Stability of Pythagorean Mean Functional Equation

Transcription

1 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: Stabilit of Pthagoea Mea Fuctioal Equatio P.Naasimma, K. Ravi, Sada Pielas 3 Deatmet of Mathematics, Thiuvalluva Uivesit College of Ats ad Sciece,Gajalaickaatti, Tiuattu , TamilNadu, Idia. Deatmet of Mathematics, Saced Heat College,Tiuattu , TamilNadu, Idia 3 Academia Milita, Deatameto de Ciêcias Eactas e Natuais, Av.Code Casto Guimaães, 70-3 Amadoa, Potugal Abstacts: I this ae, authos itoduce a ew Pthagoea mea fuctioal equatio which elates the thee classical Pthagoea mea ad ivestigate its geealized Hes -Ulam stabilit. Also, Motivated b the wok of Roma Ge [7], we deal with the geeal solutio of Pthagoea meas fuctioal equatio. We also ovide coute -eamles fo sigula cases. Ve seciall i this ae we illustate the geometical iteetatio ad alicatio of ew itoduced Pthagoea mea fuctioal equatio. Ke wods ad hases:pthagoea Meas, Aithmetic mea, Geometic mea ad Hamoic mea, Geealized Hes-Ulam stabilit. 00 Mathematics Subject Cla ssi ficatio: 39B5, 39B6, 39B7, 39B8..Itoductio Thee is a leged that oe da whe Pthagoas (c.500 BCE) was assig a blacksmith s sho, he head hamoious music igig fom the hammes. Whe he equied, he was told that the weights of the hammes wee 6, 8, 9, ad ouds. These atios oduce a fudametal ad its fouth, fifth ad octave. This was evidece that the elegace of mathematics is maifested i the hamo of atue. Retuig to music, these atios ae ideed a foudatio of music as oted b Achtus of Taetum(c.350 BCE): Thee ae thee meas i music: oe is the aithmetic, the secod is the geometic ad the thid is the subcota, which the call hamoic. The aithmetic mea is whe thee ae thee tems showig successivel the same ecess: the seco d eceeds the thid b the same amout as the fist eceeds the secod. I this ootio, the atio of the lage umbe is less, that of the smalle mumbes geate. The geometic mea is whe the secod is to the thid as the fist is to the secod; i this, the geate umbes have the same atio as the smalle umbes. The subcota, which we call hamoic, is as follows: b whateve at of itself the fist tem eceeds the secod, the middle tem eceeds the thid b the same at of the thid. I this ootio, the atio of the lage umbes is lage ad of the lowe umbes less [5]. I ode to udestad these descitios, it is ecessa to ealize that fo Geeks, a mea fo two umbes B> A, was a thid umbe C satisfig B > C > A ad a futhe oet. The above descitio of the aithmetic mea, geometic mea ad hamoic mea states that B C C A( this is" the same umeical amout " efeed to bplato), C B ( this is " the same factio of the etemes " efeed to bplato ), A C B C C A B A esectivel ad the above algebaic maiulatio echaacteizes is as follows C A B AB,, C AB C A B the aithmetic mea, geometic mea ad hamoic mea eectivel. 398 P a g e e d i t g c u b l i s h i g. c o m

2 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: Defiitio. Pthagoea Meas:[4] I Mathematics, the thee classical Pthagoea meas ae the aithmetic mea(a), the Geometic mea(g), ad the hamoic mea(h). The ae defied b A(,,..., ) ( G... (,,..., )... H (,,..., ).... ) The stabilit oblem of fuctioal equatios oigiates fom the fudametal questio: Whe is it tue that a mathematical object satisfig a cetai oet aoimatel must be close to a object satisfig the oet eactl? I coectio with the above questio, i 940, S.M. Ulam [4] aised a questio coceig the stabilit of homomohisms. Let G be a gou ad let G be a metic gou with d (.,.). Give > 0 does thee eist a > 0 such that if a fuctio f : G G satisfies the iequalit d ( f (, f ( ) f ( ) <, G, the thee is a homomohism H : G G with d ( f ( ), H( )) G? additive maigs The fist atial solutio to Ulam s questio was give b D.H. Hes [9]. He cosideed the case of aoimatel f : E E whee E ad E ae Baach saces ad f satisfies Hes iequalit f ( f ( ) f (, E, it was show that the limit eists E ad that a E E a lim ( ) f ( ) : is the uique additive maig satisfig f ( ) a( ). f is cotiuous i t fo each fied E Moeove, it was oved that if (t), the a is liea. I this case, the Cauch additive fuctioal equatio f ( f ( ) f ( is said to satisf the Hes-Ulam stabilit. I 978, Th.M. Rassias [3] ovided a geealized vesio of the theoem of Hes which emitted the Cauch diffeece to become ubouded. He oved the followig theoem. Theoem. [ Th. M. Rassias ] If a fuctio f E E : betwee Baach saces satisfies the iequalit f ( f ( ) f ( (.) fo some 0, 0 < ad, E, the thee eists a uique additive fuctio a E E : such that f ( ) a( ) E. Moeove, if f (t) is cotious i t fo each fied E, the a is liea. (.) A aticula case of Th.M. Rassias theoem egadig the Hes -Ulam stabilit of the additive maigs was oved b T. Aoki []. The theoem of Rassias was late eteded to all ad geealized b ma mathematicias (see [, 3, 5, 8, 0, 8, 9, 0]). The heomeo that was itoduced ad oved b Th.M. Rassias is called the Hes-Ulam-Rassias stabilit. The Hes-Ulam-Rassias stabilit fo vaious fuctioal equatios have bee etesivel ivestigated b umeous authos; oe ca efe to ([,, 3, 4, 6, 7]). I 994, a geealizatio of the Th.M. Rassias theoem was obtaied b P.G â vuta [6], who elaced the boud b a geeal cotol fuctio (, ). 399 P a g e e d i t g c u b l i s h i g. c o m

3 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: I , J.M.Rassias [8, 9] elaced the sum aeaed i ight had side of the equatio (.) b the oduct of owes of oms. This stabilit is called Ulam-Gavuta-Rassias stabilit ivolvig a oduct of diffeet owes of oms. Ifact, he oved the followig theoem. Theoem.3 [ J. M. Rassias ] Let E subject to the iequalit, E, whee ad ae costats with > 0 eist E ad : E E E. If > f : E E be a maig fom a omed vecto sace E ito Baach sace f ( f ( ) f ( (.3) L( ) lim ad f ( ) 0 <. The the limit L is the uique additive maig which satisfies f ( ) L( ) the iequalit (.3) holds fo eist E ad : E E E. A, E ad the limit f (.4) (.5) ( ) lim (.6) A is the uique additive maig which satisfies f ( ) A( ) (.7) Ve ecetl, J. M. Rassias elaced the sum aeaed i ight had side of the equatio (.) b the mied oduct of owes of oms i []. The ivestigatio of stabilit of fuctioal equatio ivolvig with the mied oduct of owe oms is kow as Hes-Ulam-J.M.Rassias stabilit. I 00, K. Ravi ad B.V. Sethil Kuma [] ivestigated some esults o Ulam -Gavuta-Rassias stabilit of the fuctioal equatio It was oved that the eciocal fuctio ( c ) ( ) (. ( ) ( (.8) ( is a solutio of the fuctioal equatio (.8). Defiitio.4 Pthagoea mea fuctioal equatio: Pthagoea mea fuctioal equatio is a fuctioal equatio which aises fom the elatios betwee the thee Phagoea meas of aithmetic mea, geometic mea ad hamoic mea. With the motivatio of the Pthagoea meas, that is; aithmetic mea, geometic mea, hamoic mea ad its elatios. I this ae, authos aive the Pthagoea meas fuctioal equatio of the fom with c ( ) f ( ) f ( f,, (0, ) f ( ) f ( f ad ve seciall i this ae we illustate the geometical iteetatio i Sectio. I Sectio 3, Motivated b the wok of Roma Ge [7] we deal with the solutio of Pthagoea mea fuctioal equatio (.9). I Sectio (.9) 400 P a g e e d i t g c u b l i s h i g. c o m

4 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: , we ivestigate the geealized Hes-Ulam stabilit of equatio (.9) also we ovide coute-eamles fo sigula cases. I Sectio 5, we ivestigate the alicatio of ew itoduced Pthagoea mea fuctioal equatio.. Geometical Iteetatio of Equatio (.9) Coside the cicle cete A ad the taget to the cicle fom the oit M touchig the cicle at the oit G with PM a, QM b ad a > b > 0. We ca wok out the adius of the cicle ad show the legth AM is ( a b) be the aithmetic mea of a ad b. Usig Pthagoas theoem we ca get the legth GM is ab be the geometic mea of a ad b. Usig the fact ab that the tiagles AGM ad GHM ae simila we ca get HM is be the hamoic mea of a ad b. ( a b) Fom the defiitio of Pthagoea meas ad fom the diagam, oe ca shows that the hamoic mea is elated to the GM AM aithmetic mea ad the geometic mea b HM. So GM AM HM (.) meaig the two umbes geometic mea equals the geometic mea of thei aithmetic ad hamoic meas. B ewitig the equatio (.), we get I the above diagam, if we take a ad HM GM HM. AM (.) b, ad alig i (.) we get Comaig this esult (.3) with (.9), we obtai (.9) holds good i the above geometic costuctio. f ( ). (.3). This oves that the Pthagoea mea fuctioal equatio Thoughout this ae, let us assume X be a liea sace ad Y be a Baach sace. Fo the sake of coveiece, let us deote, X. Df (, f f ( ) f ( f ( ) f ( ) 40 P a g e e d i t g c u b l i s h i g. c o m

5 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: Geeal Solutio of the Fuctioal Equatio (.9) I this sectio, motivated b the wok of Roma Ge [7], we eset the geeal solutio of the Phagoea mea fuctioal equatio i the simlest case ad also we give the diffeetiable solutio of (.9). The followig Theoem gives the solutio of (.9) i the simlest case. f ( ) Poof. Put Theoem 3. Simlest case: The ol ozeo solutio at zeo, of the equatio(.9) is of the fom i (.9) to get the equalit (0,). Settig (0,), we have as well as f :(0,) R, admittig a fiite limit of the quotiet c f ( ), (, (0, ) (0, ) f ( ) f ( ) f ( ) g ( ), lim g( ) 0 : c R g ( ) g( ), (0,). B a simle iductio, fo eve ositive itege, we also obtai the equalities g ( ( ) ) g( ), (0,), whece, fiall, g ( ) ( ) ( ) g g ( ( ) ( Cosequetl, fo eve (0,), we get ) ) c as. c f ( ) g( ) c, as claimed, because cleal c caot vaish sice, othewise, we would have f 0. The followig Theoem gives the diffeetiable solutio of the Pthagoea mea fuctioal equatio (.9). Theoem 3. Diffeetiable Solutio: Let f :(0,) R be cotiuousl diffeetiable fuctios with owhee vaishig deivatives f. The f ields a solutio to the fuctioal equatio (.9) if ad ol if thee eists ozeo eal c costats c such that f ( ), (0, ). 40 P a g e e d i t g c u b l i s h i g. c o m

6 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: Poof. Diffeetiate equatio (.9) with esect to o both side, we obtai, (0, ) (0, ) f f ( ) f ( f ( ) f ( ( ) (3.). Sice o settig i (.9) we deduce that ad, a fotioi, ( ) f ( ) f (3.) f ( ) f ( ) (3.3) (0,). Puttig (0,). The equatio (3.3) ad (3.4) gives i (3.) ad usig equatios (3.) ad (3.3), we aive f f ( 3) f ( ) 3 3 (3.4) m ( ) ( 3) f ( ) (0,). iteges, m ad due to the cotiuit of the ma f, we deive its lieait ( ) 3 f ( ) f () 3 ( ) c fo (0,). Theefoe, thee eists eal umbes c 0, d, such that f ( ) d fo (0,) ( 3) m. Note that we have to have d 0 because of the equalit f ( ) f ( ) valid ositive. Which comletes the oof. 4. Geealized Hes- Ulam Stabilit of Equatio (.9) Theoem 4. Let f : X Y be a maig satisfig whee : X Y is a fuctio such that with the coditio iequalit Df (, (, (4.) i ( ) (, ) (4.) i0 lim (, ) 0 X. The thee eists a uique eciocal-quadatic maig X Y X. i i (4.3) : which satisfies (.9) ad the ( ) f ( ) ( ) (4.4) 403 P a g e e d i t g c u b l i s h i g. c o m

7 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: Poof. Relacig b i (4.) ad multilig b, we get f ( ) f ( ) (, ) (4.5) obtai X. Now, elacig b i (5), multilig b ad summig the esultig iequalit with (5), we X i f () f ( ) (,. Poceedig futhe ad usig iductio o a ositive itege, we aive i0 i i ) multil b Allow i i i0 i f ( ) f ( ) (, i0 i i ) i (, ) (4.6) X. I ode to ove the covegece of the sequece { f ( )}, elace b i (4.6) ad, we fid that fo > m > 0 m m m m f f f f i i i,. (4.7) i0 ad usig (4.3), the ight had side of the iequalit (4.7) teds to 0. Thus the sequece { f ( )} i (4.6), we aive (4.4). To show that satisfies (.9), settig is a Cauch sequece. Allowig ( ) lim f ( ), elacig (, Allowig b (, i (4.) ad multilig b, we obtai ( D, (, ). (4.8) i (4.8), we see that satisfies (.9), X. To ove is a uique eciocal-quadatic R : X Y be aothe eciocal-quadatic maig which satisfies (.9) ad the iequalit maig satisfig (.9). Let (4.4). Cleal ( R ) R( ), ( ) ( ) ad usig (4.4), we aive X. 4.. R( ) ( ) R ( ) ( ) ( R ( ) f ( ) f ( ) ( ) ) i 4 (, ) (4.9) i0 i i Allowig i (4.9) ad usig (4.3), we fid that is uique. This comletes the oof of Theoem 404 P a g e e d i t g c u b l i s h i g. c o m

8 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: Theoem 4. Let with the coditio f : X Y be a maig satisfig (4.), whee : X Y i0 ( i) ( i) ( i) ( ) (, ) is a fuctio such that (4.0) lim (, ) 0 X. The thee eists a uique eciocal maig X Y X. : which satisfies (.9) ad the iequalit (4.) ( ) f ( ) ( ) (4.) Poof. The oof is obtaied b elacig (, i Theoem 4.. b (, ) i (4.) ad oceedig futhe b simila agumets as The followig Coollaies ae the immediate cosequeces of Theoem 4. ad 4. which gives the Hes-Ulam-Rassias stabilit, Ulam-Gavuta-Rassias stabilit ad Hes-Ulam-J.M.Rassias stabilit of the fuctioal equatio (.9). Coolla 4.3 Fo a fied c 0 ad < o >, if f X Y Df (, X, the thee eists a uique eciocal maig X Y : saitsifies, c ( ) (4.3) : such that X. Poof. If we choose (, c ( ) ad usig Theoem 4., we aive, X,, the b Theoem 4., we aive ( ) f ( ) 4c, X ad < 4c f X ad ( ) ( ), >. Now we will ovide a eamle to illustate that the fuctioal equatio (.9) is ot stable fo i Coolla 4.3. Eamle 4.4 Let :R R be a fuctio defied b a, ( ) a, if (, ) othewise 405 P a g e e d i t g c u b l i s h i g. c o m

9 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: whee a > 0 is a costat ad a fuctio :R R f b The f satisfies the fuctioal iequalit ( f ( ) 0 ) R. Df (, 8a (4.4), R. The thee do ot eist a eciocal maig :R R ad a costat > 0 such that f ( ) ( ) R. (4.5) Poof. Now ( ) a f ( ) a. 3 Theefoe we see that f is bouded. We ae goig to ove that f satisfies (4.4). If the the left had side of (4.4) is less tha a The thee eist a ositive itege such that so that <, < <, ( ). Now suose that 0 < <. (4.6) (4.7) o o >, > >, > cosequetl > >, > >. Ad agai fom (4.7), we get ( ( ) >, ( > ) ) > cosequetl ( ) >, >. Hece, we get >, >, >. > 406 P a g e e d i t g c u b l i s h i g. c o m

10 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: Theefoe fo each 0,,, ad, we have >, >, fo 0,,, ( >, ) 0. Fom the defiitio of f ad (4.6), we obtai that f ( ) f ( f f ( ) f ( ( ) 0 ( ) 3a a a 8a. Thus f satisfies (4.4), R with 0 < <. We claim that the eciocal fuctioal equatio (.9) is ot stable fo i Coolla 4.3. Suose o the cota, thee eist a eciocal maig :R R ad a costat > 0 satisfig (4.5). Thee, we have f ( ). (4.8) But we ca choose a ositive itege m with ma >., the (, ) m If (, ) 0,,, m. Fo this, we get 407 P a g e e d i t g c u b l i s h i g. c o m

11 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: f ( ) 0 ( ) m a m > 0 which cotadicts (4.8). Theefoe the eciocal fuctioal equatio (.9) is ot stable i sese of Ulam, Hes ad Rassias if, assumed i the iequalit (4.3). Coolla 4.5 Let c 0 such that f : X Y be a maig ad thee eists such that < o > a. If thee eists Df (, c ( ), X, the thee eists a uique eciocal maig X Y : satisfig the fuctioal equatio (.9) ad X. Poof. Cosideig (, c ( ) ad usig Theoem 4., we aive, X,, the b Theoem 4., we aive c ( ) f ( ), foall X ad < c ( ) f ( ), foall X ad >. Coolla 4.6 Let 3 > 0 fuctioal iequalit c ad < o > be eal umbes, ad f X Y 3 Df (, c, X. The thee eists a uique eciocal maig X Y : be a maig satisfig the : satisfig the fuctioal equatio (.9) ad X. Poof. Choosig (, c3,, X, 6c f X ad 3 ( ) ( ), < the b Theoem 4., we aive 408 P a g e e d i t g c u b l i s h i g. c o m

12 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: ad usig Theoem 4., we aive 6c3 ( ) f ( ), X ad >. Now we will ovide a eamle to illustate that the fuctioal equatio (.9) is ot stable fo i Coolla 4.6. Eamle 4.7 Let :R R be a fuctio defied b k, ( ) k, if (, ) othewise whee k > 0 is a costat ad a fuctio :R R The f satisfies the fuctioal iequalit f b ( f ( ) 0 ) R. Df (, 8k (4.9), R. The thee do ot eist a eciocal maig :R R ad a costat > 0 such that Poof. Now f ( ) ( ) R. (4.0) ( ) k f ( ) k. 3 Theefoe we see that f is bouded. We ae goig to ove that f satisfies (4.9). 0 < If the the left had side of (4.9) is less tha <. The thee eist a ositive itege such that a. Now suose that <, (4.) ( ) ad the est of the oof is same as the oof of Eamle Alicatio of Fuctioal Equatio (.9) The Paallel Cicuit ad the Fuctioal Equatio (.9): A aallel cicuit has moe tha oe esisto ad gets its ame fom havig multile aths to move alog. Also oe ca kow that the followig ule al to a aallel cicuit. that is The ivese of the total esistace of the cicuit is equal to the sum of the iveses of the idividual esistaces, R T. R R R Fo ol two esistos, the ueciocated eessio (5.) simlifies to 3 (5.) 409 P a g e e d i t g c u b l i s h i g. c o m

13 Global Joual of Mathematics Vol. 4, No., Setembe 9, 05 ISSN: R T RR R R (5.) Refe to the above Figue. If we take R, R, we get R T. (5.3) Sice electical coductace G is eciocal to esistace, theefoe total coductace of the above cicuit is G T. Now fom the equatio (5.3), we get G T. (5.4) Oe ca easil idetif the equatio (5.4) is ou mai fuctioal equatio (.9) with equatio (.9) holds good i the above cicuit. 6.Coclusio c f ( ). Hece the fuctioal I this ae authos mail achieved a ew Pthagoea mea fuctioal equatio coesodig to the elatio betwee thee classical Pthagoea meas ad obtaied the geeal solutio of Pthagoea meas fuctioal equatio fom the motivated wok of Roma Ge [7]. Also, authos ivestigated its geealized Hes -Ulam stabilit ad also ovided coute-eamles fo sigula cases. Ve seciall i this ae we illustated the geometical iteetatio ad alicatios of ou itoduced Pthagoea meas fuctioal equatio i coectio with the Paallel Cicuit. 40 P a g e e d i t g c u b l i s h i g. c o m

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