Some Additions to the Fuzzy Convergent and Fuzzy Bounded Sequence Spaces of Fuzzy Numbers

and Applied Analysis 3 Conversely, if the pair of functions α and β satisfy the conditions 1 – 4 , then there exists a unique u ∈ E1 such that u λ α− λ , β λ for each λ ∈ 0, 1 , 9 . The fuzzy number u corresponding to the pair of functions α and β is defined by u : R → 0, 1 , u x sup{λ : α x ≤ λ ≤ β x }, 5 . A sequence u uk of fuzzy numbers is a function u from the set N, the set of all positive integers, into E1, and fuzzy number uk denotes the value of the function at k and is called the kth term of the sequence. Let u, v ∈ E1 and λ ∈ R, then the operations addition and scalar multiplication are defined on E1 in terms of α-level sets by u v w ⇐⇒ w α u α v α, λu α λ u α ∀α ∈ 0, 1 . 2.2 Define a map d : E1 ×E1 → R by d u, v sup0≤α≤1d u α, v α . It is known that E1 is a complete metric space with the metric d 3 . Let us suppose that w E1 , c E1 , and ∞ E1 are set of all sequences space of all fuzzy numbers, convergent and bounded sequences of fuzzy numbers, respectively. Let us suppose that u, v ∈ E1 and G is the set of all nonnegative fuzzy numbers. The function df : E1 ×E1 → G is called fuzzy metric 6 which satisfies the following properties: 1 df u, v ≥ 0, 2 df u, v 0 if and only if u v, 3 df u, v df v, u , 4 whenever w ∈ E1, we have df u, v ≤ df u,w df w,v . In 4 , Nanda has studied the spaces of bounded and convergent sequences of fuzzy numbers and has shown that they are complete metric spaces with the metric ̂ d u, v supkd uk, vk . By using this metric, ̂ d, so many spaces of fuzzy sequences have been built and published in famous maths journals. By reviewing the literature, one can reach them easily. However, another important metric which is called as fuzzy metric is used for measuring fuzzy distances among fuzzy numbers. If df is a fuzzy metric on E1, then the pair of E1, df is called as a fuzzy metric space. For any u, v ∈ E1, the fuzzy metric of Zhang is 10–12 defined by df u, v sup λ∈ 0,1 λ [ d1 u, v , ̂ dλ u, v ] , 2.3 where ̂ dλ u, v supα∈ λ,1 d u α, v α and d1 u, v supα 1d u α, v α . Also, fuzzy metric spaces have been studied in 13 . But the given metric space definition in 13 is different from our fuzzy metric space definition. Theorem 2.1 see 10–12 . The metric df defined by equality 2.3 is a fuzzy distance of fuzzy numbers; thus, E1, df is a fuzzy metric space. Theorem 2.2. If u, v ∈ E1, then d u, v is a point of the interval determined by the fuzzy metric df u, v . 4 Abstract and Applied Analysis Proof. Clearly, if λ run from 0, then the second side of df , ̂ dλ u, v , is equal to d u, v since df u, v sup λ∈ 0,1 λ [ d1 u, v , ̂ dλ u, v ] . 2.4 Theorem 2.3 see 12 . The fuzzy metric space E1, df is complete metric space. Definition 2.4. Let λ E1 be the subset of all sequence spaces of fuzzy numbers and suppose that ‖ · ‖ : λ E1 → G is a function. The function ‖ · ‖ is called fuzzy module or fuzzy norm if it has the following properties: N1 ‖u‖ θ ⇔ u θ, N2 ‖αu‖ |α|‖u‖E1 , N3 ‖u v‖ ≤ ‖u‖ ‖v‖ If the function ‖ · ‖ : λ E1 → G satisfies N1, N2, and N3, then λ E1 is called fuzzy module sequence space of the fuzzy numbers. And if λ E1 is complete with respect to the fuzzy module, then λ E1 is called complete fuzzy module sequence space of the fuzzy numbers. Definition 2.5. The fuzzy module of the fuzzy number u is defined which corresponds to the fuzzy distance from u to 0, that is, ‖u‖E1 : sup λ∈ 0,1 λ [ d1 ( u, 0 ) , ̂ dλ ( u, 0 )] . 2.5 Proposition 2.6. The set E1 of the fuzzy numbers is fuzzy complete module space with the fuzzy module in 2.5 . Let u uk be a sequence of fuzzy numbers, and let ‖ · ‖ be a fuzzy module, then the sequence uk is said to converge fuzzy to u0 ∈ E1 with the fuzzy module ‖ · ‖ if for any given > 0, there exists an integer n0 such that ‖uk − u0‖ < , for k ≥ n0. The sequence uk is said to be fuzzy bounded in fuzzy module ‖ · ‖ if supk‖uk‖ < ∞ for all k ∈ N. We will write L∞ E1 , C E1 , and C0 E1 for the fuzzy sets of all fuzzy bounded, fuzzy convergent, fuzzy null sequences, respectively, that is, L∞ ( E1 ) : { u uk ∈ w ( E1 ) : sup k sup λ∈ 0,1 λ [ d1 ( uk, 0 ) , ̂ dλ ( uk, 0 )] < ∞ }