Some Double Sequence Spaces of Fuzzy Real Numbers of Paranormed Type

Throughout the paper, a double sequence is denoted by ⟨X nk ⟩, a double infinite array of elements X nk , where each X nk is a fuzzy real number. The initial work on double sequences is found in Bromwich [1]. Later on, it was studied by Hardy [2], Móricz [3], Tripathy [4], Basarir and Sonalcan [5], and many others. Hardy [2] introduced the notion of regular convergence for double sequences. The concept of paranormed sequences was studied by Nakano [6] and Simons [7] at the initial stage. Later on, it was studied by many others. After the introduction of fuzzy real numbers, different classes of sequences of fuzzy real numbers were introduced and studied by Tripathy and Nanda [8], Choudhary and Tripathy [9], Tripathy et al. [10–13], Tripathy and Dutta [14– 16], Tripathy and Borgogain [17], Tripathy and Das [18], and many others. Let D denote the set of all closed and bounded intervals


Introduction
Throughout the paper, a double sequence is denoted by ⟨  ⟩, a double infinite array of elements   , where each   is a fuzzy real number.
The concept of paranormed sequences was studied by Nakano [6] and Simons [7] at the initial stage.Later on, it was studied by many others.
Let  denote the set of all closed and bounded intervals  = [ 1 ,  2 ] on , the real line.For ,  ∈ , we define where A fuzzy real number  is a fuzzy set on , that is, a mapping  :  → (= [0, 1]) associating each real number  with its grade of membership ().
The set of all upper semicontinuous, normal, and convex fuzzy real numbers is denoted by (), and throughout the paper, by a fuzzy real number, we mean that the number belongs to ().
Let ,  ∈ (), and let the -level sets be , and  ∈ [0, 1]; the product of  and  is defined by
If there exists  0 ∈  such that ( 0 ) = 1, then the fuzzy real number  is called normal.
A fuzzy real number  is said to be upper semicontinuous if, for each  > 0,  −1 ([0,  + )), for all  ∈ , is open in the usual topology of .
The set  of all real numbers can be embedded in ().For  ∈ ,  ∈ () is defined by The absolute value, || of  ∈ (), is defined by (see, e.g., [19]) A fuzzy real number  is called nonnegative if () = 0, for all  < 0. The set of all nonnegative fuzzy real numbers is denoted by  * ().
The additive identity and multiplicative identity in () are denoted by 0 and 1, respectively.
A sequence (  ) of fuzzy real numbers is said to be convergent to the fuzzy real number  if, for every  > 0, there exists  0 ∈  such that (  , ) < , for all  ≥  0 .
A sequence of fuzzy numbers (  ) converges to a fuzzy number A sequence (  ) of generalized fuzzy numbers converges weakly to a generalized fuzzy number  (and we write     → ) if distribution functions (   ) converge weakly to   and (   ) converge weakly to   [21].
A double sequence (  ) of fuzzy real numbers is said to be convergent in Pringsheim's sense to the fuzzy real number  if, for every  > 0, there exists  0 ,  0 ∈  such that (  , ) < , for all  ≥  0 ,  ≥  0 .
A double sequence (  ) of fuzzy real numbers is said to be regularly convergent if it converges in Pringsheim's sense, and the following limits exist: lim   (  ,   ) = 0, for some   ∈  () , A fuzzy real number sequence (  ) is said to be bounded if sup  |  | ≤ , for some  ∈  * ().
For  ∈  and  ∈ (), we define Let  = {(  ,   ) :  ∈ ;  1 <  2 <  3 < ⋅ ⋅ ⋅ and  1 <  2 <  3 < ⋅ ⋅ ⋅} ⊆  × , and let   be a double sequence space.A K-step space of   is a sequence space A canonical preimage of a sequence ⟨     ⟩ ∈   is a sequence ⟨  ⟩ defined as follows: A canonical preimage of a step space    is a set of canonical preimages of all elements in    .A double sequence space   is said to be monotone if   contains the canonical preimage of all its step spaces.
From the above definitions, we have the following remark.
Remark 1.A sequence space   is solid ⇒   is monotone.
A double sequence space   is said to be symmetric if ( (,) ) ∈   , whenever (  ) ∈   , where  is a permutation of  × .
A double sequence space   is said to be convergence-free if (  ) ∈   , whenever (  ) ∈   , and   = 0 implies that   = 0.
Sequences of fuzzy real numbers relative to the paranormed sequence spaces were studied by Choudhary and Tripathy [9].
In this paper, we introduce the following sequence spaces of fuzzy real numbers.
Let  = ⟨  ⟩ be a sequence of positive real numbers for some  ∈  () } .
For  = 0, we get the class ( 2   ) 0 ().Also a fuzzy sequence For the class of sequences
Proof.Obviously the space ( 2 ℓ ∞ )  () is symmetric.For the other spaces, consider the following example.
The result follows from the inequality Hence, the space ( 2 ℓ ∞ )  (p) is solid.Similarly, the other spaces are also solid.
Proof.The result follows from the following example.