Density by moduli and Wijsman lacunary statistical convergence of sequences of sets

The main object of this paper is to introduce and study a new concept of f-Wijsman lacunary statistical convergence of sequences of sets, where f is an unbounded modulus. The definition of Wijsman lacunary strong convergence of sequences of sets is extended to a definition of Wijsman lacunary strong convergence with respect to a modulus for sequences of sets and it is shown that, under certain conditions on a modulus f, the concepts of Wijsman lacunary strong convergence with respect to a modulus f and f-Wijsman lacunary statistical convergence are equivalent on bounded sequences. We further characterize those θ for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{WS}_{\theta}^{f} = \mathit{WS}^{f}$\end{document}WSθf=WSf, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{WS}_{\theta}^{f}$\end{document}WSθf and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{WS}^{f}$\end{document}WSf denote the sets of all f-Wijsman lacunary statistically convergent sequences and f-Wijsman statistically convergent sequences, respectively.


Introduction
Zygmund [] was the person behind the introduction of the idea of statistical convergence. The concept of statistical convergence was formally given by Fast [] and Steinhaus []. This concept was studied by Schoenberg [] as a non-matrix summability method. For a detailed account of statistical convergence one may refer to [-], and many others.
Let N denote the set of all natural numbers. A number sequence X = (ξ k ) is said to be statistically convergent to the number l if for each ε >  the set {k ∈ N : |ξ k -l| ≥ ε} has natural density zero. The natural density of a subset E ⊂ N [] is defined by where the vertical bars indicate the number of elements in the enclosed set. Obviously we have d(E) =  provided that E is a finite set of positive integers. If a sequence (ξ k ) is statistically convergent to l, then we write it as S -lim ξ k = l or ξ k → l(S).
The concept of convergence of sequences of points has been extended by several authors [-] to the convergence of sequences of sets. In this paper we consider one such extension, namely, Wijsman convergence. Nuray and Rhoades [] extended the notion of Wijsman convergence of sequences of sets to that of Wijsman statistical convergence of sequences of sets, and gave some basic theorems. Also, they introduced the notion of Wijsman strong Cesàro summability of sequences of sets and discussed its relation with Wijsman statistical convergence. Ulusu and Nuray [] introduced the concepts of Wijsman lacunary statistical convergence of sequences of sets and Wijsman lacunary strong convergence of sequences of sets and established a relation between them. For more work on convergence of sequences of sets one may refer to [-].
Recall [, ] that a modulus f is a function from R + to R + such that (iv) f is continuous from the right at . From above properties it is easy to see that a modulus f is continuous on R + . A modulus may be unbounded or bounded. For example, f (x) = x p where  < p ≤ , is unbounded, but f (x) = x (+x) is bounded. The work related to the sequence spaces defined by a modulus may be found in, e.g., [ Before proceeding further, we first recall some definitions.

Definition . ([])
For any unbounded modulus f , the f -density of a set E ⊂ N is denoted by d f (E) and is defined by and we write it as S flim ξ k = l or ξ k → l(S f ).
It is an immediate consequence of Definition . and Remark . that every f -statistically convergent sequence is statistically convergent, but a statistically convergent sequence need not be f -statistically convergent for every unbounded modulus f . By a lacunary sequence θ = (k r ); r = , , , . . . , where k o = , we shall mean an increasing sequence of non-negative integers with h r = k rk r- → ∞ as r → ∞. The intervals determined by θ will be denoted by I r = (k r- , k r ] and the ratio k r /k r- will be denoted by q r .
The space of all lacunary strongly convergent sequences, N θ , was defined by Freedman et al.
[] as follows: There is a strong connection [] between N θ and the space w of strongly Cesàro summable sequences, which is defined by In the special case, where θ = ( r ), we have N θ = w.
In the year , the concept of strong Cesàro summability was extended to that of strong Cesàro summability with respect to a modulus by Maddox []. A sequence X = (ξ k ) is said to be strongly Cesàro summable with respect to a modulus f to l if The space of strongly Cesàro summable sequences with respect to a modulus f , is denoted by w(f ).
Furthermore, in the year , Pehlivan and Fisher [] extended the notion of lacunary strong convergence to that of lacunary strong convergence with respect to a modulus f . The space N θ (f ) of lacunary strongly convergent sequences with respect to a modulus f is defined as Fridy and Orhan [] introduced the concept of lacunary statistical convergence as follows.
Definition . Let θ = (k r ) be a lacunary sequence. A number sequence X = (ξ k ) is said to be lacunary statistically convergent to l, or S θ -convergent to l, if, for each ε > , In this case, we write S θ -lim ξ k = l or ξ k → l(S θ ).
Quite recently, Ulusu and Nuray [] introduced the notion of Wijsman lacunary strong convergence of sequences of sets and discussed its relation with Wijsman lacunary statistical convergence.
In this paper, we first extend the definition of Wijsman lacunary strong convergence to a definition of Wijsman lacunary strong convergence with respect to a modulus. It is shown that if a sequence is Wijsman lacunary strongly convergent then it is Wijsman lacunary strongly convergent with respect to a modulus, however, the converse need not be true. We also investigate the condition under which the converse is true. We also study a relationship between Wijsman lacunary strong convergence with respect to a modulus and Wijsman lacunary statistical convergence and characterize those θ for which [ Before proceeding to establish the proposed results, we pause to collect some definitions related to Wijsman convergence [, ].
Let (M, ρ) be a metric space. The distance d(x, E) from a point x to a non-empty subset E of (M, ρ) is defined to be Definition . Let (E k ) be a sequence of non-empty closed subsets of a metric space (M, ρ) and E be non-empty closed subset of M.
The set of all Wijsman convergent sequences is denoted by Wc.
In this case, we write WS -lim E k = E or E k → E(WS). The set of all sequences which are Wijsman statistically convergent is denoted by WS.
We shall denote the set of all bounded sequences of sets by L ∞ .
In this case, we write The set of all sequences which are Wijsman strongly Cesàro summable is denoted by [Ww].
In this case, we write The set of all sequences which are Wijsman lacunary strongly convergent is denoted by for each x ∈ M and for each ε > , In this case, we write for each x ∈ M and for each ε > , In this case, we write Let ε >  be given. We choose  < δ <  such that f (u) < ε for every u with  ≤ u ≤ δ. We can write Remark . The converse of the above theorem does not need to be true, which can be verified from the following example.
Example . Let M = R, ρ(x, y) = |x -y| and f (x) = log(x + ). Consider the sequence (E k ) defined by Note that (E k ) is not a bounded sequence. Then, for each x ∈ M, Maddox [] proved that for any modulus f there exists lim t→∞ t . Making use of this result we are in a position to give a condition on modulus f under which the converse holds.

Theorem . Let f be a modulus such that lim
The proof can be established using the technique of Theorem . of []. We now establish a relationship between Wijsman lacunary strong convergence with respect to a modulus and Wijsman lacunary statistical convergence.

Theorem . For any modulus f , we have [WN
from which it follows that (E k ) ∈ WS θ . We now give an example to show that the converse of the above inclusion need not hold.
Example . Let M = R, ρ(x, y) = |x -y| and f (x) = x. Consider the sequence (E k ) of subsets of M as defined in Example .. This sequence is Wijsman lacunary statistically convergent to the set E = {} because for each x ∈ M and for each ε > , But this is not Wijsman lacunary strongly convergent with respect to f .
In the next theorem, we investigate a necessary and sufficient condition on f under which the converse holds.

and only if f is bounded.
Proof Suppose that f is bounded and (E k ) ∈ WS θ . Since f is bounded, there exists a con- Taking the limit as r → ∞, we have ( Conversely, suppose that f is unbounded so that there exists a positive sequence  < p  < p  < · · · < p i < · · · such that f (p i ) ≥ h i . Define the sequence (E k ) such that E k i = {p i } for i = , , . . . and E k = {} otherwise. Then, for each x ∈ M and ε > , and so (E k ) ∈ WS θ , but, for x = , We now study a relationship between Wijsman strong Cesàro summability with respect to a modulus and Wijsman lacunary strong convergence with respect to a modulus.

Theorem . Let θ = (k r ) be a lacunary sequence and f be a modulus. If
Proof Suppose that lim inf r q r > , then there exists δ >  such that q r ≥  + δ for sufficiently large r.
. Now suppose that lim sup r q r < ∞, then there exists G >  such that q r < G for all r ≥ .
x ∈ M and ε >  we can find r  such that for every r ≥ r  We can also choose a number L >  such that N r ≤ L for all r. Now let n be any integer with k r- < n ≤ k r , where r > r  . Then, for each x ∈ M,

f -Wijsman lacunary statistical convergence
Definition . Let (M, ρ) be a metric space, f be an unbounded modulus and θ = (k r ) be a lacunary sequence. A sequence (E k ) of non-empty closed subsets of M is said to be f -Wijsman lacunary statistically convergent to a non-empty closed subset E of M, or WS f θconvergent to E, if, for each x ∈ M and for each ε > , In this case, we write WS The set of all sequences which are f -Wijsman lacunary statistically convergent is denoted by WS f θ .

Remark .
The concept of f -Wijsman lacunary statistical convergence reduces to that of Wijsman lacunary statistical convergence when modulus is the identity mapping.
Theorem . Every Wijsman convergent sequence is f -Wijsman lacunary statistically convergent, however, the converse need not be true.
Proof In view of the fact that finite sets have zero f -density, for any unbounded modulus f , it is easy to see that if a sequence is Wijsman convergent then it is f -Wijsman lacunary statistically convergent for any unbounded modulus f . For the converse part, let M = R, f (x) = x p ,  < p ≤  and (E k ) be defined as where θ = (k r ) is a lacunary sequence. This sequence is not Wijsman convergent, but, for each x ∈ M and for each ε > , and hence, (E k ) is f -Wijsman lacunary statistically convergent to the set E = {}.
Remark . In view of the above theorem it is clear that the notion of f -Wijsman lacunary statistical convergence is a generalization of the usual notion of the Wijsman convergence of sequences of sets.

Theorem . Every f -Wijsman lacunary statistically convergent sequence is Wijsman lacunary statistically convergent.
Proof Suppose (E k ) is f -Wijsman lacunary statistically convergent to E. Let x ∈ M and ε > . Then, for each positive integer m, there exists r o ∈ N such that for r ≥ r o , we have Hence, (E k ) is Wijsman lacunary statistically convergent to E.
Remark . It seems that the converse of the above theorem need not hold, but right now we are not in a position to prove it. It is, therefore, left as an open problem.
We now establish a relationship between f -Wijsman lacunary statistical convergence and Wijsman lacunary strong convergence with respect to a modulus.
Maddox [] showed the existence of an unbounded modulus f for which there is a positive constant c such that f (xy) ≥ cf (x)f (y), for all x ≥ , y ≥ . Using this we have the following.  Proof (a) (i) For any sequence (E k ), for each x ∈ M and > , by the definition of modulus function (ii) and (iii) we have   (c) This is an immediate consequence of (a) and (b).
Remark . The example given in part (a) of the above theorem shows that the boundedness condition cannot be omitted from the hypothesis of part (b).

Remark .
If we take f (x) = x in Theorem ., we obtain Theorem  of Ulusu and Nuray [].
(Necessity). Suppose that lim sup r q r = ∞. Following Lemma . of [], we can select a subsequence (k r(j) ) of lacunary sequence θ such that q r(j) > j. Define a bounded sequence (E k ) by If we take f (x) = x in Theorem . we obtain the following result which contains Theo-