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

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1 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 DOI 0.86/s R E S E A R C H Open Access Density by moduli and Wijsman lacunary statistical convergence of sequences of sets Vinod K Bhardwaj * and Shweta Dhawan 2 * Correspondence: vinodk_bhj@rediffmail.com Department of Mathematics, Kurukshetra University, Kurukshetra, 369, India Full list of author information is available at the end of the article Abstract 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 isan unbounded modulus. The definition of Wijsman lacunary strong convergence of sequences of sets is extended to a definition of Wijsman lacunary strong convergence witespect 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 witespect to a modulus f andf-wijsman lacunary statistical convergence are equivalent on bounded sequences. We further characterize those θ for which WS f θ = WSf,where WS f θ and WSf denote the sets of all f-wijsman lacunary statistically convergent sequences and f-wijsman statistically convergent sequences, respectively. MSC: 40A35; 46A45; 40G5 Keywords: modulus function; density; lacunary sequence; statistical convergence; lacunary strong convergence; Wijsman convergence Introduction Zygmund [] was the person behind the introduction of the idea of statistical convergence. The concept of statistical convergence was formally given by Fast [2] and Steinhaus [3]. This concept was studied by Schoenberg [4] as a non-matrix summability method. For a detailed account of statistical convergence one may refer to [5 ], and many others. Let N denote the set of all natural numbers. A number sequence X =(ξ k issaidtobe statistically convergent to the number l if for each ε >0theset{k N : ξ k l ε} has natural density zero. The natural density of a subset E N [2]isdefinedby d(e= n {k n : k E}, n where the vertical bars indicate the number of elements in the enclosed set. Obviously we have d(e = 0 provided that E is a finite set of positive integers. If a sequence (ξ k is statistically convergent to l,thenwewriteitass ξ k = l or ξ k l(s. The concept of convergence of sequences of points has been extended by several authors [3 20] to the convergence of sequences of sets. In this paper we consider one such extension, namely, Wijsman convergence. Nuray and Rhoades [2] 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 The Author(s 207. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 2 of 20 Wijsman strong Cesàro summability of sequences of sets and discussed its relation with Wijsman statistical convergence. Ulusu and Nuray [22] 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 [2 30]. Recall [3, 32]thatamodulusf is a function from R + to R + such that (i f (x=0if and only if x =0, (ii f (x + y f (x+f(y for x 0, y 0, (iii f is increasing, (iv f is continuous from the right at 0. From above properties it is easy to see that a modulus f is continuous on R +.Amodulus may be unbounded or bounded. For example, f (x =x p where 0 < p, is unbounded, but f (x= x (+x isbounded.theworkrelatedtothesequencespacesdefinedbyamodulus may be found in, e.g.,[24, 3, 33 37]. Aizpuru et al. [33] have recently introduced a new concept of density by moduli and consequently obtained a new concept of non-matrix convergence, namely, f -statistical convergence which is, in fact, a generalization of the concept of statistical convergence and intermediate between the ordinary convergence and the statistical convergence. This idea of replacing natural density with density by moduli has motivated us to look for some new generalizations of statistical convergence and consequently we have introduced and studied the concepts of f -statistical convergence of order α [34]andf-lacunary statistical convergence [35]. Using the notion of density by moduli Bhardwaj et al. [36] havealso introduced and studied the concept of f -statistical boundedness which is a generalization of statistical boundedness [38] and intermediate between the usual boundedness and the statistical boundedness. The notion of Wijsman statistical convergence has been extended by Bhardwaj et al. to that of f -Wijsman statistical convergence [unpublished], where f is an unbounded modulus. Before proceeding further, we first recall some definitions. Definition. ([33] For any unbounded modulus f,thef -density of a set E N is denoted by d f (Eandisdefinedby d f f ( {k n : k E} (E= n f (n in the case this it exists. Clearly, finite sets have zero f -density and d f (N E= d f (E does not hold, in general. But if d f (E=0thend f (N E=. Remark.2 For any unbounded modulus f,if E N has zero f -density then it has zero natural density, however, the converse need not be true. For example, if we take f (x = log (x +ande = {n 2 : n N},thend(E=0butd f (E = /2. Definition.3 ([33] Let f be an unbounded modulus. A number sequence X =(ξ k is said to be f -statistically convergent to l,ors f -convergent to l,if,foreachε >0, d f ({ k N : ξ k l ε } =0,

3 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 3 of 20 i.e., n f (n f ( { k n : ξ k l ε } =0, and we write it as S f ξ k = l or ξ k l(s f. It is an immediate consequence of Definition.3 and Remark.2 that every f -statistically convergent sequence is statistically convergent, but a statistically convergent sequence need notbe f -statistically convergent for every unbounded modulus f. By a lacunary sequence θ =(k r ; r =0,,2,..., where k o = 0, we shall mean an increasing sequence of non-negative integers with = k r k r as r.theintervals determined by θ will be denoted by I r =(k r, k r ]andtheratiok 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. [39] as follows: { } N θ = X =(ξ k : ξ k l =0forsomenumberl. r There is a strong connection [39] betweenn θ and the space w of strongly Cesàro summable sequences, which is defined by w = { X =(ξ k : n n } n ξ k l =0forsomenumberl. k= In the special case, where θ =(2 r, we have N θ = w. In the year 986, the concept of strong Cesàro summability was extended to that of strong Cesàro summability witespect to a modulus by Maddox [3]. A sequence X =(ξ k is said to be strongly Cesàro summable witespect to a modulus f to l if n n n f ( ξ k l =0. k= The space of strongly Cesàro summable sequences witespect to a modulus f,isdenoted by w(f. Furthermore,in the year 994,Pehlivan and Fisher [40] extended the notion of lacunary strong convergence to that of lacunary strong convergence witespect to a modulus f. The space N θ (f of lacunary strongly convergent sequences witespect to a modulus f is defined as N θ (f = { ( X =(ξ k : f ξk l } =0forsomenumberl. r Fridy and Orhan [9] introduced the concept of lacunary statistical convergence as follows. Definition.4 Let θ =(k r be a lacunary sequence. A number sequence X =(ξ k issaidto be lacunary statistically convergent to l,ors θ -convergent to l,if,foreachε >0, r { k Ir : ξ k l ε } =0. In this case, we write S θ ξ k = l or ξ k l(s θ.

4 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 4 of 20 Quite recently, Ulusu and Nuray [22] 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 witespect to a modulus. It is shown that if a sequence is Wijsman lacunary strongly convergent then it is Wijsman lacunary stronglyconvergentwithrespecttoamodulus,however,theconverseneednotbetrue.we also investigate the condition under which the converse is true. We also study a relationship between Wijsman lacunary strong convergence witespect to a modulus and Wijsman lacunary statistical convergence and characterize those θ for which [Ww f ]=[WN f θ ], where [Ww f ] is the set of all Wijsman strongly Cesàro summable sequences witespect to amodulusf and [WN f θ ] is the set of all Wijsman lacunary strongly convergent sequences witespect to a modulus f. We also introduce a new concept of f -Wijsman lacunary statistical convergence of sequences of sets which is a generalization of the concept of Wijsman lacunary statistical convergence of sequences of sets and intermediate between the usual Wijsman convergence and the Wijsman lacunary statistical convergence. It is proved that, under certain conditions on the modulus f, the concepts of Wijsman lacunary strong convergence witespect to a modulus f and f -Wijsman lacunary statistical convergence are equivalent on bounded sequences of sets. We also characterize those θ for which WS f = WS f θ, under certain restrictions on unbounded modulus f,wherewsf is the set of all f -Wijsman statistically convergent sequences of sets and WS f θ is the set of all f -Wijsman lacunary statistically convergent sequences of sets. Finally, we observe that it is possible for a sequence to have different WS f θ -its for different θ s. In Theorem 4., we investigate certain conditions under which this situation cannot occur. Before proceeding to establish the proposed results, we pause to collect some definitions related to Wijsman convergence [2, 22]. Let (M, ρ beametricspace. Thedistanced(x, E fromapointx to a non-empty subset E of (M, ρisdefinedtobe d(x, E =inf ρ(x, y. y E Definition.5 Let (E k be a sequence of non-empty closed subsets of a metric space (M, ρande be non-empty closed subset of M. (a (E k is said to be Wijsman convergent to E,if,foreachx M, (d(x, E k is convergent to d(x, E.Inthiscase,wewriteWc E k = E or E k E(Wc.Theset of all Wijsman convergent sequences is denoted by Wc. (b (E k is said to be Wijsman statistically convergent to E,orWS-convergent to E,if, for each x M, (d(x, E k is statistically convergent to d(x, E; i.e.,foreachx M and for each ε >0, n { k n : d(x, E k d(x, E ε } =0. n In this case, we write WS E k = E or E k E(WS.Thesetofallsequences which are Wijsman statistically convergent is denoted by WS. (c (E k is said to be bounded if sup k d(x, E k < for each x M. We shall denote the set of all bounded sequences of sets by L.

5 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 5 of 20 (d (E k is said to be Wijsman strongly Cesàro summable to E,if,foreachx M, (d(x, E k is strongly Cesàro summable to d(x, E; i.e.,foreachx M, n n n d(x, E k d(x, E =0. k= In this case, we write [Ww] E k = E or E k E[Ww].Thesetofallsequences which are Wijsman strongly Cesàro summable is denoted by [Ww]. (e (E k is said to be Wijsman lacunary strongly convergent to E,if,foreachx M, (d(x, E k is lacunary strongly convergent to d(x, E; i.e.,foreachx M, r d(x, Ek d(x, E =0. In this case, we write [WN θ ] E k = E or E k E[WN θ ].Thesetofallsequences which are Wijsman lacunary strongly convergent is denoted by [WN θ ]. (f (E k is said to be Wijsman lacunary statistically convergent to E,orWS θ -convergent to E,if,foreachx M, (d(x, E k is lacunary statistically convergent to d(x, E; i.e., for each x M and for each ε >0, r { k Ir : d(x, Ek d(x, E ε } =0. In this case, we write WS θ E k = E or E k E(WS θ.thesetofallsequences which are Wijsman lacunary statistically convergent is denoted by WS θ. Quite recently, Bhardwaj et al. have given the following definitions [unpublished]. Definition.6 For any an unbounded modulus f,asequence(e k ofnon-emptyclosed subsets of a metric space (M, ρissaidtobef -Wijsman statistically convergent to a nonempty closed subset E of M, or WS f -convergent to E, if,foreachx M, (d(x, E k is f - statistically convergent to d(x, E; i.e.,foreachx M and for each ε >0, n f (n f ( { k n : d(x, E k d(x, E ε } =0. In this case, we write WS f E k = E or E k E(WS f. The set of all sequences which are f -Wijsman statistically convergent is denoted by WS f. Definition.7 For any modulus f,asequence(e k ofnon-emptyclosedsubsetsofametric space (M, ρ is said to be Wijsman strongly Cesàro summable to a non-empty closed subset E of M witespect to f,if,foreachx M,(d(x, E k is strongly Cesàro summable to d(x, Ewithrespecttof ; i.e.,foreachx M, n n n f ( d(x, Ek d(x, E =0. k= In this case, we write [Ww f ] E k = E or E k E[Ww f ]. The set of all sequences which are Wijsman strongly Cesàro summable witespect to a modulus f is denoted by [Ww f ].

6 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 6 of 20 2 Wijsman lacunary strong convergence witespect to a modulus Definition 2. Let (M, ρbeametricspace,f be a modulus and θ =(k r bealacunarysequence. A sequence (E k of non-empty closed subsets of M is said to be Wijsman lacunary strongly convergent to a non-empty closed subset E of M witespect to f,if,foreach x M, (d(x, E k is lacunary strongly convergent to d(x, E withrespecttof ; i.e., foreach x M, r ( f d(x, Ek d(x, E =0. In this case, we write [WN f θ ] E k = E or E k E[WN f θ ]. The set of all sequences which are Wijsman lacunary strongly convergent witespect to a modulus f is denoted by [WN f θ ]. Remark 2.2 If we take f (x = x, the concept of Wijsman lacunary strong convergence with respect to f reduces to that of Wijsman lacunary strong convergence. We now study an inclusion relation between [WN θ ]and[wn f θ ]. To establish this relation we first recall the following proposition from [40]. Proposition 2.3 Let f be a modulus and let 0<δ <.Then, for each x δ, we have f (x 2f (δ x. Theorem 2.4 For any modulus f, we have [WN θ ] [WN f θ ]. Proof Let (E k [WN θ ], then, for each x M, N r = d(x, Ek d(x, E 0 (asr. Let ε >0begiven.Wechoose0<δ <suchthatf (u<ε for every u with 0 u δ. We can write ( f d(x, Ek d(x, E = ( f ( d(x, Ek d(x, E + ( d(x,e k d(x,e δ d(x,e k d(x,e >δ ( ε+2f (δ N r, f ( d(x, Ek d(x, E by Proposition 2.3. Therefore, (E k [WN f θ ]asr. Remark 2.5 The converse of the above theorem does not need to be true, which can be verified from the following example.

7 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 7 of 20 Example 2.6 Let M = R, ρ(x, y= x y and f (x=log(x +. Consider the sequence (E k defined by { }, ifk I r such that k = k r +, E k = {0}, otherwise. Note that (E k is not a bounded sequence. Then, for each x M, ( f d(x, Ek d ( x, {0} = f ( d(x, Ekr + d ( x, {0} = f ( d ( x, { } d ( x, {0} f ( (x hr (x 0 = f ( = log( + 0 (asr, and so (E k [WN f θ ], but, for x =0, d(x, E k d ( x, {0} = d(x, E kr + d ( x, {0} = x hr x 0 = (asr, and so (E k / [WN θ ]. f (t Maddox [4] proved that for any modulus f there exists t. Making use of this t result we are in a position to give a condition on modulus f under which the converse holds. Theorem 2.7 Let f be a modulus such that t f (t t >0,then [WN f θ ] [WN θ]. TheproofcanbeestablishedusingthetechniqueofTheorem3.5of[34]. We now establish a relationship between Wijsman lacunary strong convergence with respect to a modulus and Wijsman lacunary statistical convergence. Theorem 2.8 For any modulus f, we have [WN f θ ] WS θ. Proof Suppose that (E k [WN f θ ]. For each x M and ε >0,wehave ( f d(x, Ek d(x, E ( d(x,e k d(x,e ε f ( d(x, Ek d(x, E { k Ir : d(x, Ek d(x, E ε } f (ε, from which it follows that (E k WS θ.

8 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 8 of 20 We now give an example to show that the converse of the above inclusion need not hold. Example 2.9 Let M = R, ρ(x, y = x y and f (x =2x. Consider the sequence (E k of subsets of M as defined in Example 2.6. This sequence is Wijsman lacunary statistically convergent to the set E = {0} because for each x M and for each ε >0, r { k Ir : d(x, E k d ( x, {0} ε } = =0. r But this is not Wijsman lacunary strongly convergent witespect to f. In the next theorem, we investigate a necessary and sufficient condition on f under which the converse holds. Theorem 2.0 WS θ =[WN f θ ] if and only if f is bounded. Proof Suppose that f is bounded and (E k WS θ.sincef is bounded, there exists a constant H such that f (x H for all x 0. Now for each x M, ( f d(x, E k d(x, E = ( f ( d(x, E k d(x, E + ( d(x,e k d(x,e ε d(x,e k d(x,e <ε f ( d(x, Ek d(x, E { k Ir : d(x, E k d(x, E ε } H + f (ε. Taking the it as r,wehave(e k [WN f θ ]. Conversely, suppose that f is unbounded so that there exists a positive sequence 0 < p < p 2 < < p i < such that f (p i h i. Define the sequence (E k suchthate ki = {p i } for i =,2,...andE k = {0} otherwise. Then, for each x M and ε >0, { k Ir : d(x, E k d ( x, {0} ε } = 0 (asr, and so (E k WS θ,but,forx =0, ( f d(x, E k d ( x, {0} = f ( d(x, E kr d ( x, {0} = f (p r (asr, and so (E k / [WN f θ ]. This is a contradiction to the assumption that WS θ =[WN f θ ]. Hence, f is bounded.

9 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 9 of 20 We now study a relationship between Wijsman strong Cesàro summability witespect to a modulus and Wijsman lacunary strong convergence witespect to a modulus. Theorem 2. Let θ =(k r be a lacunary sequence and f be a modulus. If < inf r q r sup r q r <, then [WN f θ ]=[Wwf ]. Proof Suppose that inf r q r >, then there exists δ >0suchthatq r +δ for sufficiently large r.let(e k [Ww f ]. For each x M,wehave ( f d(x, Ek d(x, E = k r f ( d(x, E k d(x, E k r f ( d(x, E k d(x, E k= k= k= ( = k r k r f ( d(x, E k d(x, E ( k r k r Since = k r k r,wehave k r k r f ( d(x, E k d(x, E. (2. k= k r +δ δ and k r δ for sufficiently large r. (2.2 Using (2.2in(2., we have f ( d(x, E k d(x, E 0astheterms kr k r k= f ( d(x, E k d(x, E and kr k r k= f ( d(x, E k d(x, E bothtendsto0asr.hence,(e k [WN f θ ]. Now suppose that sup r q r <, then there exists G >0suchthatq r < G for all r. Letting (E k [WN f θ ], x M and ε >0wecanfindr 0 such that for every r r 0 N r = f ( d(x, Ek d(x, E < ε. I r We can also choose a number L >0suchthatN r L for all r. Nowletn be any integer with k r < n k r,wherer > r 0.Then,foreachx M, n n f ( d(x, Ek d(x, E k= k r k r k= = ( k r = k r < k r k I f ( r0 r= ( r0 r= f ( d(x, E k d(x, E ( d(x, Ek d(x, E + + ( f d(x, Ek d(x, E ( f d(x, Ek d(x, E + r f r=r 0 + ( f d(x, E k d(x, E + ε(k r k r0 ( d(x, Ek d(x, E

10 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 0 of 20 = (h N + h 2 N N r0 + ε(k r k r0 k r k r ( sup N k k r0 + ε(k r k r0 k r k r 0 k r < L k r 0 k r + εg, which yields (E k [Ww f ]. 3 f-wijsman lacunary statistical convergence Definition 3. Let (M, ρ beametricspace,f be an unbounded modulus and θ =(k r be a lacunary sequence. A sequence (E k ofnon-emptyclosedsubsetsofm is said to be f -Wijsman lacunary statistically convergent to a non-empty closed subset E of M,orWS f θ - convergent to E,if,foreachx M and for each ε >0, r f ( f ( { k I r : d(x, E k d(x, E ε } =0. In this case, we write WS f θ E k = E or E k E(WS f θ. The set of all sequences which are f -Wijsman lacunary statistically convergent is denoted by WS f θ. Remark 3.2 The concept of f -Wijsman lacunary statistical convergence reduces to that of Wijsman lacunary statistical convergence when modulus is the identity mapping. Theorem 3.3 Every Wijsman convergent sequence is f -Wijsman lacunary statistically convergent, however, theconverseneednotbetrue. 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,0<p and(e k bedefinedas [2, ], if k 2andk I r is a square, E k = {}, otherwise, where θ =(k r is a lacunary sequence. This sequence is not Wijsman convergent, but, for each x M and for each ε >0, f ( f ( { k I r : d(x, Ek d ( x, {} } ɛ f ( hr f ( = ( p = 0 (asr ( p ( p p/2 and hence, (E k isf -Wijsman lacunary statistically convergent to the set E = {}. Remark 3.4 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.

11 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page of 20 Theorem 3.5 Every f -Wijsman lacunary statistically convergent sequence is Wijsman lacunary statistically convergent. Proof Suppose (E k isf-wijsman lacunary statistically convergent to E. Letx M and ε > 0. Then, for each positive integer m,thereexistsr o N such that for r r o,wehave f ( { k I r : d(x, E k d(x, E ɛ } m f ( ( ( m mf hr hr = f m m and since f is increasing, we have { k Ir : d(x, Ek d(x, E } ɛ m. Hence, (E k is Wijsman lacunary statistically convergent to E. Remark 3.6 Itseemsthattheconverseoftheabovetheoremneednothold,butrightnow 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 witespect to a modulus. Maddox [3] showed the existence of an unbounded modulus f for which there is a positive constant c such that f (xy cf (xf (y, for all x 0, y 0.Usingthiswehavethe following. Theorem 3.7 Let (M, ρ be a metric space and θ =(k r be a lacunary sequence, then f (t (a For any unbounded modulus f for which t >0and there is a positive t constant c such that f (xy cf (xf (y for all x 0, y 0, (i E k E[WN f θ ] implies E k E(WS f θ, (ii [WN f θ ] is a proper subset of WSf θ. (b (E k L and E k E(WS f θ imply E k E[WN f θ ], for any unbounded modulus f. (c [WN f θ ] L = WSf θ L for any unbounded modulus f for which t f (t >0 t and there is a positive constant c such that f (xy cf (xf (y for all x 0, y 0. Proof (a (i For any sequence (E k, for each x M and ɛ > 0, by the definition of modulus function (ii and (iii we have ( f d(x, Ek d(x, E ( f d(x, Ek d(x, E ( f d(x,e k d(x,e ɛ d(x, E k d(x, E f ( { k I r : d(x, Ek d(x, E ε } ɛ c f ( { k I r : d(x, E k d(x, E ε } f (ɛ = c f ( {k I r : d(x, E k d(x, E ε} f ( f (ɛ, f ( from which it follows that (E k WS f θ as (E k [WN f θ ]and r f ( >0.

12 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 2 of 20 (ii In order to show that the inclusion [WN f θ ] WSf θ is proper, let θ =(k rbealacunary f (t sequence and f be an unbounded modulus such that t > 0 and there is a positive t constant c such that f (xy cf (xf (y for all x 0, y 0. Consider the sequence (E k such that E k to be {}, {2},...,{[ ]} at the first [ ]integersini r,ande k = {0} otherwise. Note that (E k is not bounded. Also, for each x M and ɛ >0, f ( f ( { k I r : d(x, E k d ( x, {0} ε } = f ([ hr ] f ( = f ([ ] [ ] f ( [ hr ] r f ([ ] [ ] 0 asr, because f ( [ hr ], are positive and =0. r r Thus, E k {0}(WS f θ. On the other hand, for x =0, ( f d(x, Ek d ( x, {0} = k r <k k r +([ ] f ( d(x, E k d ( x, {0} = [ f ( x x 0 + f ( x 2 x f ( x [ hr ] x 0 ] = [ f ( + f (2 + + f ( [ hr ] ] f ( [ ] = f ( [ hr ]([ ]+ 2 c f ([ ]f ( [ hr ]+ 2 ( ( f ([ hr ] f ( [ ]+ ( = c [ 2 [ hr ]( [ ]+ ] [ 2 ]+ 2 f ([ hr ] >0 asc, r [ ], f ( [ ]+ r 2 [ ]+ 2 [ hr ]( [ ]+ 2,and r are positive. Therefore, E k {0}[WN f θ ]. (b Suppose that E k E(WS f θ and(e k L,say d(x, E k d(x, E G,foreachx M and for all k N, whereg = sup k d(x, E k + d(x, E. Given ɛ > 0 and for each x M, we have ( f d(x, E k d(x, E = d(x,e k d(x,e ɛ f ( d(x, Ek d(x, E

13 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 3 of 20 + d(x,e k d(x,e <ɛ f ( d(x, Ek d(x, E { k Ir : d(x, Ek d(x, E ε } f (G+ f (ɛ. Taking the it on both sides as r,weget r ( f d(x, Ek d(x, E =0, in view of Theorem 3.5 and the fact that f is increasing. (c This is an immediate consequence of (a and (b. Remark 3.8 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 3.9 If we take f (x=x in Theorem 3.7, we obtain Theorem of Ulusu and Nuray [22]. Quite recently, Bhardwaj et al. have established the following lemmas [unpublished]. Lemma 3.0 Let (M, ρ be a metric space; f, g be unbounded moduli, (E k be a sequence of non-empty closed sets in M and E, F Mbenon-emptyclosed. We have (i The f -Wijsman statistical it is unique whenever it exists. (ii Moreover, two different methods of Wijsman statistical convergence are always compatible, which means that if WS f E k = E and WS g E k = F then E = F. Lemma 3. For any modulus f, we have [Ww] [Ww f ]. Lemma 3.2 Let f be a modulus such that t f (t t >0,then [Ww f ] [Ww]. Lemma 3.3 Let (M, ρ be a metric space, f be an unbounded modulus such that >0and there is a positive constant t f (t t csuchthat f(xy cf (xf (y for all x 0, y 0. Then for any non-empty closed subsets E, E k M: (i (E k is f -Wijsman statistically convergent to E if it is Wijsman strongly Cesàro summable to E, witespect to f. (ii If (E k is bounded and f -Wijsman statistically convergent to E then it is Wijsman strongly Cesàro summable to E witespect to f. We now study the inclusions WS f θ WSf and WS f WS f θ under certain restrictions on θ and f. Lemma 3.4 For any lacunary sequence θ, and unbounded modulus f for which f (t t >0and there is a positive constant c such that f (xy cf (xf (y for all x 0, t y 0, we have WS f WS f θ if and only if inf r q r >.

14 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 4 of 20 Proof (Sufficiency. If inf r q r >, then there exists β >0suchthatq r +β for sufficiently large r.since = k r k r,wehave β k r +β for sufficiently large r.ife k E(WS f, then, for each ɛ >0,foreachx M and sufficiently large r,wehave f (k r f ( { k k r : d(x, Ek d(x, E } ε f ( {k I r : d(x, E k d(x, E ε} f (k r = f (h ( r f ( {k Ir : d(x, E k d(x, E ε} f (k r f ( ( ( ( ( f (hr kr hr f ( {k Ir : d(x, E k d(x, E ε} = f (k r k r f ( ( ( ( ( f (hr kr β f ( {k Ir : d(x, E k d(x, E ε}. f (k r +β f ( This proves the sufficiency. (Necessity. Assume that inf r q r =. Proceeding as in Lemma 2. of [39], we can select asubsequence(k r(j ofθ satisfying k r(j k r(j <+ j and k r(j k r(j > j, wherer(j r(j +2. Define a sequence (E k as follows: {(x, y R R : x 2 +(y 2 = }, ifk I k E k = 4 r(j,forsomej =,2,3,..., {(0, 0}, otherwise. It is easy to see that the sequence (E k is bounded because for each (x, y R R, the sequence (d((x, y, E k given as below is bounded. d ( (x, y, E k = x2 + y 2, x2 +(y 2 k 4 x 2 +(y 2 + k 2, ifk I r(j,forsomej =,2,3,..., otherwise. Also, it is shown in Lemma of [22] that(e k / [WN θ ]but(e k [Ww]. Thus, in view of Theorems 2.7 and 3.7, wehave(e k / WS f θ. On the other hand, it follows from Lemmas 3. and 3.3 that (E k WS f.hence,ws f WS f θ. But this is a contradiction to the assumption that WS f WS f θ. This contradiction shows that our assumption is wrong. Hence, inf r q r >. Lemma 3.5 For any lacunary sequence θ, and unbounded modulus f for which f (t t >0and there is a positive constant c such that f (xy cf (xf (y for all x 0, t y 0, we have WS f θ WSf if and only if sup r q r <.

15 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 5 of 20 Proof (Sufficiency. If sup r q r <, then there is a K >0suchthatq r < K for all r.now, suppose that E k E(WS f θ and r f ( = l. Therefore, for given ɛ > 0 and for each x M,thereexistsr o N such that for all r > r o f ( < l + ɛ, and f ( f ( { k I r : d(x, Ek d(x, E ε } < ɛ. Let N r = {k I r : d(x, E k d(x, E ɛ}. Using this notation, we have f (N r f ( < ɛ for all r > r o. Now, let G = max{f (N, f (N 2,...,f(N ro } and let n be an integer such that k r < n k r, where r > r o then we can write f (n f ( { k n : d(x, Ek d(x, E } ε f (k r f ( { k k r : d(x, E k d(x, E ε } = f (k r f (N + N N ro + N ro N r ( f (N +f(n f (N ro +f(n ro ++ + f (N r f (k r r og f (k r + f (k r = r og f (k r + f (k r [ f (Nro ++ + f (N r ] [ f (hro + o + f (N ro + f (o + o f (h ] r f (N r f ( < r og f (k r + [( l + ɛ ɛo ( l + ɛ ] ɛ f (k r = r og f (k r + f (k r ɛ( l + ɛ [o ] = r og f (k r + f (k r ɛ( l + ɛ [k r k ro ] < r og f (k r + ɛ( l + ɛ [ kr f (k r = r og f (k r + ɛ( l + ɛ f (k r k r = r og f (k r + ɛ( l + ɛ q r f (k r k r ] k r k r < r og f (k r + ɛ( l + ɛ K, f (k r k r from which the sufficiency follows immediately, in view of the fact that r f (k r k r >0.

16 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 6 of 20 (Necessity. Suppose that sup r q r =.FollowingLemma2.2of[39], 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 {}, ifk r(j < k 2k r(j,forsomej =,2,3,..., E k = {0}, otherwise. ItisshowninLemma2of[22] that(e k [WN θ ]but(e k / [Ww]. By Theorems 2.4 and 3.7,weconcludethat(E k WS f θ,but(e k/ WS f, in view of Lemmas 3.2 and 3.3.Hence, WS f θ WSf. But this is a contradiction to the assumption that WS f θ WSf. This contradiction shows that sup r q r <. Combining Lemmas 3.4 and 3.5 we have the following. Theorem 3.6 For any lacunary sequence θ, and unbounded modulus f for which t f (t t >0and there is a positive constant c such that f (xy cf (xf (y for all x 0, y 0, we have WS f θ = WSf if and only if < inf r q r sup r q r <. 4 Uniqueness of WS f θ -it It is easy to see that, for any fixed θ, thews θ -it is unique. It is possible, however, for a sequence, even a bounded one, to have different WS θ -its for different θ s. This can be seen by applying Theorem of Ulusu and Nuray [22] tothesequence(e k definedas follows: {}, ifp(kiseven, E k = {0}, ifp(k is odd, where p(k=n,ifn!<k (n +!andθ = ((2r!, θ 2 = ((2r +!. It is observed that, for each x M, d(x, E k d ( x, {0} (2r +! (2r! = 0 (asr, + (2r +2! (2r! I r+ from which it follows that (E k [WN θ ]with[wn θ ] E k = {0},hence,WS θ E k = {0} and also d(x, Ek d ( x, {} (2r! (2r! = 0 (asr, (2r +! (2r! I r from which it follows that (E k [WN θ2 ]with[wn θ2 ] E k = {} and hence, WS θ2 E k = {}. In case of WS θ -convergence, Ulusu and Nuray [22] showed that this situation cannot if the sequence is Wijsman statistically convergent. We now establish a similar result in the case of WS f θ -convergence. First we observe that it is also possible for a sequence to have different WS f θ -its for different θ s. To illustrate this, let us take f (x=2x, θ = ((2r!, θ 2 = ((2r+! and consider the sequence (E k just defined above for which [WN θ ] E k = {0}

17 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 7 of 20 and [WN θ2 ] E k = {}. Now by applying Theorem 2.4 and Theorem 3.7, weseethat WS f θ E k = {0} and WS f θ 2 E k = {}. In the next theorem we investigate certain conditions under which this situation cannot occur. Theorem 4. For any two lacunary sequences θ and θ 2, if (E k WS f (WS f θ WS f θ 2, then WS f θ E k = WS f θ 2 E k, where f is an unbounded modulus function such that f (x f (y = f ( x y, for all x 0, y 0. To prove this theorem we need the following lemma. Lemma 4.2 For any lacunary sequence θ, if (E k WS f WS f θ, then WSf θ E k = WS f E k, where f is an unbounded modulus function such that f (x f (y = f ( x y, for all x 0, y 0. Proof Suppose WS f E k = E, WS f θ E k = F and E F.SinceE F, therefore there exists at least one x M such x E but x / F or x F but x / E. Without loss of generality we may suppose that x E but x / F. Now, clearly x cannotbeaitpointoff,because if x becomes a it point of F then x F as F isclosed.now,letε >0besuchthat0<ε < d(x,e d(x,f 2. Using the definition of modulus (iii and (ii, we have f ( {k n : d(x, E d(x, F 2ε} f (n f ( {k n : d(x, E k d(x, E ε} f (n + f ( {k n : d(x, E k d(x, F ε}. f (n Taking the it as n on both sides, we get f ( {k n : d(x, E k d(x, F ε} 0+, n f (n and hence, f ( {k n : d(x, E k d(x, F ε} =. n f (n (4. Now consider the k m th term of the sequence (( f ( { f (n k n : d(x, Ek d(x, F } ε. f (k m f ( { k k m : d(x, E k d(x, F ε } ( { m = f (k m f k I r : d(x, Ek d(x, F } ε r= ( m = f (k m f { k I r : d(x, E k d(x, F ε } r=

18 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 8 of 20 f (k m = f (k m m f ( { k I r : d(x, Ek d(x, F } ε r= m f ( f ( f ( { k I r : d(x, Ek d(x, F } ε. (4.2 r= Also, in view of the choice of unbounded modulus f,wehave m f ( =f(h +f(h f (h m r= = f (k k o +f (k 2 k + + f (k m k m = f ( k k o + f ( k 2 k + + f ( k m k m = f (k f (k o + f (k2 f (k + + f (km f (k m = f (k f (k o +f (k 2 f (k + + f (k m f (k m = f (k m. (4.3 Now, using (4.3in(4.2, we have f (k m f ( { k k m : d(x, E k d(x, F ε } m r= f ( m f ( t r, (4.4 r= where t r =(f ( f ( {k I r : d(x, E k d(x, F ε} 0becauseE k F(WS f θ. Since θ is a lacunary sequence and f being modulus is increasing, the term on the right hand side of (4.4isaregularweightedmeantransformationoft =(t r, and therefore, it, too, tends to zero as r.thus, f (k m f ( { k k m : d(x, E k d(x, F ε } 0 asm. Also, since ((f (k m f ( {k k m : d(x, E k d(x, F ε} is a subsequence of sequence ((f (n f ( {k n : d(x, E k d(x, F ε}, we conclude that ( f (n f ( { k n : d(x, Ek d(x, F ε }. Butthisisacontradictionto(4.. This contradiction shows that E = F. Proof of Theorem 4. By Lemma 4.2,wehave WS f E k = WS f θ E k and (4.5 WS f E k = WS f θ 2 E k. (4.6 Therefore, from (4.5and(4.6 we have WS f θ E k = WS f θ 2 E k.

19 Bhardwaj and Dhawan Journal of Inequalities and Applications ( :25 Page 9 of 20 If we take f (x=x in Theorem 4. we obtain the following result which contains Theorem 3 of Ulusu and Nuray [22]. Corollary 4.3 For any two lacunary sequences θ and θ 2, if (E k WS (WS θ WS θ2, then WS θ E k = WS θ2 E k. In the next theorem we show that if a sequence (E k isws f θ -convergent and WSg θ - convergent then WS f θ E k = WS g θ E k, under certain conditions on the unbounded moduli f and g. Theorem 4.4 For any lacunary sequence θ, if (E k (WS f WS f θ (WSg WS g θ, then WS f θ E k = WS g θ E k, where f and g are unbounded moduli such that ( f (x f(y = f x y and g(x g(y = g ( x y, for all x 0, y 0. Proof By Lemma 4.2,wehave WS f E k = WS f θ E k WS g E k = WS g θ E k. and (4.7 But according to Lemma 3.0,wehave WS f E k = WS g E k. (4.8 Therefore, from (4.7and(4.8, we have WS f θ E k = WS g θ E k. Competing interests The authors declare that they have no competing interests. Authors contributions VKB and SD contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details Department of Mathematics, Kurukshetra University, Kurukshetra, 369, India. 2 Department of Mathematics, KVA DAV College for Women, Karnal, 3200, India. Acknowledgements The authors wish to thank the referees for their valuable suggestions, which have improved the presentation of the paper. Received: 4 September 206 Accepted: 4 January 207 References. Zygmund, A: Trigonometric Series, 3rd edn. Cambridge University Press, London ( Fast, H: Sur la convergence statistique. Colloq. Math. 2, (95 3. Steinhaus, H: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2, (95 4. Schoenberg, IJ: The integrability of certain functions and related summability methods. Am. Math. Mon. 66, ( Et, M: Generalized Cesàro difference sequence spaces of non-absolute type involving lacunary sequences. Appl. Math. Comput. 29(7, ( Et, M, Çinar, M, Karakaş, M: On λ-statistical convergence of order α of sequences of functions. J. Inequal. Appl. 203, 204 (203

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