I-Statistical Convergent Sequence Spaces of Fuzzy Star-Shaped Numbers

Keywords: I-convergence, Fuzzy star-shaped numbers, L p -space I-yakınsaklık, Bulanık yıldız-şekilli sayılar, L p -uzayı

In this study, we acquire I-statistical convergence of sequences of fuzzy star-shaped numbers. We examine topological and algebraic features of the obtained new sequence spaces. We put forward to significant examples of these new notions. 

Introduction

Kostyrko et al. [7] proposed ideal convergence and examined significant features of this convergence concept. Then, ideal convergence of fuzzy numbers was presented by Kumar and Kumar [8]. Some implementations of ideal convergence can be seen in [6,9]. I-statistical convergence was investigated by Savaş and Das [10]. Theory of fuzzy was firstly originated by Zadeh [11]. Zadeh primarily studied the convexity feature of fuzzy sets. Some applications of fuzzy sets can be found in [11]. As a result of the significance of the star-shapedness and convexity that can be examined as a natural extension to this feature, it can be investigated in various ways ( [2,3]). Diamond [1] presented the formulation of the fuzzy star-shaped numbers and examined the features of -metric for ≥ 1 on the same study. 
Throughout the study, we denote the set of all sequences = ( ) of fuzzy star-shaped numbers in ℝ by * ( ). Significant definitions and notations which are used in present paper can be found in [4,5,10,12]. 

Material and Method

With the description in the introduction, it can be observed that this study is qualitative with grounded theory method. Papers [1] and [12] put forward to concept of fuzzy star-shaped numbers and also [4], [5] provide a fundamental survey of the convergence concepts of fuzzy star-shaped numbers. 
By utilizing the notions of statistical convergence, ideal and fuzzy star-shaped numbers, we acquire new class of I-statistical convergence of sequences of fuzzy star-shaped numbers. 

Results and Discussion

Now, we aim to present the sequence spaces () ( ), 0 () ( ) 
and ∞ () ( ) of fuzzy star-shaped numbers with regards to the -metric. We identify () Proof. Assume = ( ) and = ( ) be sequences of () ( ) which convergence to t 0 and 0 respectively and , be scalars. Then 
𝐾{𝜉, 𝜁} = {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (𝑡 𝑘 , 𝑡 0 ) ≥ 𝜉 2 | ≥ 𝜁} ∈ 𝐼, 𝐿{𝜉, 𝜁} = {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (𝑟 𝑘 , 𝑟 0 ) ≥ 𝜉 2 | ≥ 𝜁} ∈ 𝐼. 𝜌 𝑝 (𝛼𝑡 + 𝛽𝑟, 𝛼𝑡 0 + 𝛽𝑟 0 ) = (∫ 𝜌 𝐻 ([𝛼𝑡 𝑘 + 𝛽𝑟 𝑘 ] 𝜎 , [𝛼𝑡 0 + 𝛽𝑟 0 ] 𝜎 ) 𝑝 1 0 𝑑𝜎) 1 𝑝 = (∫ 𝜌 𝐻 (𝛼[𝑡 𝑘 ] 𝜎 + 𝛽[𝑟 𝑘 ] 𝜎 , 𝛼[𝑡 0 ] 𝜎 1 0 + 𝛽[𝑟 0 ] 𝜎 ) 𝑝 𝑑𝜎) 1 𝑝 ≤ (∫ 𝜌 𝐻 ([𝛼𝑡 𝑘 ] 𝜎 + [𝛼𝑡 0 ] 𝜎 ) 𝑝 1 0 𝑑𝜎) 1 𝑝 + (∫ 𝜌 𝐻 ([𝛽𝑟 𝑘 ] 𝜎 + [𝛽𝑟 0 ] 𝜎 ) 𝑝 1 0 𝑑𝜎) 1 𝑞 = |𝛼| (∫ 𝜌 𝐻 ([𝑡 𝑘 ] 𝜎 + [𝑡 0 ] 𝜎 ) 𝑝 1 0 𝑑𝜎) 1 𝑞 + |𝛽| (∫ 𝜌 𝐻 ([𝑟 𝑘 ] 𝜎 + [𝑟 0 ] 𝜎 ) 𝑝 1 0 𝑑𝜎) 1 𝑞 = |𝛼|𝜌 𝑝 (𝑡, 𝑡 0 ) + |𝛽|𝜌 𝑝 (𝑟, 𝑟 0 ). Now 𝑀{𝜉, 𝜁} = {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (𝛼𝑡 + 𝛽𝑟, 𝛼𝑡 0 + 𝛽𝑟 0 ) ≥ 𝜉| ≥ 𝜁} ⊆ {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: |𝛼|𝜌 𝑝 (𝑡, 𝑡 0 ) ≥ 𝜉 2 | ≥ 𝜁} ∪ {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: |𝛽|𝜌 𝑝 (𝑟, 𝑟 0 ) ≥ 𝜉 2 | ≥ 𝜁} = {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (𝑡, 𝑡 0 ) ≥ 𝜉 2|𝛼| | ≥ 𝜁} ∪ {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (𝑟, 𝑟 0 ) ≥ 𝜉 2|𝛽| | ≥ 𝜁} ⊆ {𝐾 { 𝜉 2|𝛼| , 𝜁} ∪ 𝐿 { 𝜉 2|𝛽|
, }} ∈ . 
This gives that ( + ) ∈ () ( ). As a result, () ( ) is a linear space. 
Theorem 3.2. The inclusions 0 () ( ) ⊂ () ( ) ⊂ ∞ () ( ) are strict. 
Proof. Obviously 0 () ( ) ⊂ () ( ). Now, to indicate that 0 () ( ) is a proper subset of () ( ), consider = ( ) ∈ * ( ) as 
𝑡 𝑘 (𝑠) = { 𝑠, 0 ≤ 𝑠 < 2 3 -𝑠, 2 ≤ 𝑠 ≤ 3 0, otherwise.
Obviously the sequence ( ) ∈ () ( ) but ( ) ∉ 0 () ( ), that is ( ) ∈ () ( )/ 0 () ( ). Now, contemplate a sequence = ( ) ∈ () ( ). Then, there is a 0 ∈ such that - = 0 , that is, 
{𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (𝑡 𝑘 , 𝑡 0 ) ≥ 𝜉| ≥ 𝜁} ∈ 𝐼.
We get ( , 0 ̅ ) ≤ ( , 0 ) + ( 0 , 0 ̅ ). 
This denotes that ( ) have to belongs to ∞ () ( ). Subsequent is an example to demonstrate the strictness of the inclusion () ( ) ⊂ ∞ () ( ). 
Example2.1. Contemplate the subsequent sequence: 
𝑡 𝑘 (𝑠) = { 1 + 2𝑠 2 , for -1 2 ≤ s ≤ 1 2 2(1 -𝑠), for 1 2 ≤ s ≤ 1 0, otherwise.
Take I as a non maximal ideal. Determine a sequence = ( ) as 
𝑟 𝑘 = { 𝑡 𝑘 , 𝑘 ∈ 𝐾 0, otherwise.
We acquire ( ) ∈ ∞ () ( ) but ( ) ∉ () ( ). 
Also, assume the sequence ( ) be identified as 
𝑡 𝑘 (𝑠) = { 1 -𝑘𝑠, 0 ≤ 𝑠 ≤ 1 𝑘 1 + 𝑘𝑠, -1 𝑘 ≤ 𝑠 ≤ 0 for 𝑘 = 2𝑚 0, otherwise.
Otherwise 
𝑡 𝑘 (𝑠) = { 𝑠 + 5, -5 ≤ 𝑠 ≤ 0 1, 0 ≤ 𝑠 ≤ 2 -𝑠 + 5, 2 ≤ 𝑠 ≤ 5 0, otherwise.
Consequently, ( ) ∈ ∞ () ( ) but ( ) ∉ () ( ). Hence, the inclusions 0 () ( ) ⊂ () ( ) ⊂ ∞ () ( ) are strict. 
Theorem 3.3. If I is not maximal then () ( ) is neither normal nor monotone. 
Proof. We examine the subsequent example. Think a sequence = ( ) ∈ * ( ) 
𝑡 𝑘 (𝑠) = { 2𝑠, if 0 ≤ s ≤ 1 2 1, if 1 2 ≤ s ≤ 3 2 -2(𝑠 -2), if 3 2 ≤ s ≤ 2 0, otherwise.
Then, (t k ) ∈ c S(I) (S n ). As I is not maximal, we identify a sequence r = (r k ) as 
𝑟 𝑘 = { 𝑡 𝑘 , 𝑘 ∈ 𝐾 0, otherwise.
such that = (r k ) exists in the canonical pre-image of (t k ) of Kstep spaces of () ( ). But ( ) ∉ () ( ). Hence, () ( ) is not monotone, so it is not normal. for every ∈ ℝ. Assume 0 be - and 0 be - . For ∈ [0,1] and , ≥ 0. 
𝜌 𝐻 ([𝑡 𝑘 ] 𝜎 [𝑟 𝑘 ] 𝜎 , [𝑡 0 ] 𝜎 [𝑟 0 ] 𝜎 ) ≤ 𝛼𝜌 𝑝 ([𝑡 𝑘 ] 𝜎 , [𝑡 0 ] 𝜎 ) + 𝛽𝜌 𝑝 ([𝑟 𝑘 ] 𝜎 , [𝑟 0 ] 𝜎 ).
Thus, we obtain ( , 0 0 ) ≤ ( , 0 ) + ( , 0 ). 
Let , > 0 be taken. Then Hence, ∩ ∈ , so that ∈ (). So (. ) ∈ () ( ) and ℎ() = ℎ()ℎ(). 
𝐾 { 𝜉 2 , 𝜁} = {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (t k , t 0 ) ≥ 𝜉 2 | ≥ 𝜁} ∈ 𝐼, 𝐿 { 𝜉 2 , 𝜁} = {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (𝑟 𝑘 , 𝑟 0 ) ≥ 𝜉 2 | ≥ 𝜁} ∈ 𝐼.

Conclusions and Recommendations

In this study, we investigate I-statistical convergence of sequences of fuzzy star-shaped numbers. We put forward to topological and algebraic features of the obtained new sequence spaces. We examine significant examples of these new notions. 

Theorem 3 . 4 .

 34 The spaces 0 () ( ), () ( ), ∞ () ( ) are sequence algebra.Proof. When K and L are fuzzy star-shaped numbers then, their product is determined as . () = sup =. min( (), ()) European Journal of Science and Technology e-ISSN: 2148-2683 


Theorem 3 . 6 .

 36 When , ∈ () ( ), then (. ) ∈ () ( ) and ℎ() = ℎ()ℎ().Proof. As , ∈ () ( ), for , > 0 the subsequent conditions supplies = { ∈ ℕ: 1 | ≤ : (, ℎ()) ≥ | < 2 } ∈ (), = { ∈ ℕ: 1 | ≤ : (, ℎ()) ≥ | < 2 } ∈ (),where , > 0 where (, 0 ̅ ) < and (, 0 ̅ ) < . Think the set = { ∈ ℕ: 1 | ≤ : (, ℎ()ℎ()) ≥ | < }and let ∈ ∩ .Now (, ℎ()ℎ()) ≤ (, ℎ()) ≤ (ℎ(), ℎ()ℎ()) ≤ (, 0) (, ℎ())+ (ℎ(), 0) (, ℎ()) 




   ( , 0 0 ) ≥ | ≥ }. 


					𝐿 𝑝 = {𝑘 ∈ ℕ:	1 𝑘	|𝑛 ≤ 𝑘: 𝜌 𝑝 (𝑟, ℎ(𝑟)) ≥ ‖𝑡 -𝑟‖| ≥ 𝜁} ∈ 𝐼.
					Therefore
					𝐾 𝑝 𝑐 = {𝑘 ∈ ℕ:	1 𝑘	|𝑛 ≤ 𝑘: 𝜌 𝑝 (𝑡, ℎ(𝑡)) ≥ ‖𝑡 -𝑟‖| < 𝜁} ∈ 𝐹(𝐼),
					𝐿 𝑝 𝑐 = {𝑘 ∈ ℕ:	1 𝑘	|𝑛 ≤ 𝑘: 𝜌 𝑝 (𝑟, ℎ(𝑟)) ≥ ‖𝑡 -𝑟‖| < 𝜁} ∈ 𝐹(𝐼).
					So 𝑀 𝑝 𝑐 = 𝐾 𝑝 𝑐 ∩ 𝐿 𝑝 𝑐 ∈ 𝐹(𝐼). Namely 𝑀 𝑝 𝑐 ≠ ∅. Let 𝑘 ∈ 𝑀 𝑝 𝑐 such
					that
					𝜌 𝑝 (ℎ(𝑡), ℎ(𝑟)) ≤ 𝜌 𝑝 (ℎ(𝑡), 𝑡) + 𝜌 𝑝 (𝑡, 𝑟) + 𝜌 𝑝 (𝑟, ℎ(𝑟))
					≤ 3‖𝑡 -𝑟‖.
	Think the set	
	𝑀{𝜉, 𝜁} = {𝑘 ∈ ℕ: |𝑛 ≤ 𝑘: It suffices to denote that 𝑀{𝜉, 𝜁} ⊆ 𝐾{𝜉, 𝜁} ∪ 𝐿{𝜉, 𝜁}. Then 1 𝑘
	{𝑘 ∈ ℕ:	1 𝑘	|𝑛 ≤ 𝑘: 𝜌 𝑝 (𝑡 𝑘 𝑟 𝑘 , 𝑡 0 𝑟 0 ) ≥ 𝜉| ≥ 𝜁} ⊆ 𝛼 {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (t k , t 0 ) ≥ ∪ 𝛽 {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (r k , r 0 ) ≥	𝜉 2 𝜉 2 | ≥ 𝜁}. | ≥ 𝜁}
	Since			
	𝑀{𝜉, 𝜁} ⊆ {𝑘 ∈ ℕ: ∪ {𝑘 ∈ ℕ: 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (t k , t 0 ) ≥ 1 𝑘 |𝑛 ≤ 𝑘: 𝜌 𝑝 (r k , r 0 ) ≥ 𝜉 | ≥ 𝜁} 2𝛼	𝜉 2𝛽	| ≥ 𝜁}.
	As a result 𝑀{𝜉, 𝜁} ⊆ 𝐾{𝜉, 𝜁} ∪ 𝐿{𝜉, 𝜁}.
	Theorem 3.5. The function ℎ: 𝑚 𝑆(𝐼) (𝑆 𝑛 ) → ℝ given by ℎ(𝑝) =
	𝐼 -𝑠𝑡𝑙𝑖𝑚𝑝 is a Lipschitz function and so uniformly continuous.
	Proof. Assume 𝑡, 𝑟 ∈ 𝑚 𝑆(𝐼) (𝑆 𝑛 ) with 𝑝 ≠ 𝑟 such that ℎ(𝑡) = 𝐼 -
	𝑠𝑡𝑙𝑖𝑚𝑡 and ℎ(𝑟) = 𝐼 -𝑠𝑡𝑙𝑖𝑚𝑟. Then
	𝐾 𝑝 = {𝑘 ∈ ℕ:	1 𝑘	|𝑛 ≤ 𝑘: 𝜌 𝑝 (𝑡, ℎ(𝑡)) ≥ ‖𝑡 -𝑟‖| ≥ 𝜁} ∈ 𝐼,

As a result, h is Lipschitz continuous. 
Acknowledge

The authors also thank to the reviewers for their comments and suggestions in this article.
Araştırma Makalesi, Ömer Kişi, Erhan Güler
I-Statistical Convergent Sequence Spaces of Fuzzy Star-Shaped Numbers