I θ -statistical and p -Cesàro summability of sequences of fuzzy numbers

1 Introduction

The concept of I -convergence was introduced by Kostyrko et al. in a metric space [8]. Later it was further studied by Dems [5], Savaş [3, 27–31] and many others. I convergence is a generalization form of statistical convergence and that is based on the notion of an ideal of the subset of positive integers N.

Among various developments of the theory of fuzzy sets (see, [37]) a progressive development has been made to find the fuzzy analogues of the classical set theory. In fact the fuzzy theory has become an area of active research for the last 50 years. It has a wide range applications in the field of science and and engineering, e.g., population dynamics, chaos control, computer programming, nonlinear dynamical systems, fuzzy physics, fuzzy topology, etc. Recently fuzzy topology proves to be a very useful tool to deal with such situation where the use of classical theories breaks down. In [12], Nanda studied on sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Statistical convergence of sequences of fuzzy numbers are introduced by Nuray and Savas (see, [14]). Nuray [13] proved the inclusion relations between the set of statistically convergent and lacunary statistically convergent sequences of fuzzy numbers. Savas [15] introduced and discussed double convergent sequences of fuzzy numbers and showed that the set of all double convergent sequences of fuzzy numbers is complete. In [9], Kwon studied statistical and p-Ceàro convergence of fuzzy numbers. Later on sequence of fuzzy numbers have been discussed by Kwon and Shim [10], Savas [16–19, 23–26], Mursaleen and Basarir [11], Karakas et al. [7], Tripathy et al. [34], Tripathy and Dutta [35], Tripathy and Das [36] and others. In [33], lacunary statistical convergence of order alpha was studied by Şengül and Et.

Before continuing with this paper we present some definitions and preliminaries which we shall use throughout this paper.

Let X be a non-empty set, then a family of sets I ⊂ 2 X (the class of all subsets of X) is called an ideal if and only if ∅ ∈ I, for each A , B ∈ I , we have A ∪ B ∈ I and for each A ∈ I and each B ⊂ A, we have B ∈ I .

A non-empty family of sets F ⊂ 2 ^X is a filter on X if and only if Φ ∉ F, for each A, B ∈ F, we have A ∩ B ∈ F and each A ∈ F and each A ⊂ B, we have B ∈ F. An ideal I is called non-trivial ideal if I ≠ Φ and X ∉ I . Obviously I ⊂ 2 X is a non-trivial ideal if and only if F = F ( I ) = { X - A : A ∈ I } is a filter on X. A non-trivial ideal I ⊂ 2 ^X is called admissible if and only if { { x } : x ∈ X } ⊂ I. Further details on ideals of 2 ^X can be found in Kostyrko et al. [8] and [21, 22].

Throughout I will stand for a proper admissible ideal of N.

Let C (R ⁿ ) ={ A ⊂ R ⁿ A compactandconvex }. The spaces C (R ⁿ ) has a linear structure induced by the operations A + B = { a + b , a ∈ A , b ∈ B } and λ A = { λ a , λ ∈ A } for A, B ∈ C (R ⁿ ) and λ ∈ R. The Hausdorff distance between A and B of C (R ⁿ ) is defined as δ ∞ ( A , B ) = max { sup a ∈ A inf b ∈ B | a - b | , sup b ∈ B inf a ∈ A | a - b | } .

It is well known that (C (R ⁿ ) , δ_∞) is a complete (not separable) metric space.

A fuzzy number is a function X from R ⁿ to [0, 1] satisfying

X is normal, i.e. there exists an x₀ ∈ R ⁿ such that X (x₀) = 1;

These properties imply that for each 0 < α ≤ 1, the α-level set X α = { x ∈ R n : X ( x ) ≥ α } is a nonempty compact convex, subset of R ⁿ , as is the support X⁰. Let L (R ⁿ ) denote the set of all fuzzy numbers. The linear structure of L (R ⁿ ) induces addition X + Y and scalar multiplication λX, λ ∈ R, in terms of α-level sets, by [ X + Y ] α = [ X ] α + [ Y ] α and [ λ X ] α = λ [ X ] α for each 0 ≤ α ≤ 1. Define for each 1≤ q < ∞ d q ( X , Y ) = { ∫ 0 1 δ ∞ ( X α , Y α ) q d α } 1 q and d ∞ ( X , Y ) = sup 0 ≤ α ≤ 1 δ ∞ ( X α , Y α ) clearly d_∞( X , Y ) = lim q → ∞ d q ( X , Y ) with d _q  ≤ d _r if q ≤ r. Moreover d _q is a complete, separable and locally compact metric space [6].

Throughout the paper, d will denote d ^q with 1≤ q ≤ ∞.

A sequence X = (X _k ) of fuzzy numbers is said to converge to a fuzzy number X₀ if for every ε > 0, there exists a positive integer n₀ such that d ¯ ( X k , X 0 ) < ε for all n ≥ n₀ .

A sequence X = (X _k ) of fuzzy numbers is said to be bounded if the set {X _k  : k ∈ N} of fuzzy numbers is bounded.

The concept of statistical convergence of fuzzy numbers was introduced by Nuray and Savas [14] in 1995. A sequence X = (X _k ) is said to be statistically convergent to the number X₀ if forevery ɛ > 0 lim n 1 n | { k ≤ n : d ( X k , X 0 ) ≥ ε } | = 0 , where by k ≤ n we mean that k = 1, 2, . . . , n and the vertical bars indicate the number of elements in the enclosed set. In this case we write st - lim X = X₀ or X _k  → X₀ (st).

A sequence X = (X _k ) of fuzzy numbers is said to be I-convergent to a fuzzy number X₀ if for each ε > 0 such that A = { k ∈ N : d ¯ ( X k , X 0 ) ≥ ε } ∈ I .

A lacunary sequence θ = (k _r ) ; r = 0, 1, 2, . . . is an increasing sequence of integers such that k₀ = 0, and h _r  = k _r  - k_r-1→ ∞ as r→ ∞. Let I _r  = (k_r-1, k _r ] and q r = k r k r - 1.

The following concept is due to Nuray [13].

Definition 1.1. Let θ = (k _r ) be a lacunary sequence; the sequence X = (X _k ) is S _θ - convergent to X₀ provided that for every ɛ > 0 lim r 1 h r | { k ∈ I r : d ( X k , X 0 ) ≥ ε } | = 0 . In this case we write S _θ  - lim X _k  = X₀ or X _k  → X₀ (S _θ ).

2 Main results

We now introduce our main definitions.

Definition 2.1. A sequence X = (X _k ) of fuzzy numbers is said to be I-statistically convergent of order α to X₀ or S ( I ) α-convergent to X₀, where 0 < α ≤ 1, if for each ɛ > 0 and δ > 0 { n ∈ N : 1 n α | { k ≤ n : d ( X k , X 0 ) ≥ ɛ } | ≥ δ } ∈ I .

In this case we write X k → X 0 ( S ( I ) α ) . The class of all I-statistically convergent sequences of order α of fuzzy numbers will be denoted by simply S ( I ) α .

Remark. If we take I = I f = { A ⊆ N : A is a finite subset}. Then I f is a non-trivial admissible ideal of N and S ( I ) α-convergence of fuzzy numbers coincides with statistical convergence of order α of fuzzy numbers. When I = I f, and α = 1, it becomes only statistical convergence of fuzzy numbers, (see, [14]).

Definition 2.2. Let θ = (k _r ) be a lacunary sequence. A sequence X = (X _k ) of fuzzy numbers is said to be I-lacunary statistically convergent of order α to X₀ or S θ ( I ) α-convergent to X₀ if for any ɛ > 0 and δ > 0 { r ∈ N : 1 h r α | { k ∈ I r : d ( X k , X 0 ) ≥ ɛ } | ≥ δ } ∈ I . In this case we write X k → L ( S θ ( I ) α ) . The class of all I-lacunary statistically convergent sequences of order α will be denoted by S θ ( I ) α.

Remark. If we take I = I f = { A ⊆ N : A is a finite subset}. Then I f is a non-trivial admissible ideal of N and S θ ( I )-convergence of fuzzy numbers coincides with lacunary statistical convergence of order α of fuzzy numbers. When I = I f, and α = 1, it becomes only lacunary statistical convergence of fuzzy numbers, (see, [13]).

Definition 2.3. Let θ be a lacunary sequence. A sequence X = (X _k ) of fuzzy numbers is said to be N θ ( I ) α-convergent to X₀, where 0 < α ≤ 1, if for any ɛ > 0 { r ∈ N : 1 h r α ∑ k ∈ I r d ( X k , X 0 ) ≥ ɛ } ∈ I. In this case we write X k → X 0 ( N θ ( I ) α ) and the class of such sequences will be denoted by simply N θ ( I ) α .

In this paper we generalize the above definition by using sequence of positive real numbers and also some inclusion theorems are proved.

Definition 2.4. Let θ = (k _r ) be a lacunary sequence and p = (p _k ) be a sequence of positive real numbers. A sequence X = (X _k ) of fuzzy numbers is said to be strongly I-lacunary convergent of order α to X₀ for the sequence p or N θ p ( I ) α-convergent to X₀ if for any ɛ > 0 { r ∈ N : 1 h r α ∑ k ∈ I r d ( X k , X 0 ) p k ≥ ɛ } ∈ I . It is denoted by X k → L ( N θ p ( I ) α ) and the class of such sequences will be denoted by simply N θ p ( I ) α .

If we take p _k  = p for all k ∈ N, N θ p ( I ) α reduce to N θ ( I ) α.

Definition 2.5. Let p = (p _k ) be a sequence of positive real numbers. A sequence X = (X _k ) of fuzzy numbers is said to be strongly I-Cesàro convergent of order α to X₀ for the sequence p or σ θ p ( I ) α-convergent to X₀ if for any ɛ > 0 { n ∈ N : 1 n α ∑ k = 1 n d ( X k , X 0 ) p k ≥ ɛ } ∈ I . In this case we can write X k → X 0 ( σ p ( I ) α ) and the class of such sequences will be denoted by simply σ p ( I ) α .

If we take p _k  = p for all k ∈ N, σ p ( I ) α reduce to σ ( I ) α.

In the following theorem we show some relations between N θ ( I ) α and S θ ( I ) α convergence.

Theorem 2.1.Let θ = (k _r ) be a lacunary sequence and let X = (X _k ) be a sequence of fuzzy numbers. Then  (a) X k → X 0 ( N θ ( I ) α ) ⇒ X k → X 0 ( S θ ( I ) α ) ,and

(b) N θ ( I ) αis a proper subset ofS θ ( I ) α

Proof. (a) Let ɛ > 0 and X k → X 0 ( N θ ( I ) α ) , we can write ∑ k ∈ I r d ( X k , X 0 ) ≥ ∑ k ∈ I r , d ( X k , X 0 ) ≥ ɛ d ( X k , X 0 ) ≥ ɛ | { k ∈ I r : d ( X k , X 0 ) ≥ ɛ } | and so 1 ɛ . h r α ∑ k ∈ I r d ( X k , X 0 ) ≥ 1 h r α | { k ∈ I r : | x k - L | ≥ ɛ } | .    Then for any δ > 0 { r ∈ N : 1 h r α | { k ∈ I r : d ( X k , X 0 ) ≥ ɛ } | ≥ δ } ⊆ { r ∈ N : 1 h r α ∑ k ∈ I r d ( X k , X 0 ) ≥ ɛ . δ } ∈ I . This proves the result.

(b) In order to establish that the inclusion N θ ( I ) α ⊆ S θ ( I ) α is proper, let θ be given and define X _k to be n dimensional triangular fuzzy numbers 1 ¯ , 2 ¯ , . . . , [ h r ¯ ] at first [ h r ¯ ] integers in I _r and X k = 0 ¯ otherwise, where each triangular fuzzy number i ¯ is assumed to have (i, . . .0) as a center and 1 as spread (that is, i ¯ is a function from R ⁿ to [0, 1] such that i ¯ ( x 1 , x 2 , . . . , x n ) = | | ( i , 0 , . . . , 0 ) - ( x 1 , x 2 , . . . , x n ) | | if || (i, 0, . . . , 0) - (x₁, x₂, . . . , x _n ) ||<1 and 0 otherwise, where || . || denote Euclidean distance). Then clearly X _k is not bounded and for every ɛ > 0, 1 h r α | { k ∈ I r : d ( X k , 0 ) ≥ ε } | ≤ [ h r α ] h r α and for any δ > 0 we get { r ∈ N : 1 h r α | { k ∈ I r : d ( X k , 0 ) ≥ ε } | ≥ δ } ⊆ { r ∈ N : [ h r α ] h r α ≥ δ } . Since the set on the right hand side is a finite set, so belongs to I , it follows that x k → 0 ( S θ ( I ) α ) .

On the other hand 1 h r α ∑ k ∈ I r d ( X k , 0 ) = 1 h r α . [ h ¯ r α ] ( [ h ¯ r α ] + 1 ) 2 . Then { r ∈ N : 1 h r α ∑ k ∈ I r d ( X k , 0 ) ≥ 1 4 } = { r ∈ N : [ h ¯ r α ] ( [ h ¯ r α ] + 1 ) h r α ≥ 1 2 } = { m , m + 1 , m + 2 , . . . } for some m ∈ N which belongs to F ( I ) since I is admissible. So X k → 0 ( N θ ( I ) α ).

Remark 4. In Theorem 3.3 of [32] it was further proved that

(ii) X ∈ l_∞ (X), where l_∞ (X) denotes the set of bounded sequences and X k → X 0 ( S θ ( I ) ) ⇒ X k → X 0 ( N θ ( I ) ,

(iii) S θ ( I ) ∩ l ∞ ( X ) = N θ ( I ) ∩ l ∞ ( X ) .

However whether these results remain true for 0 < α < 1 is not clear and we leave them as openproblems.

Theorem 2.2.Let θ = (k _r ) be a lacunary sequence and let X = (X _k ) be a sequence of fuzzy numbers, inf k p k = h and sup k p k = H . Then, X k → X 0 ( N θ p ( I ) α ) ⇒ X k → X 0 ( S θ ( I ) α ) , Proof. Assume that X k → X 0 ( N θ p ( I ) α ) and ɛ > 0 . Then, 1 h r α ∑ k ∈ I r d ( X k , X 0 ) p k = 1 h r α ∑ k ∈ I r & d ( X k , X 0 ) ≥ ɛ d ( X k , X 0 ) p k + 1 h r α ∑ k ∈ I r & d ( X k , X 0 ) < ɛ d ( X k , X 0 ) p k ≥ 1 h r α ∑ k ∈ I r & d ( X k , X 0 ) ≥ ɛ d ( X k , X 0 ) p k ≥ 1 h r α ∑ k ∈ I r & d ( X k , X 0 ) ≥ ɛ ( ɛ ) p k ≥ 1 h r α ∑ k ∈ I r & d ( X k , X 0 ) ≥ ɛ min { ( ɛ ) h , ( ɛ ) H } ≥ 1 h r α min { ( ɛ ) h , ( ɛ ) H } | { k ∈ I r : d ( X k , X 0 ) ≥ ɛ } | and { r ∈ N : 1 h r α | { k ∈ I r : d ( X k , X 0 ) ≥ ɛ } | ≥ δ } ⊆ { r ∈ N : 1 h r α ∑ k ∈ I r d ( X k , X 0 ) p k ≥ δ min ( ɛ ) h , ( ɛ ) H } ∈ I . Thus we have X k → X 0 ( S θ ( I ) α ) .

We are in position to prove the inclusion relations between the set of σ p ( I ) α and N θ ( I ) α for fuzzy numbers.

Theorem 2.3.Let I be an ideal and θ = {k _r } be a lacunary sequence, then σ p ( I ) α ⊂ N θ p ( I ) α if lim inf r q r α > 1 .

Proof. Suppose first that lim inf r q r α > 1. Then there exists σ > 0 such that q r α ≥ 1 + σ for sufficiently large r which implies that h r α k r α ≥ σ 1 + σ .

Let ɛ > 0 be given. Now observe that 1 k r α ∑ k = 1 k r d ( X k , X 0 ) p k ≥ 1 k r α ∑ k ∈ I r d ( X k , X 0 ) p k ≥ h r α k r α 1 h r α ∑ k ∈ I r d ( X k , X 0 ) p k ≥ ( σ 1 + σ ) 1 h r α ∑ k ∈ I r d ( X k , X 0 ) p k . Thus we have 1 k r α ∑ k = 1 k r d ( X k , X 0 ) p k < ɛ implies 1 h r α ∑ k ∈ I r d ( X k , X 0 ) p k < ɛ ( 1 + σ σ ) .

So we can conclude that { r ∈ N : 1 k r α ∑ k = 1 k r d ( X k , X 0 ) p k < ɛ ⊂ { r ∈ N : 1 h r α ∑ k ∈ I r d ( X k , X 0 ) p k < ɛ } 1 + σ σ } Finally, since the set defined in the first inclusion is in the filter F ( I ), then the set defined in the second inclusion is also in the filter. This proves the theorem.

Remark. The converse of this result is not clear for α < 1 and we leave it as an open problem.

For the next result we assume that the lacunary sequence θ satisfies the condition that for any set C ∈ F ( I ), ⋃ { n : k r - 1 < n < k r , r ∈ C } ∈ F ( I ).

Theorem 2.4.For a lacunary sequence θ = (k _r ) satisfying the above condition, N θ p ( I ) α ⊂ σ p ( I ) α if sup r ∑ i = 0 r - 1 h i + 1 α ( k r - 1 ) α = B ( say ) < ∞ .

Proof. If lim sup q _r < ∞ then there exists B > 0 such that q _r  < B for all r ≥ 1 . Let X ∈ N _θ  (I)  ^α and for ɛ > 0 define the sets T and R such that, T = { r ∈ N : 1 h r α ∑ k ∈ I r d ( X k , X 0 ) p k < ɛ 1 } and R = { n ∈ N : 1 n α ∑ k = 1 n d ( X k , X 0 ) p k < ɛ 2 } . Let A j = 1 h j α ∑ k ∈ I j d ( X k , X 0 ) p k < ɛ 1 for all j ∈ T . It is obvious that T ∈ F ( I ) . Choose n any integer with k_r-1 < n < k _r where r ∈ T . 1 n α ∑ k = 1 n d ( X k , X 0 ) p k ≤ 1 k r - 1 α ∑ k = 1 k r d ( X k , X 0 ) p k = 1 k r - 1 α ( ∑ k ∈ I 1 d ( X k , X 0 ) p k ) + ( ∑ k ∈ I 2 d ( X k , X 0 ) p k + . . . + ∑ k ∈ I r d ( X k , X 0 ) p k ) = k 1 α k r - 1 α ( 1 h 1 α ∑ k ∈ I 1 d ( X k , X 0 ) p k ) + ( k 2 - k 1 ) α k r - 1 α ( 1 h 2 α ∑ k ∈ I 2 d ( X k , X 0 ) p k ) + . . . + ( k r - k r - 1 ) α k r - 1 α ( 1 h r α ∑ k ∈ I r d ( X k , X 0 ) p k ) = k 1 α k r - 1 α A 1 + ( k 2 - k 1 ) α k r - 1 α A 2 + … + ( k r - k r - 1 ) α k r - 1 α A r ≤ sup j ∈ C A j . sup r ∑ i = 0 r - 1 ( k i + 1 - k i ) α k r - 1 α < ɛ 1 B Choose ɛ 2 = ɛ 1 B and in view of the fact that ⋃ {n : k_r-1 < n < k _r , r ∈ T} ⊂ R where T ∈ F (I) it follows from our assumption on θ that the set R also belongs to F ( I ) and this completes the proof of the theorem.

3 Conclusion

The concept of ideal convergence is a generalization form of statistical convergence and any concept involving lacunary statistical convergence and lacunary strongly summability plays a vital role not only in mathematics but also in other branches of science and engineering involving mathematics, especially in information theory, computer science, biological science, dynamical systems and others. In this paper, we introduce the concept of lacunary strongly p-Cesàro summability of a sequence with order α of fuzzy number by using ideal. The concept of lacunary strongly p-Cesàro summability of a sequence with order α of fuzzy number by using ideal has not been studied so far. Therefore the present paper is filled up a gab in the existing literature.