NECESSARY AND SUFFICIENT CONDITIONS FOR GEOMETRIC MEANS OF SEQUENCES IN MULTIPLICATIVE CALCULUS

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1 Miskolc Mathematical Notes HU e-issn Vol. 7 (27), No. 2, pp OI:.854/MMN.27.8 NECESSRY N SUFFICIENT CONITIONS FOR GEOMETRIC MENS OF SEQUENCES IN MUTIPICTIVE CCUUS İ. ÇNK, Ü. TOTUR, N S.. SEZER Received October, 25 bstract. I this paper we have itroduced the cocept of geometric covergece of a sequece ad determied the ecessary ad sufficiet coditio uder which covergece follows from geometric covergece of a sequece i multiplicative sese. Corollaries allow this coditio to be replaced by multiplicative aalogues of Schmidt type slow oscillatio coditio or adau type two-sided coditio. 2 Mathematics Subject Classificatio: 266; 45; 4E5 Keywords: multiplicative calculus, geometric mea. INTROUCTION N PREIMINRIES urig the 7th cetury may scholars, as Galileo, discussed the followig problem: Two estimates, ad, are proposed as the value of a horse, which estimates, if ay, deviates more from the true value of? The scholars who maitaied that deviatios should be measured by differeces cocluded that the estimate of was closer to the true value. However, Galileo evetually maitaied that the deviatios should be measured by ratios, ad he cocluded that two estimates deviated equally from the true value (see [8]). I this situatio, the questio If we measure by ratios, what kid of a calculus do we have? appears. The aswer is the mai idea of multiplicative calculus. Multiplicative calculus is alterative to the classical calculus. It provides differetiatio ad itegratio tools based o multiplicatio istead of additio. Every fact i classical calculus has a aalogue i multiplicative calculus. I recet years, multiplicative calculus has attaied oticeable importace ad popularity, due maily to its applicatios i differet areas such as biomedical image aalysis [7], physics [3], dyamical systems [2], umerical aalysis [9], complex aalysis [], sequece spaces [5] ad ecoomy [6]. Now, we give some basic properties of multiplicative calculus. c 27 Miskolc Uiversity Press

2 792 İ. ÇNK, Ü. TOTUR, N S.. SEZER et x 2 R C : The multiplicative absolute value of x (see [4]) is defied as ( jxj x; if x ; x ; if x < : Usig the above defiitio we have the followig properties of the multiplicative absolute value fuctio. For ay x;y 2 R C, the followigs are valid (see []): i) jxj, ii) j x j jxj, iii) jxj y if ad oly if y x y, iv) jxyj jxj jyj. By meas of the multiplicative absolute value, Bashirov et al. [4] itroduced the multiplicative distace betwee positive real umbers x ad y as d.x;y/ x y which satisfies the followig properties: i) d.x;y/ for all x;y 2 R C, ii) d.x;y/ if ad oly if x y, iii) d.x;y/ d.y;x/ for all x;y 2 R C, iv) d.x; / d.x;y/d.y; / for all x;y; 2 R C. s a cosequece, R C is a multiplicative metric space ad a sequece. / i R C coverges to the it 2 R C i the multiplicative sese if for all >, there exists N 2 N such that d. ;/ < for all > N; or equivaletly, d. ;/! as! : It is well kow that R C is ot complete accordig to the Euclidea metric. To emphasize the importace of this work, we should first ote that.r C ;j:j / is a complete multiplicative metric space []. lso, it should be oted that throughout this paper the cocept of covergece of a sequece is uderstood i multiplicative sese. et. / be a sequece of positive real umbers. The -th geometric mea of. / is defied by W Y k x k! C ; ;;2;:::: We say that. / coverges to by the geometric mea method, briefly: G-coverget to 2 R C if! (.)

3 NECESSRY N SUFFICIENT CONITIONS FOR GEOMETRIC MENS ad we write!.g/. If the sequece. / diverges, the we may assig a it value to. / by a differet itatio method. I this case we still say that. / diverges but, eve so, we are able to fid a fiite umber that ca be cosidered as its it value. We ca replace!, which possibly does ot exist, by! Y k x k! C 2 R C if this multiplicative it exists ad we use this value for the it of the sequece. /. This method is completely atural because it just takes geometric meas ad settles for computig a kid of mea it where a actual it fails to exist. Our cocer while studyig a ew kid of covergece is to determie whether it assigs the correct value to a sequece which is already coverget. That is, does! H) x!.g/ y method assigig a it value to a sequece is said to be regular if this is the case. I emma, we prove that if the it (.2)! exists, the the it (.) also exists. That the coverse of emma is ot true i geeral is provided by the followig example.! Q Example. The sequece. / e. /k is diverget, but it is G-coverget to p e: k Here, we defie the cocept of slowly oscillatio which is more geeral tha the covergece of a sequece i multiplicative calculus. efiitio. sequece. / of positive real umbers is said to be -slowly oscillatig if ifsup max! C! <m x ; (.3) where by we deote the itegral part of the product, i symbol W Œ, or equivaletly, if for every > there exists./ ad./ >, as close to as we wish, such that < (.4) wheever < < m.

4 794 İ. ÇNK, Ü. TOTUR, N S.. SEZER It is also easy to show that (.3) is satisfied if ad oly if xm x wheever < m!, m;!. Example 2. The sequece. /./ is -slowly oscillatig, but ot coverget. I fact, x m ; wheever < m!, m;!. Our mai goal i this paper is to obtai a ecessary ad sufficiet coditio uder which covergece of a sequece follows from covergece of its geometric meas, ad vice versa. 2. MIN RESUT We prove the followig theorem. I mai theorem, some relatios with covergece ad G-covergece of a sequece are give. lso, we preset some corollaries which iclude the multiplicative aalogues of coditios of Schmidt type ad adau type. Theorem. et. / be a sequece of positive real umbers. If. / is G- coverget to a fiite it, the. / is coverget to if ad oly if oe of the followig two coditios hold: or ifsup! C! if! sup! Y ic Y i C ; (2.) ; (2.2) where by we deote the itegral part of the product, i symbol W Œ. The proof of Theorem relies o the represetatios (3.4) ad (3.5). The ext corollaries of Theorem follows immediately. Corollary. If the sequece. / is G-coverget to ad -slowly oscillatig, the it is coverget to : Corollary 2. If. / is G-coverget to ad.. / / is bouded, where for ad x x ; the. / is coverget to :

5 NECESSRY N SUFFICIENT CONITIONS FOR GEOMETRIC MENS Corollary 3. If. / is G-coverget to ad. /! ; as! ; the. / is coverget to : 3. UXIIRY RESUTS We eed followig lemmas for the proof of our mai theorem. emma. If the sequece. / is coverget to, the the sequece. / of its geometric meas is coverget to : Proof. et! : The for all > ; there exists N 2 N such that < =2 wheever > N ; ad < M wheever N : So, we have! Y, C g! Y C x k k k! Y C x k Sice! M N C C k Y xk k k! C YN C xk M N C C. =2 / N C Y kn C M N C C =2 ; there exists N 2 N such that xk > N. Therefore, there exists N maxfn ;N g such that This completes the proof. C M N C C g < =2 wheever < for all > N: Next, we prove that if a sequece is G-coverget to a fiite it, the the so-called movig geometric meas coverge to the same it. More precisely, the followig lemma is valid.

6 796 İ. ÇNK, Ü. TOTUR, N S.. SEZER emma 2. If the sequece. / is G-coverget to a fiite it, the for each > ad for each < < > ;.x/ W < ;.x/ W Y mc Y m C! ; (3.)! ; (3.2) where by we deote the itegral part of the product, i symbol W Œ. Proof. If > ad is large eough i the sese that >, the Y, Y Y Sice mc m 2 6B 4 > ;.x/ Y m m C g g C C C C. / C g C, B! Y m C C C C ; (3.3) (3.) follows from (.). The proof of (3.2) ca be similarly obtaied. The followig two represetatios of the ratio will be used i the proof of the mai theorem. emma 3. (i) For all > ad large eough, that is, whe >, 2 3 C g 6 Y : (3.4) ic

7 NECESSRY N SUFFICIENT CONITIONS FOR GEOMETRIC MENS (ii) For all < < ad large eough, that is, whe >, x 2 3 C g 6 Y : (3.5) i C Proof. By defiitio, for every >, we have Hece, ad g Q i C Q i C Y i!.c/.c/ C C which is equivalet to (3.4). (ii) The proof of (3.5) is similar. Y ic Y ic C Y ic Y C C ic C C ; 4. PROOF Proof of Theorem. Necessity. ssume that (.) holds. Sice. / coverges to, we have : (4.) It follows by (.) that for all >,!! C : (4.2)

8 798 İ. ÇNK, Ü. TOTUR, N S.. SEZER Sice Y ic (2.) follows from (3.4), (4.), (4.2), ad (4.3). It follows by (.) that for all < <, Sice Y i C! C C ; (4.3) : (4.4) C ; (4.5) (2.2) follows from (3.5), (4.), (4.4), ad (4.5). Sufficiecy. ssume the fulfilmet of (.) ad (2.). It follows from (2.) that there exists a decreasig sequece. j / covergig to such that where j Œ j. By (3.4), we have sup j!! j Y ic j ; (4.6) sup! g sup g j j! j C j! g sup j Y j j!! ic (4.7) Takig (.) ad (4.6) ito accout, we obtai (4.8)! ssume the fulfilmet of (.) ad (2.2). It follows from (2.2) that there exists a icreasig sequece. j / covergig to such that where j Œ j. sup j!! Y i j C j ; (4.9)

9 NECESSRY N SUFFICIENT CONITIONS FOR GEOMETRIC MENS By (3.5), we have sup! g sup g! j C j j!! g j sup Y j j!! i j C (4.) Takig (.) ad (4.9) ito accout, we obtai (4.)! Combiig (.) ad (4.8) or (4.), i either case we coclude that. / coverges to. Sice -slow oscillatio implies both (2.) ad (2.2), the proof of Corollary follows from Theorem. However, the proof of Corollary seems to be fudametal, we prove it. CKNOWEGEMENT The authors would like to thak the referee for very helpful suggestios ad useful commets which improved the mauscript. REFERENCES [] M. bbas, B. li, ad Y. I. Suleima, Commo fixed poits of locally cotractive mappigs i multiplicative metric spaces with applicatio. It. J. Math. Math. Sci., vol. 25, o. rticle I 28683, p. 7 pages, 25, doi:.55/25/ [2]. iszewska ad M. Rybaczuk, alysis of the multiplicative orez system. Chaos Solitos Fractals, vol. 25, o., pp. 79 9, 25, doi:.6/j.chaos [3]. iszewska ad M. Rybaczuk, Chaos i Multiplicative Systems, i CHOTIC SYSTEMS: THEORY N PPICTIONS, Skiadas, CH ad imotikalis, I, Ed., doi:.42/ PO BOX 28 FRRER R, SINGPORE 928, SINGPORE: WOR SCIENTIFIC PUB CO PTE T, 2, Proceedigs Paper, pp. 9 6, 2d Chaotic Modelig ad Simulatio Iteratioal Coferece (CHOS29), Chaia, GREECE, JUN - 5, 29. [4]. E. Bashirov, E. M. Kurpıar, ad. Özyapıcı, Multiplicative calculus ad its applicatios. J. Math. al. ppl., vol. 337, o., pp , 28, doi:.6/j.jmaa [5]. F. Çakmak ad F. Başar, Some ew results o sequece spaces with respect to o-newtoia calculus. J. Iequal. ppl., vol. 22, p. 7, 22, doi:.86/29-242x [6].. Filip ad C. Piatecki, o-newtoia examiatio of the theory of exogeous ecoomic growth. Math. etera, vol. 4, o. 2, pp. 7, 24. [7]. Florack ad H. Va sse, Multiplicative calculus i biomedical image aalysis. J. Math. Imagig Vis., vol. 42, o., pp , 22, doi:.7/s [8] M. Grossma ad R. Katz, No-Newtoia calculus. Pigeo Cove, Mass.: ee Press. VIII, 94 p. $ 6. (972)., 972. [9] E. Misirli ad Y. Gurefe, Multiplicative dams Bashforth-Moulto methods. Numer. lgorithms, vol. 57, o. 4, pp , 2, doi:.7/s

10 8 İ. ÇNK, Ü. TOTUR, N S.. SEZER []. Uzer, Multiplicative type complex calculus as a alterative to the classical calculus. Comput. Math. ppl., vol. 6, o., pp , 2, doi:.6/j.camwa uthors addresses İ. Çaak Ege Uiversity, epartmet of Mathematics, 35 Izmir, Turkey address: ibrahim.caakege.edu.tr Ü. Totur da Mederes Uiversity, epartmet of Mathematics, 9 ydi, Turkey address: utoturadu.edu.tr S.. Sezer Istabul Medeiyet Uiversity, epartmet of Mathematics, 35 Izmir, Turkey address: sefaail.sezermedeiyet.edu.tr