A Study of Fuzzy Sequence Spaces

The purpose of this chapter is to introduce and study some new ideal convergence sequence spaces F S J θ Tð Þ , F S J θ 0 Tð Þ and F S J θ ∞ Tð Þ on a fuzzy real number F defined by a compact operator T . We investigate algebraic properties like linearity, solidness and monotinicity with some important examples. Further, we also analyze closedness of the subspace and inclusion relations on the said spaces.


Introduction
The concepts of fuzzy sets were initiated by Zadeh [1], since then it has become an active area of researchers. Matloka [2] initiated the notion of ordinary convergence of a sequence of fuzzy real numbers and studied convergent and bounded sequences of fuzzy numbers and some of their properties, and proved that every convergent sequence of fuzzy numbers is bounded. Nanda [3] investigated some basic properties for these sequences and showed that the set of all convergent sequences of fuzzy real numbers form a complete metric space. Alaba and Norahun [4] studied fuzzy Ideals and fuzzy filters of pseudocomplemented semilattices Moreover, Nuray and Savas [5] extended the notion of convergence of the sequence of fuzzy real numbers to the notion of statistical convergence.
Fast [6] introduced the theory of statistical convergence. After that, and under different names, statistical convergence has been discussed in the ergodic theory, Fourier analysis and number theory. Furthermore, it was examined from the sequence space point of view and linked with summability theory. Esi and Acikgoz [7] examined almost λ-statistical convergence of fuzzy numbers. Kostyrko et al. [8] introduced ideal J Àconvergence which is based on the natural density of the subsets of positive integers. Kumar and Kumar [9] extended the theory of ideal convergence to apply to sequences of fuzzy numbers. Khan et al. [10][11][12] studied the notion of J -convergence in intuitionistic fuzzy normed spaces. Subsequently, Hazarika [3] studied the concept of lacunary ideal convergent sequence of fuzzy real numbers. Where a lacunary sequence is an increasing integer sequence θ ¼ k r ð Þ such that k 0 ¼ 0 and h r ¼ k r À k rÀ1 ! ∞ as r ! ∞. The intervals are determined by θ and defined by I r ¼ k rÀ1 , k r ð . We outline the present work as follows. In Section 2, we recall some basic definitions related to the fuzzy number, ideal convergent, monotonic sequence and compact operator. In Section 3, we introduce the spaces of fuzzy valued lacunary ideal convergence of sequence with the help of a compact operator and prove our maim results. In Section 4, we state the conclusion of this chapter.

Preliminaries
In this section, we recall some basic notion, definitions and lemma that are required for the following sections.
Þis a real fuzzy number on the set  associating real number s with its grade of membership F s ð Þ. Let C denote the set of all closed and bounded intervals It can be easily seen that " ≤ " is a partial order on C and C, ρ ð Þ is a complete metric space. The absolute value |F| of F ∈  λ ð Þ is defined by Hence, ρ defines a metric on  λ ð Þ. The multiplicative and additive identity in  λ ð Þ are denoted by 1 and 0, respectively. Definition 2.2. A family of subsets J of the power set P  ð Þ of the natural number  is known as an ideal if and only if the following conditions are satisfied [8] i. ∅ ∈ J , ii. for every A 1 , A 2 ∈ J one obtain A 1 ∪ A 2 ∈ J , iii. for every A 1 ∈ J and every A 2 ⊆ A 1 one obtain A 2 ∈ J .
An ideal J is known as non-trivial if J 6 ¼ P  ð Þ and non-trivial ideal is said to be an admissible if J ⊇ n f g : n ∈  f g . Lemma 1. If ideal J is maximal, then for every A ⊂  we have either A ∈ J or nA ∈ J [8].
Definition 2.3. A family of subsets H of the power set P  ð Þ of the natural number  is known as filter in if and only if following condition are satisfied [8].
iii. for every A 1 ∈ H and A 2 ⊇ A 1 one have A 2 ∈ H. Remark 1. Filter associated with the ideal J is defined by the family of sets Definition 2.4. A sequence F k ð Þ of fuzzy real numbers is known as J À convergent to fuzzy real numbers F 0 if for each ε > 0, the set [9].
Definition 2.5. A sequence F k ð Þ is known as J À null if there exists a fuzzy real numbers 0 such that for each ε > 0 [9], Definition 2.6. A sequence F k ð Þ of fuzzy real numbers is known as J -Cauchy if there exists a subsequence F l ε ð Þ of F k ð Þ in such a way that for every ε > 0 [13], Definition 2.7. A sequence F k ð Þ is known as J À bounded if there exists a fuzzy real numbers M > 0 so that, the set [9].
The canonical pre-image of a sequence x k i ð Þ∈ λ  K is a sequence y k À Á ∈ ω defined as follows: K : Definition 2.9. A sequence space  is known as monotone, if it is contains the canonical pre-images of it is step space [12].
That is, if for all infinite K ⊆  and x k ð Þ∈  the sequence α k x k ð Þ, where α k ¼ 1 for k ∈ K and α k ¼ 0 otherwise, belongs to .
Definition 2.10. [12] A sequence space  is known as convergent free, if x k ð Þ∈  whenever y k À Á ∈  and y k À Á ¼ 0 implies that x k ð Þ ¼ 0 for all k ∈ . Lemma 2.1. Every solid space is monotone [12]. Definition 2.11. Suppose U and V are normed spaces. An operator T : U ! V is known as compact linear operator if [12].

Main results
In this section, we introduce the spaces of fuzzy valued lacunary ideal convergence of sequence with the help of a compact operator and investigate some topological and algebraic properties on these spaces. We denote ω F the class of all sequences of fuzzy real numbers and J be an admissible ideal of the subset of the natural numbers .
Suppose α and β be scalars, and assume that Now, let ∈ H J ð Þ: be such that A c 1 , A c 2 ∈ J . Therefore, the set Thus, the sets on right hand side of (2) belong to H J ð Þ. Therefore A c 3 belongs to J . Therefore, α F k ð Þ þ β G k ð Þ∈ F S J θ T ð Þ. Hence F S J θ T ð Þ is linear space. In similar manner, one can easily prove that F S J θ 0 T ð Þ and F S J ∞ T ð Þ are linear spaces.
For the proof of the theorem, we consider the subsequent example.
( It can be easily seen that Then for every ε > 0, the set Which holds for all N ε ∈ B ε . Hence On Contrary, assume that That is Then, the set . If we fix an ε > 0, then we have C ε ∈ H J ð Þ as well as C ε 2 ∈ H J ð Þ. Hence C ε 2 ∩ C ε ∈ H J ð Þ. This implies that Where the diam of I denotes the length of interval I. Continuing in this way, by induction, we get the sequence of closed intervals.
With the property that Then there exists a ξ ∈ ∩ I k where k ∈  in such a way that ξ ¼ J À lim T F k ð Þ. Therefore, the result holds.
Then there exists a number F 0 ∈  such that That is, the set On the both sides, taking the supremum over k of the above equation, we obtain F k ∈ F S J θ T ð Þ. Therefore inclusion holds.
Example 3.4. Suppose fuzzy number Then G k s ð Þ ∈ F S J θ T ð Þ. Applying Lemma 1, there exists a subset K of  in such a way that K ∉ J and nK ∉ J . Determine a sequence G ¼ G k ð Þ as & Therefore G k ð Þ belong to the canonical pre-image of the K-step spaces of ð Þ is not normal. Theorem 3.6. The sequence space F S J θ 0 T ð Þ is solid and monotone.
Suppose that sequence of scalars α k ð Þ with the property |α k | ≤ 1 ∀ k ∈ . Therefore Hence, from the Eq. (5) and above inequality, one obtain Þ, then for each ε > 0 there exists a q such that converges to a. J À lim F ¼ a: Since F q k À Á be a Cauchy sequence in F S I θ T ð Þ. Then for each ε > 0 there exists n 0 ∈ , such that for all q, s ≥ n 0 : For a given ε > 0, Assume n 0 ∈ B c . Then, for every q, s ≥ n 0 it has Hence, a q is a Cauchy sequence of scalars in ℂ, so there exists scalar a ∈ ℂ in such a way that a q À Á ! a as q ! ∞. For this step, let 0 < α < 1 be given. Therefore it proved that whenever Since F q k À Á ! F, there exists q 0 ∈  so that implies that P c ∈ J . The number, q o can be chosen together with Eq. (7), it have Which implies that Q c ∈ J . Since Then it has a subset S such that S c ∈ J , where The right hand side of Eq. (8) belongs to H J ð Þ. Hence, the sets on the left hand side of Eq. (8) belong to H J ð Þ. Therefore its complement belongs to J . Thus, J À lim ρ a q 0 , a À Á ¼ 0. □ In the following example to prove that F S J θ T ð Þ is closed subspace of F S θ ∞ T ð Þ. Example 3.5. Suppose that sequence of fuzzy number determine by F k s ð Þ ¼ 2 À1 1 þ s ð Þ, for s ∈ À1, 1 ½ 2 À1 3 À s ð Þ, for s ∈ 1, 3 ½ 0, otherwise: Hence, ρ F k , 0 À Á ¼ supρ F k , 0 À Á . Therefore F k ð Þ∈ F S J θ T ð Þ and L ¼ 0. So, it can be easily seen that F S J θ T ð Þ is closed subspace of F S θ ∞ T ð Þ.

Conclusion
The spaces of fuzzy valued lacunary ideal convergence of sequence with the help of a compact operator and investigate algebraic and topological properties together with some examples on the given spaces. We proved that the new introduced sequence spaces are linear. Some spaces are convergent free and we also proved that space F S J θ T ð Þ is closed subspace of F S θ ∞ T ð Þ. These new spaces and results provide new tools to help the authors for further research and to solve the engineering problems.