This paper presents three definitions which are natural combination of the definitions of asymptotic equivalence, statistical convergence, lacunary statistical convergence, and Wijsman convergence. In addition, we also present asymptotically equivalent (Wijsman sense) analogs of theorems in Patterson and Savaş (2006).

1. 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 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.

2. Definitions and NotationsDefinition 1 (see Marouf [<xref ref-type="bibr" rid="B17">3</xref>]).

Two nonnegative sequences x=(xk) and y=(yk) are said to be asymptotically equivalent if
(1)limkxkyk=1
(denoted by x~y).

Definition 2 (see Fridy [<xref ref-type="bibr" rid="B11">4</xref>]).

The sequence x=(xk) is said to be statistically convergent to the number L if for every ɛ>0,
(2)limn1n|{k≤n:|xk-L|≥ɛ}|=0.
In this case we write st-limxk=L.

The next definition is natural combination of Definitions 1 and 2.

Definition 3 (see Patterson [<xref ref-type="bibr" rid="B19">1</xref>]).

Two nonnegative sequences x=(xk) and y=(yk) are said to be asymptotically statistical equivalent of multiple L provided that for every ɛ>0(3)limn1n|{k≤n:|xkyk-L|≥ɛ}|=0
(denoted by x~SLy) and simply asymptotically statistically equivalent if L=1.

By a lacunary sequence we mean an increasing integer sequence θ={kr} such that k0=0 and hr=kr-kr-1→∞ as r→∞. Throughout this paper the intervals determined by θ will be denoted by Ir=(kr-1,kr], and ratio kr/kr-1 will be abbreviated by qr.

Definition 4 (see Patterson and Savaş [<xref ref-type="bibr" rid="B20">2</xref>]).

Let θ be a lacunary sequence; the two nonnegative sequences x=(xk) and y=(yk) are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every ɛ>0(4)limr1hr|{k∈Ir:|xkyk-L|≥ɛ}|=0
(denoted by x~SθLy) and simply asymptotically lacunary statistically equivalent if L=1.

Definition 5 (see Patterson & Savaş [<xref ref-type="bibr" rid="B20">2</xref>]).

Let θ be a lacunary sequence; two nonnegative number sequences x=(xk) and y=(yk) are strongly asymptotically lacunary equivalent of multiple L provided that
(5)limr1hr∑k∈Ir|xkyk-L|=0
(denoted by x~NθLy) and strongly simply asymptotically lacunary equivalent if L=1.

Let (X,ρ) be a metric space. For any point x∈X and any nonempty subset A of X, we define the distance from x toA by
(6)d(x,A)=infa∈Aρ(x,A).

Definition 6 (see Baronti & Papini [<xref ref-type="bibr" rid="B2">5</xref>]).

Let (X,ρ) be a metric space. For any nonempty closed subset A,Ak⊆X, we say that the sequence {Ak} is Wijsman convergent to A if
(7)limk→∞d(x,Ak)=d(x,A)
for each x∈X. In this case we write W-limAk=A.

Definition 7 (see Nuray & Rhoades [<xref ref-type="bibr" rid="B18">6</xref>]).

Let (X,ρ) a metric space. For any nonempty closed subset A,Ak⊆X, we say that the sequence {Ak} is Wijsman statistically convergent to A if {d(x,Ak)} is statistically convergent to d(x,A); that is, for ɛ>0 and for each x∈X,
(8)limn→∞1n|{k≤n:|d(x,Ak)-d(x,A)|≥ɛ}|=0.
In this case we write st-limWAk=A or Ak→A(WS).

Also the concept of bounded sequence for sequences of sets was given by Nuray and Rhoades.

Definition 8 (see Nuray & Rhoades [<xref ref-type="bibr" rid="B18">6</xref>]).

Let (X,ρ) be a metric space. For any nonempty closed subset Ak of X, we say that the sequence {Ak} is bounded if
(9)supkd(x,Ak)<∞
for each x∈X. In this case we write {Ak}∈L∞.

Definition 9 (see Ulusu & Nuray [<xref ref-type="bibr" rid="B25">7</xref>]).

Let (X,ρ) be a metric space and let θ={kr} be a lacunary sequence. For any non-empty closed subset A,Ak⊆X, we say that the sequence {Ak} is Wijsman lacunarily statistically convergent to A if {d(x,Ak)} is lacunarily statistically convergent to d(x,A); that is, for ɛ>0 and for each x∈X,
(10)limr1hr|{k∈Ir:|d(x,Ak)-d(x,A)|≥ɛ}|=0.
In this case we write Sθ-limW=A or Ak→A(WSθ).

Following these results we introduce three new notions that are asymptotically statistical equivalent (Wijsman sense) of multiple L, asymptotically lacunary statistical equivalent (Wijsman sense) of multiple L, and strongly asymptotically lacunary equivalent (Wijsman sense) of multiple L.

Definition 10.

Let (X,ρ) be a metric space. For any non-empty closed subset Ak, Bk⊆X such that d(x,Ak)>0 and d(x,Bk)>0 for each x∈X. We say that the sequences {Ak} and {Bk} are asymptotically equivalent (Wijsman sense) if for each x∈X,
(11)limkd(x,Ak)d(x,Bk)=1
(denoted by Ak~Bk).

As an example, consider the following sequences of circles in the (x,y)-plane:
(12)Ak={(x,y):x2+y2+2kx=0},Bk={(x,y):x2+y2-2kx=0}.
Since
(13)limkd(x,Ak)d(x,Bk)=1,
the sequences {Ak} and {Bk} are asymptotically equivalent (Wijsman sense); that is, Ak~Bk.

Definition 11.

Let (X,ρ) be a metric space. For any non-empty closed subset Ak, Bk⊆X such that d(x,Ak)>0 and d(x,Bk)>0 for each x∈X. We say that the sequences {Ak} and {Bk} are asymptotically statistically equivalent (Wijsman sense) of multiple L if for every ɛ>0 and for each x∈X,
(14)limn1n|{k≤n:|d(x,Ak)d(x,Bk)-L|≥ɛ}|=0
(denoted by {Ak}~WSL{Bk}) and simply asymptotically statistical equivalent (Wijsman sense) if L=1.

As an example, consider the following sequences of circles in the (x,y)-plane:
(15)Ak={{(x,y):x2+y2+2ky=0},ifkis a square integer,{(1,1)},ifkis a squotherwise,Bk={{(x,y):x2+y2-2ky=0},ifkis a square integer,{(1,1)},otherwise.

Since
(16)limn1n|{k≤n:|d(x,Ak)d(x,Bk)-1|≥ɛ}|=0,
the sequences {Ak} and {Bk} are asymptotically statistically equivalent (Wijsman sense); that is, {Ak}~WS1{Bk}.

Definition 12.

Let (X,ρ) be a metric space and let θ be a lacunary sequence. For any non-empty closed subset Ak, Bk⊆X such that d(x,Ak)>0 and d(x,Bk)>0 for each x∈X. We say that the sequences {Ak} and {Bk} are asymptotically lacunary equivalent (Wijsman sense) of multiple L if for each x∈X,
(17)limr1hr∑k∈Ird(x,Ak)d(x,Bk)=L
(denoted by {Ak}~WNθL{Bk}) and simply asymptotically lacunarily equivalent (Wijsman sense) if L=1.

Definition 13.

Let (X,ρ) be a metric space and let θ be a lacunary sequence. For any non-empty closed subset Ak, Bk⊆X such that d(x,Ak)>0 and d(x,Bk)>0 for each x∈X. We say that the sequences {Ak} and {Bk} are strongly asymptotically lacunary equivalent (Wijsman sense) of multiple L if for each x∈X,
(18)limr1hr∑k∈Ir|d(x,Ak)d(x,Bk)-L|=0
(denoted by {Ak}~[WN]θL{Bk}) and simply strongly asymptotically lacunarily equivalent (Wijsman sense) if L=1.

As an example, consider the following sequences:
(19)Ak:={{(x,y)∈ℝ2:(x-k)2k+y22k=1},ifkr-1<k<kr-1+[hr]{(1,1)},otherwise,Bk:={{(x,y)∈ℝ2:(x+k)2k+y22k=1},ifkr-1<k<kr-1+[hr]{(1,1)},otherwise.

Since
(20)limr1hr∑k∈Ir|d(x,Ak)d(x,Bk)-1|=0,
the sequences {Ak} and {Bk} are strongly asymptotically lacunarily equivalent (Wijsman sense); that is, {Ak}~WNθ1{Bk}.

Definition 14.

Let (X,ρ) be a metric space and let θ be a lacunary sequence. For any non-empty closed subset Ak,Bk⊆X such that d(x,Ak)>0 and d(x,Bk)>0 for each x∈X. We say that the sequences {Ak} and {Bk} are asymptotically lacunarily statistical equivalent (Wijsman sense) of multiple L if for every ɛ>0 and each x∈X,
(21)limr1hr|{k∈Ir:|d(x,Ak)d(x,Bk)-L|≥ɛ}|=0
(denoted by {Ak}~WSθL{Bk}) and simply asymptotically lacunarily statistically equivalent (Wijsman sense) if L=1.

As an example, consider the following sequences:
(22)Ak:={{(x,y)∈ℝ2:x2+(y-1)2=1k},ifkr-1<k<kr-1+[hr],kis a square integer,{(0,0)},otherwise,Bk:={{(x,y)∈ℝ2:x2+(y+1)2=1k},ifkr-1<k<kr-1+[hr],kis a square integer,{(0,0)},otherwise.
Since
(23)limr1hr|{k∈Ir:|d(x,Ak)d(x,Bk)-1|≥ɛ}|=0,
the sequences {Ak} and {Bk} are asymptotically lacunarily statistically equivalent (Wijsman sense); that is, {Ak}~WSθ1{Bk}.

3. Main ResultsTheorem 15.

Let (X,ρ) be a metric space, let θ={kr} be a lacunary sequence, and let Ak, Bk be non-empty closed subsets of X:

(a){Ak}~[WN]θL{Bk}⇒{Ak}~WSθL{Bk}

(b)[WN]θL is a proper subset of WSθL,

{Ak}∈L∞ and {Ak}~WSθL{Bk}⇒{Ak}~[WN]θL{Bk},

WSθL∩L∞=[WN]θL∩L∞,

where L∞ denotes the set of bounded sequences of sets.

Proof.

(i)-(a). Let ɛ>0 and {Ak}~[WN]θL{Bk}. Then we can write
(24)∑k∈Ir|d(x,Ak)d(x,Bk)-L|≥∑k∈Ir|d(x,Ak)/d(x,Bk)-L|≥ɛ|d(x,Ak)d(x,Bk)-L|≥ɛ·|{k∈Ir:|d(x,Ak)d(x,Bk)-L|≥ɛ}|
which yields the result.

(ii)-(b). Suppose that [WN]θL⊂WSθL. Let {Ak} and {Bk} be the following sequences:
(25)Ak={{k},ifkr-1<k≤kr-1+[hr]r=1,2,…{0},otherwise,Bk={0}∀k.

Note that {Ak} is not bounded. We have, for every ɛ>0 and for each x∈X,
(26)1hr|{k∈Ir:|d(x,Ak)d(x,Bk)-1|≥ɛ}|=[hr]hr⟶0asr⟶∞.
That is, {Ak}~WSθ1{Bk}. On the other hand,
(27)1hr∑k∈Ir|d(x,Ak)d(x,Bk)-L|↛0asr⟶∞.
Hence {Ak}≁[WN]θL{Bk}.

(ii) Suppose that {Ak}∈L∞ and {Ak}~WSθL{Bk}. Then we can assume that
(28)|d(x,Ak)d(x,Bk)-L|≤M
for each x∈X and all k.

Given ɛ>0, we get
(29)1hr∑k∈Ir|d(x,Ak)d(x,Bk)-L|=1hr∑k∈Ir|d(x,Ak)/d(x,Bk)-L|≥ɛ|d(x,Ak)d(x,Bk)-L|+1hr∑k∈Ir|d(x,Ak)/d(x,Bk)-L|<ɛ|d(x,Ak)d(x,Bk)-L|≤Mhr|{k∈Ir:|d(x,Ak)d(x,Bk)-L|≥ɛ}|+ɛ.
Therefore {Ak}~[WN]θL{Bk}.

This is an immediate consequences of (i)and(ii).

Theorem 16.

Let (X,ρ) be a metric space and let Ak, Bk be non-empty closed subsets of X. If θ={kr} is a lacunary sequence with liminfrqr>1, then
(30){Ak}~WSL{Bk}⇒{Ak}~WSθL{Bk}.

Proof.

Suppose first that liminfrqr>1, then there exists a λ>0 such that qr≥1+λ for sufficiently large r, which implies that
(31)hrkr≥λ1+λ.
If {Ak}~WSL{Bk}, then for every ɛ>0, for each x∈X, and for sufficiently large r, we have
(32)1kr|{k≤kr:|d(x,Ak)d(x,Bk)-L|≥ɛ}|≥1kr|{k∈Ir:|d(x,Ak)d(x,Bk)-L|≥ɛ}|≥λ1+λ·(1hr|{k∈Ir:|d(x,Ak)d(x,Bk)-L|≥ɛ}|).
This completes the proof.

Theorem 17.

Let (X,ρ) be a metric space and let Ak, Bk be non-empty closed subsets of X. If θ={kr} is a lacunary sequence with limsuprqr<∞, then
(33){Ak}~WSθL{Bk}⇒{Ak}~WSL{Bk}.

Proof.

Let limsuprqr<∞. Then there is an M>0 such that qr<M for all r≥1. Let {Ak}~WSθL{Bk} and ε>0. There exists R>0 such that for every j≥R(34)Aj=1hj|{k∈Ij:|d(x,Ak)d(x,Bk)-L|≥ɛ}|<ɛ.
We can also find H>0 such that Aj<H for all j=1,2,…. Now let t be any integer satisfying kr-1<t≤kr, where r>R.

Then we can write
(35)1t|{k≤t:|d(x,Ak)d(x,Bk)-L|≥ɛ}|≤1kr-1|{k≤kr:|d(x,Ak)d(x,Bk)-L|≥ɛ}|=1kr-1{|{k∈I1:|d(x,Ak)d(x,Bk)-L|≥ɛ}|}+1kr-1{|{k∈I2:|d(x,Ak)d(x,Bk)-L|≥ɛ}|}+⋯+1kr-1×{|{k∈Ir:|d(x,Ak)d(x,Bk)-L|≥ɛ}|}=k1kr-1k1|{k∈I1:|d(x,Ak)d(x,Bk)-L|≥ɛ}|+k2-k1kr-1(k2-k1)×|{k∈I2:|d(x,Ak)d(x,Bk)-L|≥ɛ}|+⋯+kR-kR-1kr-1(kR-kR-1)×|{k∈IR:|d(x,Ak)d(x,Bk)-L|≥ɛ}|+⋯+kr-kr-1kr-1(kr-kr-1)×|{k∈Ir:|d(x,Ak)d(x,Bk)-L|≥ɛ}|=k1kr-1A1+k2-k1kr-1A2+⋯+kR-kR-1kr-1AR+kR+1-kRkr-1AR+1+⋯+kr-kr-1kr-1Ar≤{supj≥1Aj}kRkr-1+{supj≥BAj}kr-kRkr-1≤HkBkr-1+ɛM.
This completes the proof.

Combining Theorems 16 and 17 we have the following.

Theorem 18.

Let (X,ρ) be a metric space and let Ak, Bk be non-empty closed subsets of X. If θ={kr} is a lacunary sequence with 1<liminfrqr≤limsuprqr<∞, then
(36){Ak}~WSθL{Bk}={Ak}~WSL{Bk}.

Proof.

This is an immediate consequence of Theorems 16 and 17.

PattersonR. F.On asymptotically statistical equivalent sequencesPattersonR. F.SavaşE.On asymptotically lacunary statistically equivalent sequencesMaroufM. S.Asymptotic equivalence and summabilityFridyJ. A.On statistical convergenceBarontiM.PapiniP. L.Convergence of sequences of setsNurayF.RhoadesB. E.Statistical convergence of sequences of setsUlusuU.NurayF.Lacunary statistical convergence of sequence of sets