More informationASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES ABSTRACT
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying
More informationInternational Journal of Mathematical Archive-5(10), 2014, Available online through ISSN
International Journal of Mathematical Archive-5(10), 2014, 217-224 Available online through www.ijma.info ISSN 2229 5046 COMMON FIXED POINT OF WEAKLY COMPATIBLE MAPS ON INTUITIONISTIC FUZZY METRIC SPACES
More informationDrazin Invertibility of Product and Difference of Idempotents in a Ring
Filomat 28:6 (2014), 1133 1137 DOI 10.2298/FIL1406133C Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Drazin Invertibility of
More informationON COMMON FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN INTUITIONISTIC FUZZY METRIC SPACES
IJRRAS 9 (1) October 11 www.arpapress.com/volumes/vol9issue1/ijrras_9_1_9.pdf ON COMMON FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN INTUITIONISTIC FUZZY METRIC SPACES Kamal Wadhwa 1 & Hariom Dubey
More informationA Generalized Contraction Mapping Principle
Caspian Journal of Applied Mathematics, Ecology and Economics V. 2, No 1, 2014, July ISSN 1560-4055 A Generalized Contraction Mapping Principle B. S. Choudhury, A. Kundu, P. Das Abstract. We introduce
More informationMoore-Penrose-invertible normal and Hermitian elements in rings
Moore-Penrose-invertible normal and Hermitian elements in rings Dijana Mosić and Dragan S. Djordjević Abstract In this paper we present several new characterizations of normal and Hermitian elements in
More informationJordan Journal of Mathematics and Statistics (JJMS) 8(3), 2015, pp THE NORM OF CERTAIN MATRIX OPERATORS ON NEW DIFFERENCE SEQUENCE SPACES
Jordan Journal of Mathematics and Statistics (JJMS) 8(3), 2015, pp 223-237 THE NORM OF CERTAIN MATRIX OPERATORS ON NEW DIFFERENCE SEQUENCE SPACES H. ROOPAEI (1) AND D. FOROUTANNIA (2) Abstract. The purpose
More informationSome Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph
J o u r n a l of Mathematics and Applications JMA No 39, pp 81-90 (2016) Some Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph Hamid
More informationA Common Fixed Point Theorem For Occasionally Weakly Compatible Mappings In Fuzzy Metric Spaces With The (Clr)-Property
Advances in Fuzzy Mathematics. ISSN 0973-533X Volume 11, Number 1 (2016), pp. 13-24 Research India Publications http://www.ripublication.com A Common Fixed Point Theorem For Occasionally Weakly Compatible
More informationSOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES. S. S. Dragomir
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 5: 011), 151 16 DOI: 10.98/FIL110151D SOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR
More informationBi-parameter Semigroups of linear operators
EJTP 9, No. 26 (2012) 173 182 Electronic Journal of Theoretical Physics Bi-parameter Semigroups of linear operators S. Hejazian 1, H. Mahdavian Rad 1,M.Mirzavaziri (1,2) and H. Mohammadian 1 1 Department
More informationSOME GENERALIZED SEQUENCE SPACES OF INVARIANT MEANS DEFINED BY IDEAL AND MODULUS FUNCTIONS IN N-NORMED SPACES
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 39 208 579 595) 579 SOME GENERALIZED SEQUENCE SPACES OF INVARIANT MEANS DEFINED BY IDEAL AND MODULUS FUNCTIONS IN N-NORMED SPACES Kuldip Raj School of
More informationC.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series
C.7 Numerical series Pag. 147 Proof of the converging criteria for series Theorem 5.29 (Comparison test) Let and be positive-term series such that 0, for any k 0. i) If the series converges, then also
More informationI K - convergence in 2- normed spaces
Functional Analysis, Approximation and Computation 4:1 (01), 1 7 Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/faac I K - convergence
More information1 Introduction and preliminaries
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 5: (0), 5 3 DOI: 0.98/FIL05D MULTIVALUED GENERALIZATIONS OF THE KANNAN FIXED POINT THEOREM
More informationOn the spectrum of linear combinations of two projections in C -algebras
On the spectrum of linear combinations of two projections in C -algebras Julio Benítez and Vladimir Rakočević Abstract In this note we study the spectrum and give estimations for the spectral radius of
More informationTHE NEARLY ADDITIVE MAPS
Bull. Korean Math. Soc. 46 (009), No., pp. 199 07 DOI 10.4134/BKMS.009.46..199 THE NEARLY ADDITIVE MAPS Esmaeeil Ansari-Piri and Nasrin Eghbali Abstract. This note is a verification on the relations between
More informationA Common Fixed Point Theorems in Menger(PQM) Spaces with Using Property (E.A)
Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 4, 161-167 A Common Fixed Point Theorems in Menger(PQM) Spaces with Using Property (E.A) Somayeh Ghayekhloo Member of young Researchers club, Islamic
More informationQuintic Functional Equations in Non-Archimedean Normed Spaces
Journal of Mathematical Extension Vol. 9, No., (205), 5-63 ISSN: 735-8299 URL: http://www.ijmex.com Quintic Functional Equations in Non-Archimedean Normed Spaces A. Bodaghi Garmsar Branch, Islamic Azad
More informationOn Fixed Point Results for Matkowski Type of Mappings in G-Metric Spaces
Filomat 29:10 2015, 2301 2309 DOI 10.2298/FIL1510301G Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Fixed Point Results for
More informationMATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:
MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is
More informationTHE SEMI ORLICZ SPACE cs d 1
Kragujevac Journal of Mathematics Volume 36 Number 2 (2012), Pages 269 276. THE SEMI ORLICZ SPACE cs d 1 N. SUBRAMANIAN 1, B. C. TRIPATHY 2, AND C. MURUGESAN 3 Abstract. Let Γ denote the space of all entire
More informationFixed point theorems of nondecreasing order-ćirić-lipschitz mappings in normed vector spaces without normalities of cones
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 18 26 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Fixed point theorems of nondecreasing
More informationORLICZ - PETTIS THEOREMS FOR MULTIPLIER CONVERGENT OPERATOR VALUED SERIES
Proyecciones Vol. 22, N o 2, pp. 135-144, August 2003. Universidad Católica del Norte Antofagasta - Chile ORLICZ - PETTIS THEOREMS FOR MULTIPLIER CONVERGENT OPERATOR VALUED SERIES CHARLES SWARTZ New State
More informationA Common Fixed Point Theorem for Sub Compatibility and Occasionally Weak Compatibility in Intuitionistic Fuzzy Metric Spaces
Gen. Math. Notes, Vol. 21, No. 1, March 2014, pp. 73-85 ISSN 2219-7184; Copyright ICSRS Publication, 2014 www.i-csrs.org Available free online at http://www.geman.in A Common Fixed Point Theorem for Sub
More informationOn Positive Solutions of Boundary Value Problems on the Half-Line
Journal of Mathematical Analysis and Applications 259, 127 136 (21) doi:1.16/jmaa.2.7399, available online at http://www.idealibrary.com on On Positive Solutions of Boundary Value Problems on the Half-Line
More informationTHE HUTCHINSON BARNSLEY THEORY FOR INFINITE ITERATED FUNCTION SYSTEMS
Bull. Austral. Math. Soc. Vol. 72 2005) [441 454] 39b12, 47a35, 47h09, 54h25 THE HUTCHINSON BARNSLEY THEORY FOR INFINITE ITERATED FUNCTION SYSTEMS Gertruda Gwóźdź- Lukawska and Jacek Jachymski We show
More informationA fixed point method for proving the stability of ring (α, β, γ)-derivations in 2-Banach algebras
Journal of Linear and Topological Algebra Vol. 06, No. 04, 07, 69-76 A fixed point method for proving the stability of ring (α, β, γ)-derivations in -Banach algebras M. Eshaghi Gordji a, S. Abbaszadeh
More informationarxiv: v1 [math.gm] 20 Oct 2014
arxiv:1410.5748v1 [math.gm] 20 Oct 2014 A CHARACTERIZATION OF CAUCHY SEQUENCES IN FUZZY METRIC SPACES AND ITS APPLICATION TO FUZZY FIXED POINT THEORY MORTAZA ABTAHI Abstract. We introduce the concept of
More informationA Common Fixed Point Theorem in M-Fuzzy Metric Spaces for Integral type Inequality
International Journal of Theoretical & Applied Sciences Special Issue-NCRTAST 81: 113-1192016 ISSN No Print: 0975-1718 ISSN No Online: 2249-3247 A Common Fixed Point Theorem in M-Fuzzy Metric Spaces for
More informationSPACES ENDOWED WITH A GRAPH AND APPLICATIONS. Mina Dinarvand. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69, 1 (2017), 23 38 March 2017 research paper originalni nauqni rad FIXED POINT RESULTS FOR (ϕ, ψ)-contractions IN METRIC SPACES ENDOWED WITH A GRAPH AND APPLICATIONS
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics LACUNARY STATISTICAL CONVERGENCE JOHN ALBERT FRIDY AND CIHAN ORHAN Volume 160 No. 1 September 1993 PACIFIC JOURNAL OF MATHEMATICS Vol. 160, No. 1, 1993 LACUNARY STATISTICAL
More informationFixed Point Theorems for -Monotone Maps on Partially Ordered S-Metric Spaces
Filomat 28:9 (2014), 1885 1898 DOI 10.2298/FIL1409885D Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Fixed Point Theorems for
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationA FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS
Fixed Point Theory, (0), No., 4-46 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS A. ABKAR AND M. ESLAMIAN Department of Mathematics,
More informationSUBDOMINANT POSITIVE SOLUTIONS OF THE DISCRETE EQUATION u(k + n) = p(k)u(k)
SUBDOMINANT POSITIVE SOLUTIONS OF THE DISCRETE EQUATION u(k + n) = p(k)u(k) JAROMÍR BAŠTINECANDJOSEFDIBLÍK Received 8 October 2002 A delayed discrete equation u(k + n) = p(k)u(k) with positive coefficient
More informationarxiv: v1 [math.fa] 2 Jan 2017
Methods of Functional Analysis and Topology Vol. 22 (2016), no. 4, pp. 387 392 L-DUNFORD-PETTIS PROPERTY IN BANACH SPACES A. RETBI AND B. EL WAHBI arxiv:1701.00552v1 [math.fa] 2 Jan 2017 Abstract. In this
More informationSome results on the reverse order law in rings with involution
Some results on the reverse order law in rings with involution Dijana Mosić and Dragan S. Djordjević Abstract We investigate some necessary and sufficient conditions for the hybrid reverse order law (ab)
More information1.4 The Jacobian of a map
1.4 The Jacobian of a map Derivative of a differentiable map Let F : M n N m be a differentiable map between two C 1 manifolds. Given a point p M we define the derivative of F at p by df p df (p) : T p
More informationMohamed Akkouchi WELL-POSEDNESS OF THE FIXED POINT. 1. Introduction
F A S C I C U L I M A T H E M A T I C I Nr 45 2010 Mohamed Akkouchi WELL-POSEDNESS OF THE FIXED POINT PROBLEM FOR φ-max-contractions Abstract. We study the well-posedness of the fixed point problem for
More informationOn Neutrosophic Soft Metric Space
International Journal of Advances in Mathematics Volume 2018, Number 1, Pages 180-200, 2018 eissn 2456-6098 c adv-math.com On Neutrosophic Soft Metric Space Tuhin Bera 1 and Nirmal Kumar Mahapatra 1 Department
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationOn Multivalued G-Monotone Ćirić and Reich Contraction Mappings
Filomat 31:11 (2017), 3285 3290 https://doi.org/10.2298/fil1711285a Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Multivalued
More informationChapter IV Integration Theory
Chapter IV Integration Theory This chapter is devoted to the developement of integration theory. The main motivation is to extend the Riemann integral calculus to larger types of functions, thus leading
More informationREVERSALS ON SFT S. 1. Introduction and preliminaries
Trends in Mathematics Information Center for Mathematical Sciences Volume 7, Number 2, December, 2004, Pages 119 125 REVERSALS ON SFT S JUNGSEOB LEE Abstract. Reversals of topological dynamical systems
More informationYuqing Chen, Yeol Je Cho, and Li Yang
Bull. Korean Math. Soc. 39 (2002), No. 4, pp. 535 541 NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY Yuqing Chen, Yeol Je Cho, and Li Yang Abstract. In this paper, we introduce the concept of lower semicontinuous
More informationOn absolutely almost convergence
An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) Tomul LXIII, 2017, f. 1 On absolutely almost convergence Hüseyin Çakalli Emine Iffet Taylan Received: 24.VII.2012 / Revised: 30.III.2013 / Accepted: 24.IV.2013
More informationBEST PROXIMITY POINT RESULTS VIA SIMULATION FUNCTIONS IN METRIC-LIKE SPACES
Kragujevac Journal of Mathematics Volume 44(3) (2020), Pages 401 413. BEST PROXIMITY POINT RESULTS VIA SIMULATION FUNCTIONS IN METRIC-LIKE SPACES G. V. V. J. RAO 1, H. K. NASHINE 2, AND Z. KADELBURG 3
More informationEVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS
EVANESCENT SOLUTIONS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS Cezar Avramescu Abstract The problem of existence of the solutions for ordinary differential equations vanishing at ± is considered. AMS
More informationSOME REMARKS ON SUBDIFFERENTIABILITY OF CONVEX FUNCTIONS
Applied Mathematics E-Notes, 5(2005), 150-156 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ SOME REMARKS ON SUBDIFFERENTIABILITY OF CONVEX FUNCTIONS Mohamed Laghdir
More informationNotions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy
Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.
More informationWhere is matrix multiplication locally open?
Linear Algebra and its Applications 517 (2017) 167 176 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Where is matrix multiplication locally open?
More informationA FIXED POINT THEOREM FOR UNIFORMLY LOCALLY CONTRACTIVE MAPPINGS IN A C-CHAINABLE CONE RECTANGULAR METRIC SPACE
Surveys in Mathematics and its Applications A FIXED POINT THEOREM FOR UNIFORMLY LOCALLY CONTRACTIVE MAPPINGS IN A C-CHAINABLE CONE RECTANGULAR METRIC SPACE Bessem Samet and Calogero Vetro Abstract. Recently,
More informationFuzzy Dot Subalgebras and Fuzzy Dot Ideals of B-algebras
Journal of Uncertain Systems Vol.8, No.1, pp.22-30, 2014 Online at: www.jus.org.uk Fuzzy Dot Subalgebras and Fuzzy Dot Ideals of B-algebras Tapan Senapati a,, Monoranjan Bhowmik b, Madhumangal Pal c a
More informationOn EP elements, normal elements and partial isometries in rings with involution
Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012 Article 39 2012 On EP elements, normal elements and partial isometries in rings with involution Weixing Chen wxchen5888@163.com Follow this
More informationASYMPTOTICALLY I 2 -LACUNARY STATISTICAL EQUIVALENCE OF DOUBLE SEQUENCES OF SETS
Journal of Inequalities and Special Functions ISSN: 227-4303, URL: http://ilirias.com/jiasf Volume 7 Issue 2(206, Pages 44-56. ASYMPTOTICALLY I 2 -LACUNARY STATISTICAL EQUIVALENCE OF DOUBLE SEQUENCES OF
More informationON MATCHINGS IN GROUPS
ON MATCHINGS IN GROUPS JOZSEF LOSONCZY Abstract. A matching property conceived for lattices is examined in the context of an arbitrary abelian group. The Dyson e-transform and the Cauchy Davenport inequality
More informationAn extension of Knopp s core theorem for complex bounded sequences
Mathematical Communications 13(2008, 57-61 57 An extension of Knopp s core theorem for complex bounded sequences Şeyhmus Yardimic Abstract. In this paper, using the idea used by Choudhary, we extend previously
More informationCommon Fixed Point Theorems for Generalized Fuzzy Contraction Mapping in Fuzzy Metric Spaces
Volume 119 No. 12 2018, 14643-14652 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Common Fixed Point Theorems for Generalized Fuzzy Contraction Mapping in Fuzzy Metric Spaces 1 R.
More informationOn Semicontinuity of Convex-valued Multifunctions and Cesari s Property (Q)
On Semicontinuity of Convex-valued Multifunctions and Cesari s Property (Q) Andreas Löhne May 2, 2005 (last update: November 22, 2005) Abstract We investigate two types of semicontinuity for set-valued
More informationA Fixed Point Theorem for G-Monotone Multivalued Mapping with Application to Nonlinear Integral Equations
Filomat 31:7 (217), 245 252 DOI 1.2298/FIL17745K Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Fixed Point Theorem for G-Monotone
More informationNormed Vector Spaces and Double Duals
Normed Vector Spaces and Double Duals Mathematics 481/525 In this note we look at a number of infinite-dimensional R-vector spaces that arise in analysis, and we consider their dual and double dual spaces
More informationFixed Point Theorems in Strong Fuzzy Metric Spaces Using Control Function
Volume 118 No. 6 2018, 389-397 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Fixed Point Theorems in Strong Fuzzy Metric Spaces Using Control Function
More information