Lacunary summable sequence spaces of fuzzy ...

Lacunary summable sequence spaces of fuzzy numbers defined by ideal convergence and an Orlicz function Ayhan Esi & Bipan Hazarika

Afrika Matematika ISSN 1012-9405 Volume 25 Number 2 Afr. Mat. (2014) 25:331-343 DOI 10.1007/s13370-012-0117-3

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Author's personal copy Afr. Mat. (2014) 25:331–343 DOI 10.1007/s13370-012-0117-3

Lacunary summable sequence spaces of fuzzy numbers defined by ideal convergence and an Orlicz function Ayhan Esi · Bipan Hazarika

Received: 2 December 2011 / Accepted: 10 October 2012 / Published online: 6 November 2012 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2012

Abstract In this paper we introduce some certain new sequence spaces via ideal convergence and an Orlicz function and study some basic topological and algebraic properties of these spaces. Also we investigate the relations related to these spaces. Keywords Ideal, I -Convergence, Orlicz function, Fuzzy number, Multiplier space, Lacunary sequence. Mathematics Subject Classification (2010) 03E72

Primary 40A99; Secondary 40A05 · 46A45 ·

1 Introduction The concept of ideal convergence as a generalization of statistical convergence, and any concept involving statistical convergence plays a vital role not only in pure mathematics but also in other branches of science involving mathematics, especially in information theory, computer science, biological science, dynamical systems, geographic information systems, population modelling, and motion planning in robotics. The concept of fuzzy set was initially introduced by Zadeh [31]. It has a wide range of applications in almost all the branches of studied in particular in science, where mathematics is used. Now the notion of fuzziness are using by many researchers for cybernetics, artificial intelligence, expert system and fuzzy control, pattern recognition, operation research,

A. Esi Science and Art Faculty, Department of Mathematics, Adiyaman University, Adiyaman 02040, Turkey e-mail: [email protected] B. Hazarika (B) Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791112, Arunachal Pradesh, India e-mail: [email protected]

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decision making, image analysis, projectiles, probability theory, agriculture, weather forecasting etc. The fuzziness of all the subjects of mathematical sciences has been investigated. It attracted many workers on sequence spaces and summability theory to introduce different types of fuzzy sequence spaces and study their different properties. Our studies are based on the linear spaces of sequences of fuzzy numbers which are very important for higher level studies in quantum mechanics, particle physics and statistical mechanics etc. Different classes of sequences of fuzzy real numbers have been discussed by Nanda [20], Nuray and Savas [21], Matloka [19] and many others. The notion of I -convergence initially introduced by Kostyrko et al.[14]. Later on, it was futher investigated from the sequence space point of view and linked with the summability ˘ theory by Salát et al. [24,25], Tripathy and Hazarika [27–29], Kumar and Kumar [16,17], Hazarika and Savas [6], Khan and Ebdullah [10,11], Khan et al. [12], Khan and Tabassum [13], Das et al. [1], and many other authors. Goes and Goes [5] initially introduced the differential sequence space d E and the integrated sequence space ∈ E for a given sequence space E, by using the multiplier sequences (k −1 ) and (k) respectively. A multiplier sequence can be sued to accelerate the convergence of the sequences in some spaces. In some sense, it can be viewed as a catalyst, which is used to accelerate the process of chemical reaction. Sometimes the associated multiplier sequence delays the rate of the convergence of a sequence. This is also covers a larger class of sequences for study. Tripathy and Mahanta [30] used a general multiplier sequence  = (λk ) of nonzero scalars for all k ∈ N . Let  = (λk ) be a sequence of non-zero scalars. Then for a given sequence space E, the multiplier sequence space E() associated with multiplier sequence  is defined by (for details see [30]) E() = {(xk ) : (λk xk ) ∈ E}. Let X be a non-empty set, then a family of sets I ⊂ 2 X (the class of all subsets of X ) is called an ideal if and only if for each A, B ∈ I, we have A ∪ B ∈ I and for each A ∈ I and each B ⊂ A, we have B ∈ I. A non-empty family of sets F ⊂ 2 X is a filter on X if and only if  ∈ / F, for each A, B ∈ F, we have A ∩ B ∈ F and each A ∈ F and each A ⊂ B, we have B ∈ F. An ideal I is called non-trivial ideal if I  =  and X ∈ / I. Clearly I ⊂ 2 X is a non-trivial ideal if and only if F = F(I ) = {X − A : A ∈ I } is a filter on X . A non-trivial ideal I ⊂ 2 X is called admissible if and only if {{x} : x ∈ X } ⊂ I . A non-trivial ideal I is maximal if there cannot exists any non-trivial ideal J  = I containing I as a subset. Further details on ideals of 2 X can be found in Kostyrko et al. [14]. Recall in [15] that an Orlicz function M is continuous, convex , nondecreasing function define for x > 0 such that M(0) = 0 and M(x) > 0. If convexity of Orlicz function is replaced by M(x + y) ≤ M(x) + M(y) then this function is called the modulus function and characterized by Ruckle [23]. An Orlicz function M is said to satisfy 2 − condition for all values of u, if there exists K > 0 such that M(2u) ≤ K M(u), u ≥ 0. Lemma 1.1 Let M be an Orlicz function which satisfies 2 − condition and let 0 < δ < 1. Then for each t ≥ δ, we have M(t) < K δ −1 M(2) for some constant K > 0. Two Orlicz functions M1 and M2 are said to be equivalent if there exist positive constants α, β and x0 such that M1 (α) ≤ M2 (x) ≤ M1 (β)

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Lindenstrauss and Tzafriri [18] studied some Orlicz type sequence spaces defined as follows:     ∞  |xk | M = (xk ) ∈ w : M < ∞, for some ρ > 0 . ρ k=1

The space M with the norm



|| x || = inf ρ > 0 :

∞  k=1

 M

|xk | ρ



 ≤1

becomes a Banach space which is called an Orlicz sequence space. The space M is closely related to the space p which is an Orlicz sequence space with M(t) = |t| p , for 1 ≤ p < ∞. By a lacunary sequence θ = (kr ), where k0 = 0 , we shall mean an increasing sequence of non-negative integers with kr − kr −1 → ∞ as r → ∞. The intervals determined by θ will be denoted by Ir = (kr −1 , kr ] and we let h r = kr − kr −1 . The space of lacunary strongly convergent sequences Nθ was defined by Freedman et al. [4] as follows: ⎧ ⎫ ⎨ ⎬ 1  |xk − L| = 0, for some L Nθ = x = (xr ) : lim r hr ⎩ ⎭ k∈Ir

In the later stage different classes of Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [22], Esi [2], Esi and Et [3], Tripathy and Sarma [26] and many others. Throughout the article w F denote the class of all fuzzy real-valued sequence space. Also N and R denote the set of positive integers and set of real numbers, respectively.

2 Definitions and notations We now give here a brief introduction about the sequences of real numbers. Let D denote the set of all closed and bounded intervals X = [x1 , x2 ] on the real line R. For X, Y ∈ D, we define X ≤ Y if and only if x1 ≤ y1 and x2 ≤ y2 , d(X, Y ) = max{|x1 − y1 |, |x2 − y2 |}, where X = [x1 , x2 ] and Y = [y1 , y2 ]. Then it can be easily seen that d defines a metric on D and (D, d) is a complete metric space (see [8]). Also the relation “≤” is a partial order on D. A fuzzy number X is a fuzzy subset of the real line R i.e. a mapping X : R → J (= [0, 1]) associating each real number t with its grade of membership X (t). A fuzzy number X is convex if X (t) ≥ X (s) ∧ X (r ) = min{X (s), X (r )}, where s < t < r. If there exists t0 ∈ R such that X (t0 ) = 1, then the fuzzy number X is called normal. A fuzzy number X is said to be upper semicontinuous if for each > 0, X −1 ([0, a + )) for all a ∈ [0, 1] is open in the usual topology in R. Let R(J ) denote the set of all fuzzy numbers which are upper semicontinuous and have compact support, i.e. if X ∈ R(J ) the for any α ∈ [0, 1], [X ]α is compact, where [X ]α = {t ∈ R : X (t) ≥ α, if α ∈ [0, 1]}, [X ]0 = closure of ({t ∈ R : X (t) > α, if α = 0}).

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The set R of real numbers can be embedded in R(J ) if we define r ∈ R(J ) by

1, if t = r ; r (t) = 0, if t  = r. The additive identity and multiplicative identity of R(J ) are defined by 0 and 1, respectively. The arithmetic operations on R(J ) are defined as follows: (X ⊕ Y )(t) = sup {X (s) ∧ Y (t − s)}, t ∈ R, (X Y )(t) = sup {X (s) ∧ Y (s − t)}, t ∈ R, t (X ⊗ Y )(t) = sup {X (s) ∧ Y ( )}, t ∈ R, s   X (t) = sup {X (st) ∧ Y (s)}, t ∈ R. Y Let X, Y ∈ R(J ) and the α− level sets be [X ]α = [x1α , x2α ], [Y ]α = [y1α , y2α ], α ∈ [0, 1]. Then the above operations can be defined in terms of α-level sets as follows: [X ⊕ Y ]α = [x1α + y1α , x2α + y2α ],

[X Y ]α = [x1α − y1α , x2α − y2α ],   α α α α α [X ⊗ Y ] = min xi yi , max xi yi , i∈{1,2} i∈{1,2}  α −1 α −1  α −1 α [X ] = (x2 ) , (x1 ) , xi > 0, for each 0 < α ≤ 1.

For r ∈ R and X ∈ R(J ), the product r X is defined as follows:

X (r −1 t), if r  = 0; r X (t) = 0, if r = 0. The absolute value , |X | of X ∈ R(J ) is defined by (for details see [8])

max {X (t), X (−t)}, if t ≥ 0; |X |(t) = 0, if t < 0. Define a mapping d¯ : R(J ) × R(J ) → R+ ∪ {0} by ¯ d(X, Y ) = sup d([X ]α , [Y ]α ). 0≤α≤1

¯ is a complete metric space (for details see [8]). A metric on R(J ) It is known that (R(J ), d) us said to be translation invariant if ¯ + Z , Y + Z ) = d(X, ¯ d(X Y ), f or X, Y, Z ∈ R(J ). A sequence X = (X k ) of fuzzy numbers is said to converge to a fuzzy number X 0 if for ¯ k , X 0 ) < for all n ≥ n 0 . A every > 0, there exists a positive integer n 0 such that d(X sequence X = (X k ) of fuzzy numbers is said to be bounded if the set {X k : k ∈ N} of fuzzy numbers is bounded. A sequence X = (X k ) of fuzzy numbers is said to be I -convergent to a fuzzy number X 0 if for each > 0, ¯ k , X 0 ) ≥ } ∈ I. A = {k ∈ N : d(X

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The fuzzy number X 0 is called I -limit of the sequence (X k ) of fuzzy numbers and we write I − lim X k = X 0 . A sequence X = (X k ) of fuzzy numbers is said to be I -bounded if there exists M > 0, ¯ k , 0) ¯ > M} ∈ I. {k ∈ N : d(X Let E F be denote the sequence space of fuzzy numbers. A sequence space E F is said to be ¯ k , 0) ≤ d(X ¯ k , 0) for all k ∈ N. solid ( or normal) if (Yk ) ∈ E F whenever (X k ) ∈ E F and d(Y A sequence space E F is said to be symmetric if (X k ) ∈ E F implies (X π(k) ) ∈ E F where π is a permutation of N. A sequence space E F is said to be monotone if E F contains the canonical pre-images of all its step spaces. Example 2.1 If we take I = I f = {A ⊆ N : A is a finite subset }. Then I f is a nontrivial admissible ideal of N and the corresponding convergence coincide with the usual convergence. Example 2.2 If we take I = Iδ = {A ⊆ N : δ(A) = 0} where δ(A) denote the asyptotic density of the set A. Then Iδ is a non-trivial admissible ideal of N and the corresponding convergence coincide with the statistical convergence. Lemma 2.1 A sequence space E F is normal implies E F is monotone. (For the crisp set case, one may refer to Kamthan and Gupta [9], page 53). Lemma 2.2 (Kostyrko, et.,al. [14], Lemma 5.1) If I ⊂ 2N is a maximal ideal, then for each A ⊂ N we have either A ∈ I or N − A ∈ I.

3 Some new sequence spaces of fuzzy numbers The following well-known inequality will be used throughout the article. Let p = ( pk ) be any sequence of positive real numbers with 0 ≤ pk ≤ supk pk = G, H = max{1, 2G−1 } then |ak + bk | pk ≤ H (|ak | pk + |bk | pk ) for all k ∈ N and ak , bk ∈ C. Also |ak | pk ≤ max{1, |a|G } for all a ∈ C. The main aim of this article to introduce the following sequence spaces and examine topological and algebraic properties of the resulting sequence spaces. Let I be an admissible ideal of N and let p = ( pk ) be a sequence of positive real numbers for all k ∈ N. Let θ = (kr ) be a lacunary sequence, M be an Orlicz function,  = (λk ) be a sequence of non-zero scalars and X = (X k ) be a sequence of fuzzy numbers, we define the following sequence spaces as: ⎧ ⎨

⎧ ⎨

1 I (F) wθ [M, , p] = (X k ) ∈ w F : ∀ ε > 0, r ∈ N : ⎩ ⎩ hr

 k∈Ir

⎫ ¯  pk ⎬ d (λk X k , X 0 ) ≥ ε ∈ I, M ⎭ ρ

⎫ ⎬ for some ρ > 0 and X 0 ∈ R(J ) , ⎭ ⎧ ⎧ ⎫      pk ⎨ ⎨ ⎬ d¯ λk X k , 0¯ 1  I (F) F ≥ ε ∈ I, M wθ [M, , p]0 = (X k ) ∈ w : ∀ ε > 0, r ∈ N : ⎩ ⎩ ⎭ hr ρ k∈Ir ⎫ ⎬ for some ρ > 0 , ⎭

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⎧ ⎨

1 I (F) wθ [M, , p]∞ = (X k ) ∈ w F : ∃K > 0 s.t. r ∈ N : ⎩ ⎩ hr for some ρ > 0

⎫ ⎬





k∈Ir

⎫     pk ⎬ d¯ λk X k , 0¯ M ≥ K ∈ I, ⎭ ρ

⎭

and

⎧ ⎫      pk ⎨ ⎬ ¯ λk X k , 0¯  d 1 wθF [M, , p]∞ = (X k ) ∈ w F : sup < ∞ , for some ρ > 0 . M ⎩ ⎭ ρ r hr k∈Ir

Some classes are obtained by specializing θ = (kr ), M,  = (λk ) and p = ( pk ). Here are some examples: (i) If θ = (kr ) = (2r ) , then we obtain ⎧ ⎧ ⎫  pk n   ¯ ⎨ ⎨ ⎬  d X , X 1 ) (λ 0 k k M w I (F) [M, , p] = (X k ) ∈ w F : ∀ ε > 0, n ∈ N : ≥ ε ∈ I, ⎩ ⎩ ⎭ n ρ k=1 ⎫ ⎬ for someρ > 0 and X 0 ∈ R(J ) , ⎭ ⎧ ⎧ ⎫      pk n ⎨ ⎨ ⎬ d¯ λk X k , 0¯ 1 I (F) F [M, , p]0 = (X k ) ∈ w : ∀ ε > 0, n ∈ N : ≥ ε ∈ I, M w ⎩ ⎩ ⎭ n ρ k=1 ⎫ ⎬ for some ρ > 0 , ⎭ ⎧ ⎧ ⎫      pk n ⎨ ⎨ ⎬ d¯ λk X k , 0¯ 1 I (F) F [M, , p]∞ = (X k ) ∈ w : ∃K > 0 s.t. n ∈ N : ≥ K ∈ I, M w ⎩ ⎩ ⎭ n ρ k=1 ⎫ ⎬ for some ρ > 0 ⎭

and w F [M, , p]∞ =

⎧ ⎨

n 1 (X k ) ∈ w F : sup ⎩ n n k=1

 M

    pk d¯ λk X k , 0¯ ρ

< ∞, for some ρ > 0

which were defined and studied by Hazarika and Esi [7]. (ii) If M(x) = x, then we obtain, ⎧ ⎨

⎧ ⎨

1 I (F) wθ [, p] = (X k ) ∈ w F : ∀ ε > 0, r ∈ N : ⎩ ⎩ hr ⎫ ⎬ for some X 0 ∈ R(J ) , ⎭ ⎧ ⎧ ⎨ ⎨ 1 I (F) F wθ [, p]0 = (X k ) ∈ w : ∀ ε > 0, r ∈ N : ⎩ ⎩ hr ⎧ ⎧ ⎨ ⎨ I (F) wθ [, p]∞ = (X k ) ∈ w F : ∃K > 0 s.t. r ∈ N : ⎩ ⎩

123



d¯ (λk X k , X 0 )

 pk

≥ε

k∈Ir



d¯ (λk X k , X 0 )

 pk

≥ε

⎫ ⎬ ⎭

⎫ ⎬ ⎭

k∈Ir

⎭

∈ I,

∈I

⎫ ⎬

⎭ ⎫ ⎬ p 1  ¯ d (λk X k , X 0 ) k ≥ K ∈ ⎭ hr

k∈Ir

⎫ ⎬

, ⎫ ⎬ I

⎭

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⎧ ⎨

⎫ ⎬    1 p = (X k ) ∈ w F : sup d¯ (λk X k , X 0 ) k < ∞ . ⎩ ⎭ r hr k∈Ir

(iii) If  = (λk ) = (1, 1, 1, ...) , then we obtain ⎧ ⎨

⎧ ⎨

1 I (F) wθ [M, p] = (X k ) ∈ w F : ∀ ε > 0, r ∈ N : ⎩ ⎩ hr

 k∈Ir

⎫  pk ¯ ⎬ d (X k , X 0 ) M ≥ ε ∈ I, ⎭ ρ

⎫ ⎬ for some ρ > 0 and X 0 ∈ R(J ) , ⎭ ⎧ ⎧ ⎫      pk ⎨ ⎨ ⎬ d¯ X k , 0¯ 1  I (F) F ≥ ε ∈ I, M wθ [M, p]0 = (X k ) ∈ w : ∀ ε > 0, r ∈ N : ⎩ ⎩ ⎭ hr ρ k∈Ir ⎫ ⎬ for some ρ > 0 , ⎭ ⎧ ⎧ ⎫      pk ⎨ ⎨ ⎬ d¯ X k , 0¯ 1  I (F) F ≥ K ∈ I, M wθ [M, p]∞ = (X k ) ∈ w : ∃K > 0 s.t. r ∈ N : ⎩ ⎩ ⎭ hr ρ k∈Ir

for some ρ > 0}

and wθF [M, p]∞

⎧ ⎨

⎫      pk ⎬ ¯ ¯  , 0 d X 1 k = (X k ) ∈ w F : sup < ∞, for some ρ > 0 . M ⎩ ⎭ ρ r hr k∈Ir

(iv) If p = ( pk ) = (1, 1, 1, ...), then we obtain ⎧ ⎨

⎧ ⎨

1 I (F) wθ [M, ] = (X k ) ∈ w F : ∀ ε > 0, r ∈ N : ⎩ ⎩ hr

 k∈Ir

⎫ ¯  ⎬ d (λk X k , X 0 ) M ≥ ε ∈ I, ⎭ ρ

⎫ ⎬ for some ρ > 0 and X 0 ∈ R(J ) , ⎭ ⎫ ⎧ ⎧      ⎬ ⎨ ⎨ d¯ λk X k , 0¯ 1  I (F) F ≥ ε ∈ I, M wθ [M, ]0 = (X k ) ∈ w : ∀ ε > 0, r ∈ N : ⎭ ⎩ ⎩ hr ρ k∈Ir ⎫ ⎬ for some ρ > 0 , ⎭ ⎧ ⎧ ⎫      ⎨ ⎨ ⎬ d¯ λk X k , 0¯ 1  I (F) F M ≥ K ∈ I, wθ [M, ]∞ = (X k ) ∈ w : ∃K > 0 s.t. r ∈ N : ⎩ ⎩ ⎭ hr ρ k∈Ir ⎫ ⎬ for some ρ > 0 ⎭

and wθF [M, ]∞

⎧ ⎨

⎫      ⎬ ¯ λk X k , 0¯  d 1 = (X k ) ∈ w F : sup M < ∞, for some ρ > 0 . ⎩ ⎭ ρ r hr k∈Ir

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4 Main results In this section we examine the basic topological and algebraic properties of these spaces and obtain the inclusion relation between these spaces. I (F)

Theorem 4.1 wθ

I (F)

[M, , p], wθ

I (F)

[M, , p]0 , and wθ

[M, , p]∞ are linear spaces.

I (F)

Proof We will proved the result for the space wθ [M, , p]0 only and the others can be I (F) proved in similar way. Let X = (X k ) and Y = (Yk ) be two elements in wθ [M, , p]0 . Then there exist ρ1 > 0 and ρ2 > 0 such that ⎫ ⎧  ¯  pk ⎨ ¯ d(λk X k , 0) 1  ε⎬ ∈I M A 2ε = r ∈ N : ≥ ⎩ hr ρ1 2⎭ k∈Ir

and

⎫ ⎧  ¯  pk ⎨ ¯ ε⎬ d(λk Yk , 0) 1  B 2ε = r ∈ N : ≥ ∈ I. M ⎩ hr ρ2 2⎭ k∈Ir

Let α, β be two scalars. By the continuity of the function M the following inequality holds:  ¯  pk ¯ 1  d(λk (α X k + βYk , 0)) M hr |α|ρ1 + |β|ρ2 k∈Ir   pk ¯ ¯ |α| 1  d(λk X k , 0) ≤H M hr |α|ρ1 + |β|ρ2 ρ1 k∈Ir   pk ¯ ¯ |β| 1  d(λk Yk , 0) +H M hr |α|ρ1 + |β|ρ2 ρ2 k∈Ir     ¯  pk p k ¯ k X k , 0) ¯ ¯ d(λ d(λk Yk , 0) 1  1  M M ≤ HK + HK , hr ρ1 hr ρ2 k∈Ir

 G  G  |β| , . where K = max 1, |α|ρ1|α| +|β|ρ2 |α|ρ1 +|β|ρ2

k∈Ir

From the above relation we obtain the following: ⎫ ⎧  ¯  pk ⎬ ⎨ ¯ d(λk (α X k + βYk , 0)) 1  M r ∈N: ≥ε ⎭ ⎩ hr |α|ρ1 + |β|ρ2 k∈Ir ⎧ ⎫  ¯  pk ⎨ ¯ ε⎬ HK  d(λk X k , 0) ⊆ r ∈N: ≥ M ⎩ hr ρ1 2⎭ k∈Ir ⎧ ⎫  ¯  pk ⎨ ¯ HK  ε⎬ d(λk Yk , 0) ∪ r ∈N: ≥ ∈ I. M ⎩ hr ρ2 2⎭ k∈Ir

This completes the proof. Remark 4.1 It is easy to verify that the space wθF [M, , p]∞ is a linear space.

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Theorem 4.2 The space wθF [M, , p]∞ is a paranormed space (not totally paranormed) with the paranorm g defined by    pk  ¯ ¯ pr d(λk X k , 0) ≤ 1, for some ρ > 0, r = 1, 2, 3, ... , g (X ) = inf ρ E : sup M ρ k   where E = max 1, supk pk . Proof Clearly g (−X ) = g (X ) and g (θ ) = 0. Let X = (X k ) and Y = (Yk ) be two elements in wθF [M, , p]∞ . Then for ρ > 0 we put

¯   ¯ d(λk X k , 0) A1 = ρ > 0 : sup M ≤1 ρ k and

¯   ¯ d(λk Yk , 0) A2 = ρ > 0 : sup M ≤1 . ρ k

Let ρ1 ∈ A1 and ρ2 ∈ A2 . If ρ = ρ1 + ρ2 then we obtain the following ¯ ¯    ¯ ¯ ¯ ¯ d(λk X k , 0) d(λk Yk , 0) ρ1 ρ2 d(λk (X k + Yk ), 0) M M ≤ + . M ρ ρ1 + ρ2 ρ1 ρ1 + ρ2 ρ2 Thus we have

and



¯  pk ¯ d(λk (X k + Yk ), 0) ≤1 sup M ρ k   pr g (X + Y ) = inf (ρ1 + ρ2 ) E : ρ1 ∈ A1 , ρ2 ∈ A2

pr 

pr  ≤ inf ρ1E : ρ1 ∈ A1 + inf ρ2E : ρ2 ∈ A2 = g (X ) + g (Y ).

Let tk → t where tk , t ∈ C and let g (X k − X ) → 0 as k → ∞. To prove that g (tk X k − t X ) → 0 as k → ∞. We put    ¯  pk ¯ d(λk X k , 0) A3 = ρk > 0 : sup M ≤1 ρk k and



 ¯  pk ¯ d(λk Yk , 0) ≤1 . A4 = ρl > 0 : sup M ρl k 

By the continuity of the function M we observe that  ¯ ¯ d(λk (tk X k − t X ), 0) M |tk − t|ρk + |t|ρl ¯  ¯  ¯ ¯ d(λk (tk X k − t X k ), 0) d(λk (t X k − t X ), 0) ≤M +M |tk − t|ρk + |t|ρl |tk − t|ρk + |t|ρl  ¯ ¯  ¯ ¯ |t|ρl |tk − t|ρk d(λk X k , 0) d(λk Yk , 0) + ≤ M M . |tk − t|ρk + |t|ρl ρk |tk − t|ρk + |t|ρl ρl

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From the above inequality it follows that  ¯  pk ¯ d(λk (tk X k − t X ), 0) sup M ≤1 |tk − t|ρk + |t|ρl k and consequently

  pr g (tk X k − t X ) = inf (|tk − t|ρk + |t|ρl ) E : ρk ∈ A3 , ρl ∈ A4     pk pr pr pr ≤ |tk − t| E inf (ρk ) E : ρk ∈ A3 + |t| E inf (ρl ) H : ρl ∈ A4     pr pr (1) ≤ max 1, |tk − t| E g (X k ) + max 1, |t| E g (X k − X ).

Note that g (X k ) ≤ g (X ) + g (X k + X ) for all k ∈ N. Hence by our assumption the right hand side of the relation (1) tends to 0 as k → ∞ and the result follows. This completes the proof. Theorem 4.3 Let M and S be Orlicz functions. Then the following hold: I (F)

I (F)

(i) wθ [S, , p]0 ⊆ wθ [M.S, , p]0 , provided p = ( pk ) be such that G 0 = inf pk > 0. I (F) I (F) I (F) (ii) wθ [M, , p]0 ∩ wθ [S, , p]0 ⊆ wθ [M + S, , p]0 .   Proof (i) Let ε > 0 be given. Choose ε1 > 0 such that max ε1G , ε1G 0 < ε. Choose 0 < δ < 1 such that 0 < t < δ implies that M(t) < ε1 . Let X = (X k ) be any element in I (F) wθ [S, , p]0 . Put ⎧ ⎫  ¯  pk ⎨ ⎬  ¯ 1 d(λk X k , 0) ≥ δG . S Aδ = r ∈ N : ⎩ ⎭ hr ρ k∈Ir

Then by the definition of ideal we have Aδ ∈ I. If n ∈ / Aδ we have  pk  pk  ¯    d(λ ¯ k X k , 0) ¯ ¯ 1  d(λk X k , 0) S S < δG ⇒ < hr δ G hr ρ ρ k∈Ir k∈Ir  pk  ¯ ¯ d(λk X k , 0) ⇒ S < δ G , for all k ∈ Ir ρ ¯  ¯ d(λk X k , 0) ⇒S (2) < δ G , for all k ∈ Ir . ρ Using the continuity of the function M from the relation (2) we have  ¯  ¯ d(λk X k , 0) M S < ε1 , for all k ∈ Ir . ρ Consequently we get 

 pk  ¯   ¯ d(λk X k , 0) < h r . max ε1G , ε1G 0 < h r ε M S ρ k∈Ir  pk   ¯ ¯ d(λk X k , 0) 1  < ε. M S ⇒ hr ρ k∈Ir

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This implies that ⎫ ⎧  pk   ¯ ⎬ ⎨ ¯ 1  d(λk X k , 0) ≥ ε ⊆ Aδ ∈ I. r ∈N: M S ⎭ ⎩ hr ρ k∈Ir

This completes the proof. I (F)

I (F)

(ii) Let X = (X k ) ∈ wθ [M, , p]0 ∩wθ [S, , p]0 . Then by the following inequality the result follows:  pk  pk  ¯  ¯ ¯ ¯ 1  1  d(λk X k , 0) d(λk X k , 0) ≤H (M + S) M hr ρ hr ρ k∈Ir k∈Ir    pk ¯ k X k , 0) ¯ d(λ 1  S +H . hr ρ k∈Ir

The proof of the following theorems are easy and so omitted.   Theorem 4.4 et 0 < pk ≤ qk and qpkk is bounded, then I (F)

wθ

I (F)

[M, , q]0 ⊆ wθ

[M, , p]0 .

Theorem 4.5 For any two sequences p = ( pk ) and q = (qk ) of positive real numbers, then the following holds: (i) (ii) (iii) (iv)

I (F)

I (F)

wθ [M, , p]0 ∩ wθ [M, , q]0  = φ; I (F) I (F) wθ [M, , p] ∩ wθ [M, , q]  = φ; I (F) I (F) wθ [M, , p]∞ ∩ wθ [M, , q]∞  = φ; F F wθ [M, , p]∞ ∩ wθ [M, , q]∞  = φ. I (F)

Proposition 4.1 The sequence spaces wθ as well as monotone.

I (F)

[M, , p]0 and wθ

[M, , p]∞ are normal

I (F)

Proof We shall give the prove of the theorem for wθ [M, , p]0 only. Let X = (X k ) ∈ I (F) ¯ k , 0) ≤ d(X ¯ k , 0) for all k ∈ N. Then for wθ [M, , p]0 and Y = (Yk ) be such that d(Y given ε > 0 we have ⎫ ⎧  pk  ¯ ⎬ ⎨ ¯ 1  d(λk X k , 0) M ≥ ε ∈ I. B= r ∈N: ⎭ ⎩ hr ρ k∈Ir

 Again the set B1 = I (F)

r ∈N:

1 hr

 k∈Ir

M

¯

¯ d(λk Yk ,0) ρ

! pk

 ≥ε

⊆ B. Hence B1 ∈ I and

I (F)

so Y = (Yk ) ∈ wθ [M, , p]0 . Thus the space wθ [M, , p]0 is normal. Also from the I (F) Lemma 2.2, it follows that wθ [M, , p]0 is monotone. I (F)

Proposition 4.2 If I is not a maximal ideal, then the space wθ nor monotone.

[M, , p] is neither normal

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A. Esi, B. Hazarika I (F)

Proof We first prove that the space wθ [M, , p] is not monotone. Let us consider a sequence X = (X k ) of fuzzy numbers defined by ⎧ −1 if t ∈ [−1, 1]; ⎨ 2 (1 + t), X k (t) = 2−1 (−t + 3), if t ∈ [1, 3]; ⎩ 0, otherwise. I (F)

Then (X k ) ∈ wθ [M, , p]. Since I is not maximal, so by Lemma 2.2, there exists a subset K in N such that K ∈ / I and N − K ∈ / I. Let us define a sequence Y = (Yk ) by

X , if k ∈ K ; Yk = ¯ k 1, otherwise. Then Y = (Yk ) belongs to the canonical pre-image of the K -step space of (X k ) ∈ I (F) I (F) I (F) / wθ [M, , p]. Hence wθ [M, , p] is not monotone. wθ [M, , p]. But (Yk ) ∈ I (F) Therefore by the Lemma 2.2, it follows that the space wθ [M, , p] is not normal. I (F)

Proposition 4.3 If I is neither maximal nor I = I f then the spaces wθ I (F) wθ [M, , p]0 are not symmetric.

[M, , p] and

Proof Let us consider a sequence X = (X k ) of fuzzy real numbers defined by ⎧ ⎨ 1 + t − 2t, if t ∈ [2k − 1, 2k]; X k (t) = 1 − t + 2k, if t ∈ [2k, 2k + 1]; ⎩ 0, otherwise. for k ∈ A ⊂ I an infinite set. I (F) Then (X k ) ∈ wθ [M, , p]. Let K ⊆ N be such that K ∈ / I and N − K ∈ / I (the set K exists by the Lemma 2.2, as I is not maximal). Consider a sequence Y = (Yk ) a rearrangement of the sequence (X k ) defined as follows:

X , if k ∈ K ; Yk = ¯ k 1, otherwise. I (F)

I (F)

/ wθ [M, , p]. Hence wθ [M, , p] is not symmetric. Then (Yk ) ∈ I (F) Similarly we can prove that the space wθ [M, , p]0 is not symmetric. I (F)

Proposition 4.4 If I is neither maximal nor I = I f then the space wθ symmetric.

[M, , p]∞ is not

Proof Let us consider a sequence X = (X k ) of fuzzy real numbers defined by ⎧ ⎨ 1 + t − 3t, if t ∈ [3k − 1, 3k]; X k (t) = 1 − t + 3k, if t ∈ [3k, 3k + 1]; ⎩ 0, otherwise. ¯ for k ∈ A ⊂ I an infinite set. Otherwise X k = 1. Since I is not maximal, so by Lemma 2.2, there exists a subset K in N such that K ∈ / I and N − K ∈ / I. Let f : K → A and h : N − K → N − A be bijections. Consider a sequence Y = (Yk ) a rearrangement of the sequence (X k ) defined as follows:

X f (k) , if k ∈ K ; Yk = X h(k) , otherwise. I (F)

Then (Yk ) ∈ / wθ

123

I (F)

[M, , p]∞ . Hence wθ

[M, , p]∞ is not symmetric.

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Acknowledgments The authors expresses their heartfelt gratitude to the anonymous reviewer for such excellent comments and suggestions which have enormously enhanced the quality and presentation of this paper.

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