СОЈУЗ НА МАТЕМАТИЧАРИТЕ НА Р. МАКЕДОНИЈА МАТЕМАТИЧКИ БИЛТЕН BULLETIN MATHÉMATIQUE

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1 СОЈУЗ НА МАТЕМАТИЧАРИТЕ НА Р. МАКЕДОНИЈА ISSN X (prit) ISSN (olie) МАТЕМАТИЧКИ БИЛТЕН BULLETIN MATHÉMATIQUE КНИГА 40 (LXVI) TOME No. 3 РЕДАКЦИСКИ ОДБОР Малчески Алекса Главен уредник Маркоски Ѓорѓи Секретар Mushkarov Oleg (Bugarija) Oberguggeberger Michael (Avstrija) Пилиповиќ Стефан (Србија) Erdal Karapiar (Turcija) Ристо Малчески (Македонија) Костадин Тренчевски (Македонија) Bala Vladimir (Romaija) Čoba Mitrofa (Moldavija) Felloris Argyris (Grcija) Scarpalezos Dimitrios (Fracija) Škrekovski Riste (Sloveija) Valov Vesko (Kaada) Vidas Jasso (Belgija) Ромео Мештровиќ (Црна Гора) Aa Vukelić (Hrvatska ) Сава Гроздев (Бугарија) Riste Ataasov (USA) Violeta Vasilevska (USA) COMITÉ DE RÉDACTION Malcheski Aleksa Rédactuer e chef Markoski Gorgi Secrétaire Mushkarov Oleg (Bugarie) Oberguggeberger Michael (Autriche) Pilipovic Stefa (Serbie) Erdal Karapiar (Turquie) Risto Malcheski (Macédoie) Kostadi Trechevski (Macédoie) Bala Vladimir (Roumaie) Čoba Mitrofa (Moldavie) Felloris Argyris (Grèce) Scarpalezos Dimitrios (Frace) Škrekovski Riste (Slovéie) Valov Vesko (Caada) Vidas Jasso (Belgique) Romeo Meshtrovic, (Motéégro) Aa Vukelić (Croatie) Sava Groydev (Bugarie) Riste Ataasov (USA) Violeta Vasilevska (USA) СКОПЈЕ SKOPJE 016

2 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 три 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 10, 1000 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 three time i 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 Makedoija Bul. Aleksadar Makedoski bb, PF 10, 1000 Skopje Republic of Macedoia

3 Математички Билтен Vol. 40(LXVI) No (3) Скопје, Македонија ISSN X (prit) ISSN (olie) C O N T E N T S 1. Lazo Dimov ABOUT THE TYPES OF HOMOGENOUS LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER AND THEIR SOLUTIONS 5. Marti Lukarevski, Samoil Malčeski SEQUENTIALLY CONVERGENT MAPPINGS AND COMMON FIXED POINTS OF MAPPINGS IN -BANACH SPACES Slagjaa Brsakoska ABOUT A TWIN SOLUTION OF THE VEKUA EQUATION 3 4. Kateria Aevska, Risto Malčeski REMARK ABOUT CHARACTERIZATION OF -INNER PRODUCT 9 5. Slagjaa Brsakoska ABOUT THE ACCORDANCE BETWEEN THE CANONICAL VEKUA DIFFERENTIAL EQUATION AND THE GENERALIZED HOMOGENEOUS DIFFERENTIAL EQUATION Soja Čalamai, Dočo Dimovski, Marzaa Sewery-Kuzmaovska ON (3,,)-S-K-METRIZABLE SPACES Lidija Goračiova-Ilieva, Emilija Spasova Kamčeva ALGEBRAIC REPRESENTATION OF A CLASS OF HOMOGENOUS STEINER QUADRUPLE SYSTEMS Stela Çeo, Gladiola Tigo ASYMMETRIC INNER PRODUCT AND THE ASYMMETRIC QUASI NORM FUNCTION 61 3

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5 Математички Билтен ISSN X (prit) Vol. 40(LXVI) No. 3 ISSN (olie) 016 (5-1) UDC: Скопје, Македонија ABOUT THE TYPES OF HOMOGENOUS LINEAR DIFFERENTIAL EQUATIONS OF SECOND ORDER AND THEIR SOLUTIONS Lazo Dimov Abstract I the paper, types of liear differetial equatios of secod order are defied i the sese of the kow classificatio of the types of liear partial differetial equatios of secod order. The the shapes of their solutios are examied ad some classes differetial equatios are solved. 1. PART 1 For the liear partial differetial equatios of secod order z z z z z x xy y x y A B C D E Fz G 0, where A, B, C, D, E, F ad G are real fuctios from the two variables x ad y, depedig o the value of the determiat A B B C, we have the followig classificatio: 1. If 0, it is a PDE of hyperbolic type.. If 0, it is a PDE of parabolic type. 3. If 0, it is a PDE of elliptic type. Here, we will make a similar classificatio for the homogeous liear differetial equatios of secod order. It is well kow that the equatio y Ay 0 (1) 010 Mathematics Subject Classificatio. 34A05, 34A30. Key words ad phrases. classificatio, elliptic type, parabolic type, hyperbolic type, differetial equatio with costat coefficiets.

6 6 Lazo Dimov where A cost (equatio.9 o page 365 i []), has a solutio which ca be writte with the formula y( x) c si Ax c cos Ax, for A 0 1 y( x) c x c, for A 0 () 1 y( x) c sh Ax c ch Ax, for A 0. 1 By aalogy with the classificatio of the liear PDE of secod order that we made, we ca make the followig classificatio for the equatio (1) ad its solutio (): 1. If A 0, it is a equatio of hyperbolic type.. If A 0, it is a equatio of parabolic type. 3. If A 0, it is a equatio of elliptic type. If A a( x) is a give fuctio, the we ca make the followig classificatio of the equatio (1), which ow has the form y a( x) y 0 (3) ad its solutio as well, i.e.: 1. A equatio of hyperbolic type, if ax ( ) 0.. A equatio of parabolic type, if ax ( ) A equatio of elliptic type, if ax ( ) 0. For the solutio of the equatio (3) for every type, we examie the followig formulas: y( x) c1 sia( x) x c cos a( x) x, for a( x) 0 y( x) c x c, for a( x) 0 (4) 1 y( x) c1 sha( x) x c ch a( x) x, for a( x) 0. Here, the fuctios si a( x) x, cos a( x) x, sh a( x) x, cha( x) x are defied ad determied i [3]. It is well kow that if i the differetial equatio y f ( x) y g( x) y 0 (5) we itroduce a ew fuctio give with 1 f ( x) dx (6) y( x) e z( x) it will trasform i the differetial equatio z a( x) z 0 (7) kow as caoical equatio, where 1 1 a( x) g( x) f ( x) f ( x) 4. (8)

7 About the types of homogeous liear 7 Now, accordig to the previous classificatio we made, we get the followig classificatio for the equatio (5): 1. The equatio (5) is of hyperbolic type, if ax ( ) 0.. The equatio (5) is of parabolic type, if ax ( ) The equatio (5) is of elliptic type, if ax ( ) 0. Here, the solutio of the equatio (5), we get with the formulas: 1 f ( x) dx y( x) e [ c1 sia( x) x c cos a( x) x], for a( x) 0 1 f ( x) dx y( x) e [ c x c ], for a( x) 0 (9) 1 f ( x) dx 1 y( x) e [ c sh x c ch x], for a( x) 0. 1 a( x) a( x) Example. The differetial equatio where ad k y ( x k) y ( x kx k k 1) y 0 f ( x) ( x k), g( x) x kx k k 1, cost has a caoical equatio z ( k k) z 0, so, accordig to the formulas () has a solutio 1 z( x) c si k k x c cos k k x, if k,0 1, z( x) c x c, if k 0 ad k 1 1 kk x kk x 1 z( x) c e c e, if k 0,1. Now, the give equatio accordig to the formulas (9) has a solutio: x kx 1 y( x) e [ c si k k x c cos k k x], if k,0 1, x y( x) e [ c x c ], if k 0 x x 1 y( x) e [ c x c ], if k 1 1 x kx 1 k k x k k x y( x) e [ c e c e ], if k 0,1.

8 8 Lazo Dimov. PART If we itroduce a ew idepedet variable i the equatio (5) with the relatio x x() t, i order to get the equatio (5) shaped as (1), we get that the fuctios that appear i the equatio, satisfy the coditio g( x) f ( x) g( x) 0 (10) ad we get the ew variable from the relatio x ( t ) A gx ( ). (11) From (11) we have the followig: 1. for gx ( ) 0 we have A 0, so the equatio becomes a equatio from elliptic type, where the coectio betwee the ew ad the old variable is At g( x) dx. g x we have A 0, so the equatio becomes a equatio from. for 0 hyperbolic type, where the coectio betwee the ew ad the old variable is At g( x) dx. 3. if gx ( ) 0, we ca say that the equatio is from parabolic type. Actually, the equatio is y f ( x) y 0 ad its solutio is give with the formula f ( x) dx y( x) c1 e dx c. Now, for the geeral solutio of this kid of equatio accordig to the formula (), we get the followig formula: g( x) dx g( x) dx y( x) c1e ce, for g( x) 0 f ( x) dx y( x) c1 e dx c, for g( x) 0 (1) y( x) c si g( x) dx c cos g( x) dx, for g( x) 0. 1 Example. I the differetial equatio y 1 k 0, x y x y where the fuctios f( x) 1 ad gx ( ) k satisfy the coditio (10). So, x x accordig to the formulas (1), its solutio is

9 About the types of homogeous liear 9 kx kx 1 y( x) c e c e, for kx 0 y( x) c x c, for k 0 1 (1 ) y( x) c1si( kx) ccos( kx), for kx PART 3 Here we will explore the differetial equatio of secod order with positive costat coefficiets y ( t) ay( t) b y( t) 0, (13) (i [5] it is explored with dimesioal costats). The characteristic equatio of the equatio (13) is Its solutios are r ar b 0. r1 a a b. It is useful to write them i the followig shape a a a b b b r1 ( ( ) 1) b ( k l ) b, k, l k 1. Now, depedig o the value of l, we make the followig classificatio of the equatio (13): 1. for l 0 i.e. k 1, the equatio is a equatio from elliptic type,. for l 0 i.e. k 1, the equatio is a equatio from parabolic type, 3. for l 0 i.e. k 1, the equatio is a equatio from hyperbolic type. Its solutio is give with the formulas: btk y( x) e ( c1cos lbt csi lbt), for l 0, i.e. k 1 bt y( x) e ( c1 ct), for l 0, i.e. k 1 (14) btm bt y( x) c1e ce, for l 0, i.e. k 1, where m k l, k l. Now, if i the differetial equatio (5) we itroduce ew idepedet variable with the relatio x x() t, i order to get the equatio (5) shaped as the differetial equatio (13), we get that the followig coditios have to be satisfied ( ) ( ) x() t f x x t x() t a (15) g( x) x ( t) b. Elimiatig t from (15) ad (16) we get (16)

10 10 Lazo Dimov g( x) f ( x) g( x) a k. (17) 4 g( x) g( x) b Now, if we fid the derivative of (17) we get the fuctioal coectio: f ( x) g( x) g( x) 4 g ( x) f ( x) g( x) g ( x) 3 g( x) 0 (18) which i fact is the coditio that has to satisfy the fuctios f( x ) ad gx ( ) i order the equatio (5) to be able to trasform ito (13). From (16) we come to the coectio betwee the old ad the ew idepedet variable: So, we have prove the followig theorem: bt g( x) dx. (19) Theorem. The differetial equatio (5) ca be trasformed ito a differetial equatio of type (13) if the fuctios f( x ) ad gx ( ) satisfy the coditio (18) ad the ew idepedet variable is give with the relatio (19). Here, we get the geeral solutio accordig to the formulas (14) i which we substitute bt with (19) ad the ratio a k is give with the formula (17). b For the geeral solutio we have the formulas: k g ( x ) dx 1 y e c cos l g( x) dx c si l g( x) dx, for 0 k 1, l 1 k. g( x) dx y e ( c c g( x) dx), for k 1, 1 m g( x) dx g( x) dx y c1e c e, for k 1, where m k l, k l. Example 1. I the differetial equatio 1 x y ( c x ) y xy 0, the fuctios f ( x) c x 1 ad g( x) x satisfy the coditio (18), for k x from (17) we get k c. So, accordig to the formula for the substitute, we have g( x) dx xdx x. 3 Depedig o the value of c we have the followig forms for the solutio of the differetial equatio c 3 x y e ( c1cos 4 c x c 3 si 4 c x ), for 0 c. 3 3

11 About the types of homogeous liear x y ( c 3 1 cx ) e, for c, 3 3 c c 4 c c 4 x x 3 3 y c1e c e, for c. Example. The Euler differetial equatio writte i shape (5) is x y cxy y 0, c x 1 0 x y y y. The fuctios f( x) c ad gx ( ) 1 satisfy the coditio (18) ad for k x x from (17) we get k c1. So, accordig to the formula for the substitute, we have ( ) 1 g x dx dx l x. x Depedig o the value of c we have the followig forms for the solutio of the differetial equatio c y e ( c cos 3 c c l x c si 3 c c l x), for 1 c 3, 1 1 x y ( c c l x), for c 3, 1c c c3 1c c c3 y c1e c e, for c 3. Note. I [] o page 383, ex..75 is prove that the differetial equatio (5) with similar substitutio ca be trasformed ito the differetial equatio with costat coefficiets y ay y 0, which is a special case of the equatio (13). Refereces [1] Аинс, Объкновеньıе дифференциалъньıе уравнения, ИЛ Москва, 1953 [] Е. Камке, Справочник по объкновеньıм дифференциалъньıем уравнениям, ГИ Москва, 1951

12 1 Lazo Dimov [3] Л. Стефановска, Нови квадратурни аспекти на теоријата на линеарните диференцијални равенки, PhD thesis, Skopje, 1994 [4] Д. Димитровски, J. Митевска, Линеарни тригонометрии од четврти и шести ред, ПМФ - Скопје, Посебни изданија [5] B. Vujaović, D. Spasić, Metodi optimizacije, Uiverzitet u Novom Sadu, Fakultet tehičkih auka, 1997 [6] Л. Димов, За решавањето на една диференцијална равенка од втор ред, Математички билтен, Скопје, 1996 [7] Л. Димов, За обликот на решението на една линеарна диференцијална равенка од втор ред со функционални коефициенти, Осми македонски симпозиум по диференцијални равенки, Охрид, [8] Л. Димов, За обликот на решението на една линеарна диференцијална равенка од трети ред со функционални коефициенти, Четврти конгрес на математичари на Македонија, Струга, [9] Л. Димов, За обликот на решението на една линеарна диференцијална равенка од втор ред со функционални коефициенти, (втор сепарат), Математички билтен 011 Ss. Cyril ad Methodius Uiversity Faculty of Mechaical Egieerig, Skopje, R. Macedoia lazo.dimov@mf.edu.mk

13 Математички Билтен ISSN X (prit) Vol. 40(LXVI) No. 3 ISSN (olie) 016 (13-) UDC: : Скопје, Македонија SEQUENTIALLY CONVERGENT MAPPINGS AND COMMON FIXED POINTS OF MAPPINGS IN -BANACH SPACES Marti Lukarevski 1, Samoil Malčeski Abstract. I the past few years, the classical results about the theory of fixed poit are trasmitted i -Baach spaces, defied by A. White (see [3] ad [8]). Several geeralizatios of Kaa, Chatterjea ad Koparde-Waghmode theorems are give i [1], [4], [5] ad [7]. I this paper, several geeralizatios of already kow theorems about commo fixed poits of mappigs i -Baach spaces, are prove, by usig the sequetially coverget mappigs. 1. INTRODUCTION I 1968 White ([3]) itroduces -Baach spaces. -Baach spaces are beig studied by several authors, ad certai results ca be see i [8]. Further, aalogously as i ormed space P. K. Hatikrisha ad K. T. Ravidra i [6] are itroducig the term cotractio mappig to -ormed space as follows. Defiitio 1 ([6]). Let ( L,, ) be a real vector -ormed space. The mappig S : L L is cotractio if there is [0,1) such that Sx Sy, z x y, z, for all x, y, z L. Regardig cotractio mappig Hatikrisha ad Ravidra i [6] proved that cotractio mappig has a uique fixed poit i closed ad restricted subset of -Baach space. Further, i [1], [4], [5] ad [7] are prove more results related to fixed poits of cotractio mappig of -Baach spaces, ad i [7] are prove several results for commo fixed poits of cotractio mappig defied o the same -Baach space. 010 Mathematics Subject Classificatio. 46J10, 46J15, 47H10 Key words ad phrases. -ormed spaces, -Baach spaces, fixed poit, cotractio mappigs 13

14 14 M. Lukareski, S. Malčeski I our further cosideratios, we will give some geeralizatios of the above results for commo fixed poits of mappig defied o the same -Baach space. Thus, the metioed geeralizatios we will do with the help of so-called sequetially coverget mappigs which are defied as follows. Defiitio. Let ( L,, ) be a -ormed space. A mappig T : L L is said to be sequetially coverget if, for every sequece { y }, if { Ty } is coverget the { y } also is coverget.. COMMON FIXED POINTS OF MAPPING OF THE KANNAN TYPE Theorem 1. Let ( L,, ) be a - Baach space, S1, S: L L ad mappig T : L L is cotiuous, ijectio ad sequetially coverget. If 0, 0 are such that 1 ad TS1x TS y, z ( Tx TS1x, z Ty TSy, z ) Tx Ty, z, (1) for each x, y, z L, the S 1 ad S have a uique commo fixed poit z L. Proof. Let x 0 be a arbitrary poit of L ad let the sequece { x } be defied x S x, x S x, for 0,1,,.... If there is 0 such with that x x1 x, the it is easy to prove that u x is a commo fixed S. Therefore, let's assume that there do ot exist three poit for S 1 ad differet cosecutive equal members of the sequece{ x }. So, usig iequalities (1), it is easy to prove that for each 1 ad for each z L the followig holds true Tx Tx, z ( Tx Tx, z Tx Tx, z ) Tx Tx, z ad Tx1 Tx, z ( Tx Tx1, z Tx1 Tx, z ) Tx Tx1, z, from which it follows that Tx Tx, z Tx Tx, z, () for each 0,1,,..., where that Now from iequality () it follows 1 Tx 1 Tx, z Tx1 Tx0, z, (3)

15 SEQUENTIALLY CONVERGENT MAPPINGS AND COMMON 15 for each z L ad for each 0,1,,.... But, the from iequality (3) follows that for each m, N, m ad for each z L the followig holds true m 1 Tx Tx, z Tx Tx, z, m 1 0 which meas that the sequece { Tx } is Cauchy ad because space L is - Baach we get that the sequece { Tx } is coverget. Further, the mappig T : L L is sequetially coverget ad because the sequece { Tx } is coverget, from defiitio follows that the sequece { x } is coverget, i.e. exists u L such that lim x u. Now from the cotiuity of T follows that lim Tx Tu. The, for each z L the followig holds true Tu TS1u, z Tu Tx, z Tx TS1u, z Tu Tx, z TSx1 TS1u, z Tu Tx, z ( Tu TS1u, z Tx1 TSx1, z ) Tu Tx1, z Tu Tx, z ( Tu TS1u, z Tx1 Tx, z ) Tu Tx1, z. If i the last iequality we take that, for each z L the followig holds true Tu TS1u, z Tu TS1u, z, ad sice 1, we coclude that TS1u Tu, z 0, for each z L, i.e. TS1u Tu. But, T is ijectio, so S 1 u u, i.e. u is fixed poit o S 1. Aalogously ca be proved that u is fixed poit of S. Let v L is aother fixed poit of S, i.e. Sv v. The, for each z L the followig holds true Tu Tv, z TS1u TSv, z ( Tu TSv, z Tv TS1u, z ) Tu Tv, z ( ) Tu Tv, z, ad as 1 we get that for each z L the followig holds true Tu Tv, z 0, from which follows Tu Tv. But, T is ijectio, sou v. Corollary 1. Let ( L,, ) be a - Baach space, S1, S: L L ad mappig T : L L is cotiuous, ijectio ad sequetially coverget. If 0, 0 are such that 1 ad

16 16 M. Lukareski, S. Malčeski TxTS1x, z TyTSy, z 1 TxTS1x, z TyTSy, z TS x TS y, z Tx Ty, z, for each x, y, z L, z 0, the S 1 ad S have a uique commo fixed poit z L. Proof. From iequality of coditio follows iequality (1). Now the assertio follows from Theorem 1. Corollary. Let ( L,, ) be a - Baach space, S1, S: L L ad mappig T : L L is cotiuous, ijectio ad sequetially coverget. If 0 1 ad TS 3 1x TS y, z Tx TS1x, z Ty TSy, z Tx Ty, z, for each x, y, z L, the S 1 ad S have a uique commo fixed poit z L. Proof. From the iequality betwee the arithmetic ad geometric mea follows that d( TS1x, TS y) ( d( Tx, TS 3 1x) d( Ty, TSy) d( Tx, Ty)). Now the assertio follows from Theorem 1 for. 3 p q Corollary 3. Let ( L,, ) be a - Baach space, S1, S : L L, pqn, ad mappig T : L L is cotiuous, ijectio ad sequetially coverget. If 0, 0 are such that 1 ad TS p 1 x TS q y, z ( Tx TS p 1 x, z Ty TS q y, z ) Tx Ty, z, for each x, y, z L L.. The S 1 ad S have a uique commo fixed poit u p q Proof. From Theorem 1 follows that mappigs S 1 ad S have a uique commo fixed poit u L. That meas S1 u u, so S1u S1( S1 u) S1 ( S1u ), p ad Su 1 is fixed poit of S 1. Aalogously, we ca prove that Su is fixed q q poit of S. But, from the proof of Theorem 1 follows that mappigs S ad p 1 S have uique fixed poit, so u Su ad u S1u. Accordig to that, u L is a uique commo fixed poit of S 1 ad S. Clearly, if v L is aother uique commo fixed poit of S 1 ad S, the it is a commo fixed poit of S 1 q p q ad S. But, S 1 ad S have a uique commo fixed poit, so v u. p p p p

17 SEQUENTIALLY CONVERGENT MAPPINGS AND COMMON 17 Remark 1. Mappig T : L L defied by Tx x, x L is sequetially coverget. Therefore, if i theorem 1 ad the corollaries 1, ad 3 we take that Tx x follows Theorem 4 ad corollaries 6, 7 ad 8, [7]. 3. COMMON FIXED POINTS OF MAPPINGS OF CHATTERJEA TYPE Theorem. Let ( L,, ) be a - Baach space, S1, S: L L ad mappig T : L L is cotiuous, ijectio ad sequetially coverget. If 0, 0, are such that 1 ad TS1x TS y, z ( Tx TSy, z Ty TS1x, z ) Tx Ty, z, (4) for each x, y, z L, the S 1 ad S have a uique commo fixed poit u L. Proof. Let x 0 is arbitrary poit from L ad the sequece { x } is defied with x S x, x S x, for 0,1,,.... If there is 0 such that x x1 x, the u x is commo fixed poit of S 1 ad S. Therefore, let's assume that there are three differet cosecutive equal members of the sequece{ x }. The, from equality (4) follows that for every z L ad for every 1 the followig holds true Tx Tx, z ( Tx Tx, z Tx Tx, z ) ad Tx Tx1, z, Tx1 Tx, z ( Tx Tx1, z Tx1 Tx, z ) Tx Tx1, z, so for each z L ad for each 0,1,,... the followig holds true Tx1 Tx, z Tx Tx1, z, where 1. The, for each z L 1 followig holds true Tx 1 Tx, z Tx1 Tx0, z ad for each 0,1,,... the. (5) Furthermore, usig the iequality (5), i the same way as i the proof of Theorem 1 ca be proved that the sequece { Tx } is coverget, from where it follows that the sequece { x } is coverget, i.e. there is u L such that lim x u ad lim Tx Tu. We will prove that u is a fixed poit of S 1. For each z L we have

18 18 M. Lukareski, S. Malčeski Tu TS1u, z Tu Tx, z Tx TS1u, z Tu Tx, z TSx1 TS1u, z Tu Tx, z ( Tx1 TS1u, z Tu TSx1, z ) Tu Tx1, z Tu Tx, z ( Tx1 TS1u, z Tu Tx, z ) Tu Tx1, z, L ad if i the last iequality we take we get that for each z the followig holds true Tu TS1u, z Tu TS1u, z, ad how 1, from the last iequality follows TS1u Tu, z 0, for each z L. Now, as i the proof of Theorem 1 we ca coclude that u is fixed poit of S 1. Aalogously ca be proved that u is fixed poit of S. Let v L is aother fixed poit of S, i.e. Sv v. For each z L the followig holds true Tu Tv, z TS u TS v, z 1 ( Tu TS v, z Tv TS u, z ) Tu Tv, z 1 ( ) Tu Tv, z. Sice 1 from the last iequality it follows that for every z L the followig holds true Tu Tv, z 0, from which follows that Tu Tv. But, T is ijectio, so u v. Corollary 4. Let ( L,, ) be a -Baach space, S1, S: L L ad the mappig T : L L is cotiuous, ijectio ad sequetially coverget. If 0, 0 are such that 1 ad TxTSy, z TyTS1x, z 1 TxTSy, z TyTS1x, z TS x TS y, z Tx Ty, z, for each x, y, z L, z 0, the S 1 ad S have a uique commo fixed poit u L. Proof. From iequality of coditio follows iequality (4). Now the assertio follows from Theorem. Corollary 5. Let ( L,, ) be a -Baach space, S1, S: L L ad mappig T : L L is cotiuous, ijectio ad sequetially coverget. If 0 1 ad TS 3 1x TS y, z Tx TSy, z Ty TS1x, z Tx Ty, z, for each x, y, z L, the S 1 ad S have a uique commo fixed poit z L.

19 SEQUENTIALLY CONVERGENT MAPPINGS AND COMMON 19 Proof. From the iequality betwee the arithmetic ad geometric mea follows that d( TS x, TS y) ( d( Tx, TS y) d( Ty, TS x) d( Tx, Ty)) Now the assertio follows from Theorem for. 3 p q Corollary 6. Let ( L,, ) be a -Baach space, S1, S : L L, pqn, ad mappig T : L L is cotiuous, ijectio ad sequetially coverget. If 0, 0 are such that 1 ad TS p 1 x TS q y, z ( Tx TS q y, z Ty TS p 1 x, z ) Tx Ty, z, for each x, y, z L L.. The S 1 ad S have a uique commo fixed poit u Proof. The proof is idetical to the proof of the corollary 5. Remark. The mappig T : L L determied by Tx x, x L is sequetially coverget. Therefore, if i Theorem ad corollaries 4, 5 ad 6 we take Tx x, follows the correctess of Theorem 5 ad corollaries 9, 10 и 11, [7]. 4. COMMON FIXED POINTS OF MAPPINGS OF KOPARDE-WAGHMODE TYPE Theorem 3. Let ( L,, ) be a -Baach space, S1, S: L L ad mappig T : L L is cotiuous, ijectio ad sequetially coverget. If 0, 0, 1 ad TS1x TS y, z ( Tx TS1x, z Ty TSy, z ) Tx Ty, z L., (6) for each x, y, z L, the S 1 ad S have a uique commo fixed poit u Proof. Let x 0 be a arbitrary poit of L ad let the sequece { x } is defied x S x, x S x, for 0,1,,.... If there is a 0 such with that x x1 x, the u x is a commo fixed poit for S 1 ad S. Therefore, let's assume that there do ot exist three cosecutive equal members of the sequece { x }. The, from iequality (6) follows that for each 1 ad for each z L the followig holds true ad Tx1 Tx, z ( Tx Tx1, z Tx1 Tx, z ) Tx Tx1, z,

20 0 M. Lukareski, S. Malčeski Tx1 Tx, z ( Tx Tx1, z Tx1 Tx ) Tx Tx1, z, from which it follows that for each 0,1,,... ad for each z L the followig holds true Tx Tx, z Tx Tx, z, (7) where Now from iequality (7) follows 1 Tx 1 Tx, z Tx1 Tx0, z, (8) for each 0,1,,... ad for each z L. Furthermore, from iequality (8), i the same way as i the proof of Theorem 1 it follows that the sequece { Tx } is coverget, ad therefore the sequece { x } is coverget also, i.e. exists u X such that lim x u ad lim Tx Tu. We will prove that u is fixed poit of S 1. We have Tu TS u, z Tu Tx, z Tx TS u, z 1 1 Tu Tx, z TS1u TSx1, z Tu Tx, z ( Tu TS1u, z Tx1 TSx1, z ) Tu Tx1, z Tu Tx, z ( Tu TS1u, z Tx1 Tx, z Tu Tx1, z for each N ad for each z L. If i the last iequality we take we get that Tu TS1u, z d Tu TS1u, z, for each z L ad how 1, it follows that Tu TS1u, z 0. Now, agai as i the proof of Theorem 1 we coclude that u is fixed poit of S 1. Aalogously it ca be proved that u is fixed poit of S. Let v L be aother fixed poit of S, i.e. Sv v. The, for each z L the followig holds true Tu Tv, z TS1u TSv, z ( Tu TS1u, z Tv TSv, z ) Tu Tv, z Tu Tv, z, ad how 0 1 we get that Tu Tv, z 0, from where it follows that Tu Tv. But, T is ijectio, so u v.

21 SEQUENTIALLY CONVERGENT MAPPINGS AND COMMON 1 p q Corollary 7. Let ( L,, ) be a -Baach space, S1, S : L L, pqn, ad mappig T : L L is cotiuous, ijectio ad sequetially coverget. If 0, 0 are such that 1 ad TS p 1 x TS q y, z ( Tx TS p 1 x, z Ty TS q y, z ) Tx Ty, z, for each x, y, z L L.. The S 1 ad S have a uique commo fixed poit u Proof. The proof is idetical to the proof of the corollary 6. Remark 3. The mappig T : L L determied by Tx x, x L is sequetially coverget. Therefore, if i Theorem 3 ad corollary 7 we take Tx x, it follows the correctess of Theorem 6 ad corollary 1, [7]. CONFLICT OF INTEREST No coflict of iterest was declared by the authors. AUTHOR S CONTRIBUTIONS All authors cotributed equally ad sigificatly to writig this paper. All authors read ad approved the fial mauscript. Refereces [1] A. Malčeski, R. Malčeski, A. Ibrahimi, Extedig Kaa ad Chatterjea Theorems i Baach Spaces by usig Sequetialy Coverget Mappigs, Matematichki bilte, Vol. 40, No. 1 (016), 9-36 [] A. Malčeski, S. Malčeski, K. Aevska, R. Malčeski, New Extesio of Kaa ad Chatterjea Fixed Poit Theorems o Complete Metric Spaces, British Joural of Mathematics & Computer Sciece, Vol. 17, No., (016), 1-10 [3] A. White, -Baach Spaces, Math. Nachr. Vol. 4 (1969), [4] M. Kir, H. Kiziltuc, Some New Fixed Poit Theorems i -Normed Spaces, It. Joural of Math. Aalysis, Vol. 7 No. 58 (013), [5] P. Chouha, N. Malviya, Fixed Poits of Expasive Type Mappigs i - Baach Spaces, Iteratioal Juoral of Aalysis ad Applicatios, Vol. 3 No. 1 (013), 60-67

22 M. Lukareski, S. Malčeski [6] P. K. Hatikrisha, K. T. Ravidra, Some Properties of Accretive Operators i Liear -Normed Spaces, Iteratioal Mathematical Forum, Vol. 6 No. 59 (011), [7] R. Malčeski, A. Ibrahimi, Cotractio Mappigs ad Fixed Poit i - Baach Spaces, IJSIMR, Vol. 4, Issue 4, pp [8] R. Malčeski, K. Aevska, About the -Baach spaces, Iteratioal Joural of Moder Egieerig Research (IJMER), Vol. 4 Iss. 5 (014), 8-3 [9] R. Malčeski, A. Malčeski, K. Aevska, S. Malčeski, Commo Fixed Poits of Kaa ad Chatterjea Types of Mappigs i a Complete Metric Space, British Joural of Mathematics & Computer Sciece, Vol. 18, No. 1, (016), ) Faculty of iformatics, Uiversity Goce Delchev, Shtip, Macedoia address: marti.lukarevski@ugd.edu.mk ) Cetre for research ad developmet of educatio, Skopje, Macedoia address: samoil.malcheski@gmail.com

23 Математички Билтен ISSN X (prit) Vol. 40(LXVI) No. 3 ISSN (olie) 016 (3-7) UDC: Скопје, Македонија ABOUT A TWIN SOLUTION OF THE VEKUA EQUATION Slagjaa Brsakoska Abstract. I the paper mai object of research is the Vekua equatio. Тwo types of fuctios are foud that are strogly coected to each other because it will be prove that if oe of them is a solutio of the Vekua equatio, so will be the other oe with a correspodig coditio. Three differet cases are cosidered. 1. INTRODUCTION G. V. Kolosov i 1909 [1], whe he was solvig a problem from the theory of elasticity, itroduced the expressios 1 ˆ [ u v i ( v u) dw ad x y x y dz 1 ˆ [ u v i ( v u) dw x y x y dz kow as operatory derivatives of a complex fuctio W W( z) u( x, y) iv( x, y) from a complex variable z x iy ad z x iy, respectively. The operator rules for these derivatives are give i the moograph of Г. Н. Положиǔ [] (pages 18-31). I the metioed moograph, are also defied the so called operatory itegrals f ( z ) dz ad f ( z) dz by z x iy ad z x iy, respectively, from the complex fuctio f f () z i the area D, where their operatory rules are prove as well, page Mathematics Subject Classificatio. 34M45, 35Q74. Key words ad phrases. areolar derivative, areolar equatio, solutio, aalytic fuctio, Vekua equatio.

24 4 Slagjaa Brsakoska. FORMULATION OF THE PROBLEM Mai object of this research is the Vekua equatio ˆdW dz АW BW F (1) where the fuctios A A() z, B B() z, F F() z are arbitrary fuctios from complex variable without ay limitatio or coditio that they have to fulfill. Because i geeral case there is o method for fidig its geeral solutio, we explore the idea to fid some solutio of the Vekua equatio (1) i the followig form: W W( ( z), ( z)) () where ( z) is atyaalytic fuctio ad () z is aalytic fuctio. 3. MAIN RESULT ( z ) Case 1. Let W ( z) be a solutio of the equatio (1) (ad is from the form ()), i.e. it is a ratio from oe atyaalytic ad oe aalytic fuctio. That meas that this fuctio satisfies the equatio (1), so if we fid the operator derivative by z from W ad replace it i (1) we get: ˆ ˆ ( ) ˆ dw d z 1 d ( z ) ( ) dz dz ( z) ( z) dz ˆ 1 d ( z ) ( ) ( ) ( ) ( z ( ) ) ( z A B z dz z ( z) ) F dˆ ( z ) ( z) A ( z) ( z) B ( z) ( z) F ( z) ( z) dz If we make oe more trasformatio, we ca get a proof to oe more iterestig ˆ ( ) dz statemet. If we add o both sides the expressio ( ) d z z ad give ito cosideratio that the, we get So, if d ˆ( ( z ) ( z )) ˆ ( ) ˆ ( ) d z ( z) ( z) d z dz dz dz dˆ( ( z ) ( z )) dˆ ( z ) ( z) A( ( z) ( z)) B ( z) ( z) F ( z) ( z) dz dz

25 About a twi solutio of the Vekua equatio 5 ˆ ( ) dz d z ( z) ( z) 1 ad ( z ) 0 (3) we get aother solutio to the Vekua equatio (1), i.e. the fuctio W1 ( z) ( z) which is ot from the form (). It is a atyaalytic fuctio. If we wat a solutio that 0 d coditio i (3) is ˆ ( z) 0. dz W, the z 0 So, ow we ca formulate the prove fact as a theorem., which meas that the secod Theorem 1. Let ( z) be a atyaalytic fuctio ad () z be a aalytic fuctio. If ( z ) W ( z) is a solutio to the Vekua equatio (1), the W1 ( z) ( z) is also a solutio to the Vekua equatio (1), if the coditios (3) are satisfied. ( z) Case. Let W ( z ) be a solutio of the equatio (1) (ad is from the form ()), i.e. it is a ratio from oe aalytic ad oe atyaalytic fuctio. That meas that this fuctio satisfies the equatio (1), so if we fid the operator derivative by z from W ad replace it i (1) we get: ˆ dw ˆ ˆ ( z) ˆ 1 ( z) d ( z ) d ( ) ( z) d ( ) dz dz ( z ) dz ( z ) ( z ) dz ( z) dˆ ( z ) ( z) ( z) ( z ) dz ( z ) ( z ) A( ) B( ) F ( z) dˆ ( z ) ( ) z A ( z) ( z ) B ( z) ( z) F( z) z z dz If we make oe more trasformatio, we ca get a proof to aother iterestig statemet. Here we expect a ew solutio of the Vekua equatio to be the fuctio W1 ( z) ( z). It is aalytic fuctio, so its areolar derivative ˆdW 1 is 0. So, if we add o both sides the expressio dz, the we get ( z) dˆ ( z ) dˆ( ( z) ( z)) ( ) z A( ( z) ( z)) B ( z) ( z) F( z) ( z) z dz dz So, if

26 6 Slagjaa Brsakoska ( z) dˆ ( z ) ( z) ( z) 1 ad ( z) 0 (4) ( z ) dz we get aother solutio to the Vekua equatio (1), i.e. the fuctio W1 ( z) ( z) which is ot from the form (). It is a aalytic fuctio. d Agai, the secod coditio i (4), meas that ˆ ( z) 0. So, ow we ca formulate the prove fact as a theorem. Theorem. Let ( z) be a atyaalytic fuctio ad () z be a aalytic fuctio. If dz ( z) W ( z ) is a solutio to the Vekua equatio (1), the W1 ( z) ( z) is also a solutio to the Vekua equatio (1), if the coditios (4) are satisfied. Case 3. Let W ( z) ( z) is a solutio of the equatio (1) (ad is from the form ()), i.e. it is a product from oe atyaalytic ad oe aalytic fuctio. That meas that this fuctio satisfies the equatio (1), so if we fid the operator derivative by z from W ad replace it i (1) we get: dw ˆ dˆ ˆ ( ) ( ( ) ( )) ( ) d z z z z dz dz dz dˆ ( z ) ( z) A( ( z) ( z)) B( ( z) ( z)) F dz ˆ 1 d ( z ) ( z ) ( ) A B z F 1 ( z ) dz ( z) ( z ) ( z) ( z ) If we make oe more trasformatio, we ca get a proof to oe more iterestig ( z ) dˆ ( z ) statemet. If we add o both sides the expressio ad give ito ( z ) dz cosideratio that ˆ ˆ ˆ ˆ dˆ ( ( z ) 1 ( ) ( ) 1 ( ) ( ) ( ) ( ) ) ( d z ( ) ( ) d z ) d z z d z z z dz z ( ) ( z) dz dz z dz ( z) dz the, we get ˆ dˆ ( ) ( ) ( ) 1 ( ) ( ) ( z z z z d z ) A( ) B F dz ( z ) ( z ) ( z ) ( z) ( z ) ( z ) dz So, if ( z ) dˆ ( z ) ( z) ( z) 1 ad 0 (5) ( z ) dz

27 About a twi solutio of the Vekua equatio 7 we get aother solutio to the Vekua equatio (1), i.e. the fuctio d which is ot from the form (). Agai, (5) meas that ˆ ( z) 0. dz 1 ( z ) W ( z ) So, ow we ca formulate the prove fact as a theorem. Theorem 3. Let ( z) be a atyaalytic fuctio ad () z be a aalytic fuctio. If W ( z) ( z) is a solutio to the Vekua equatio (1), the 1 ( z ) W ( z ) is also a solutio to the Vekua equatio (1), if the coditios (5) are satisfied. Note. As we ca see i all three cases, the secod twi solutio is ot from the form (). The fuctios are differet, but the coditios are similar. Refereces [1] Г. В. Колосов, Об одном приложении теории функции комплесного переменного к плоское задаче математическои упругости, 1909 [] Г. Н. Положии, Обопштение теории аналитических фукции комплесного переменного, Издателство Киевского Университета, 1965 [3] D. S. Mitriovic, Kompleksa aaliza, Grageviska kjiga,beograd, 1971 [4] Д.Димитровски: Прилог кон теоријата на обопштените аналитички функции, Годишен зборник на ПМФ, Скопје, књ. 0, 1970 [5] М. Чанак, Неки доприноси теорији полианалитичких диференцијалних једначина, Математички весник, Београд, 001 [6] S. Brsakoska, B. Ilievski, Two theorems for the Vekua equatio, MASSEE Iteratioal Cogress o Mathematics MICOM 009, , Ohrid, Republic of Macedoia Ss. Cyril ad Methodius Uiversity Faculty of Natural Scieces ad Mathematics, Arhimedova 3, 1000 Skopje, R. Macedoia sbrsakoska@gmail.com

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29 Математички Билтен ISSN X (prit) Vol. 40(LXVI) No. 3 ISSN (olie) 016 (9-36) UDC Скопје, Македонија REMARK ABOUT CHARACTERIZATION OF -INNER PRODUCT Kateria Aevska 1 ad Risto Malčeski Abstract. Characterizatio of -ier product is focus of iterest of may mathematicias. I this paper proofs of two characterizatios of - ier product, which are actually cosequeces of the Theorem 1 [15], are give. Also, geeralizatios of already kow Hayashi (see [4], pg. 97) ad Zaratoello ([5]) iequalities are fully elaborated. 1. INTRODUCTION The cocepts of -orm ad -ier product are two-dimesioal aalogies to the cocepts of orm ad ier product. S. Gähler ([13]), 1965, gave the term of -orm ad R. Ehret ([11]), 1969, proved the followig: If ( L,(, )) is a -pre-hilbert space, the 1/ x, y ( x, x y), (1) for all x, y L, defies a -orm. So, we get the -ormed space ( L,, ) ad furthermore for all x, y, z ϵ L the followig equalities are satisfied: ( x, y z) xy, z xy, z, () 4 x y, z x y, z ( x, z y, z ), (3) The equality (3) is aalogue to the parallelogram equality, ad it is said to be parallelepiped equality. Moreover, -ormed space L is -pre-hilbert if ad oly if the equality (3) is satisfied for all x, y, z L. The papers [1]-[3], [6], [1], [14]-[16] cosist of may prove characterizatios about -ier product. 010 Mathematics Subject Classificatio. 46C50, 46C15, 46B0 Key words ad phrases. -orm, -ier product, parallelepiped equality 9

30 30 K. Aevska, R. Malčeski Theorem 1 ([15]). Let ( L,, ) be a real -ormed space. The, L is a -pre- Hilbert space if ad oly if the followig coditio is satisfied: if 3, x1, x,..., x, z L ad a1, a,..., a are real umbers such that аi 0 i1, the аi xi, z аiа j xi x j, z i1 1i j. (4). CHARACTERIZATION OF -PRE-HILBERT SPACE The characterizatio of -ieer product by applyig the Euler-Lagrage type of equality is elaborated i [6] or i other words geeralizatio of Corollary. [8], is elaborated. The followig theorem is oe other proof of the above stated geeralizatio. Theorem ([6]). Let ( L,, ) be a real -ormed space. The -orm is geerated by -ier product if ad oly if the followig equality is satisfied axby, z bxay, z xz, y, z, (5) for all x, y, z L ad for all abr,,, 0, a b. Proof. Let L be a real -ormed space such that for all x, y, z L ad for all abr,,, 0, a b the equality (5) is satisfied. For a b 1, the equality (5) is trasformed to a equality which is equivalet to the parallelepiped equality, (3), what actually meas that L is - pre-hilbert space i which the -ier product is defied as () ad moreover (1) holds true. Coversely, let -ier product, which determies the -orm, exist ad let x, y, z L ad abr,,, 0 be such that a b is satisfied. For a1 a, a b, a3 ab, x1 x, x y, x3 0 theorem 1 is trasformed as the followig Further, for axby, z a( ab) b( ab) ab x, z y, z x y, z. (6)

31 Remark about characterizatio of -ier product 31 b a ba 1,, 3, x x y, x3 0 a a a 1 x, theorem 1 is trasformed as the followig bx ay, z b( b a ) a( b a ) ab x, z y, z x y, z. (7) Fially, if we summarize the equalities (6) ad (7) ad have also o mid that a b we get the followig axby, z bx ay, z a( ab) b( b a ) ( ) xz, i.e. the equality (5) is satisfied. a b a b xz, b( ab) a( ba) yz ( ), x, z y, z yz,, The followig theorem is actually geeralizatio of M. S. Moslehia ad J. M. Rassias (Corollary., [9]) result. Theorem 3.A real - ormed space ( L,, ) is - pre-hilfert space if ad oly if for each ad for all x1, x,..., x, z L the equality (8) is satisfied x1 ai xi, z ( x1, z ai xi, z ) ai{ 1,1} i ai{ 1,1} i. (8) Proof. Let (8) be satisfied for each ad for all x1, x,..., x, z L. For, x 1 x ad x y the equality (8) is trasformed to the parallelepiped equality (3). That actually meas that L is -pre-hilbert space i which the - ier product is defied as () ad furthermore (1) holds true. Coversely, let a -ormed space ( L,, ) be a -pre-hilbert space, ad x1, x,..., x, z L. For a1 (1 ai) ad x 1 0, Theorem 1 is trasformed as the followig: k

32 3 K. Aevska, R. Malčeski 1 x1 ai xi, z x1 ai xi, z i i (1 ak )( x1, z ai xi, z ) k i ai x1 xi, z аiа j xi x j, z i i j xi, z x1, z ak ak ai xi, z i1 k ki ik ai x1 xi, z аiа j xi x j, z i i j 1 ad sice ai { 1,1}, for i,3,...,, we get equalities of the above type. By summarizig the such obtaied equalities, we get the followig. 1 x1 ai xi, z xi, z ak x1, z ai{ 1,1} i i1 ak{ 1,1} k akai xi, z ak, ai{ 1,1} ki ik i.e. the equality (8) holds true. a x x, z i ak, ai{ 1,1} i ak, ai{ 1,1} i j 1 xi z i1, i 1 i а а x x, z i j i j ( x, z a x, z ), 1 i a{ 1,1} i i 3. GENERALIZATION OF HAYASHI AND ZARANTONELLO INEQUALITIES The followig theorems, are actually geeralizatio of two already kow equalities, obtaied by usig theorem 1. Thus, we will firstly give a geeralizatio of Hayashi (see [4], pg. 97) iequality for complex umbers.

33 Remark about characterizatio of -ier product 33 Theorem 4. Let ( L,, ) be a real -ormed space. The (9) x x1, z x x, z x1 x, z x1 x, z x x3, z x3 x1, z cyclic for all x, x1, x, x3, z L. The iequality is trasformed to a equality, if at least oe of the sets { x x1, z},{ x x, z},{ x x3, z} is liearly depedet or more over if the set x x, z x x, z x x, z { x x 1 3 1, z ( x x ) x x, z ( x x ) xx3, z ( x x ), z} is liearly depedet. Proof. Let at least oe of the sets { x x1, z},{ x x, z},{ x x3, z} be liearly depedet. With o loose of the geerality, let { x x1, z} be such the set, i.e. x x z. The, the properties of -orm imply the followig 1 x x1, z x x, z x1 x, z x x, z x x3, z x x3, z cyclic x1 z x, z x1 z x3, z x x3, z x1 x, z x x3, z x3 x1, z, The above meas that (9) is a equality. Let s suppose that the sets { x x1, z},{ x x, z},{ x x3, z} are liearly idepedet. For x, x1, x, x3, z L 3 a4 а i1 i ad x4 a, a, ad for all 1 a3 R the equality x i Theorem 1, we get that for all аi xi x аi, z ( аi ) ( аi x xi, z ) аiа j xi x j, z i1 i1 i1 i1 1i j3 holds true. The right side of the above equality is oegative. Therefore, for all x, x1, x, x3, z L ad for all a1, a, a3 R the iequality (10) holds true For 3 3 ( аi ) ( аi x xi, z ) аiа j xi x j, z i1 i1 1i j3. (10) x x3, z 3 1, 1, 1,, x x z,, x x z x x z x x z 3 xx, z a a a 1 3 the iequality (10) is trasformed as the followigs xi x j, z x j xk, z xk xi, z,,, x xk, z x i x j z x x k z xxi, z xx j, z x i x j z i jki i jki i jki

34 34 K. Aevska, R. Malčeski xix j, z xxk, z i jki i jki x x, z x x, z i j k x x, z x x, z x x, z xx1, z xx, z xx3, z xix j, z x1x, z x x3, z x3 x1, z xxk, z xx1, z xx, z xx3, z i jki. i jki x x, z x x, z i j k Clearly, the last iequality is equivalet to the iequality (9). The proof implies that the iequality (9) might be trasformed to a equality if (10) is a equality, 3 3 i.e. if the set { а x x а, z} is liearly depedet, that is if the set i i i i1 i1 x x, z x x, z x x, z { x x 1 3 1, z ( x x ) x x, z ( x x ) xx3, z ( x x ), z} is liearly depedet. O the ed of our cosideratios we will geeralize the Zaratoello ([5]), iequality, i.e. we will prove the followig theorem. Theorem 5. Let L be a real -pre-hilbert space ad f : L L be a fuctio such that f ( x) f ( y), z x y, z, (11) holds true, for all x, y, x L, The for all a1, a,..., a 0, such that ad for all y1, y,..., y, z L ai f ( yi ) f ( ak yk ), z aiak ( yi yk, z f ( yi ) f ( yk ), z ) i1 k1 1i k ai 1 i1 (1) holds true. Proof. For xi f ( yi), i 1,,..,., x1 f ( ai yi ) i1 ad a 1 1, i Theorem 1 ad the by usig the iequality (11) ad the ai 1 i1 properties of -orm, we get that for all a1, a,..., a 0 such that ad for all y1, y,..., y, z L, the followig holds true

35 Remark about characterizatio of -ier product 35 ai f yi f ak yk z i1 k1 ai f ( yi ) f ( ak yk ), z аiаk f ( yi ) f ( yk ), z i1 k1 1ik ai yi ak yk, z аiаk f ( yi ) f ( yk ), z i1 k1 1ik ai ak yi ak yk, z аiаk f ( yi ) f ( yk ), z i1 k1 k1 1ik ai ak ( yi yk ), z аiаk f ( yi ) f ( yk ), z. i1 k1 1ik xk yi yk, k 1,,...,, x 0 ad a 1 1 i ( ) ( ), O the other had, for 1 Theorem 1 ad also by usig that ai 0, for i 1,,...,, we get that ak ( yi yk ), z ak yi yk, z аiа j yi y j, z k1 k1 1i j ak yi yk, z. k1 Fially, the last two iequalities imply the iequality (1). CONFLICT OF INTEREST No coflict of iterest was declared by the authors. AUTHOR S CONTRIBUTIONS All authors cotributed equally ad sigificatly to writig this paper. All authors read ad approved the fial mauscript. Refereces [1] C. Dimiie, A. White, -Ier Product Spaces ad Gâteaux partial derivatives, Commet Math. Uiv. Caroliae 16(1) (1975), [] C. Dimiie, A. White, A characterizati of -ier product spaces, Math. Japo. No. 7 (198),

36 36 K. Aevska, R. Malčeski [3] C. Dimiie, S. Gähler, A. White, -Ier Product Spaces, Demostratio Mathematica, Vol. VI (1973), [4] D. S. Mitriović, J. E. Pečarić, V. Voleec, Recet Advaces i Geometric Iequalities, Kluwer Academic Publisher, 1989 [5] E. H. Zaratoello, Projectios o covex sets i Hilbert spaces ad spectral theory, Cotributios to Noliear Fuctioal Aalysis, Acad. Press., New York, 1971, [6] K. Aevska, R. Malčeski, Characterizatio of -ier product usig Euler- Lagrage type of equality, Iteratioal Joural of Sciece ad Research (IJSR), ISSN , Vol. 3 Issue 6 (014), [7] M. Fréchet, Sur la defiitio axiomatique d ue classe d espaces vectoriels distaciés applicable vectorillemet sur l espace de Hilbert, A. of Math. Vol. 36 (3) (1935), [8] M. S. Moslehia, J. M. Rassias, A characterizatio of ier product spaces cocerig a Euler-Lagrage idetity, Commu. Math. Aal. Vol. 8, o. (010), 16-1 [9] M. S. Moslehia, J. M. Rassias, A characterizatio of ier product spaces, Kochi. J. Math., Vol. 6 (011), [10] P. Jorda, J. vo Neuma, O ier products i liear, metric spaces, A. of Math. Vol. 36 (3) (1935), [11] R. Ehret, Liear -Normed Spaces, Doctoral Diss., Sait Louis Uiv., 1969 [1] R. Malčeski, K. Aevska, Characterizatio of -ier product by strictly covex -orm of modul c, Iteratioal Joural of Mathematical Aalysis, Vol. 8, o. 33 (014), [13] S. Gähler, Lieare -ormierte Räume, Math. Nachr. 8 (1965), 1-4 [14] S. Malčeski, A. Malčeski, K. Aevska, R. Malčeski, Aother characterizatios of -pre-hilbert space, IJSIMR, e-issn , p- ISSN X, Vol. 3, Issue (015), pp [15] S. Malčeski, K. Aevska, R. Malčeski, New characterizatio of -pre- Hilbert Space, Мatematički bilte, Vol. 38(XLIV) o. 1 (014), [16] Y. J. Cho, S. S. Kim, Gâteaux derivatives ad -Ier Product Spaces, Glasik Math., Vol. 7 (47) (199), 71-8 [17] D. S. Mariescu, M. Moea, M. Opimcariu, M. Stroe, Some Equivalet Characterizatios of Ier Product Spaces ad Their Cosequeces, Filomat 9 (7) (015), ) Faculty of iformatics, FON Uiversity, Skopje, Macedoia address: aevskak@gmail.com ) Faculty of iformatics, FON Uiversity, Skopje, Macedoia address: risto.malceski@gmail.com

37 Математички Билтен ISSN X (prit) Vol. 40(LXVI) No. 3 ISSN (olie) 016 (37-4) UDC: : Скопје, Македонија ABOUT THE ACCORDANCE BETWEEN THE CANONICAL VEKUA DIFFERENTIAL EQUATION AND THE GENERALIZED HOMOGENEOUS DIFFERENTIAL EQUATION Slagjaa Brsakoska Abstract. I the paper two equatios, the caoical Vekua differetial equatio ad the geeralized homogeeous differetial equatio, are cosidered. The mai result is the theorem with the coditio for the accordace betwee this two equatios. 1. INTRODUCTION The equatio ˆdW dz AW BW F (1) where A A() z, B B() z ad F F() z are give complex fuctios from a complex variable zd is the well kow Vekua equatio [1] accordig to the ukow fuctio W W() z u iv. The derivative o the left side of this equatio has bee itroduced by G.V. Kolosov i 1909 []. Durig his work o a problem from the theory of elasticity, he itroduced the expressios ad 1 ˆ [ u v i ( v u)] dw () x y x y dz 1 ˆ [ u v i ( v u)] dw x y x y dz (3) 010 Mathematics Subject Classificatio. 34M45, 35Q74. Key words ad phrases. areolar derivative, areolar equatio, aalytic fuctio, Vekua equatio, geeralized homogeeous differetial equatio.

38 38 Slagjaa Brsakoska kow as operator derivatives of a complex fuctio W W( z) u( x, y) iv( x, y ) from a complex variable z x iy ad z x iy correspodig. The operatig rules for this derivatives are completely give i the moograph of Г. Н.Положиǔ [3] (page18-31). I the metioed moograph are defied so cold operator itegrals f ( z) dz ad f ( z) dz from z x iy ad z x iy correspodig (page 3-41). As for the complex itegratio i the same moograph is emphasized that it is assumed that all operator itegrals ca be solved i the area D. I the Vekua equatio (1) the ukow fuctio W W() z is uder the sig of a complex cojugatio which is equivalet to the fact that B B() z is ot idetically equaled to zero i D. That is why for (1) the quadratures that we have for the equatios where the ukow fuctio W W() z is ot uder the sig of a complex cojugatio, stop existig. This equatio is importat ot oly for the fact that it came from a practical problem, but also because depedig o the coefficiets A, B ad F the equatio (1) defies differet classes of geeralized aalytic fuctios. For example, for F F( z) 0 i D the equatio (1) i.e. ˆdW AW BW (4) dz which is called caoical Vekua equatio, defies so cold geeralized aalytic fuctios from fourth class; ad for A 0 ad F 0 i D, the equatio (1) i.e. the equatio ˆdW BW defies so cold geeralized aalytic fuctios from dz third class or the (r+is)-aalytic fuctios [3], [4]. Those are the cases whe B 0. But if we put B 0, we get the followig special cases. I the case A 0, B 0 ad F 0 i the workig area D the equatio (1) takes the followig expressio dw ˆ 0 ad this equatio, i the dz class of the fuctios W u( x, y) iv( x, y) whose real ad imagiary parts have ubroke partial derivatives u x, u y, v x ad v y i D, is a complex writig of the Cauchy - Riema coditios. I other words it defies the aalytic fuctios i the sese of the classic theory of the aalytic fuctios. I the case B 0 i D i.e. ˆdW AW F is the so cold areolar liear differetial equatio [3] (page dz 39-40) ad it ca be solved with quadratures by the formula:

39 About the accordace betwee the caoical Vekua 39 A( z) dz A( z) dz W e [ ( z) F( z) e dz ]. Here () z is a arbitrary aalytic fuctio i the role of a itegral costat.. FORMULATION OF THE PROBLEM AND MAIN RESULT I the paper [5], the followig lemma is proved. Lemma. The equatios ad where df ˆ 0 dw dw ˆ f ( z, W ) (5) dz dw ˆ g( z, W ) (6) dz, have commo solutios if ad oly if df ˆ df ˆ dg ˆ dg ˆ dg ˆ g f g dz dw dz dw dw. (7) It is assumed that the operator derivatives i (7) exist ad that they are cotiuous fuctios i the workig area D from the complex plae. I this paper we are examiig the accordace betwee the caoical Vekua equatio (4), o oe side ad the geeralized homogeeous differetial equatio dw ˆ ( W ) (8) dz z o the other side, where z is a give complex fuctios from a complex variable zd, such that equatio of type (6) where dˆ 0 dw. The caoical Vekua equatio (4) is a g( z, W) AW BW (9) ad the geeralized homogeeous differetial equatio (8) is a equatio of type (5), where W z f ( z, W) ( ). (10)

40 40 Slagjaa Brsakoska Here, the fuctio f is a aalytic fuctio accordig to W, which meas that df ˆ 0 dw. That is the oly coditio to be accomplished, so that we ca use the metioed lemma. If we calculate all the derivatives i (7), we get that df ˆ dˆ df ˆ ˆ ˆ, d ( W d ) 1, dz dz dw dw z dw z dg ˆ da ˆ db ˆ dg ˆ dg ˆ W W, A, B. dz dz dz dw dw Ad if we put them i (7) we get that d ˆ d ˆ 1 ˆ ˆ ( AW BW ) da W db W A B( AW BW ). dz dw z dz dz Now we write the last equatio i the followig form dˆ ˆ ˆ ˆ ˆ ( A da d d ) ( B db W B W AB) A 0. dw z dz dw z dz dz This liear combiatio is true oly if the followig system of equatio is satisfied dˆ A da ˆ B 0 dw z dz dˆ B db ˆ AB 0. dw z dz dˆ A 0 dz (11) ˆd If we elimiate the derivative dw from the first ad the secod equatio i the system (11), we get dˆ z ˆ ( da ) dw B A dz z ˆ ˆ ( da B ) B db AB 0 A dz z dz da ˆ db ˆ B A B B B A 0 dz dz Because of the fact that dˆ 1 ˆ ˆ ( A) ( da B AdB), we get that dz B B dz dz ˆ B d ( A) B( B A ) 0 dz B or ˆ d A B A dz B B ( ) 0 d ˆ ( A ) B A A 0 (1) dz B B