p -Bounded variation fuzzy real-valued double sequence space

. In this paper we have introduced the notion of fuzzy real-valued p -bounded variation double sequence space pF


Introduction
Fuzzy set is a mathematical model of vague qualitative or quantitative data, frequently generated by the means of natural language.It is based on the generalization of the classical concepts of set and its characteristic function.The theory of fuzzy set was given by Zadeh (1965), since then many major theoretical breakthroughs have been established and successfully applied to many industrial applications.Numerous research workers are involving to develop and extend it in different directions.The introduction of fuzzy number opened many new dimensions in the field of Mathematics.With the idea of fuzzy set theory and fuzzy real numbers, existing notions in different branches of mathematics are generalized.In the field of pure mathematics the use of fuzzy set and fuzzy real numbers are very remarkable.Specifically, we may mention that these notions are used extensively in studying properties of sequence spaces.Since our work is based on fuzzy real numbers, we begin with some basic ideas on it for easy understanding of the work.
A fuzzy real number X is a fuzzy set on R, more precisely a mapping X: R → I (= [0, 1]), which associate each real number t, with its grade of membership X (t).
The α-level set of a fuzzy real number X is defined by A fuzzy real number X is said to be upper-semicontinuous if for each ε > 0, X -1 ([0, a+ε)), for all a∈I is open in the usual topology of R.
If there exists t∈R such that X(t) = 1, then the fuzzy real number X is called normal.
A fuzzy real number X is said to be convex, if X(t) ≥ X(s)∧X(r) = min(X (s), X (r)), where s < t < r.
We denote the class of all upper-semicontinuous, normal and convex fuzzy real numbers by R(I) and that of all positive fuzzy real numbers by R * (I).
For X, Y∈R(I), X ≤ Y if and only if X α ≤ Y α for α∈[0,1] and "≤" is a partial order in R(I).
The absolute value of X∈R(I) is defined by The set of real numbers R can be embedded into R(I), for r∈R, r ∈R(I) is defined by The additive identity and multiplicative identity of R(I) are denoted by 0 and 1 respectively.
For any X, Y, Z∈ R(I), the linear structure of R(I) induces addition X + Y and scalar multiplication λX, λ∈R in terms of α-level set, defined as I) is said to be bounded above if there exist a fuzzy real number μ such that X ≤ μ for every X∈E.We called μ as the upper bound of E and it is called least upper bound if μ ≤ μ * for all upper bound μ * of E. A lower bound and greatest is defined similarly.The set E is said to be bounded if it is both bounded above and bounded below.
Let D be the set of all closed bounded intervals We consider the function It is well established that (R(I), d ) is a complete metric space.

Preliminaries and background
In this section we discuss some fundamental concepts and properties related to the subject matter of the article.
A sequence X = (X k ) of fuzzy real number is a function X from the set of positive integer into R(I).The fuzzy number X k is called the k th term of the sequence.
The set E F of sequences taken from R(I) is said to be a sequence space of fuzzy real number if, for (X k ), (Y k )∈E F , r∈R i.e.X k , Y k ∈ R(I), and for all k∈N, where: Works on double sequence started in the early nineties.Initially it was studied by Hardy (1917), Bromwich (1965) and some others.In recent years the theory was further developed by Moričz (1991), Basarir and Solancan (1999), Savas (2007), Savas (1996), Savas and Mursaleen (2004), Tripathy and Dutta (2007, 2008, 2010), Tripathy and Sarma (2008aand b, 2009, 2011), and some others.
The notion of difference sequence in complex terms was introduced by Kizmaz (1981) and defined by Tripathy and Sarma (2008a) studied it for double sequence spaces.In terms of fuzzy real numbers it was studied by Basarir and Mursaleen (2003), Tripathy and Dutta (2008), Tripathy and Borgohain (2010, 2011), and others. Hardy (1917) introduced the notions of regular convergence of double sequences and the notion of bounded variation of double sequences as follows: Definition 2.1.A double sequence > < nk a is said to converge regularly if it converges in Pringsheim's sense and in addition the following limits holds: x nk is for every fixed value of n and k, of bounded variation in n or k.
A fuzzy real-valued double sequence is a double infinite array of fuzzy real numbers.We denote a fuzzy real-valued double sequence by <X nk >, where X nk are fuzzy real numbers for each n, k∈N.
Definition 2.3.A fuzzy real-valued double sequence <X nk > is said to be convergent in Pringsheim's sense to the fuzzy real number X, if for every ε > 0, there exists Definition 2.5.A fuzzy real-valued double sequence <X nk > is said to be regularly convergent if it is convergent in Pringsheim's sense and the followings hold: For a given ε>0, there exists n 0 = n 0 (ε, k) and for all n ≥ n 0 , for some L k ∈R(I) for each k∈N, and The notion of double difference sequences of fuzzy real numbers was introduced by Tripathy and Dutta (2008) as follows: where ΔX nk = X nk -X n+1,k -X ,n k+1 + X n+1,k+1 , for all n, k∈N.
The class of fuzzy real-valued bounded variation double sequences 2 F bv was introduced by Tripathy and Dutta (2010) as follows: , where , for all n, k∈N.
The class of sequence bv p (F) was introduced and studied by Talo and Basar (2008).In this article we introduce the class of p-bounded variation fuzzy real-valued double sequence p F bv 2 as follows: Where ΔX nk = X nk -X n+1,k -X n, k+1 + X n+1,k+1 , for all n, k∈N.
We use the following inequality throughout the article, wherever it is applicable.
Let p = (p k ) be a positive sequence of real numbers with 0<p k <sup p k = H and D = max(1, 2 H-1 ).Then for all a k , b k ∈C; ( ) , for all k∈N.

Results and discussion
Theorem 3.1.The class of double sequence Proof: It is easily to verify that ρ , defined by ( 1) This imply that Since R(I) is a complete metric space by the metric _ d , so , for each n∈N.
Similarly from (2) we have ( i k X 1 ) is a Cauchy sequence and hence convergent, so converges for each n, k∈N.

Let us consider
are convergent and hence Proceeding in this way, we get, , for all n, k∈N.Now taking limit as j → ∞ in (2), we have ρ (X i , X)< ε for all i ≥ n 0 .Thus for all i ≥ n 0 , we have For all n odd,

Hence we conclude that
The matrix representation of the <X nk > is given by Then <ΔX nk > is represented by For n ≠ k and n odd, Then <ΔY nk > is represented by Example 3.2.Consider the double sequence (X nk ) defined as follows: and Then <ΔY nk > is defined as follows and This implies that <Y nk >∉ Example 3.3.Let p>1.Consider the double sequence <X nk > defined by, − , for n ≥ 2 and X nk = 0 , otherwise.The matrix representation of <X nk > is given by The matrix determined by <ΔX nk > is represented as Then we have the following matrix representations.
This completes the proof of the theorem., for 1 ≤ q < p < ∞ and the inclusion is strict.
Proof: (a) Let us consider the double sequence It is easy to verify that the inclusion is strict.
This completes the proof of the theorem.For (n, k)∈K = N × N -{(n, k): n ≤ n 0 , k ≤ k 0 } and for 0 < q < p < ∞, we have It is easy to verify that the inclusion is strict.Hence we conclude that

Conclusion
The concept of double sequence in terms of fuzzy real numbers is a very recent development.Soon after it many researchers have introduced different classes of double sequences and studied some algebraic and topological properties.The class of sequence introduced here has its importance from the point of view of its structure and norm.We have verified some important properties for the class of double sequence with some concrete examples.
The proof follows from the following example:Example 3.1.Let p>1.Consider the double sequence <X nk > defined as follows: The result follows from the following example.
for some M n ∈R(I) for each n∈N.Definition 2.6.A fuzzy real-valued double sequence space E F is said to be normal (or solid) if <Y nk >∈E F , whenever |Y nk | ≤ |X nk | for all n, k∈N and <X nk >∈E F .Definition 2.7.A fuzzy real-valued double sequence space E F is said to be symmetric if <X π(n,k) >∈E F , whenever <X nk >∈E F , where π is a permutation of N. Definition 2.8.A fuzzy real-valued double sequence space E F is said to be convergence free if <X nk >∈E F whenever <Y nk >∈E F and Y