СОЈУЗ НА МАТЕМАТИЧАРИТЕ НА Р. МАКЕДОНИЈА МАТЕМАТИЧКИ БИЛТЕН BULLETIN MATHÉMATIQUE

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2 ISSN X СОЈУЗ НА МАТЕМАТИЧАРИТЕ НА Р. МАКЕДОНИЈА МАТЕМАТИЧКИ БИЛТЕН BULLETIN MATHÉMATIQUE КНИГА 38 (LXIV) TOME No. РЕДАКЦИСКИ ОДБОР Малчески Алекса(Македонија) Маркоски Ѓорѓи Секретар Bala Vladimir (Romaija) Čoba Mitrofa (Moldavija) Felloris Argyris (Grcija) Манова Ераковиќ Весна (Македонија) Musharov Oleg (Bugarija) Oberguggeberger Michael (Avstrija) Пилиповиќ Стефан (Србија) Scarpalezos Dimitrios (Fracija) Šreovsi Riste (Sloveija) Valov Veso (Kaada) Vidas Jasso (Belgija) COMITÉ DE RÉDACTION Malchesi Alesa (Macédoie) Marosi Gorgi Secrétaire Bala Vladimir (Roumaie) Čoba Mitrofa (Moldavie) Felloris Argyris (Grèce) Maova Eraovic Vesa (Macédoie) Musharov Oleg (Bugarie) Oberguggeberger Michael (Autriche) Pilipovic Stefa (Serbie) Scarpalezos Dimitrios (Frace) Šreovsi Riste (Slovéie) Valov Veso (Caada) Vidas Jasso (Belgique) СКОПЈЕ SKOPJE 04

3 МАТЕМАТИЧКИОТ БИЛТЕН е наследник на БИЛТЕНОТ НА ДРУШТВАТА НА МАТЕМАТИЧАРИТЕ И ФИЗИЧАРИТЕ ОД МАКЕ- ДОНИЈА. Објавува оригинални научни трудови од сите области на математиката и нејзините примени на македонски јазик или на едено од јазиците: англиски, француски, германски или руски. Излегува два пати годишно. Статиите за печатење, напишани во TEX или WORD (пожелно е во TEX), треба да бидат не поголеми од десет страни. Се доставува во електронска форма на Секоја статија треба да има апстракт, главен дел и резиме. Ако статијата е напишана на македонски јазик, тогаш резимето треба да биде на англиски јазик, ако пак статијата е напишана на еден од погоре наведените јазици, тогаш резимето треба да биде на македонски јазик. Во почетна фаза трудовите може да се достават и во печатена верзија, во два примероци на адреса МАТЕМАТИЧКИ БИЛТЕН Сојуз на математичари на Македонија Бул. Александар Македонски бб, ПФ 0, 000 Скопје Р. Македонија MATEMATIČKI BILTEN BULLETIN MATHÈMATIQUE is the successor of BILTEN NA DRUŠTVATA NA MATEMATIČARITE I FIZIČARITE OD MAKEDONIJA. It publishes origial papers of all braches of mathematics ad its applicatios. Cotributios are i Macedoia laguage or i Eglish, Frech, Russia or Germa. It is published twice a year. Papers should be writte i TEX or Word (preferably i TEX) ad should ot exceed about te pages. They should be submitted i electroic form o matbilte@gmail.com Every paper should cotai abstract, mai part ad summary. If the paper is writte i Macedoia, the summary should be writte i Eglish ad if the paper is writte i oe of the above metioed laguages, the the summary should be writte i Macedoia laguage. I the iitial phase the papers ca be submitted i two copies i prited versio too, o the followig address MATEMATIČKI BILTEN Sojuz a matematičari a Maedoija Bul. Alesadar Maedosi bb, PF 0, 000 Sopje Republic of Macedoia *** Издавањето на Математичкиот билтен го финансира Министерството за образование и наука

4 Математички Билтен ISSN X Vol. 38(LXIV) No. 04 (3) Скопје, Македонија С О Д Р Ж И Н А. Soja Čalamai ad Dočo Dimovsi ON CONTINUITY OF A (3,,ρ)- METRIC 5. Elida Hoxha, Kastriot Zoto ad Pada Sumati Kumari NONLINEAR CONTRACTIONS AND FIXED POINTS IN COMPLETE DISLOCATED AND b-dislocated METRIC SPACES 3. Elida Hoxha, Erem Aljimi ad Valdete Lou WEIGHTED NORLUND-EULER A-STATISTICAL CONVERGENCE FOR SEQUENCES OF POSITIVE LINEAR OPERATORS 4. Samoil Malčesi, Kateria Aevsa ad Risto Malčesi NEW CHARACTERIZATION OF -PRE-HILBERT SPACE Slagjaa Brsaosa ABOUT COMPATIBILITY OF THE VEKUA EQUATION WITH SOME OTHER EQUATIONS Lazo A. Dimov, Boro M. Piperevsi ad Elea Hadzieva ON THE EXACT SOLUTION ON A LINEAR DIFFERENTIAL EQUATION OF FIRST ORDER 5 7. Dhurata Valera ad Ivi Dylgjeri THE SPECIAL ROLE OF THE g -FUNCTIONS 57 3

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6 Математички Билтен ISSN X Vol. 38(LXIV) No. UDC: 55.4: (5-0) Скопје, Македонија ON CONTINUITY OF A (3,, ) METRIC Soja Čalamai, Dočo Dimovsi Abstract. A give (3,, ) metric d o a set M, iduces more tha oe topology o M. I geeral the map d from the third power of (M,) to the real umbers with the usual topology is ot cotiuous. I this paper we cosider oe of the topologies o M ad some additioal coditios that will imply the cotiuity of d.. INTRODUCTION The geometric properties, their axiomatic classificatio ad the geeralizatio of metric spaces have bee cosidered i a lot of papers. We will metio some of them: K. Meger ([]), V. Nemytzi, P.S. Alesadrov ([3], []), Z. Mamuzic ([0]), S. Gähler ([8]), A. V. Arhagelsii, M. Choba, S. Nedev ([], [3], [4]), R. Kopperma ([9]), J. Usa ([5]), B. C. Dhage, Z. Mustafa, B. Sims ([5], []). The otio of ( m,, ) metric is itroduced i [6]. Coectios betwee some of the topologies iduced by a (3,, ) metric d ad topologies iduced by a pseudo-o-metric, o-metric ad symmetric are give i [7]. For a give (3,, ) metric d o set M, j {,}, seve topologies (G,d), (H,d), (D,d), (N,d), (W,d), (S,d) ad (K,d) o M, iduced by d, are defied i [4], ad several properties of these topologies are show. I this paper we cosider oly the topology ( Gd, ) iduced by a (3,, ) metric d. For ( Gd, ), we will state two coditios for d ad show that these coditios imply the cotiuity of d, as a map from the third power of the topological space ( M, ) to the real umbers with the usual topology. We recall the basic otios. Let M be a oempty set, ad let coditios for such a map. ( M 0) d( x, x, x) 0, for ay xm; 3 d : M R0 [0, ). ( P ) d( x, y, z) d( x, z, y) d( y, x, z) for ay x, y, z M; ad M d x, y, z d x, y, a d x, a, z d a, y, z, for ay x, y, z, a M. We state three 00 Mathematics Subject Classificatio. 54A0, 54E35, 54E99 Key words ad phrases. (3,, ) metric, (3,, ) metrizable spaces, cotiuity of (3,, ) metric d 5

7 6 S. Čalamai, D. Dimovsi For a map d as above let 3 {( x, y, z) ( x, y, z) M, d( x, y, z) 0}. The set is a (3,) -equivalece o M, as defied ad discussed i [6], [4]. The sets {( x, x, x) x M} ad {( x, x, y) x, y M} are (3,)-equivaleces o M. The coditio ( M 0) implies that. 3 Defiitio. Let d : M R0 ad be as above. If d satisfies ( M 0), ( P ) ad ( M ) we say that d is a (3,, ) metric o M. Let d be a (3,, ) metric o M, x, y M ad 0. As i [4], we cosider the followig ball, as subset of M : B( x, y, ) { z zm, d( x, y, z) } - -ball with ceter at ( xy, ) ad radius. Amog the others, a (3,, ) metric d o M iduces the topology ( Gd, ) - geerated by all the -balls B( x, y, ), i.e. the topology whose base is the set of the fiite itersectios of -balls B( x, y, ), (see as [4]).. CONTINUITY OF A (3,, ) METRIC d FOR ( Gd, ) Propositio. Let d be a (3,, ) metric o M, let ( Gd, ), ad let d satisfies the followig two coditios: (A) For each x, x, x 3 of M there is permutatio i, i, i 3 of,,3 such that d( x, x, x ) d( x, x, x ) d( x, x, x ) 0 ad i i i i i i3 i i3 i3 (B) For each two poits uv, of M ad each 0 there are ope sets Uu, Uv such that u U u, v Uv ad for each x Uu ad y U v : d( u, x, y) ad d( v, x, y). The, the (3,, ) metric d is a cotiuous fuctio. Proof. Let u, v, t be poits of M ad 0. Usig (A), w.l.o.g. we ca set x u, x t ad x3 v. Thus: d( u, u, t) d( u, u, v) d( t, v, v) 0. () For uv, of M ad 0, the coditio (B) implies that there are ope eighborhoods Uu, U v of u ad v, such that for each x of U u ad each y of U v we have d( x, y, u) / 6 ad d( x, y, v) / 6. () Let U u ad U v be the ope sets defied by: Uu B( u, v, / 6) B( u, t, / 6) Uu ad Uv B( t, v, / 6 ) Uv. (3)

8 O cotiuity of a (3,,ρ)-metric 7 The, () implies that of u U u ad v U v. This together with (3), implies that for each x U u ad each y of U v, we have d( u, v, x) / 6 d( u, t, x) / 6ad d( t, v, y) / 6. (4) For ut, of M ad 0, the coditio (B) implies that there are ope eighborhoods U u of u ad U t of t such that for each x of U u ad each z of U t we have d( u, x, z) / 6 ad d( t, x, z) / 6. (5) For tv, of M ad 0, the coditio (B) implies that there are ope eighborhoods U t of t ad Let U v of v such that for each z of ' u u u u, U U U U these ope sets implies that U t ad each y of d( t, z, y) / 6 ad d( v, z, y) / 6. ' v v Uv v ad U U U ' u U u, ' v U v ad U v we have (6) ' Ut Ut Ut. The costructio of ' t U t. Moreover, for each x of ' U u, y ' ' of U v ad z of U t, usig (), (4), (5) ad (6), ad the tetrahedral iequality (M) several times we obtai the followig iequalities: d( u, t, v) d( u, t, x) d( u, x, v) d( x, t, v) / 6 / 6 d( x, t, v) / 3 d( x, t, y) d( x, y, v) d( y, t, v) / 3 d x, t, y / 6 / 6 / 3 d( x, t, z) d( x, z, y) d( z, t, y) / 3 / 6 d( x, z, y) / 6 d( x, z, y), d( x, z, y) d( x, z, t) d( x, t, y) d( t, z, y) / 6 / 6 d( x, t, y) / 3 d( x, t, v) d( x, v, y) d( v, t, y) / 3 d( x, t, v) / 6 / 6 / 3 d( x, t, u) d( x, u, v) d( u, t, v) / 3 / 6 d( u, t, v) / 6 d( u, t, v). Next, (7) ad (8) imply that: d( u, t, v) d( x, z, y). All this shows that d is a cotiuous fuctio from 3 M to R with the usual topology. With the ext example we show the existece of a cotiuous (3,, ) metric d satisfyig the coditio (A), but ot satisfyig the coditio (B) as i Propositio. Example. Let M ( p) ( p) whereas ( p ) ad ( p ) are parallel lies ad let 3 d : M R0 be defied by: d x, y, z 0, x y z or x y z x ad x, y, z ( p ),,, i other cases 0, x, y ( p ),, or x ( p), y ( p) d( x, x, y), x ( p), y ( p). (7) (8)

9 8 S. Čalamai, D. Dimovsi It is easy to show that d is a (3,, ) metric o M with For x y M {( x, x, y) x ( p ), y ( p ) or x, y ( p ),,} {( x, y, z) x y z xad x, y, z ( p ),,}. ad 0, ad for xy M ad 0 ( p), x y, x, y ( p),,, B( x, y, ) { x}, x y, x ( p), y ( p), M,, ( p), x ( p), B( xx,, ) M, x ( p ),. From this it follows that ( G, d) D( p ) { ( p) V V ( p )} where D is the discrete topology o ( p ). First we show that the (3,, ) metric d satisfies the coditio (A). - If x, y, z ( p ),,, the d( x, x, y) d( x, x, z) d( y, z, z) 0. - If x, y ( p), z ( p), the d( y, y, x) d( y, y, z) d( x, z, z) 0. - If x ( p ), y, z ( p), the d( x, x, y) d( x, x, z) d( y, z, z) 0. - If x z ( p ), y( p ), the d( x, x, x) d( x, y, y) 0 ad d ( y, y, y ) d ( y, y, x ) 0. We will show that the (3,, ) metric d is a cotiuous fuctio. For each xy, of M we defie the map fxy, : M R by fxy, ( z) d( x, y, z)., ( p ) Let U be a ope set i R with the usual topology, such that U ad 0 U. The We cosider the followig cases: - if x y ad x, y ( p ), the - if x y ad x, y ( p), the fx, y( U) { z fx, y( z) U} { z d( x, y, z) }. fxy, ( U) ( p), fxy, ( U) ( p ), - if x y ad x ( p ), y ( p ), the fxy, ( U) ( p ) \{ x} ( p ), - if x y ad x ( p ), the - if x y ad x ( p), the xx f, ( U), xx, f ( U) ( p ). Let V be a ope set i R such that 0 V, V. The We cosider the followig cases: - if if x y ad x, y ( p ), the fx, y( V ) { z fx, y( z) V} { z d( x, y, z) 0}. fxy, ( V ) ( p),

10 O cotiuity of a (3,,ρ)-metric 9 - if x y ad x, y ( p), the fxy, ( V ) ( p), - if x y ad x ( p ), y ( p ), the fxy, ( V ) { x}, - if x y ad x ( p ), the xx f, ( V ) M, - if x y ad x ( p), the fxx, ( V ) ( p). Let W be a ope set i R such that 0, W. The fx, y( W) { z fx, y( z) W} { z d( x, y, z) 0 or d( x, y, z) } M. All this implies that d is a cotiuous fuctio. Next, we show that d does ot satisfy the coditio (B) from the above propositio. Let u, v M, ad let u( p ), v ( p) ad 0. For each ope eighborhoods U u of u ad v of, U v for x u ad each y v we have d( u, x, y) d( u, u, v) 0 ad d( v, x, y) d( u, v, v). Hece, the coditio (B) is ot satisfied. Refereces [] П.С.Александров, В.В.Немыцкий, Условия метризуемости топологических пространств и аксиома симетрии, Мат. сб. 3:3 (938), [] А.В. Архангельский, О поведении метризуемости при факторных тображениях, ДАН 64, Nо (965), [3] М. Чобан, О метрисуемыих пространствах, Вестн.Моск. Ун-та, сер.матем., мех, Nо 3 (959), [4] S. Čalamai, D. Dimovsi, Topologies iduced by (3,,ρ)-metrics ad (3,,ρ)- metrics, Ieratioal mathematical forum, Vol.9, o.-4, (04) [5] B.C. Dhage, Geeralized metric spaces ad topological structure II, Pure Appl.Math. Sci., 40 (-) (994), 37-4 [6] D. Dimovsi, Geeralized metrics - (,m,r)-metrics, Mat. Bilte, 6, Sopje (99), [7] D. Dimovsi, (3,,ρ)-metrizable topological spaces, Math. Macedoica, 3 (005), [8] S. Gähler, -metrische räume ud ihre topologische Strutur, Math. Nachr. 6 (963), 5-48 [9] R. Kopperma, All topologies come from geeralized metrics, A. Math. Moth. V 95 i, [0] Z. Mamuziċ, Abstract distace ad eighborhood spaces, Proc. Prague Symp. (96), 6-66 [] K. Meger, Utersuchuge über allgemeie Metri, Math. A. 00 (98), 75-63

11 0 S. Čalamai, D. Dimovsi [] Z. Mustafa, B. Sims, A ew approach to geeralized metric spaces, Joural of Noliear ad Covex Aalysis, Vol. 7, Number (006), [3] V. Nemytzi, O the third axiom of metric spaces, Tr. Amer. Math. Soc. 9 (97), [4] С. Недев, o-метризуемыe пространства, Тр. Моск. Мат. Общ. Том 4 (97), 0-36 [5] J. Usa, <Nm,E>-seti s (+)-rastojaiem, Review of Research, PMF, Novi Sad, Ser. Mat. 7 (989),, ) Faculty of techical scieces, Uiv. St. Klimet Ohridsi, Bitola, Macedoia address: scalamai@yahoo.com ) Faculty of Natural Scieces & Mathematics, Ss. Cyril ad Methodius, Sopje, Macedoia address: doco@pmf.uim.m

12 Математички Билтен ISSN X Vol. 38(LXIV) No. UDC: : (-9) Скопје, Македонија NONLINEAR CONTRACTIONS AND FIXED POINTS IN COMPLETE DISLOCATED AND b-dislocated METRIC SPACES Elida Hoxha, Kastriot Zoto ad Pada Sumati Kumari 3 Abstract. I this paper, we cotiue the study of complete dislocated ad b-dislocated metric spaces ad established some commo fixed poit theorems for oe ad two mappigs. Our results geeralizes ad exted some existig results i the literature i a class effectively larger such as b-dislocated metric spaces, where the self distace for a poit may ot be equal to zero.. INTRODUCTION The cocept of b -metric space was itroduced by Bahti [4] ad extesively used by Czerwi i [0]. After that, several iterestig results about the existece of a fixed poit for sigle-valued ad multi-valued operators i b -metric spaces have bee obtaied. Recetly there are a umber of geeralizatios of metric space. Some of them are the otios of dislocated metric spaces ad b -dislocated metric spaces where the distace of a poit i the self may ot be zero. These spaces was itroduced ad studied by Hitzler ad Seda [5], Nawab Hussai et.al [7]. Also i [7] are preseted some topological aspects ad properties of b -dislocated metrics. Subsequetly, several authors have studied the problem of existece ad uiqueess of a fixed poit for sigle-valued ad set-valued mappigs ad differet types of cotractios i these spaces. The purpose of this paper is to uify ad geeralize some recet results i the settig of dislocated ad b -dislocated metric spaces usig a class of cotiuous fuctios G 4.. PRELIMINARIES Defiitio. [6]. Let X be a oempty set ad a mappig d : X X [0, ) is called a dislocated metric (or simplyd l -metric) if the followig coditios hold for ay x, y, z X : i. If dl ( x, y) 0, the x y ii. d ( x, y) d ( y, x) l l 00 Mathematics Subject Classificatio. Primary: 47H0 Secodary: 55M0 Key words ad phrases. dislocated metric, b -dislocated metric, fixed poit, cotractio. l

13 E. Hoxha, K. Zoto, P. S. Kumari iii. d ( x, y) d ( x, z) d ( z, y) l l l The pair ( Xd, l ) is called a dislocated metric space (or d -metric space for short). Note that whe x y, dl ( x, y ) may ot be 0. Example.. If X R, the d( x, y) x y defies a dislocated metric o X. Defiitio.3 [6]. A sequece ( x ) i dl -metric space ( Xd, l ) is called: () a Cauchy sequece if, for give 0, there exists 0 N such that for all m, 0, we have dl ( xm, x ) or m, lim d ( x, x ) 0, l m () coverget with respect to d l if there exists x X such that dl( x, x) 0 as. I this case, x is called the limit of ( x ) ad we write x x. A dl -metric space X is called complete if every Cauchy sequece i X coverges to a poit i X. Defiitio.4[8]. Let X be a oempty set ad a mappig b : X X [0, ) is called a b -dislocated metric (or simply coditios hold for ay x, y, z X ad s : a. If bd ( x, y) 0, the x y, b. b ( x, y) b ( y, x), d d c. bd ( x, y) s[ bd ( x, z) bd ( z, y)]. d bd -dislocated metric) if the followig The pair ( Xb, d ) is called a b -dislocated metric space. Ad the class of b -dislocated metric space is larger tha that of dislocated metric spaces, sice a b -dislocated metric is a dislocated metric whe s. I [8] was showed that each bd -metric o X geerates a topology b d whose base is the family of ope bd -balls Bb ( x, ) { y X : bd( x, y) } d Also i [8] are preseted some topological properties of bd -metric spaces Defiitio.5. Let ( Xb, d ) be a bd -metric space, ad ( x ) be a sequece of poits i X. A poit x X is said to be the limit of the sequece ( x ) if lim bd( x, x) 0 ad we say that the sequece ( x ) is bd -coverget to x ad deote it by. x x as The limit of a bd -coverget sequece i a bd -metric space is uique [8, Propositio.7].

14 Noliear cotractios ad fixed poits i complete dislocated... 3 Defiitio.6. A sequece ( x ) i a bd -metric space ( Xb, d ) is called a bd -Cauchy sequece iff, give 0, there exists 0 N such that for all, m 0, we have bd ( x, xm) or lim b ( x, x ) 0, m d m space is a bd -Cauchy sequece.. Every bd -coverget sequece i a bd -metric Remar.7. The sequece ( x ) i a bd -metric space ( Xb, d ) is called a bd -Cauchy sequece iff lim bd ( x, x p) 0 for all p N m, Defiitio.8. A bd -metric space ( Xb, d ) is called complete if every bd -Cauchy sequece i X is bd -coverget. I geeral a bd -metric is ot cotiuous, as i Example.3 i [8] showed. Example.9. Let X R {0} ad ay costat 0. Defie the fuctio dl : X X [0, ) by dl ( x, y) ( x y). The, the pair ( Xd, l ) is a dislocated metric space. Lemma.0. Let ( Xb, d ) be a b -dislocated metric space with parameter s. Suppose that ( x ) ad ( y ) are bd -coverget to x, y X, respectively. The we have b (, ) lim if (, ) lim sup (, ) (, ) d x y bd x y bd x y s bd x y s I particular, if bd ( x, y) 0, the we have lim bd ( x, y) 0 bd ( x, y). Moreover, for each z X, we have bd ( x, z) lim if bd ( x, z) lim sup bd ( x, z) sbd ( x, z) s I particular, if bd ( x, z) 0, the we have lim bd ( x, z) 0 bd ( x, z). Some examples i the literature shows that i geeral a b -dislocated metric is ot cotiuous. Example.. If X R {0}, the o X with parameter s. bd ( x, y ) ( x y ) defies a b -dislocated metric 3. MAIN RESULT We cosider the set G 4 of all cotiuous fuctios followig properties: 4 g :[0, ) [0, ) with the

15 4 E. Hoxha, K. Zoto, P. S. Kumari a) g is o-decreasig i respect to each variable b) g( t, t, t, t) t, t [0, ) Some examples of these fuctios are as follows: g : g( t, t, t3, t4) max{ t, t, t3, t4} g : g( t, t, t3, t4) max{ t t, t t3, t t3, t3 t4} g : g( t, t, t, t ) [max{ t t, t t, t t, t t }] p p p p g : g( t, t, t, t ) [max{ t, t, t, t }], p 0. Theorem 3.. Let Xd, be a complete b -dislocated metric space with parameter s ad T, S : X X two mappigs satisfyig the followig cotractive coditio (, ) [ (, ), (, ), (, ), d( x, Sx) d( y, Ty) sd Sx Ty c g d x y d x Sx d y Ty d( x, y) ] () for all x, y X where g G4 ad 0c. The T ad S have a uique commo fixed poit ad if u is a commo fixed poit of S ad T, the d( u, u) 0. Proof. Let x 0 be a arbitrary poit i X. Defie the sequece ( x ) as follows: x S( x0), x T( x),..., x T( x), x S( x),... if we assume that for some N, x x the x x Sx ad also usig the cotractive coditio of theorem we will have that x x is a fixed poit of T. Thus we assume that for N, x x. By coditio () we have: p sd( x, x) sd( Sx, Tx) d ( x, Sx ) d ( x, Tx) cg[ d( x, x), d( x, Sx ), d( x, Tx), ] d ( x, x) d ( x, x) d ( x, x) cg[ d( x, x), d( x, x), d( x, x), d ( x, x ) ] cd( x, x). Thus Similarly by coditio () have: sd( x, x) sd( Tx, Sx ) sd( Sx, Tx) Thus c s () d( x, x ) d( x, x ) cg d x x d x Sx d x Tx d ( x, Sx ) d ( x, Tx ) [ (, ), (, ), (, ), ] d ( x, x) d ( x, x) d ( x, x ) cg[ d( x, x), d( x, x), d( x, x ), ] d ( x, x ) cd( x, x). c s. (3) d( x, x ) d( x, x )

16 Noliear cotractios ad fixed poits i complete dislocated... 5 Geerally by coditios (), (3) ad deotig Sice 0 c, we have s d( x, x) d( x, x)... d( x0, x), taig limit for we have for. d( x, ) 0 x. (4) m with Now, we prove that ( x ) is a b d -Cauchy sequece, ad to do this let be, 0 m, ad usig defiitio.4 (c) we have bd ( x, xm) s[ bd ( x, x) bd ( x, xm)] 3 sbd ( x, x) s bd ( x, x) s bd ( x, x3)... 3 s bd ( x0, x) s bd ( x0, x) s bd ( x0, x)... 3 s b d x0, x [ s ( s) ( s)...] s b ( d x0, x). s O taig limit for mwe, have bd ( x, xm) 0 as s. Therefore ( x ) is a bd -Cauchy sequece i complete b-dislocated metric space ( Xb, d ). So there is some u X such that ( x ) dislocated coverges to u. Therefore the subsequeces { Sx } u ad { Tx} u. Sice T, S : X X are cotiuous mappigs we get: Su u ad Tu u. Thus, u is a commo fixed poit of T ad S. If cosider that T is cotiuous ad S ot cotiuous we have that Tu u. Usig the cotractive coditio of theorem we have, d ( u, Su) d ( x, Tx) d ( u, x) d( u, Su) d ( x, x) cg d u x d u Su d x Tx d( u, x) sd( Su, Tx ) cg[ d( u, x ), d( u, Su), d( x, Tx ), ] [ (, ), (, ), (, ), ]. Taig i upper limit as, usig lemma.0, property of g ad result (4) we get s d( u, Su) cg[0, d( u, Su),0,0]. s This iequality implies d( u, Su) cd( u, Su) that meas d( u, Su) 0. Thus Su u ad u is a fixed poit of S. If cosider (c) we have that, u is a commo fixed poit of S ad T. Usig the cotractive coditio of theorem, we obtai sd( u, u) sd( Su, Tu) d ( u, u) d ( u, u) d( u, u) cg[ d( u, u), d( u, u), d( u, u), ] cd( u, u). The iequality above implies that d( u, u) d( u, u). So d( u, u) 0, sice 0 c s. Uiqueess. Let suppose that u ad v are two commo fixed poits of TS. ; From coditio () we have:

17 6 E. Hoxha, K. Zoto, P. S. Kumari Replacig v u i (5) we get: sd( u, v) sd( Su, Tv) d ( u, Su) d ( v, Tv) cg[ d( u, v), d( u, Su), d( v, Tv), ] d ( u, v) d ( u, u) d ( v, v) cg[ d( u, v), d( u, u), d( v, v), ]. d ( u, v) (, ) [ (, ), (, ), (, ), d( u, u) d( u, u) sd u u cg d u u d u u d u u d( u, u) ] cd ( u, u ), c s replacig u v i (5), we obtai d( v, v) 0. Agai from (5) have d( u, v) d( u, v) i.e. d( u, u) d( u, u) d( u, u). Sice 0 we obtai d( u, u) 0. Similarly sice 0 get d( u, v) 0, which implies u v. Thus fixed poit is uique. (5) Corollary 3.. Let ( Xd, ) be a complete b -dislocated metric space with parameter s ad T, S : X X two mappigs satisfyig the followig cotractive coditio sd( Sx, Ty) c g[ d( x, y), d( x, Sx), d( y, Ty)] for all x, y X where g G3 ad 0c. The T ad S have a uique commo fixed poit ad if u is a commo fixed poit of S ad T, the d( u, u) 0. Corollary 3.3. Let ( Xd, ) be a complete dislocated metric space ad T, S : X X two mappigs satisfyig the followig cotractive coditio (, ) [ (, ), (, ), (, ), d( x, Sx) d( y, Ty) d Sx Ty c g d x y d x Sx d y Ty d( x, y) ], for all x, y X where g G4 ad 0c. The T ad S have a uique commo fixed poit ad if u is a commo fixed poit of S ad T, the d( u, u) 0. The followig example supports our theorem. Example 3.4. Let X [0,] ad d( x, y) x y, for all x, y X. It is clear that d is a dislocated metric o X. We defie the self mappigs S, T : X X as follows x, x[0,) 8 5 x, x[0,) Sx ad Tx 6, x 3, x. Note that S ad T are discotiuous maps. Now we will show that the cotractive coditio of 3.3 is satisfied for costat c [0,) ad taig the fuctio g( t, t, t, t ) max{ t, t, t, t } We have the followig cases: Case. Note that for all xy, [0,), we have y y x x Case. Note that for x y, we have d( Sx, Ty) d(, ) ( x y) d( x, y)

18 Noliear cotractios ad fixed poits i complete dislocated Case 3. for x [0,) ad y, we have d( Sx, Ty) d( S, T) d(, ) d( x, y). x x Case 4. For all y [0,) ad x, we have d( Sx, Ty) d(, ) ( x ) d( x, y). y y 56y d( Sx, Ty) d(, ) ( y) d( x, y). Thus all coditios of corollary 3.3 are satisfied ad x 0 is a uique commo fixed poit of S ad T. Also we ote that this theorem is ot available i a usual metric space if d( x, y) x y ad i b -metric space x y we will have because if cosider poits d( x, y) x y d( S, T) cd(,) 0 d( S, T) ( ) cd(,) So the cotractive coditio is failed i two cases. Corollary 3.5. Let ( Xd, ) be a complete dislocated metric space ad S : X X a selfmappig satisfyig the followig cotractive coditio (, ) [ (, ), (, ), (, ), d( x, Sx) d( y, Sy) d Sx Sy c g d x y d x Sx d y Sy d( x, y) ] for all x, y X where g G4 ad 0c. The, S has a uique fixed poit ad d u, u 0 Example 3.6. Let 0,0 d is a X ad d( x, y) ( x y), for all x, y X. It is clear that dislocated metric o X ad ( Xd, ) is complete. Also d is ot a metric o X. We defie the self-mappig S : X X by x, x0 Sx 0, x 0 ad tae the fuctio g( t, t, t3, t4) max{ t, t, t3, t4} ad also choose the costat c 9. For xy, {0,,...,0}, we have the followig cases. 0 Case. For x y 0 have d( Sx, Sy) d(0,0) 0 Case. If x y 0, the Case 3. If x y 0, the d( Sx, Sy) d( x, x ) x x d( x, y). 9 x d( Sx, Sy) d( x,0) ( x ) d( x, y).

19 8 E. Hoxha, K. Zoto, P. S. Kumari Case 4. If x y 0, the d( Sx, Sy) d( x, y ) ( x y ) 9 ( x y) 9 d( x, y). 0 0 Thus all coditios of theorem are satisfied ad S has a uique fixed poit i X. Also we ote that for x ad y 0 the cotractive coditio is failed i the usual metric. Theorem 3.7. Let ( Xd, ) be a complete b -dislocated metric space ad T, S : X X two self-mappigs satisfyig the coditio: sd( Sx, Ty) c max{ d( x, y) d( x, Sx), d( x, Sx) d( y, Ty), d( x, Sx) d( y, Ty) d( x, y) d( x, y) d( y, Ty), d( y, Ty) } for all x, y X ad 0 c. The T ad S have a uique commo fixed poit i X. Proof. This theorem is corollary of theorem 3. if we use the fuctio g G4. Theorem 3.8. Let ( Xd, ) be a complete b -dislocated metric space ad T, S : X X two self mappigs satisfyig the coditio: p p p p p d( x, Sx) d( y, Ty) p s d ( Sx, Ty) c max{ d ( x, y), d ( x, Sx), d ( y, Ty),( ) }, d( x, y) for all x, y X ad 0c.The T ad S have a uique commo fixed poit i X. Proof. This theorem is tae as a corollary of theorem, if we use the fuctio g4 G4. Theorem 3.9. Let ( Xd, ) be a complete b -dislocated metric space ad T, S : X X two self-mappigs satisfyig the coditio: (, ) max{ (, ) (, ), (, ) (, ), (, ) (, ), (, ) d( x, Sx) d( y, Ty) s d Sx Ty c d x y d x Sx d x Sx d y Ty d x y d y Ty d y Ty d( x, y) } for all x, y X ad 0 c. The T ad S have a uique commo fixed poit i X. Proof. This theorem is corollary of theorem, if we use the fuctio g 3 G 4. Remar 3.0. Results of the above theorems ad corollaries are exteded ad uified of some classical fixed poit results i metric spaces ad geeralizatio of results of the authors [,,9,0,8,9] ad other results i dislocated metric spaces. Refereces [] C. T. Aage, J. N. Salue, The results o fixed poits i dislocated ad dislocated quasimetric space, Appl. Math. Sci.,(59), (008), [] C. T. Aage, J. N. Salue, Some results of fixed poit theorem i dislocated quasi-metric spaces, Bulleti of Marathwada Mathematical Society, 9(008),-5

20 Noliear cotractios ad fixed poits i complete dislocated... 9 [3] A. Beiravad, S. Moradi, M. Omid, H. Pazadeh, Two fixed poit theorems for special mappig, arxiv: v [math.fa]. [4] I. A. Bahti, The cotractio mappig priciple i quasimetric spaces, Fuct. Aal., Uiaows Gos. Ped. Ist. 30, (989), 6-37 [5] P. Hitzler, A. K. Seda, Dislocated topologies, J. Electr. Egi, 5(/S):3:7, 000. [6] R. Shrivastava, Z. K. Asari ad M. Sharma, Some results o Fixed Poits i Dislocated ad Dislocated Quasi-Metric Spaces, Joural of Advaced Studies i Topology, Vol. 3, No., (0) [7] N. Hussai, J. R. Rosha, V. Parvaeh ad M. Abbas, Commo fixed poit results for wea cotractive mappigs i ordered b-dislocated metric spaces with applicatios, Joural of iequalities ad Applicatios, /486, (03) [8] M. A. Kutbi, M. Arshad, J. Ahmad, A. Azam, Geeralized commo fixed poit results with applicatios, Abstract ad Applied Aalysis, volume 04, article ID 36395, 7 pages [9] K. Zoto, E. Hoxha, Fixed poit theorems i dislocated ad dislocated quasi-metric space, Joural of Advaced Studies i Topology; Vol. 3, No., (0). [0] Czerwi, S: Cotractio mappigs i b-metric spaces. Acta Math. Iform. Uiv. Ostrav., 5- (993) [] L. B. Ciric, A geeralizatio of Baach s cotractio priciple, Prooceedigs of the America Mathematical Society, vol. 45, (974), [] K. M. Das, K. V. Nai, Commo fixed poit theorems for commutig maps o metric spaces. Proc Am Math Soc., 77, (979), [3] M. Arshad, A. Shoaib, I. Beg, Fixed poit of a pair of cotractive domiated mappigs o a closed ball i a ordered dislocated metric space, Fixed poit theory ad applicatios, vol. 03, article 5, 03 [4] M. A. Alghmadi, N. Hussai, P. Salimi, Fixed poit ad coupled fixed poit theorems o b-metric-lie spaces, Joural of iequalities ad applicatios, vol. 03, article 40, 03 [5] M. Arshad, A. Shoaib, P. Vetro; Commo fixed poits of a pair of Hardy Rogers type mappigs o a closed ball i ordered dislocated metric spaces, Joural of fuctio spaces ad applicatios,vol 03, article id 6388 [6] R. Yijie, L. Julei, Y. Yarog, Commo fixed poit theorems for oliear cotractive mappigs i dislocated metric spaces, Abstract ad Applied Aalysis vol 03, article id [7] K. Zoto, P. S. Kumari, E. Hoxha. Some Fixed Poit Theorems ad Cyclic Cotractios i Dislocated ad Dislocated Quasi-Metric Spaces, America Joural of Numerical Aalysis,.3 (04), [8] M. Kir, H. Kiziltuc, O Some Well Kow Fixed Poit Theorems i b-metric Spaces, Turish Joural of Aalysis ad Number Theory,. (03), 3-6. [9] M. P. Kumar, S. Sachdeva, S. K. Baerjee, Some Fixed Poit Theorems i b-metric Space, Turish Joural of Aalysis ad Number Theory. (04), 9-. ) Faculty of Natural Scieces, Uiversity of Tiraa, Tiraa, Albaia ) Faculty of Natural Scieces, Uiversity of Gjiroastra, Gjiroastra, Albaia address: zotoastriot@yahoo.com 3) Departmet of Mathematics, K L Uiversity, Gree Fields, A. P, Idia

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22 Математички Билтен ISSN X Vol. 38(LXIV) No. UDC: 57.5: (-33) Скопје, Македонија WEIGHTED NORLUND-EULER A-STATISTICAL CONVERGENCE FOR SEQUENCES OF POSITIVE LINEAR OPERATORS Elida Hoxha, Erem Aljimi ad Valdete Lou 3 Abstract. We itroduce the otio of weighted Norlud Euler A-Statistical Covergece of a sequece, where A represets the oegative regular matrix. We also prove the Korovi approximatio theorem by usig the otio of weighted Norlud-Euler A-statistical covergece. Further, we give a rate of weighted Norlud-Euler A-statistical covergece.. BACKGROUND, NOTATIONS AND PRELIMINARIES Suppose that E N {,, } ad E { : E}. The E lim E is called the atural desity of E provided that the limit exist, where. represets the () umber of elemets i the eclosed set. The term statistical covergece was first preseted by Fast [] which is geeralizatio of the cocept of ordiary covergece. Actually, a root of the otio of statistical covergece ca be detected by Zygmud [] (also see [3]), where he used the term almost covergece which tured out to be equivalet to the cocept of statistical covergece. The otio of Fast was further ivestigated by Schoeberg [4], Salat [5], Fridy [6], ad Coer [7]. The followig otio is due to Fast []. A sequece x ( x ) is said to be statistically coverget to L if ( K ) 0 for every 0, where equivaletly, K { N : x L } () lim { : x L } 0. (3) I symbol, we will write S lim x L. We remar that every coverget sequece is statistically coverget but ot coversely. Let X ad Y be two sequece spaces ad let A ( a, ) be a ifiite matrix. If for each x ( x ) i X the series 00 Mathematics Subject Classificatio. Primary: 40G05,40G5 Key words ad phrases. A-statistical covergece, oegative regular matrix

23 E. Hoxha, E. Aljimi, V. Lou A x a, x a, x (4) coverges for each N ad the sequece Ax A x belogs to Y, the we say the matrix A maps X to Y. By the symbol ( XY, ) we deote the set of all matrices which map X ito Y. A matrix A (or a matrix map A ) is called regular if A ( c, c), where the symbol c deotes the spaces of all coverget sequeces ad lim Ax lim x (5) for all x c. The well-ow Silverma-Toeplitz theorem (see [8]) assert that A ( a, ) is regular if ad oly if i) lim a, 0 for each ; ii) lim a, ; iii) sup a,. Kol [9] exteded the defiitio of statistical covergece which the help of oegative regular matrix A ( a, ) callig it A -statistical covergece. The defiitio of A -statistical covergece is give by Kol as follows. For ay oegative regular matrix A, we say that a sequece is A -statistically coverget to L provided that for every 0 we have lim a, L (6) : x L I 009, the cocept of weighted statistical covergece was defied ad studied by Karaaya ad Chishti [0] ad further modified by Mursalee et al. [] i 0. I 03, Bele ad Mohiuddie [] preseted a geeralizatio of this otio through de la Vallee-Poussi mea i probabilistic ormed spaces. Let x be a give ifiite series with sequece of its 0 ( E,) trasform is defied as ad we say that this summability method is coverget if case we say the series Ad we will write S S E, E th partial sum S. If ( ) S (7) 0 E S as. I this x is ( E,) summable to a defiite umber S. (Hardy [3]). 0 as. Let ( p ) ad ( q ) be the two sequeces of o-zero real costats such that

24 Weighted Norlud-Euler A-Statistical Covergece for.. 3 P p p p P p 0, 0 0, 0 Q q q q Q q For the give sequeces ( p ) ad ( q ), covolutio p* q is defied by: *. (8) 0 R p q p q The series x or the sequece S is summable to S by geeralized Norlud 0 method ad it is deoted by S S( N, p, q) if p, q t R pvqvsv (9) v 0 teds to S as. Let us use i cosideratio the followig method of summability: If p, q, E R R v 0 0 v0 (0) t p q E p q ( ) S v p, q, E t S as, the we say that the series x or the sequece S is 0 summable to S by Norlud-Euler method ad it is deoted by S S N, p, q ( E,). Remar. If p, q, the we get Euler summability method. Now we are able to give the defiitio of the weighted statistical covergece related to the ( N, p, q)( E,) summability method. We say that E have weighted desity, deoted by NE ( E ), if NE E lim { : } R R E. () A sequece x ( x ) is said to be weighted Norlud-Euler statistical coverget (or S coverget) if for every 0 : NE R I these case we write lim v v () v0 lim { R : p q ( ) x L } 0 L SNE st x. I the other had, let us recall that C[ a, b ] is the space of all fuctios f cotiuous o [ ab, ]. We ow that f C[ a, b] is Baach spaces with orm f sup f ( x), f C[ a, b] (3) x[ a, b]

25 4 E. Hoxha, E. Aljimi, V. Lou Suppose that L is a liear operator from C[ a, b ] ito C[ a, b ]. It is clear that if f 0 implies Lf 0, the the liear operator L is positive o C[ a, b ]. We deote the value of Lf at a poit x [ a, b] by L( f ; x ). The classical Korovi approximatio theorem states the followig [4]. Theorem. Let ( T ) be a sequece of positive liear operators from C[ a, b ] ito C[ a, b ]. The, for all C[ a, b ] if oly if i where f x x i ad i 0,,. lim T ( f ; x) f ( x) 0 (4) lim T fi; x fi( x) 0 (5) May mathematicias exteded the Korovi-type approximatio theorems by usig various test fuctios i several setups, icludig Baach spaces, abstract Baach lattices, fuctio spaces, ad Baach algebras. Firstly, Gadjiev ad Orha [5] established classical Korovi theorem through statistical covergece ad display a iterestig example i support of our result. Recetly, Korovi-type theorems have bee obtaied by Mohiuddie [6] for almost covergece. Korovi-type theorems were also obtaied i [7] for -statistical covergece. The authors of [8] established these types of approximatio theorem i weighted L p spaces, where p, through A -summability which is stroger tha ordiary covergece. For these types of approximatio theorems ad related cocepts, oe ca be referred to [9 9] ad refereces therei.. KOROVKIN-TYPE THEOREMS BY WEIGHTED NORLUND-EULER A-STATISTICAL CONVERGENCE Kol [9] itroduced the otio of A -statistical covergece by cosiderig oegative regular matrix A istead of Cesáro matrix i the defiitio of statistical covergece due to Fast. Ispired from the wor of S. A. Mohiuddie, Abdullah Alotaibi, ad Bipa Hazaria [30] we itroduce the otio of weighted Norlud-Euler A -statistical covergece of a sequece ad the we establish some Korovi-type theorems by usig this otio. Defiitio 3. Let A ( a, ) be a oegative regular matrix. A sequece x ( x ) of real or complex umbers is said to be weighted Norlud Euler A -statistical covergece, deoted by NE SA coverget, to L if for every 0

26 Weighted Norlud-Euler A-Statistical Covergece for.. 5 where I symbol, we will write lim a, 0 (6) E( p, ) v v (7) v0 E( p, ) { N : p q ( ) x L } NE SA lim x L. Remar 4. Note that covergece sequece implies weighted Norlud-Euler A - statistical coverget to the same value but coverse is ot true i geeral. For example, tae p, q for all ad defie a sequece x ( x ) by, if x 0, otherwise where N. The this sequece is statistically coverget to 0 but ot coverget; i this case, weighted Norlud-Euler A -statistical covergece of a sequece coicides with statistical covergece. Theorem 5. Let A ( a, ) be a oegative regular matrix. Cosider a sequece of positive liear operators ( M ) from C[ ab, ] ito itself. The, for all f C[ a, b] bouded o whole real lie, if oly if (8) NE SA lim M f ; x f ( x) 0 (9) NE SA lim M ; x 0, NE SA lim M v; x x 0, NE SA lim M( v ; x) x 0 Proof. Equatio (0) directly follows from (9) because each of (0), xx, belogs to C[ a, b ]. Cosider a fuctio f C[ a, b]. The there is a costat C 0 such that f ( x) C for all x(, ). Therefore, f f( ),, v x C v x, () Let 0 be give. By hypothesis there is a ( ) 0 such that v Solvig () ad () ad the substitutig f f ( x), v x () v C f x, oe obtais Ω( v) ( v x) f ( ) Ω. (3) Equatio (3) ca be also writte by as

27 6 E. Hoxha, E. Aljimi, V. Lou Operatig (; ) v CΩ f f x CΩ. (4) M x to (4) sice M ( f ; x ) his liear ad mooto, oe obtais M (; )( CΩ) (; )( ( ) ( )) (; )( C x M Ω) x f v f x M x Note that x is fixed, so f( x ) is costat umber. Thus, we obtai from (5) that (5) C C The term '' M( f ; x) f ( x) M(; x)'' i (6) ca also writte as M (; x) M (Ω; x) M ( f ; x) f ( x) M (; x) M (; x) M (Ω; x) (6) M ( f ; x) f ( x) M (; x) M ( f ; x) f ( x) f ( x)[ M (; x) ] (7) M f x f x M x i (6), we get that Now substitutig the value of ( ; ) ( ) (; ) C M ( f ; x) f ( x) M (; x) M (Ω; x) f ( x)[ M (; x) ] (8) We ca rewrite the term '' M (Ω; x )'' i (8) as follows: [ M (ν ; x) x xm ( x x x M x M (Ω; x) M (( x) ; x) M (ν ; x) xm (ν; x) x M (; x) ] ν; ) [ (; ) ] Equatio (8) with the above value of M (Ω; x ) becomes M ( ; ) ( ) (; ) C f x f x M x {[ M (ν ; x) x ] x[ M (ν; x) x] x [ M(; x) ]} f ( x)[ M(; x) ] [ M (; ) ] C x {[ M (ν ; x) x ] x[ M (ν; x) x] x [ M(; x) ]} f ( x)[ M(; x) ] Therefore, M ( ; ) ( ) ( Cb f x f x C) M (; ) x C M (ν ; ) x x 4Cb M (ν; x) x where b max x. Taig supremum over x [ a, b], oe obtais or M ( ; ) ( ) ( Cb f x f x C) M (; ) x C M (ν ; ) x x 4Cb M (ν; x) x M( f ; x) f ( x) T{ M(; x) M (ν ; x) x M (ν; x) x } (9) (30) (3) (3) (33)

28 Weighted Norlud-Euler A-Statistical Covergece for.. 7 where Hece Cb C C 4 Cb T max{,, }. (34) pq ( ) ( ; ) ( ) { ( ) (; ) v M f x f x T p q v M x 0 0 p q ( ) (ν ; ) v M x x 0 p q ( ) (ν; ) } v M x x 0 For give 0, choose 0 such that, ad we will defie the followig sets: It easy to see that E { N : pq ( v ) M ( f ; x) f ( x) } 0 E { N : p q ( ) (, ) } v M x 3T 0 E { N : p q ( ) (ν; ) } v M x x 3T 0 E3 { N : p q ( ) (ν ; ) } v M x x 3T 0 (35) (36) E EE E3 (37) Thus, for each N, we obtai from (35) that a a a a (38),,,, E E E E 3 Taig limit i (38) ad also (0) gives that lim a, 0. (39) E These yields that for all f C[ a, b]. NE SA lim M( f ; x) f ( x) 0 (40) We also obtai the followig Korovi-type theorem for weighted Norlud-Euler statistical covergece istead of oegative regular matrix A i Theorem 5. Theorem 6. Cosider a sequece of positive liear operators ( M ) from C[ a, b ] ito itself. The, for all f C[ a, b] bouded o whole real lie, SNE lim M ( f ; x) f ( x) 0 (4)

29 8 E. Hoxha, E. Aljimi, V. Lou if oly if SNE lim M (; x) 0 (4) S lim M ( ; x) x 0 (43) NE SNE lim M ( ; x) x 0 (44) Proof. Followig the proof of Theorem 5, oe obtais E EE E3 (45) ad so NE ( E) NE ( E ) NE ( E) NE ( E3) (46) Equatios (4)-(44) give that SNE lim M ( f ; x) f ( x) 0. (47) Remar 7. By the Theorem of [3], we have that if a sequece x ( x ) is weighted Norlu-Euler statistically coverget to L, the it is strogly ( N, p, q)( E,) summable to L, provided that pq ( ) v x L is bouded; that is, there exist 0 a costat C such that for all N. We write v 0 p q ( ) x L C (,, )(,) { : lim ( N p q E x x p R q ) 0 for some } 0 v xv L L 0 (48) the set of all sequeces x ( x ) which are strogly ( N, p, q)( E,) summable to L. Theorem 8. Let M : C[ a, b] C[ a, b] be a sequece of positive liear operators which satisfies (43)-(44) of Theorem 6 ad the followig coditios holds: The, lim M ; x 0. (49) v 0 0 for ay f C[ a, b]. (50) lim p q ( ) M ( f ; x) f ( x) 0, R Proof. It follow from (49) that M ( f ; x) C ', for some costat C ' 0 ad for all N. Hece for f C[ a, b], oe obtais

30 Weighted Norlud-Euler A-Statistical Covergece for.. 9 pq ( ) ( ; ) ( ) ( )( (; ) ) v M f x f x pq v f M x f 0 0 (5) pq ( ) ( ' ). v C C 0 Right had side of (5) is costat, so pq ( ) ( ; ) ( ) v M f x f x 0 is bouded. Sice (49) implies (4), by Theorem 6 we get that SNE lim M ( f ; x) f ( x) 0. (5) By remar 7, (5) ad (5) together give the desired result. 3. RATE OF WEIGHTED NORLUND-EULER A-STATISTICAL CONVERGENCE First we defie the rate of weighted Norlud-Euler A-statistical coverget sequece as follows. Defiitio 9. Let A ( a, ) be a oegative regular matrix ad let ( a ) be a positive o icreasig sequece. The, a sequece x ( x ) is weighted Norlud-Euler A- statistical coverget to L with the rate of oa ( ) if for each 0 where I symbol, we will write lim, 0 a a (53) E ( p, ) v (54) 0 E p, { N : p q ( ) x L } NE x L SA o( a ) as (55) We will prove the followig auxiliary result by usig the above defiitio. Lemma 0. Let A ( a, ) be a oegative regular matrix. Suppose that ( a ) ad ( b ) are two positive oicreasig sequeces. Let x ( x ) ad y ( y ) be two sequeces such that The, NE x L SA o( a ) ad y L SA NE o( b ). NE (i) ( x L) ( y L) SA o( c ), NE (ii) ( x L) ( y L) SA o( ab ),

31 30 E. Hoxha, E. Aljimi, V. Lou NE (iii) ( x L ) SA o( a ), for ay scala r, where c max { a, b }. Proof. Suppose that Give 0, defie It easy to see that These yields that NE NE x L SA o( a ), y L SA o( b ) (56) E ' { N : pq ( ) v ( x L) ( y L) } 0 '' { : E N p q ( ) } v x L 0 ''' { : E N p q ( ) v y L } 0 (57) E ' E '' E ''' (58) a, a, a c, c c E ' E '' E ''' holds for all N. Sice c max { a, b }, (59) gives that (59) c a,,, a a b a E ' E '' E ''' Taig limit i (60) together with (56), we obtai Thus, (60) lim, 0 c a (6) E' NE ( x L) ( y L) SA o( c ) (6) Similarly, we ca prove (ii) ad (iii). Now, we recall the otio of modulus of cotiuity of f i C[ a, b ] is defied by It is well ow that ( f, ) sup{ f ( x) f ( y) : x, y [ a, b], x y } (63) xy. (64) f ( x) f ( y) ( f, )( ) Theorem. Let A ( a, ) be a oegative regular matrix. If the sequece of positive liear operators M : C[ a, b] C[ a, b] satisfies the coditios NE (i) M (; x) SA o( a ), (ii) NE A x x ( f, ) S o( b ), with M ( ; x) ad ( y) ( y x),

32 Weighted Norlud-Euler A-Statistical Covergece for.. 3 where ( a ) ad ( b ) are two positive oicreasig sequeces, the for all f C[ a, b], where c max { a, b }. NE M ( f ; x) f ( x) SA o( c ) (65) Proof. Equatio (7) ca be reformed ito the followig form: M ( f ; x) f ( x) M ( f ( x) f ( y) ; x) f ( x) M (; x) yx M( ; x) ( f, ) f ( x) M(; x) ( yx) x M ( ; x) ( f, ) f ( x) M (; x) ( M (; x) M ( ; x)) ( f, ) f ( x) M (; x) M (; x) ( f, ) f ( x) M (; x) ( f, ) M ( ; ) (, ) x x f Choosig M ( x; x), oe obtais M ( f ; x) f ( x) T M (; x) ( f, ) M (; x) ( f, ) (67) T f. For give 0, we will defie the followig sets: where ' E { N : pq ( v ) M ( f ; x) f ( x) } 0 ' E { N : p q ( ) (, ) } v M x 3T 0 ' E3 { N : p q ( ) (, ) } v f 6 0 ' E4 { N : p q ( ) (, ) (; ) }. v f M x 3 0 It follow from (67) that a, a, a c c c, a c, ' ' ' ' E E E3 E4 (66) (68) (69) holds for N. Sice c max{ a, b }, we obtai from (69) that c a, a a, b a, c a,. (70) ' ' ' ' E E E3 E4 Taig limit i (70) together with Lemma 0 ad our hypotheses (i) ad (ii), oe obtais These yields lim, 0 c a (7) E ' NE M ( f ; x) f ( x) SA o( c ) (7)

33 3 E. Hoxha, E. Aljimi, V. Lou Refereces [] H. Fast, Sur la covergece statistique, Colloquium Mathematicum,vol., pp. 4 44, 95. [] A. Zygmud, Trigoometric Series, vol. 5 of Moografje Matematycze, Warszawa- Lwow, 935. [3] A. Zygmud, Trigoometric Series, CambridgeUiversity Press,Cambridge, UK, 959. [4] I. J. Schoeberg, The itegrability of certai fuctios ad related summability methods, The America Mathematical Mothly, vol. 66, pp , 959. [5] T. Šalát, O statistically coverget sequeces of real umbers, Mathematica Slovaca, vol. 30, o., pp , 980. [6] J. A. Fridy, O statistical covergece, Aalysis, vol. 5, o. 4, pp , 985. [7] J. S. Coor, The statistical ad strog p -Cesaro covergece of sequeces, Aalysis, vol. 8, o. -, pp , 988. [8] R. G. Cooe, Ifiite Matrices ad Sequece Spaces, Macmilla,Lodo, UK, 950. [9] E. Kol, Matrix summability of statistically coverget sequeces, Aalysis, vol. 3, o. -, pp , 993. [0] V. Karaaya, T. A. Chishti, Weighted statistical covergece, Iraia Joural of Sciece ad Techology A, vol. 33, o.3, pp. 9 3, 009. [] M. Mursalee, V. Karaaya, M. Ertür, F. Gürsoy, Weighted statistical covergece ad its applicatio to Korovi type approximatio theorem, Applied Mathematics ad Computatio, vol. 8, o. 8, pp , 0. [] C. Bele, S. A. Mohiuddie, Geeralized weighted statistical covergece ad applicatio, Applied Mathematics ad Computatio, vol. 9, o. 8, pp , 03. [3] A. Esi, Statistical summability through de la Vallée-Poussi mea i probabilistic ormed space, Iteratioal Joural of Mathematics ad Mathematical Scieces, vol. 04, Article ID 67459, 5 pages, 04. [4] P. P. Korovi, Liear Operators ad Approximatio Theory, Hidusta Publishig, New Delhi, Idia, 960. [5] A. D. Gadjiev ad C. Orha, Some approximatio theorems via statistical covergece, The Rocy Moutai Joural of Mathematics, vol. 3, o., pp. 9 38, 00. [6] S. A. Mohiuddie, A applicatio of almost covergece i approximatio theorems, Applied Mathematics Letters, vol. 4, o., pp , 0. [7] O. H. H. Edely, S. A. Mohiuddie, A. Noma, Korovi type approximatio theorems obtaied through geeralized statistical covergece, Applied Mathematics Letters, vol. 3,o., pp , 00. [8] T. Acar, F. Diri, Korovi-type theorems i weighted p L -spaces via summatio process, The Scietific World Joural, vol.03, Article ID , 6 pages, 03. [9] N. L. Braha, H. M. Srivastava, S. A. Mohiuddie, A Korovi s type approximatio theorem for periodic fuctios via the statistical summability of the geeralized de la Vallée Poussi mea, Applied Mathematics ad Computatio, vol. 8, pp. 6 69, 04. [0] O. Duma, M. K. Kha, C. Orha, A statistical covergece of approximatig operators, Mathematical Iequalities & Applicatios, vol. 6, o. 4, pp , 003. [] O. Duma, C. Orha, Statistical approximatio by positive liear operators, Studia Mathematica, vol. 6, o., pp , 004.

34 Weighted Norlud-Euler A-Statistical Covergece for.. 33 [] M. Mursalee, A. Kiliçma, Korovi secod theorem via B-statistical A-summability, Abstract ad Applied Aalysis, vol. 03, Article ID , 6 pages, 03. [3] S. A. Mohiuddie, A. Alotaibi, Statistical covergece ad approximatio theorems for fuctios of two variables, Joural of Computatioal Aalysis ad Applicatios, vol. 5, o., pp.8 3, 03.8 The ScietificWorld Joural [4] S. A. Mohiuddie, A. Alotaibi, Korovi secod theorem via statistical summability (C, ), Joural of Iequalities ad Applicatios, vol. 03, article 49, 9 pages, 03. [5] H. M. Srivastava, M. Mursalee, A. Kha, Geeralized equi-statistical covergece of positive liear operators ad associated approximatio theorems, Mathematical ad Computer Modellig, vol. 55, o. 9-0, pp , 0. [6] O. H. H. Edely, M. Mursalee, A. Kha, Approximatio for periodic fuctios via weighted statistical covergece, Applied Mathematics ad Computatio, vol. 9, o. 5, pp , 03. [7] V. N. Mishra, K. Khatri, L. N. Mishra, Statistical approximatio by Katorovich-type discrete q-beta operators, Advaces i Differece Equatios, vol. 03, article 345, 03. [8] S. N. Berstei, Démostratio du théoréme de Weierstrass fodée sur le calcul des probabilités, Commuicatios of the Kharov Mathematical Society, vol. 3, o., pp.. [9] E. A. Aljimi, E, Hoxha, V. Lou, Some Results of Weighted Norlud-Euler Statistical Covergece, Iteratioal Mathematical Forum, Vol. 8, 03, o. 37, HIKARI Ltd. [30] S. A. Mohiuddie, A. Alotaibi, B, Hazaria, Weighted A Statistical Covergece for Sequeces of Positive Liear Operators. Hidawi Publishig Corporatio, The Scietific World Joural Volume 04, Article ID , 8 pages [3] G. Hardy, Diverget series, first editio, Oxford Uiversity Press, [3] E. A. Aljimi, V. Lou, Geeralized Weighted Norlud-Euler Statistical Covergece, It. Joural of Math. Aalysis, Vol. 8, 04, o. 7, HIKARI Ltd. Departmet of Mathematics, Uiversity of Tiraa, Albaia. address: hoxhaelida@yahoo.com Departmet of Mathematics, Uiversity of Tiraa, Albaia. address:eremhalimii@yahoo.co.u 3 Departmet of Computer Scieces ad Applied Mathematics, College, Vizioi per Arsim, Ahmet Kaciu, Nr=3, Ferizaj

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36 Математички Билтен ISSN X Vol. 38(LXIV) No. UDC: (35-43) Скопје, Македонија NEW CHARACTERIZATION OF -PRE-HILBERT SPACE Samoil Malčesi, Kateria Aevsa ad Risto Malčesi 3 Abstract. The problem of fidig ecessary ad sufficiet coditios a -ormed space to be treated as -pre-hilbert space is the focus of iterest of may mathematicias. Few characterizatios of -ier product are give i [], [3], [5], [6], [8] ad [9]. I this paper a ew ecessary ad sufficiet coditio for existece of -ier product ito - ormed space is give.. INTRODUCTION Let L be a real vector space with dimesio greater tha ad, be a real fuctio o L L such that: a) xy, 0, for all x, y L ad xy, 0 if ad oly if the set { xy, } is liearly depedet, b) x, y y, x, for all x, y L, c) x, y x, y, for all x, y L ad for each R, ad d) x y, z x, z y, z, for all x, y, z L. The fuctio, is said to be -orm of L, ad ( L,, ) is said to be vector -ormed space ([7]). The iequality i the axiom d) is said to be parallelepiped iequality. Let be a positive iteger, L be a real vector space, diml ad (, ) be a real fuctio over LL L such that: i) ( x, x y) 0, for all x, y L ad ( x, x y) 0 if ad oly if x ad y are liearly depedet, ii) ( x, y z) ( y, x z), for all x, y, z L, iii) ( x, x y) ( y, y x), for all x, y L, iv) ( x, y z) ( x, y z), for all x, y, z L. ad for each R, ad v) ( x x, y z) ( x, y z) ( x, y z), for all x, x, y, z L. The fuctio (, ) is said to be -ier product, ad ( L,(, )) is said to be -pre- Hilbert space ([3]). 00 Mathematics Subject Classificatio. 46C50, 46C5, 46B0 Key words ad phrases. -orm, -ier product, parallelepiped equality 35