tigated by Esi [1], Matloka [4], Subrahmanyam [5], Syau [6], Tripathy and Baruah. [7,8,9], Tripathy and Borgohain [10], Tripathy and Dutta [11,12], Trpathy and.
Bol. Soc. Paran. Mat. c
SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm
(3s.) v. 32 2 (2014): 35–43. ISSN-00378712 in press doi:10.5269/bspm.v32i2.19868
p-absolutely Summable Type Fuzzy Sequence Spaces by Fuzzy Metric Paritosh Chandra Das
∗
abstract: In this article we introduce the notion of p-absolutely summable class of sequences of fuzzy real numbers with fuzzy metric, ℓF ∞ , for 1 ≤ p < ∞. We study some of its properties like completeness, solidness, symmetricity, sequence algebra and convergence free. Also we study some inclusion results.
Key Words: Fuzzy real number, solid space, symmetricity, convergence free, sequence algebra, fuzzy metric. Contents 1 Introduction
35
2 Definitions and Preliminaries
36
3 Main Results
39 1. Introduction
The concept of fuzzy set, a set whose boundary is not sharp or precise has been introduced by L.A. Zadeh in 1965. It is the origin of new theory of uncertainty, distinct from the notion of probability. After the introduction of fuzzy set, the scope for studies in different branches of pure and applied mathematics increased widely. The notion of fuzzy set theory has been applied to introduce the notion of fuzzy real numbers which help in constructing the sequence of fuzzy real numbers. Different types of sequence spaces of fuzzy real numbers have been studied under classical metric by several workers in the last half century. Till now very few works have been done on fuzzy norm, which has relationship with fuzzy metric and there is a lot to be explored on sequences of fuzzy real numbers, those can be examined by fuzzy metric. A fuzzy real number X is a fuzzy set on R, i.e. a mapping X : R → I(= [0, 1]) associating each real number t with its grade of membership X(t). A fuzzy real number X is called convex if X(t) ≥ X(s) ∧ X(r) = min (X(s), X(r)), where s < t < r. ∗ The work of the author is supported by University Grants Commission of India vide project No. F. 5-54/2007- 08 (MRP/NERO)/6198, dated- 31 MARCH, 2008. 2000 Mathematics Subject Classification: 40A05, 40D25.
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Typeset by BSP style. M c Soc. Paran. de Mat.
36
Paritosh Chandra Das
If there exists t0 ∈ R such that X(t0 ) = 1, then the fuzzy real number X is called normal. A fuzzy real number X is said to be upper-semi continuous if, for each ε > 0, X −1 ([0, a + ε)), for all a ∈ I is open in the usual topology of R. The set of all upper-semi continuous, normal, convex fuzzy real numbers is denoted by R(I). Throughout the article, by a fuzzy real number we mean that the number belongs to R(I). The α-level set [X]α of the fuzzy real number X, for 0 < α ≤ 1, defined as [X] = {t ∈ R : X(t) ≥ α}. If α = 0, then it is the closure of the strong 0cut.(The strong α -cut of the fuzzy real number X, for 0 ≤ α ≤ 1 is the set {t ∈ R : X(t) > α}). α
Let X, Y ∈ R(I) and α-level sets be α α α α [X]α = [aα 1 , b1 ], [Y ] = [a2 , b2 ], α ∈ [0, 1].
Then the arithmetic operations on R(I) in terms of α-level sets are defined as follows:
α α α [X ⊕ Y ]α = [aα 1 + a2 , b 1 + b 2 ] , α α α [X ⊖ Y ]α = [aα 1 − b 2 , b 1 − a2 ] , α α α b , max a b [X ⊗ Y ]α = min aα i j i j i,j∈{1,2} i,j∈{1,2} i h and [1 ÷ Y ]α = b1α , a1α , 0 ∈ / Y. 2
2
The absolute value, |X| of X ∈ R(I) is defined by (see for instance Kaleva and Seikkala [3] ) |X|(t) =
(
max(X(t), X(−t)), f or t ≥ 0, 0, f or t < 0.
A fuzzy real number X is called non-negative if X(t) = 0, for all t < 0. The set of all non-negative fuzzy real numbers is denoted by R∗ (I). 2. Definitions and Preliminaries The notion of fuzzy set has been applied for introducing different classes of sequences. Their different algebric and topological properties have been investigated.
p-absolutely summable type fuzzy sequence spaces
37
In the recent years different classes of sequences of fuzzy numbers have been investigated by Esi [1], Matloka [4], Subrahmanyam [5], Syau [6], Tripathy and Baruah [7,8,9], Tripathy and Borgohain [10], Tripathy and Dutta [11,12], Trpathy and Sarma [13,14], Tripathy and Nanda [15] and many others. A fuzzy real number sequence (Xk ) is said to be bounded if |Xk | ≤ µ, for some µ ∈ R∗ (I). A class of sequences E F is said to be normal (or solid) if (Yk ) ∈ E F , whenever |Yk | ≤ |Xk |, for all k ∈ N and (Xk ) ∈ E F . Let K = {k1 < k2 < k3 . . .} ⊆ N and E F be a class of sequences. A K-step set F of E F is a class of sequences λE = {(Xkn ) ∈ wF : (Xn ) ∈ E F }. k A canonical pre-image of a sequence (Xkn ) ∈ λE k defined as follows: ( Xn , f or n ∈ K, Yn = ¯ 0, otherwise.
F
is a sequence (Yn ) ∈ wF ,
F
A canonical pre-image of a step set λE is a set of canonical pre-images of all k F EF elements in λk , i.e., Y is in canonical pre-image λE if and only if Y is canonical k F pre-image of some X ∈ λE . k A class of sequences E F is said to be monotone if E F contains the canonical pre-images of all its step sets. From the above definitions we have the following well known Remark. Remark 2.1. A class of sequences E F is solid ⇒ E F is monotone. A class of sequences E F is is said to be symmetric if (Xπ(n) ) ∈ E F , whenever (Xk ) ∈ E F , where π is a permutation of N . A class of sequences E F is is said to be sequence algebra if (Xk ⊗ Yk ) ∈ E F , whenever (Xk ), (Yk ) ∈ E F . A class of sequences E F is is said to be convergence free if (Yk ) ∈ E F , whenever 0 implies Yk = ¯0. (Xk ) ∈ E F and Xk = ¯ Throughout the article 0 and 1 represent the additive and multiplicative identities for fuzzy numbers respectively.
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Paritosh Chandra Das
Definition 2.2. Let d be a mapping from R(I) × R(I) into R∗ (I) and let the mappings L, M : [0, 1] × [0, 1] → [0, 1] be symmetric, non-decreasing in both arguments and satisfy L(0, 0) = 0 and M (1, 1) = 1. Denote [d(X, Y )]α = [λα (X, Y ), ρα (X, Y )], for X, Y ∈ R(I) and 0 < α ≤ 1. The quadruple (R(I), d, L, M ) is called a fuzzy metric space and d a fuzzy metric, if (1) d(X, Y ) = 0 if and only if X = Y, (2) d(X, Y ) = d(Y, X) for all X, Y ∈ X, (3) for all X, Y, Z ∈ R(I), (i) d(X, Y )(s + t) ≥ L(d(X, Z)(s), d(Z, Y )(t)) whenever s ≤ λ1 (X, Z), t ≤ λ1 (Z, Y ) and (s + t) ≤ λ1 (X, Y ), (ii) d(X, Y )(s + t) ≤ M (d(X, Z)(s), d(Z, Y )(t)) whenever s ≥ λ1 (X, Z), t ≥ λ1 (Z, Y ) and (s + t) ≥ λ1 (X, Y ). Using the concept of fuzzy metric, we introduce the following class of sequences. ℓF p
=
(
) ∞ ∞ X X p p X = (Xk ) ∈ w : λ(Xk , 0) < ∞ and ρ(Xk , 0) < ∞ . F
k=1
k=1
We procure the following classes of sequences of fuzzy numbers defined with respect to the concept of fuzzy metric,those will be used in this article. ℓF ∞ =
X = (Xk ) ∈ wF : sup λ(Xk , 0) < ∞ and supλ(Xk , 0) < ∞ . k
k
cF = X = (Xk ) ∈ wF : λ(Xk , T ) → 0 and ρ(Xk , T ) → 0, as k → ∞, for some T ∈ R(I)} .
F cF 0 = X = (Xk ) ∈ w : λ(Xk , 0) → 0 and ρ(Xk , 0) → 0, as k → ∞ . F F F Throughout wF , ℓF denote the classes of all, bounded, p∞ , ℓp , c and c0 absolutely summable, convergent and null sequences of fuzzy real numbers respectively.
p-absolutely summable type fuzzy sequence spaces
39
3. Main Results Theorem 3.1. The class of sequences ℓF p , 1 ≤ p < ∞ is a complete metric space with the fuzzy metric d∗ defined by " p1 ∞ p1 # ∞ P P p p ∗ [d (X, Y )]α = {λα (Xk , Yk )} , , {ρα (Xk , Yk )} k=1
where X = (Xk ), Y = (Yk ) ∈
k=1
ℓF p
and 0 < α ≤ 1.
Proof: Let (X (n) ) be a Cauchy sequence in ℓF p, (n) (n) (n) (n) (n) where X = Xk = X1 , X2 , X3 , . . . ∈ ℓ F p , for all n ∈ N and
op op ∞ n ∞ n P P (n) (n) λ Xk , 0 ρ Xk , 0 < ∞ and < ∞.
k=1
k=1
Then, for each ε > 0, there exists a positive integer n0 such that for all m, n ≥ n0 , d∗ (X (n) , X (m) ) < ε. ∞ n op p1 op p1 ∞ n P P (n) (m) (n) (m) i.e., λ Xk , Xk < ε and ρ Xk , Xk < ε . . .(1) k=1 k=1 (n) (m) (n) (m) ⇒ λ Xk , Xk < ε and ρ Xk , Xk < ε. (n) Now Xk , for all k ∈ N is a Cauchy sequences in R(I). Since R(I) is (n) complete, so Xk , for all k ∈ N is convergent in R(I).
(n)
Let lim Xk n→∞
= Xk , for all k ∈ N.
We shall prove that, X (n) → X, where X = (Xk ) and X ∈ ℓF p. Now fix n ≥ n0 and let m → ∞ in (1), we have
∞ n op p1 op p1 ∞ n P P (n) (n) < ε and ρ Xk , Xk < ε . . .(2) λ Xk , Xk k=1
k=1 ∗
⇒ d (X
(n)
, X) < ε, for all n ≥ n0
⇒ X (n) → X as n → ∞.
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Paritosh Chandra Das
Again, we have from (2) op op ∞ n ∞ n P P (n) (n) λ Xk , Xk < ε and < ε. ρ Xk , Xk
k=1
(n) (Xk )
Since X (n) = ∈ ℓF p , so o n ∞ p P (n) < ∞. ρ Xk , 0
k=1 ∞ P
k=1
n op (n) λ Xk , 0 < ∞ and
k=1
Now for all n ≥ n0 we have, ∞ P
k=1
and ∞ P k=1
op P op ∞ n ∞ n p P (n) (n) λ Xk , 0 λ Xk , Xk + =
