On Asymptotically Lacunary Statistical Equivalent Set Sequences

In 1993, Marouf presented definitions for asymptotically equivalent and asymptotic regular matrices. In 2003, Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. In 2006, Patterson and Savaş extended the definitions presented in [1] to lacunary sequences. In addition to these definitions, natural inclusion theorems were presented. The concept of Wijsman statistical convergence is implementation of the concept of statistical convergence to sequences of sets presented by Nuray and Rhoades in 2012. Similar to this concept, the concept of Wijsman lacunary statistical convergence was presented by Ulusu and Nuray in 2012. This paper extends the definitions presented in [2] to Wijsman statistical convergent sequences and Wijsman lacunary statistical convergent sequences. In addition to these definitions, natural inclusion theorems will also be presented.


Introduction
In 1993, Marouf presented definitions for asymptotically equivalent and asymptotic regular matrices.In 2003, Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices.In 2006, Patterson and Savas ¸extended the definitions presented in [1] to lacunary sequences.In addition to these definitions, natural inclusion theorems were presented.The concept of Wijsman statistical convergence is implementation of the concept of statistical convergence to sequences of sets presented by Nuray and Rhoades in 2012.Similar to this concept, the concept of Wijsman lacunary statistical convergence was presented by Ulusu and Nuray in 2012.This paper extends the definitions presented in [2] to Wijsman statistical convergent sequences and Wijsman lacunary statistical convergent sequences.In addition to these definitions, natural inclusion theorems will also be presented.
Definition 2 (see Fridy [4]).The sequence  = (  ) is said to be statistically convergent to the number  if for every  > 0, In this case we write  − lim   = .
The next definition is natural combination of Definitions 1 and 2.
Definition 3 (see Patterson [1]).Two nonnegative sequences  = (  ) and  = (  ) are said to be asymptotically statistical equivalent of multiple  provided that for every  > 0 (denoted by    ∼ ) and simply asymptotically statistically equivalent if  = 1.
By a lacunary sequence we mean an increasing integer sequence  = {  } such that  0 = 0 and ℎ  =   −  −1 → ∞ as  → ∞.Throughout this paper the intervals determined by  will be denoted by   = ( −1 ,   ], and ratio   / −1 will be abbreviated by  .
for each  ∈ .In this case we write  − lim   = .
Also the concept of bounded sequence for sequences of sets was given by Nuray and Rhoades.
Definition 8 (see Nuray & Rhoades [6]).Let (, ) be a metric space.For any nonempty closed subset   of , we say that the sequence for each  ∈ .In this case we write {  } ∈  ∞ .
Definition 9 (see Ulusu & Nuray [7]).Let (, ) be a metric space and let  = {  } be a lacunary sequence.For any nonempty closed subset ,   ⊆ , we say that the sequence In this case we write   − lim  =  or   → (  ).
Following these results we introduce three new notions that are asymptotically statistical equivalent (Wijsman sense) of multiple , asymptotically lacunary statistical equivalent (Wijsman sense) of multiple , and strongly asymptotically lacunary equivalent (Wijsman sense) of multiple .
As an example, consider the following sequences of circles in the (, )-plane: ∼ {  }) and simply asymptotically statistical equivalent (Wijsman sense) if  = 1.
As an example, consider the following sequences of circles in the (, )-plane: otherwise, As an example, consider the following sequences: As an example, consider the following sequences: is a square integer, {(0, 0)} , otherwise, is a square integer, {(0, 0)} , otherwise. (

Main Results
Theorem 15.Let (, ) be a metric space, let  = {  } be a lacunary sequence, and let   ,   be non-empty closed subsets of : where  ∞ denotes the set of bounded sequences of sets. Proof otherwise, Note that {  } is not bounded.We have, for every  > 0 and for each  ∈ , = (  ) are said to be asymptotically lacunary statistical equivalent of multiple  provided that for every  > 0 Let (, ) be a metric space.For any point  ∈  and any nonempty subset  of , we define the distance from  to  by [2]inition 4 (see Patterson and Savas ¸[2]).Let  be a lacunary sequence; the two nonnegative sequences  = (  ) and Wijsman lacunarily statistically convergent to  if {(,   )} is lacunarily statistically convergent to (, ); that is, for  > 0 and for each  ∈ , :      (,   ) −  (, )     ≥ }     = 0.