Research Article Invariant Statistical Convergence of Sequences of Sets with respect to a Modulus Function

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1 Hidawi Publishig Corporatio Abstract ad Applied Aalysis, Article ID 88020, 5 pages Research Article Ivariat Statistical Covergece of Sequeces of Sets with respect to a Modulus Fuctio Nimet Pacaroglu ad Fatih Nuray Departmet of Mathematics, Afyo Kocatepe Uiversity, Afyokarahisar, Turkey Correspodece should be addressed to Fatih Nuray; furay@aku.edu.tr Received 5 December 203; Accepted 2 February 204; Published 23 March 204 Academic Editor: Irea Rachůková Copyright 204 N. Pacaroglu ad F. Nuray. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We itroduce ad study the cocept of ivariat covergece for sequeces of sets with respect to modulus fuctio f ad give some iclusio relatios.. Itroductio The cocept of statistical covergece for sequeces of real umbers was itroduced by Fast [] adstudiedbyšalát [2] ad others. Let K N ad K ={k :k K}.Thethe atural desity of K is defied by δ(k) = K,ifthe it exists, where K deotes the cardiality of K. Asequecex = (x k ) complex umbers is said to be statistically coverget to L if, for each ε > 0, {k : x k L ε} =0. () Covergece cocept for sequeces of set had bee studied by Beer [3], Aubi ad Frakowska [4], ad Baroti ad Papii [5]. The cocept of statistical covergece of sequeces of set was itroduced by Nuray ad Rhoades [6]i 202.Ulusu ad Nuray [7] itroduced the cocept of Wijsma lacuary statistical covergece of sequeces of set. Similarly, the cocepts of Wijsma ivariat statistical ad Wijsma lacuary ivariat statistical covergece were itroduced by Pacaroglu ad Nuray [8]i 203. Afuctiof : [0, ) [0, ) is called a modulus, if () f(x) = 0 if ad if oly if x=0; (2) f(x+y) f(x)+f(y); (3) f is icreasig; (4) f is cotiuous from the right at 0. A modulus may be ubouded (e.g., f(x) = x p, 0<p< ) or bouded (e.g., f(x)=x/(x+)). Modulus fuctio was itroduced by Nakao [9]i 953. Ruckle [0] usediideaofmodulusfuctiof to costruct a class of FK spaces. Cosider L(f)={x=(x k ): f( x k )<}. (2) The space L(f) is closely related to the space l which is a L(f) space with f(x) = x,forallrealx 0. Maddox [] defied the followig spaces by usig a modulus fuctio f: w 0 (f) = {x s : w(f)={x s: f( x k )=0}, f( x k L )=0for some L},

2 2 Abstract ad Applied Aalysis w (f) = {x s : sup f( x k )<}, where s is space of all complex sequeces. Later, Coor [2] exteded his defiitio by replacig the Cesaro matrix with a arbitrary oegative matrix summability method B=(b k ) as follow: w (B, f) ={x s: 2. Defiitios ad Notatios (3) b,k f( x k L )=0for some L}. Let σ beamappigofthepositiveitegersitoitself.a cotiuous liear fuctioal φ o l, the space of real bouded sequeces, is said to be a ivariat mea or a σ mea, if ad oly if, (4) () φ(x) 0, forallsequecesx=(x ) with x 0for all ; (2) φ(e) =,wheree=(,,,...); (3) φ(x σ() )=φ(x)for all x l. The mappigs φ areassumedtobeoe-to-oesuchthat σ m () = for all positive itegers ad m, whereσ m () deotes the mth iterate of the mappig σ at.thus,φexteds the it fuctioal o c, the space of coverget sequeces, i the sese that φ(x) = x, forallx c.icaseσ is traslatio mappig σ() = +, theσ mea is ofte called a Baach it ad V σ, the set of bouded sequeces all of whose ivariat meas are equal, is the set of almost coverget sequeces. Itcabeshowthat Nuray ad Savaş [3] defied the followig sequece spaces by usig a modulus fuctio f ad a oegative regular matrix B=(b k ): w 0 (B σ,f)={x s: w(b σ,f) ={x s: b k f( x σ k (m) )=0 uiformly i m}, w (B σ,f)={x s:sup,m b k f( x σ k (m) L )=0 for some L, uiformly i m}, b k f( x σ k (m) )<}. Defiitio (see [4]). A set E of positive itegers is said to have uiform ivariat desity of zero if {k : E {σ (m),σ2 (m),...,σ k (m)}} =0 (8) uiformly i m. By usig uiform ivariat desity, the followig defiitio was give. Defiitio 2 (see [0]). A complex umber sequece x=(x k ) is said to be σ-statistically coverget to L if, for every ε>0, {0 k : x σ k (m) L ε} =0 (9) uiformly i m =, 2,.... This will be deoted by S σ x=lor x k L(S σ ). Let (X, ρ) be a metric space. For ay poit x X ad oempty subset A of X, we defie the distace from x to A by (7) d (x, A) = if a A ρ (x, A). (0) V σ = {x=(x ): t m (x) =L uiformly i m, L = σ x}, (5) The cocept of Wijsma covergece was itroduced by Wijsma [7] as follows. Let (X, ρ) be a metric space. For ay oempty closed subsets A, A k X, we say that the sequece {A k } is Wijsma coverget to A if where, k d(x,a k)=d(x, A), () t m (x) = x m +x σ(m) +x σ 2 (m) + +x σ (m). (6) + for each x X. This will be deoted by W A k =A. Covergece cocept for sequeces of set had bee studied by Beer [3], Aubi ad Frakowska [4], ad Baroti ad

3 Abstract ad Applied Aalysis 3 Papii [5]. The cocepts of Wijsma statistical covergece ad Wijsma strog Cesaro summability were itroduced by Nuray ad Rhoades [6] as follows. Let (X, ρ) be a metric space. For ay oempty closed subsets A, A k X,thesequece{A k } is said to be Wijsma strogly Cesaro summable to A if, for each x X, d(x,a k) d(x, A) =0. (2) Let (X, ρ) be a metric space. For ay oempty closed subsets A, A k X,thesequece{A k } is said to be Wijsma statistically coverget to A if, for ε > 0ad each x X, {k : d (x, A k) d(x, A) ε} =0. (3) I this case we write st W A k =Aor A k A(WS). 3. Mai Result The purpose of this paper is, by usig a modulus fuctio, to itroduce ad study ew sequece spaces of sequeces of sets. The followig three defiitios were give i [8]. Defiitio 3. Let (X, ρ) be a metric space. For ay oempty closed subsets A, A k X, we say that the sequece {A k } is Wijsma ivariat coverget to A, if, for each x X, uiformly i m. d (x, A σ k (m)) =d(x, A) (4) I this case, we write A k A(WV σ ) ad the set of all Wijsma ivariat coverget sequeces of sets will be deoted WV σ. Defiitio 4. Let (X, ρ) be a metric space. For ay oempty closed subsets A, A k X, we say that the sequece {A k } is Wijsma strogly ivariat coverget to A, ifforeachx X, d(x,a σ (m)) d(x, A) k =0 (5) uiformly i m. I this case, we write A k A([WV σ ]) adthesetofall Wijsma strogly ivariat coverget sequeces of sets will be deoted [WV σ ]. Defiitio 5. Let (X, ρ) be a metric space. For ay oempty closed subsets A, A k X, we say that the sequece {A k } is Wijsma ivariat statistically coverget to A, if, for each ε>0ad for each x X, {0 k : d(x,a σ k (m) ) d(x, A) ε} =0 (6) uiformly i m. I this case, we write A k A(WS σ ) ad the set of all Wijsma ivariat statistically coverget sequeces of sets will be deoted WS σ. Let (X, ρ) be metric space. For ay oempty closed subsets A, A k Xad x X, we defie the sequeces of sets space [WV σ ] as follows: [WV σ ] ={{A k } sup m, d(x,a σ k (m) ) <}. (7) Now, by usig a modulus fuctio, we itroduce the followig ew sequece spaces of sequeces of sets. Defiitio 6. Let (X, ρ) be a metric space ad f be a modulus fuctio. For ay oempty closed subsets A, A k X,we say that the sequece {A k } is Wijsma strogly ivariat coverget to A with respect to the modulus f, if, for each x X, uiformly i m. f ( d (x, A σ k (m) ) d(x, A) ) =0 (8) I this case we write A k A([WV σ (f)]) ad the set of all Wijsma strogly ivariat coverget sequeces of sets with respect to the modulus f will be deoted [WV σ (f)]. Let (X, ρ) be metric space ad f be a modulus fuctio. For ay oempty closed subsets A, A k Xad x X,we defie the sequeces of sets space [WV σ (f)] as follows: [WV σ (f)] ={{A k }: sup m, f( d(x,a σ k (m) ) )<}. (9) If f(x) = x, the the spaces [WV σ (f)] ad [WV σ (f)] reduce to [WV σ ] ad [WV σ ],respectively. Now we study the relatio betwee WS σ ad [WV σ (f)] covergece. Theorem 7. Let (X, ρ) be a metric space. For ay oempty closed subsets A, A k X.The (i) A k A([WV σ (f)]) implies A k A(WS σ ); (ii) f is bouded ad A k A(WS σ ) implies A k A([WV σ (f)]); (iii) WS σ =[WV σ (f)],iff is bouded. Proof. (i) Let ε>0ad A k A([WV σ (f)]). Theweca write f( d(x,a σ k (m) ) d(x, A) ) f( d(x,a σ k (m) ) d(x, A) ) d(x,a σ k (m) ) d(x,a) ε ε {k : d(x,a σ k (m) ) d(x, A) ε}, which yields the result. (20)

4 4 Abstract ad Applied Aalysis (ii) Suppose that A k A(WS σ ) ad f is bouded, for each m,set G=sup f( k,m d(x,a σ k (m) ) + d (x, A) ). (2) Proof. Let {A k } [WV σ ].Thewecawrite S m = d(x,a σ k (m) ) d(x, A) 0 ( ) (25) Let ε>0ad select N ε such that {k : d(x,a σ k (m) ) d(x, A) ε 2 } < ε 2G, (22) for all m ad > N ε,adsetl(, m, x) = {k : d(x, A σ k (m) ) d(x, A) ε/2}.now,forallm ad >N ε,we have that f( d(x,a σ k (m) ) d(x, A) ) = f( d(x,a σ k (m) ) d(x, A) ) k L(,m,x) < + f( d(x,a σ k (m) ) d(x, A) ) k L(,m,x) ( ε 2G )G+ ( ε 2 )= ε 2 + ε 2 =ε. (23) Hece, {A k } is strogly ivariat coverget to A with respect to the modulus fuctio f. (iii) This is a immediate cosequece of (i) ad (ii). This completes the proof of the theorem. Theorem 8. Let f be a modulus fuctio. The [WV σ (f)] [WV σ (f)]. Proof. Suppose that, {A k } [WV σ (f)].thewecawrite f( d(x,a σ k (m) ) ) = f( d(x,a σ k (m) ) d(x, A) +d(x, A) ) f( d(x,a σ k (m) ) d(x, A) )+ f ( d (x, A) ) f( d(x,a σ k (m) ) d(x, A) )+M f (), (24) where M is a iteger such that d(x, A) < M. Therefore, {A k } [WV σ (f)]. Theorem 9. If f is a modulus fuctio ad {A k } is strogly ivariat coverget to A, the{a k } is strogly ivariat coverget to A with respect to the modulus f;thatis,[wv σ ] [WV σ (f)]. uiformly i m. Let ε>0ad choose δ with 0<δ<such that f(t) < ε for 0 t δ.writea km = d(x,a σ k (m) ) d(x,a), f(a km )=Σ +Σ 2, (26) where the first summatio is over a km δad secod over a km >δ.theσ εad, for a km >δ, f(a km )< a km δ <+[a km ], (27) δ where [z] deotes the iteger part of z. By defiitio modulus fuctio, we have, for a km >δ, f(a km ) (+[ a km δ ]) f () 2f() a km δ. (28) Hece, Σ 2 2f()(a km /δ), which together with Σ ε yields [WV σ ] [WV σ (f)]. Lemma 0 (see [5]). Let f be a modulus fuctio. Let α>0 be give costat. The, there is a costat c > 0 such that f(x)>cx(0<x<α). Theorem. Let {A k } be a bouded sequece ad f be a modulus fuctio. The {A k } is Wijsma strogly ivariat coverget to A with respect to modulus f ifadolyif{a k } is Wijsma strogly ivariat coverget to A;that is, [WV σ (f)] L = [WV σ ], (29) where L deotes the set of bouded sequeces of sets. Proof. TheproofofthetheoremfollowsfromTheorem 9 ad Lemma 0. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Refereces [] H. Fast, Sur la covergece statistique, Colloquium Mathematicae,vol.2,o.3-4,pp ,95. [2] T. Šalát, O statistically coverget sequeces of real umbers, Mathematica Slovaca,vol.30,o.2,pp.39 50,980. [3] G. Beer, Covergece of cotiuous liear fuctioals ad their level sets, Archiv der Mathematik, vol.52,o.5,pp , 989. [4] J.P.AubiadH.Frakowska,Set Valued Aalysis,Birkhäuser, Bosto, Mass, USA, 990.

5 Abstract ad Applied Aalysis 5 [5] M. Baroti ad P. L. Papii, Covergece of sequeces of sets, i Methods of Fuctioal Aalysis i Approximatio Theory,vol. 76, pp , Birkhäuser, Basel, Switzerlad, 986. [6] F. Nuray ad B. E. Rhoades, Statistical covergece of sequeces of sets, Fasciculi Mathematici, o.49, pp.87 99, 202. [7] U. Ulusu ad F. Nuray, Lacuary statistical covergece sequeces of sets, Progress i Applied Mathematics, vol. 4, o. 2, pp , 202. [8] N. Pacaroglu ad F. Nuray, O ivariat statistically covergece ad lacu-ary ivariat statistically covergece of sequeces of sets, Progress i Applied Mathematics, vol. 5, o. 2, pp , 203. [9] H. Nakao, Cocave modulars, Joural of the Mathematical Society of Japa,vol.5,o.,pp.29 49,953. [0] W. H. Ruckle, FK spaces i which the sequece of coordiate vectors is bouded, Caadia Joural of Mathematics, vol.25, pp , 973. [] I. J. Maddox, Sequece spaces defied by a modulus, Mathematical Proceedigs of the Cambridge Philosophical Society,vol. 00, o., pp. 6 66, 986. [2] J. Coor, O strog matrix summability with respect to a modulus ad statistical covergece, Caadia Mathematical Bulleti, vol. 32, o. 2, pp , 989. [3] F. Nuray ad E. Savaş, Some ew sequece spaces defied by a modulus fuctio, Idia Joural of Pure ad Applied Mathematics,vol.24,o.,pp ,993. [4] E. Savaş, Strogly σ-coverget sequeces, Bulleti of the Calcutta Mathematical Society,vol.8,o.4,pp ,989. [5] E. Savaş, O strog almost A-summability with respect to a modulus ad statistical covergece, Idia Joural of Pure ad Applied Mathematics,vol.23,o.3,pp ,992.

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