ULAM STABILITY OF THE RECIPROCAL FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN FIELDS. 1. Introduction

Transcription

1 Acta Math. Uiv. Comeiaae Vol. LXXXV, (06, pp ULAM STABILITY OF THE RECIPROCAL FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN FIELDS A. BODAGHI, P. NARASIMMAN, J. M. RASSIAS ad K. RAVI Abstract. I this paper, we itroduce a ew geeralized reciprocal fuctioal equatio ad stud its Hers-Ulam-Rassias stabilit. We also provide the couter eamples for some cases, Ulam-Găvruta-Rassias stabilit ad Hers-Ulam-Rassias stabilit i o-archimedea fields.. Itroductio The stabilit problem of fuctioal equatios origiates from the fudametal questio: Whe is it true a mathematical object satisfig a certai propert approimatel must be close to a object satisfig the propert eactl? I coectio with this questio, i 940, Ulam [0] raised a questio cocerig the stabilit of groups homomorphisms. The first partial solutio to Ulam s questio was give b Hers [8] for additive mappigs subject to the Hers coditio f( + f( f( δ o approimatel additive mappigs f : X Y for a fied δ 0 ad all, X, where X is a real ormed space ad Y a real Baach space. I this case, the Cauch additive fuctioal equatio f( + = f( + f( is said to satisf the Hers- Ulam stabilit. I 978, Th. M. Rassias [9] provided a geeralized versio of the theorem of Hers which permitted the Cauch differece to become ubouded. A particular case of Th. M. Rassias theorem regardig the Hers-Ulam stabilit of the additive mappigs was proved b T. Aoki []. The theorem of Rassias was later eteded to all p ad geeralized b ma mathematicias (see [7, 9, 3]. The pheomeo was itroduced ad proved b Th. M. Rassias is called the Hers-Ulam-Rassias stabilit. The Hers-Ulam-Rassias stabilit for various fuctioal equatios have bee etesivel ivestigated b umerous authors; oe ca refer to [, 3, 0, 5, 4, 8, ]. I 994, a geeralizatio of the Th. M. Rassias theorem was obtaied b P. Găvruta [6], who replaced the boud θ ( p + p b a geeral cotrol fuctio φ(,. I , J. M. Rassias [, ] replaced φ b the product of powers of orms. This stabilit is called Ulam-Găvruta-Rassias stabilit ivolvig a product of differet powers of orms. Received March 8, 05; revised Jul 9, Mathematics Subject Classificatio. Primar 39B8, 39B5, 46S0, 47S0. Ke words ad phrases. No-Archimedea field; reciproal mappig; Ulam-Hers stabilit.

2 4 A. BODAGHI, P. NARASIMMAN, J. M. RASSIAS ad K. RAVI I 00, K. Ravi ad B. V. Sethil Kumar [6] ivestigated some results o Ulam-Găvruta-Rassias stabilit of the fuctioal equatio ( r( + = r(r( r( + r(, where r : X (0, is a mappig o the space of o-zero real umbers ad for all, X. It is eas to check the reciprocal fuctio f( = c is a solutio of the fuctioal equatio (. Later, Ravi et al., [7] itroduced the reciprocal differece fuctioal equatio ( + ( r r( + = r(r( r( + r( ad the reciprocal adjoit fuctioal equatio ( + (3 r + r( + = 3r(r( r( + r( ad ivestigated the Hers-Ulam stabilit of the followig equatios ( ad (3, where r : X (0, is a mappig o the space of o-zero real umbers ad for all, X. Further i the same paper, it was proved the fuctioal equatios (, ( ad (3 are equivalet (for some forms of a quadratic reciprocal fuctioal equatio, see [4] ad [5]. With the idea of the above fuctioal equatios, i the curret work, we itroduce a ew geeralized reciprocal fuctioal equatio of the form (4 ( + f + mf ( + = (m + f(f(, m, Z, 0, f( + f( ad discuss the geeral solutio i Sectio. I Sectio 3, we ivestigate its Hers- Ulam-Rassias stabilit ad Ulam-Găvruta-Rassias stabilit i o-archimedea fields. Also, we give some couter eamples for the Hers-Ulam-Rassias stabilit of the reciprocal fuctioal equatio for special cases.. Geeral solutio of equatio (4 I this sectio, we fid the geeral solutio of equatio (4 as follows: Propositio.. Let m, Z with 0, ad let X ad Y be sets of ozero real umbers. If a fuctio f : X Y satisfies the fuctioal equatio (, the f satisfies the fuctioal equatio (4. The coverse is true if =. Proof. Assume f satisfies the fuctioal equatio (. Lettig = i (, we get (5 f( = f( for all X. Repalcig (, b (, i ( ad usig (5, we have f(3 = f( for all X. The above process ca be repeated to obtai 3 (6 f( = f(

3 RECIPROCAL FUNCTIONAL EQUATION 5 for all, X. The equalit (6 implies ( (7 f = f( for all X. We substitute (, b (, i ( ad the appl (7, we deduce (8 ( + f = f(f( f( + f( for all, X. Now, it follows from ( ad (8 the equalit (4 holds. Coversel, suppose f satisfies (4 for the fied iteger m ad =. That is ( + (m + f(f( (9 f + mf( + = f( + f( for all, X. Iterchagig (, ito (, i (9, we get ( (0 f = f( for all X. Now, it follows from (9 ad (0 the equalit ( holds. 3. Hers-Ulam-Rassias stabilit of equatio (4 We firstl recall some defiitios ad results i the settig of o-archimedea fields. A o-archimedea field is a field K equipped with a fuctio (valuatio from K ito [0, such r = 0 if ad ol if r = 0, rs = r s, ad r + s ma{ r, s for all r, s K. Clearl = = ad for all N. We alwas assume, i additio, is o-trivial, i.e., there eists a a 0 K such a 0 0,. A eample of a o-archimedea valuatio is the mappig takig everthig but 0 ito ad 0 = 0. This valuatio is called trivial. Aother eample of a o-archimedea valuatio o a field K is the mappig 0, if r = 0 r = r, if r > 0 r, if r < 0 for a r K. Let p be a prime umber. For a o-zero ratioal umber = p r m i which m ad are coprime to the prime umber p, cosider the p-adic absolute value p = p r o Q. It is eas to check is a o-archimedea orm o Q. The completio of Q with respect to which is deoted b Q p is said to be the p-adic umber field. Note if p >, the = for all itegers. From ow o, for a o-archimedea field K, we put K = K {0. Throughout the paper, we assume K ad M are a o-archimedea field ad a complete

4 6 A. BODAGHI, P. NARASIMMAN, J. M. RASSIAS ad K. RAVI o-archimedea field, respectivel. Let f : K M be a mappig. For the sake of coveiece, we deote ( + (m + f(f( R m f(, = f + mf ( + f( + f( for all, K, m N. I the et theorem, we prove the stabilit of the fuctioal equatio (4 i the spirit of Hers, Ulam ad Rassias. I the et theorem, we prove the Ulam-Hers stabilit of the fuctioal equatio (4 i o-archimedea fields. ( Theorem 3.. Let φ: K K M be a fuctio such φ(, lim = 0 for all, K. Suppose f : K M is a mappig satisfig the iequalit ( R m f(, φ(, for all, K, where m is a iteger. The there eists a reciprocal mappig R: K M such (3 f( R( m φ( for all K, where (4 Moreover, if φ( = sup { φ( j, j : j N {0. j { φ( j lim lim ma, j k : k j < + k = 0, j the R is the uique reciprocal mappig satisfig (3. Proof. Settig (, b (, i ( ad i the resultat replacig b, we arrive f( (5 f( φ(, m for all K. Replacig b i (5, we get f( (6 for all K. f( m φ(, It follows from (6 ad ( the sequece Cauch. Sice M is complete, we coclude { f( f( R( := lim. Usig iductio, oe ca show f( ( f( { ψ( k m sup, k (7 : j N {0 k { f( is is coverget. Set

5 RECIPROCAL FUNCTIONAL EQUATION 7 for all m N ad all K. Takig to approach ifiit i (7 ad usig (4, oe obtais (3. Replacig ad b ad, respectivel, i (, we get ( f (+ + m f( ( + (m + f( f( φ(, (8. f( +f( Lettig the limit as ad usig (, we have ( + R + mr ( + (m + R(R( =, m N, R( + R( for all, K. If R is aother reciprocal mappig satisfig (3, the R( R ( = k R( k R ( k k ma { R( k f( k, f( k R ( k { ( φ j lim m lim ma, j k : k j < + k = 0 j for all K. Therefore, R is uique. Corollar 3.. Let ρ: [0, [0, be a fuctio satisfig ρ( t ρ( ρ(t (t 0, ρ( <. Let δ > 0 ad let f : K M be a mappig satisfig the iequalit R m f(, δ(ρ( + ρ( for all, K. The there eists a uique reciprocal mappig R: K M such (9 f( R( m δρ( for all K. Proof. Defiig φ: K K [0, b φ(, := δ(ρ( + ρ(, we have φ(, ( ρ( φ(, lim lim = 0 for all K. O the other had, { ψ( φ( j = lim ma, j : 0 j < j Also, { φ( j lim lim ma, j k : k j < + k j = φ(,. φ( k, k = lim k = 0. k Now, b applig Theorem 3., we coclude the required result.

6 8 A. BODAGHI, P. NARASIMMAN, J. M. RASSIAS ad K. RAVI The classical eample of the fuctio ρ i the Corrollar 3. is the mappig ρ(t = t p (t [0,, where p > with the etra assumptio <. We have the followig result which is aalogous to Theorem 3. for the fuctioal equatio (4. We iclude the proof. Theorem 3.3. Let φ: K K M be a fuctio such lim ( φ, (0 = 0 for all, K. Suppose f : K M is a mappig satisfig the iequalit ( R m f(, φ(, for all, K, where m is a iteger. The there eists a reciprocal mappig R: K M such ( f( R( m φ( for all K, where (3 φ( = sup { ( j ψ j+, j+ : j N {0. Moreover, if { ( lim lim ma k j φ j+, j+ : k j < + k the R is the uique reciprocal mappig satisfig (3. Proof. Iterchagig (, ito (, i (, we have f( ( f ( m φ, (4 for all K. Replacig b ( (5 f i (4, we get ( f m ( φ, = 0, for all K. The relatios (5 ad (0 impl the sequece { f ( is Cauch. Due to the completess of M, we coclude { f ( is coverget. Set R( := lim f (. B iductio we ca show ( f f( { ( m ma k φ k+, (6 k+ : 0 k < for all N ad all K. Lettig to approach ifiit i (6 ad usig (3, oe ca obtai (. Replacig ad b ad, respectivel, i (, we deduce ( f ( + + m ( f ( + (m + f ( f ( f ( + f ( (7 ( φ,.

7 RECIPROCAL FUNCTIONAL EQUATION 9 Takig the limit as ad usig (0, we get ( + (m + R(R( R + mr ( + =, m Z, R( + R( for all, K. The rest of the proof is doe b similar argumets as i Theorem 3.. Corollar 3.4. Let ρ: [0, [0, be a fuctio satisfig ρ( t ρ( ρ(t (t 0, ρ( <. Let δ > 0, ad let f : K M is a mappig satisfig the iequalit R m f(, δ(ρ( + ρ( for all, K. The, there eists a uique reciprocal mappig R: K M such (8 f( R( m δρ( for all K. Proof. Similar to the proof of Corollar 3., defiig φ: K K [0, via φ(, := δ(ρ( + ρ(, we have ( lim φ, ( ( φ(, lim ρ = 0 for all K. Also, { ( φ( = lim j ψ j+, j+ : 0 j < = φ(,. ad { ( lim lim ma k j φ j+, j+ :, k j < + k = 0. Now, we ca obtai the desired result b applig Theorem 3.3. Corollar 3.5. Let ε, p be a o-egative real umbers such p. Suppose f : K M is a mappig satisfig the iequalit (9 R m f(, ε( p + p for all, K. The, there eists a uique reciprocal mappig R: K M such ε m p p > f( R( ε m p p p < for all K ad m N. Proof. Let φ(, := ( p + p for all K. B Theorems 3. ad 3.3, oe ca obtai the desired result.

8 0 A. BODAGHI, P. NARASIMMAN, J. M. RASSIAS ad K. RAVI Now we shall provide a eample to illustrate the fuctioal equatio (4 is ot stable for p = i Corollar 3.5. Eample. Let a be a fied positive real umber ad φ: R R be a fuctio defied b a, if (, φ( = a, otherwise. Defie the fuctio f : R R via f( = We firstl ote for each R, we have f( φ( ( R. φ( = a = a. Hece f is bouded. We are goig to prove f satisfies ( R m f(, (8 + 6ma + (30 for all, R. If +, the the left had side of (30 is less tha (6+ma. Now suppose 0 < + <. The there eists a positive iteger r such (3 r+ + < r, so r <, r <, or equivaletl, >, r >. Cosequetl, r ( + > ad ( + r >. Therefore, for each = 0,,..., r, we have ad,, ( +, ( + >, ( ( + ( φ + mφ( + r > or r > >, (m + φ ( ( φ ( φ + φ ( = 0 r >

9 RECIPROCAL FUNCTIONAL EQUATION for = 0,,..., r. From the defiitio of f ad (3, we obtai ( + f (m + f(f( + mf( + f( + f( ( ( ( φ ( + ( + mφ( (m + φ φ + ( ( φ + φ ( ( ( φ ( + ( + mφ( (m + φ φ + + ( ( =r φ + φ ( 4a + 3ma = 4 + 3m a r =r ( = (4 + 3ma + ( = (8 + 6ma +. Thus f satisfies (30 for all, R with 0 < + <. We claim the reciprocal fuctioal equatio (4 is ot stable for p = i Corollar 3.5. Suppose o the cotrar, there eists a reciprocal mappig R: R R ad a costat β > 0 satisfig f( R( β (3 for all R. The, we have f( (β + (33. But we ca choose a positive iteger m with ma > β +. Now, if (, m, the (, for all = 0,,..., m. For such, we get f( = φ( m a = m a > (β + which cotradicts (33. Therefore, the reciprocal fuctioal equatio (4 is ot stable i sese of Ulam, Hers ad Rassias if p =, assumed i the iequalit (9. The followig Corollaries are the immediate cosequeces of Theorem 3. ad 3.3 which gives the Hers-Ulam-Rassias ad Ulam-Găvruta-Rassias stabilit of the fuctioal equatio (4. Sice the proofs are similar to the proof of Corollar 3.5, we omit them. Corollar 3.6. Let ε, p be o-egative real umbers such p. Suppose f : K M is a mappig satisfig the iequalit R m f(, ε( p p

10 A. BODAGHI, P. NARASIMMAN, J. M. RASSIAS ad K. RAVI for all, K. The, there eists a uique reciprocal mappig R: K M such ε m p p > f( R( ε m p p p < for all K ad m N. Corollar 3.7. Let ε, p be o-egative real umbers such p. Suppose f : K M is a mappig satisfig the iequalit ] R m f(, ε [ p + p + ( p p for all, K. The there eists a uique reciprocal mappig R: K M such 3 ε m p p > f( R( 3ε m p p p < for all K ad m N. I aalog with Eample, we idicate the followig eample to show the fuctioal equatio (4 ca ot be stable whe p = i Corollar 3.7. Eample. Let φ: R R be a fuctio defied b k, if (, φ( = k, otherwise, where k > 0 is a costat ad a fuctio f : R R defied via φ( f( = for all R. The f satisfies the fuctioal iequalit ( R m f(, (8 + 6mk + (34 + for all, R, while there does ot eist a reciprocal mappig R: R R ad a costat β > 0 such f( r( β (35 for all R. I other words, the fuctio f is bouded because φ( k f( = = k.

11 RECIPROCAL FUNCTIONAL EQUATION 3 If + +, the the left had side of (34 is less tha (6 + ma. Now suppose 0 < + + <. The there eists a positive iteger r such (36 r+ + + < r. The rest of the proof is same as the proof of Eample. Refereces. Aoki T., O the stabilit of the liear trasformatio i Baach spaces, J. Math. Soc. Japa (950, Bodaghi A., Cubic derivatios o Baach algebras, Acta Math. Vietam. 8 (03, Bodaghi A. ad Alias I. A., Approimate terar quadratic derivatios o terar Baach algebras ad C -terar rigs, Adv. Differece Equ. 0, Art. No. (0, doi:0.86/ Bodaghi A. ad Kim S. O., Approimatio o the quadratic reciprocal fuctioal equatio, J. of Fuct. Spaces. Volume 04, Article ID 53463, 5 pages. 5. Bodaghi A., Rassias J. M. ad Park C., Fudametal stabilities of a alterative quadratic reciprocal fuctioal equatio i o-archimedea fields, Proc. Jag. Math. Soc. 8(3 (05, Găvruta P., A geeralizatio of the Hers-Ulam-Rassias Stabilit of approimatel additive mappig, J. Math. Aal. Appl. 84 (994, Gruber P. M., Stabilit of isometries, Tras. Amer. Math. Soc. 45 (978, Hers D. H., O the stabilit of the liear fuctioal equatio, Proc. Nat. Acad. Sci. U.S.A. 7 (94, Isac G. ad Rassias Th. M., Stabilit of ψ additive mappigs: Applicatios to oliear aalsis, It. J. Math. Math. Sci., 9( (996, Jug, S. M. A fied poit approach to the stabilit of the equatio f( + = f(f( f(+f(, Aust. J. Math. Aal. Appl., 6(8 (009, 6.. Rassias J. M., O approimatio of approimatel liear mappigs b liear mappigs, J. Fuct. Aal. 46 (98, Rassias J. M., O approimatio of approimatel liear mappigs b liear mappigs, Bull. Sci. Math. 08 (984, Rassias J. M., Solutio of a problem of Ulam, J. Appro. Theor, 57 (989, Ravi K., Arukumar M. ad M. Rassias J., O the Ulam stabilit for a orthogoall geeral Euler-Lagrage tpe fuctioal equatio, It. J. Math. Stat. 3, No. A08 (008, Ravi K. ad Narasimma P., Stabilit of Geeralized Quadratic Fuctioal Equatio i No-Archmedea L-Fuzz Normed Space, It. J. of Math. Aalsis, 6(4 (0, Ravi K. ad Sethil Kumar B. V., Ulam-Gavruta-Rassias stabilit of Rassias Reciprocal fuctioal equatio, Global J. of App. Math. Math. Sci. 3(- (00, Ravi K., M. Rassias J. ad Sethil Kumar B. V., Ulam stabilit of Reciprocal Differece ad Adjoit Futioal Equatios, Aust. J. Math. Aal. Appl. 8( (0, Ravi K., Thadapai E. ad Sethil Kumar B. V., Stabilit of reciprocal tpe fuctioal equatios, Pa America Math. J. ( (0, Rassias Th. M., O the stabilit of the liear mappig i Baach spaces, Proc. Amer. Math. Soc. 7 (978, Ulam S. M., A collectio of mathematical problems, Itersciece Publishers, Ic. New York, 960.

12 4 A. BODAGHI, P. NARASIMMAN, J. M. RASSIAS ad K. RAVI. Yag S. Y., Bodaghi A. ad Mohd Ata K. A., Approimate cubic -derivatios o Baach -algebras, Abst. Appl. Aal. Volume 0, Article ID 68479, pages, doi:0.55/0/ A. Bodaghi, Departmet of Mathematics, Garmsar Brach, Islamic Azad Uiversit, Garmsar, Ira, abasalt.bodaghi@gmail.com P. Narasimma, Departmet of Mathematics, Jeppiaar Istitute of Techolog, Kuam, Sriperumbudur(TK, Cheai , Idia, simma gtec@ahoo.co.i J. M. Rassias, Sectio of Mathematics ad Iformatics, Pedagogical Departmet, Natioal ad Capodistria Uiversit of Athes, 4, Agamemoos Street, Aghia Paraskevi, 534 Athes, Greece, jrassias@primedu.uoa.gr K. Ravi, Departmet of Mathematics, Sacred Heart College, Tirupattur , TamilNadu, Idia, shckravi@ahoo.co.i