λ-Statistical convergence of fuzzy numbers and fuzzy functions of order θ

Soft Comput (2014) 18:1027–1032 DOI 10.1007/s00500-013-1125-4

METHODOLOGIES AND APPLICATION

λ-Statistical convergence of fuzzy numbers and fuzzy functions of order θ P. D. Srivastava · Sarita Ojha

Published online: 15 September 2013 © Springer-Verlag Berlin Heidelberg 2013

Abstract In this paper, we introduce the concept of λ-statistical convergence of order θ and strong λ-summability of order θ for the sequence of fuzzy numbers. Further the same concept is extended to the sequence of fuzzy functions θ ( fˆ). Some incluand introduce the spaces like Sλθ ( fˆ) and ωλp sion relations in those spaces and also the underlying relation between these two spaces are also obtained. Keywords Fuzzy real numbers · Fuzzy sequences · Statistical convergence · Strong summability of order β

1 Introduction The concept of fuzzy sets is first introduced by Zadeh (1965). Later on, the sequences of fuzzy numbers is discussed by several mathematicians such as Matloka (1986), Nanda (1989), Savas (2000) etc and they have introduced the sequence spaces of fuzzy numbers. The concepts such as statistical convergence, summability etc. are generalized for fuzzy numbers by Nuray and Savas (1995). As the set of all real numbers can be embedded in the set of all fuzzy numbers, statistical convergence in reals can be considered as a special case of those fuzzy numbers. However since the set of all fuzzy numbers is partially ordered and does not carry a group structure, most of the results known for the

Communicated by T. Allahviranloo. P. D. Srivastava (B) · S. Ojha Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, West Bengal, India e-mail: [email protected] S. Ojha e-mail: [email protected]

sequences of real numbers may not be valid in fuzzy setting. Therefore, the theory is not a trivial extension of what has been known in real case. The concept of statistical convergence was introduced by Steinhaus (1951) and Fast (1951) and later reintroduced by Schoenberg (1959) independently. Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy (1985), Savas (2000), Connor (1988), Mursaleen (2000) and many others. The statistical convergence with degree 0 < β < 1 was given by Gadijiev and Orhan (2002) for a real or complex sequence. After then the statistical convergence of order α and summability of order α were studied by Colak (2010) for sequences of real or complex numbers. Existing work on statistical convergence appears to have been restricted to real or complex sequences, but Nuray and Savas (1995) extended the idea to apply to sequences of fuzzy numbers and also introduced and discussed the concept of statistically Cauchy sequences of fuzzy numbers. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. Et et al. (2013) introduced and examined the concepts of pointwise λ-statistical convergence of order α and pointwise [V, λ]-summability of order α of sequences of real valued functions. In an analogous way, Savas (2000) have introduced and studied the concept of λ-statistical convergence of sequence of fuzzy real numbers. This has motivated us to define the concept of λ-statistical convergence of fuzzy numbers and fuzzy functions of order θ and obtain various topological and algebraic properties of these spaces. In the present paper, we have extended these results to introduce the concept of λ-statistical convergence of order θ for the sequence of fuzzy numbers as well as strongly

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P. D. Srivastava, S. Ojha

λ-summability of order θ for fuzzy numbers and the same for fuzzy functions.

2 Definitions and preliminaries A sequence x = (xk ) is said to be statistically convergent to a number l if for every ε > 0, 1 |{k ≤ n : |xk − l| ≥ ε}| = 0 n→∞ n where the vertical bars indicate the number of elements in the enclosed set. Next Colak (2010) generalized this concept to statistical convergence of order β where 0 < β ≤ 1. A sequence x = (xk ) of real or complex numbers is said to be statistically convergent of order β if there is a complex number l such that 1 |{k ≤ n : |xk − l| ≥ ε}| = 0 lim n→∞ n β Fuzzy sequences and their properties were first developed by Matloka (1986). Later on, Nanda (1989) introduced l p space of sequences of fuzzy numbers. A fuzzy real number X : R → [0, 1] is a fuzzy set which satisfies the following conditions:

of order α. Next Et et al. (2013) generalized this concept of order α for real sequences of functions. In the present paper, we have extended this notion of λ-statistical convergence of order θ , 0 < θ < 1 for sequences of fuzzy real numbers as well as fuzzy sequences of functions to get the inclusion relations.

3 λ-Statistical convergence of fuzzy numbers of order θ

lim

(i) normal i.e. X (t) = 1 for some t ∈ R. (ii) fuzzy convex i.e. X (t) ≥ min{X (s), X (r )} where s < t < r. (iii) X is upper semicontinuous. (iv) The closure of X 0 = {t ∈ R : X (t) > 0} is compact. Clearly, R is embedded in L(R), the set of all fuzzy numbers satisfying (i)–(iv), in this way: for each r ∈ R, r ∈ L(R) is defined as,  1, t = r r (t) = 0, t = r For, 0 < α ≤ 1, α-cut of a fuzzy number X is defined by X (α) = {t ∈ R : X (t) ≥ α}. Now for any two fuzzy numbers X, Y , Matloka (1986) (α) − introduced, d(X (α) , Y (α) ) = max{|X (α) − Y (α) |, |X (α) (α) Y |} where X (α) and X are the lower and upper bound of the α-cut and d(X, Y ) = sup d(X

(α)

,Y

(α)

)

0≤α≤1

and showed that d is a metric on the space L(R). Let Λ be the class of all those sequences of positive numbers λ = (λn ) which are non-decreasing and tend to ∞ satisfying λn+1 ≤ λn +1, λ1 = 1. Colak and Bektas (2011) introduced strongly λ-summability and λ-statistical convergence

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In this section, we introduce and study the concept of λ-statistical convergence of order θ in case of fuzzy numbers. Definition 1 A sequence X = (X k ) of fuzzy numbers is said to be λ-statistically convergent of order θ for 0 < θ ≤ 1 or Sλθ -convergent to a fuzzy number X 0 if for every ε > 0, 1 |{k ∈ In : d(X k , X 0 ) ≥ ε}| = 0 n→∞ λθn lim

where In = [n − λn + 1, n] where (λn ) ∈ Λ. We shall denote it as, Sλθ − lim X k = X 0 . For θ = 1 and λn = n, λ-statistical convergence of order θ is the same as statistical convergence. Remark 1 Statistical convergence of order θ for θ > 1 is not well defined. For an example, for θ > 1 and for the sequence (X k ) defined as  1, k = 2n Xk = 0, k = 2n we get, limn→∞

1 |{k λθn

∈ In : d(X k , 1) ≥ ε}| ≤ limn→∞

= 0 as well as, limn→∞ n] limn→∞ [λ 2λθn

= 0.

1 |{k λθn

[λn ] 2λθn

∈ In : d(X k , 0) ≥ ε}| ≤

Which leads to Sλθ − lim X k = 1 as well as Sλθ − lim X k = 0, an absurd. Now we give an example in support of Definition 1. √ Example 1 Let us take λn = n, θ = 1 and the sequence (X k ) as follows:  Xk =

A, k = n 3 0, k = n 3

where A is a positive fuzzy number. Then, for any 0 ≤ α ≤ 1 and ε > 0, we have  α sup0≤α≤1 |A |, k = n 3 d(X k , 0) = 0, k = n 3 √ 1 [ n]1/3 ∴ √ |{k ∈ In : d(X k , 0) ≥ ε}| ≤ √ →0 n n as n → ∞. So, Sλ − lim X k = 0.

λ-Statistical convergence of fuzzy numbers and fuzzy functions of order θ

Theorem 1 Let 0 < θ ≤ 1 and X = (X k ), Y = (Yk ) be two sequences of fuzzy numbers. Then we have, (i) If Sλθ − lim X k = X 0 and c ∈ R, c = 0, then Sλθ − lim cX k = cX 0 . (ii) If Sλθ − lim X k = X 0 and Sλθ − lim Yk = Y0 , then Sλθ − lim(X k + Yk ) = X 0 + Y0 .

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Then, for any 0 ≤ α ≤ 1, we have,   1 − α  1 = d(X k , 0) = sup  k  k 0≤α≤1 n 1  lim √ [d(X k , X 0 )] p = 0. n→∞ n k=1

1 2

Proof Proof is straight forward, so we omit it. Definition 2 A sequence X = (X k ) of fuzzy numbers is λ-statistically Cauchy sequence of order θ, (0 < θ ≤ 1) if for every ε > 0, ∃ a number N (= N (ε)) such that, 1 lim θ |{k ∈ In : d(X k , X N ) ≥ ε}| = 0. n→∞ λn Theorem 2 If a sequence X = (X k ) of fuzzy numbers is λ-statistically convergent of order θ , then it is λ-statistically Cauchy of order θ . Proof Let, Sλθ − lim X k = X 0 . Then by definition, for a pre-assigned positive ε > 0, we have, d(X k , X 0 ) < ε/2 for almost all k. So, we can choose N such that d(X N , X 0 ) < ε/2. Since we have, d(X k , X N ) ≥ ε ⇒ d(X k , X 0 ) + d(X N , X 0 ) ≥ ε ⇒ d(X k , X 0 ) ≥ ε/2 ∴ (X k ) is statistically Cauchy of order θ . Definition 3 Let p be a positive real number. A sequence θ -summable if there is a X = (X k ) is said to be strongly ωλp fuzzy real number X 0 such that, 1  [d(X k , X 0 )] p = 0. lim θ n→∞ λn k∈In

θ − lim(X ) = X . In this case, we write ωλp k 0

Example 2 Let p > 1 be any real number and λn = n, θ = 1 2 . We take the sequence X = (X k ) whose membership function is given by,  kt + 1, − k1 ≤ t ≤ 0 X k (t) = −kt + 1, 0 ≤ t ≤ k1

Then, ωλp − lim(X k ) = 0. 4 λ-Statistical convergence of fuzzy sequence of functions of order θ Let X, Y be the universal sets and f : X → Y is a function. Let F(X ), F(Y ) be the respective universes of fuzzy sets, identified with their membership functions, i.e. F(X ) = {A : X → [0, 1]} and similarly for F(Y ). By the extension principle f induces a function fˆ : F(X ) → F(Y ) (Perfilieva 2011) such that for all A ∈ F(X ),  supx∈ f −1 (y) A(x), f −1 (y) = φ ˆ f (A)(y) = 0, f −1 (y) = φ Definition 4 Let { f k } be a sequence of functions f k : X → X and the induced fuzzy functions be fˆk . Let 0 < θ ≤ 1 is given. Then the sequence ( fˆk ) = ( fˆk (A)), A ∈ F(X ) of fuzzy functions is said to be Sλθ ( fˆ)-statistical convergence to the function fˆ if for every ε > 0, lim

n→∞

1 |{k ∈ In : d( fˆk (A), fˆ(A)) ≥ ε ∀ A ∈ F(X )}| = 0 λθn

In this case, we write Sλθ − lim fˆk (A) = fˆ(A) on F(X ). Theorem 3 Let λ = (λn ) and μ = (μn ) be two sequences in Λ such that λn < μn for all n ≥ n 0 , 0 < θ ≤ β ≤ 1 and ( fˆk ) be a sequence of fuzzy functions defined on F(X ). (i) If lim inf

n→∞

λθn

β

μn

> 0,

(1)

β then Sμ ( fˆ) ⊆ Sλθ ( fˆ). (ii) If the limit

lim (μn − λn ) exists and finite,

n→∞

(2)

β

then Sλθ ( fˆ) ⊆ Sμ ( fˆ). Proof Since λn < μn for all n ≥ n 0 , then In ⊂ Jn where In = [n − λn + 1, n] and Jn = [n − μn + 1, n].

123

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P. D. Srivastava, S. Ojha θ ( fˆ)-summable if ( fˆk ) is said to be strongly ωλp

(i) Also, λn < μn ∀ n ≥ n 0 ⇒ λθn < μθn ≤ μβn ∀ n ≥ n 0 ⇒

λθn

β

μn



1 n→∞ λθn lim

< 1 ∀ n ≥ n0.

[d( fˆk (A), fˆ(A))] p = 0.

k∈In A∈F (X )

θ − lim fˆ (A) = fˆ(A) on F(X ). In this case, we write ωλp k The set of all strongly ωθ ( fˆ)-summable sequences of funcλp

Thus the limit in (1) is always finite. Now for every ε > 0, we have {k ∈ Jn : d( fˆk (A), fˆ(A)) ≥ ε ∀ A ∈ F (X )} ⊃ {k ∈ In : d( fˆk (A), fˆ(A)) ≥ ε ∀ A ∈ F (X )} i.e. |{k ∈ Jn : d( fˆk (A), fˆ(A)) ≥ ε ∀ A ∈ F (X )}| ≥ |{k ∈ In : d( fˆk (A), fˆ(A)) ≥ ε ∀ A ∈ F (X )}| 1 i.e. β |{k ∈ Jn : d( fˆk (A), fˆ(A)) ≥ ε ∀ A ∈ F (X )}| μn λθ 1 ≥ nβ θ |{k ∈ In : d( fˆk (A), fˆ(A)) ≥ ε ∀ A ∈ F (X )}| μn λn

∀ n ≥ n 0 . Now taking n → ∞ in the above inequality, β we get Sμ ( fˆ) ⊆ Sλθ ( fˆ). (ii) Let Sλθ − lim fˆk (A) = fˆ(A) on F(X ) and (2) holds. Then for every ε > 0,

θ ( fˆ). For θ = 1, we write ω ( fˆ) tions will be denoted by ωλp λp instead of ωθ ( fˆ). λp

Theorem 4 Let λ = (λn ) and μ = (μn ) be two sequences in Λ such that λn < μn ∀ n ≥ n 0 , 0 < θ ≤ β ≤ 1, 0 < p < ∞ and ( fˆk ) be a sequence of fuzzy functions defined on F(X ). Then, β

θ ( fˆ) ∀ A ∈ F(X ). (i) If (1) holds, then ωμp ( fˆ) ⊆ ωλp θ ( fˆ) ⊆ ωβ ( fˆ) where (ii) If (2) holds, then B(F(X )) ∩ ωλp μp B(F(X )) is the class of all bounded sequences in F(X ).

Proof (i) Let (1) holds. Since In ⊂ Jn , we have,  k∈In A∈F (X )

β μn

|{k ∈ Jn : d( fˆk (A), fˆ(A)) ≥ ε ∀ A ∈ F (X )}|

=

1

|{n − μn +1 ≤ k ≤ n − λn : d( fˆk (A), fˆ(A)) ≥ ε β

∀ A ∈ F (X )}| +

≤

β μn

μn

μn − λn β

μn

1 β

μn

|{k ∈ In : d( fˆk (A), fˆ(A)) ≥ ε

∀ A ∈ F (X )}| 1 + θ |{k ∈ In : d( fˆk (A), fˆ(A)) ≥ ε λn ∀A ∈ F (X )}|

∀ n ≥ n 0 . Taking n → ∞, we obtain that the right hand side of the above inequality tends to 0. β ⇒ Sλθ ( fˆ) ⊆ Sμ ( fˆ). Remark 2 In case λn = μn , the proof of the above theorem can be written without the conditions (1) and (2).

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[d( fˆk (A), fˆ(A))] p

k∈In A∈F (X )

[d( fˆk (A), fˆ(A))] p

k∈Jn A∈F (X )

β Since (1) holds, ∴ taking n → ∞, we get ωμp ( fˆ) ⊆ θ ωλp ( fˆ). θ ( fˆ). Since ( fˆ (A)) ∈ (ii) Let ( fˆk (A)) ∈ B(F(X )) ∩ ωλp k B(F(X )), then there exists M > 0 such that d( fˆk (A), fˆ(A)) ≤ M ∀ k ∈ N and ∀ A ∈ F(X ). Let (2) holds. Then,



1 β

μn



1 β

μn

+

[d( fˆk (A), fˆ(A))] p

k∈Jn −In A∈F (X )

1



[d( fˆk (A), fˆ(A))] p

β

μn

 ≤

[d( fˆk (A), fˆ(A))] p

k∈Jn A∈F (X )

=

5 Strongly λ-summability of fuzzy sequence of functions of order θ Let the sequence λ = (λn ) ∈ Λ, θ ∈ (0, 1] be any real number and p be a positive real number. A sequence of functions





1

≤

[d( fˆk (A), fˆ(A))] p

k∈Jn A∈F (X )

λθn 1 β θ μn λn

i.e. 1



[d( fˆk (A), fˆ(A))] p ≤

k∈In A∈F (X )

μn − λn β

μn



Mp +

1 λθn

 k∈In A∈F (X )

[d( fˆk (A), fˆ(A))] p

λ-Statistical convergence of fuzzy numbers and fuzzy functions of order θ β

θ ( fˆ) ⊆ ω ( fˆ). ∴ Taking n → ∞, we get B(F(X )) ∩ ωλp μp

Remark 3 In case λn = μn , the proof of the above theorem can be written without the conditions (1) and (2). Theorem 5 Let θ and β be fixed real numbers such that 0 < θ ≤ β ≤ 1, 0 < p < ∞, λn < μn ∀ n ≥ n 0 and ( fˆk ) be a sequence of fuzzy functions defined on F(X ). Then, 1. Let (1) holds, if a sequence of fuzzy functions defined β on F(X ) is strongly ωμp ( fˆk )-summable to fˆ, then it is Sλθ ( fˆ)-statistically convergent to fˆ. 2. Let (2) holds and ( fˆk ) be a sequence of bounded fuzzy functions defined on F(X ), then if a sequence strongly θ ( fˆ)-summable to fˆ, then it is S β ( fˆ)-statistically conωλp μ vergent to fˆ.

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Now taking n → ∞ in both sides of the above inequality, the relation follows. (ii) Let ( fˆk (A)) ∈ Sλθ ( fˆ) and ( fˆk (A)) ∈ B(F(X )). Then there exists M > 0 such that d( fˆk (A), fˆ(A)) ≤ M ∀ k. Let (2) holds. Then for every ε > 0, we have,  k∈In A∈F (X )

=



k∈Jn ,A∈F (X ) d( fˆk (A), fˆ(A))≥ε

1 λθn

[d( fˆk (A), fˆ(A))] p



i.e.

[d( fˆk (A), fˆ(A))] p

k∈Jn A∈F (X )

1

≥ β |{k ∈ In : d( fˆk (A), fˆ(A)) ≥ ε ∀ A ∈ F(X )}|ε p μn λθ 1 = nβ θ |{k ∈ In : d( fˆk (A), fˆ(A)) μn λn ≥ ε ∀ A ∈ F(X )}|ε . p

[d( fˆk (A), fˆ(A))] p

k∈In A∈F (X )

≥

Thus, we have

β

[d( fˆk (A), fˆ(A))] p

−M p (μn − λn )

A ∈ F(X )}|ε p

μn



k∈Jn −In A∈F (X )

≥ |{k ∈ In : d( fˆk (A), fˆ(A)) ≥ ε ∀



[d( fˆk (A), fˆ(A))] p

Thus we have, [d( fˆk (A), fˆ(A))] p

k∈In ,A∈F (X ) d( fˆk (A), fˆ(A))≥ε

1



≥ |{k ∈ Jn : d( fˆk (A), fˆ(A)) ≥ ε}|ε p

k∈In ,A∈F (X ) d( fˆk (A), fˆ(A))