Stability of Quadratic and Cubic Functional Equations in Paranormed Spaces

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1 IOSR Joua o Matheatics IOSR-JM e-issn 8-578, p-issn Voue, Issue Ve. IV Ju - Aug. 05, - Stabiit o uadatic ad ubic Fuctioa Equatios i aaoed Spaces Muiappa, Raa S Depatet o Matheatics, Adhiaaa oege o Egieeig, ai Nadu, Idia Depatet o Matheatics, Eode Ats ad Sciece oege, ai Nadu, Idia Abstact I this pape, we pove Hes-Ua Stabiit o a cass o uadatic ad ubic uctioa equatios i aaoed spaces. Kewods Hes-Ua Stabiit, uadatic Fuctioa Equatios, ubic Fuctioa Equatios, aaoed Spaces I. Itoductio he cocept o stabiit o uctioa equatios aises whe we epace the uctioa equatio b a iequait which acts as a petubatio o the equatio. I 90, S. M. Ua [] asked the questio coceig the stabiit o goup hooophiss. Net ea Hes [] gave the ist positive aswe to the questio o Ua o Baach spaces. I 978 h.m. Rassias [] povided a geeaizatio o the Hes theoe which aows the auch dieece to be ubouded. Ate this esut a o atheaticias wee attacted ad otivated to ivestigate the stabiit pobes o uctioa equatios. I paticua, the Stabiit pobees o dieet uctioa equatios have bee ivestigated i vaious spaces. Recet.ak ad D..Shi[] peseted Hes-Ua Stabiit o a cass o uadatic, ubic ad uatic uctioa equatios i paaoed spaces. he uctioa equatio. is a quadatic uctioa equatio ad eve soutio o the quadatic uctioa equatio is said to be a quadatic uctio. he Fuctioa equatio 5 5. is a cubic uctioa equatio ad eve soutio o the cubic uctioa equatio is said to be a cubic uctio. I this pape, we ivestigate the Hes-Ua Stabiit o the uadatic Equatio. ad ubic equatio. i aaoed spaces. his pape is ogaized as oows I Sectio, we pove the Hes-Ua stabiit o quadatic uctioa equatio. i paaoed space. I Sectio, we pove the Hes-Ua stabiit o cubic uctioa equatio. i paaoed space. II. eiiaies houghout this pape, we assue that, is a Fechet space ad that, is a Baach Space. Deiitio. A Noed Space ove K is a pai V,, whee V is a vecto space ove K ad V R, such that i 0 i 0 ii iii o a K ad v V o a, V Deiitio.[] Let be a vecto space. A paao [0, is a uctio o such that i 0 0 ii iii iage Iequait I t is a sequece o scaa with t t ad with 0, iv the t t 0, is caed a paaoed space. hods. he the pai DOI / age

2 Stabiit o uadatic ad ubic Fuctioa Equatio i aaoed Spaces DOI / age Deiitio. he aao is caed tota i i additio we have = 0 ipies = 0 Deiitio. he Fechet space is tota ad copete paaoed space. III. Hes - ua stabiit o quadatic uctioa equatios I this sectio, we dea with the stabiit pobe o the oowig quadatic uctioa equatio i paaoed spaces. heoe. Let, be positive ea ubes with, ad et be a appig satisig 0 = 0 ad. o a., he thee eists a uique quadatic appig such that. o a. oo uttig = i., we get o a. So o a., Hece. o a oegative iteges ad with > ad o a. It oows o. that the sequece is a auch sequece o a. Sice is copete, the sequece coveges. So oe ca deie the appig b i. o a. Moeove, ettig = 0 ad passig the iit i., we get..

3 Stabiit o uadatic ad ubic Fuctioa Equatio i aaoed Spaces DOI / age It oows o. that 0 i i i o a., Hece o a., ad so the appig is quadatic. Now et be aothe quadatic appig satisig.. he we have..8 which teds to zeo as o a. So we ca cocude that o a. his poves the uiqueess o. hus the appig is a uique quadatic appig satisig.. heoe. Let be a ea positive ube with, ad et be a appig satisig 0 = 0 ad.5 o a,. he thee eists a uique quadatic appig such that.6 o a oo Lettig = i.5, we get ad so

4 o a. Siia o a. Hece Stabiit o uadatic ad ubic Fuctioa Equatio i aaoed Spaces o a oegative iteges ad with > ad o a. It oows o.6 that the sequece is a auch sequece o a. Sice is copete, the sequece coveges. So oe ca deie the appig b i.8 o a. Moeove, ettig = 0 ad passig the iit i.7, we get.6. It oows o.5 that i DOI / age i i 0 o a,. hus o a, ad so the appig is quadatic. Now et be aothe quadatic appig satisig.6. he we have.. 8..

5 Stabiit o uadatic ad ubic Fuctioa Equatio i aaoed Spaces DOI / age which teds to zeo as o a. So we ca cocude that o a. his poves the uiqueess o. hus the appig is a uique quadatic appig satisig.5. IV. Hes ua stabiit o cubic uctioa equatio I this sectio we pove the Hes Ua stabiit o the oowig cubic uctioa equatio i paaoed spaces. 5 5 heoe. Let, be positive ea ubes with, ad et be a appig such that 5 5. o a., he thee eists a uique cubic appig such that. o a. ROOF uttig = 0 i., we get o a. So o a. Now o a. Hece.. o a oegative iteges ad with > ad o a. It oows o. that the sequece is a auch sequece o a. Sice is copete, the sequece coveges. So oe ca deie the appig b i o a.

6 Stabiit o uadatic ad ubic Fuctioa Equatio i aaoed Spaces DOI / age Moeove, ettig = 0 ad passig the iit i., we get.. It oows o. that i 0 i 5 5 i o a., Hece 5 5 o a, ad so the appig is cubic. Now et be aothe quadatic appig satisig.. he we have.. which teds to zeo as o a. So we ca cocude that o a. his poves the uiqueess o. hus the appig is a uique cubic appig satisig.. heoe. Let be a ea positive ube with, ad et be a appig such that 5 5. o a., he thee eists a uique quadatic appig such that.5 o a. oo Lettig = 0 i., we get o a ad so o a. Hece

7 Stabiit o uadatic ad ubic Fuctioa Equatio i aaoed Spaces.6 o a oegative iteges ad with > ad o a. It oows o.6 that the sequece is a auch sequece o a. Sice is copete, the sequece coveges. So oe ca deie the appig b i o a. Moeove, ettig = 0 ad passig the iit i.6, we get.5. It oows o. that 5 5 i DOI / age i i 0 o a,. hus 5 5 o a, ad so the appig is cubic. Now et be aothe quadatic appig satisig.5. he we have.. which teds to zeo as o a. So we ca cocude that o a. his poves the uiqueess o. hus the appig is a uique cubic appig satisig.5. Reeeces []..ak ad D..Shi, Fuctioa Equatios i aaoed Spaces, advaces i dieece equatios 0. []. hoewa, W Reaks o the stabiit o uctioa equatios, Aequatioes Math., []. zewik, S O the stabiit o the quadatic appig i oed spaces, Abh. Math. Se. Uiv. Habug 6, []. Jug, S Hes-Ua-Rassias stabiit o uctioa equatios i atheatica aasis, Hadoic ess Ic., a Habo, Foida 00. [5]. Rassias, M Fuctioa Equatios ad Iequaities, Kuwe Acadeic, Dodecht 000. [6]. D.H. Hes, O the stabiit o the iea uctioa equatio, oc. Nat. Acad.Sci [7]. Rassias, M, O the stabiit o uctioa equatios i baach spaces, J. Math.Aa.App.5, [8]. Rassias, M, O the stabiit o uctioa equatio ad a pobe o Ua. Acta Math.App, 6, [9]. Ju, K, Ki, H he geaized Hes-Ua-Rassias stabiit o a cubic uctioa equatio J.Math.Aa.App,

8 Stabiit o uadatic ad ubic Fuctioa Equatio i aaoed Spaces [0]. h.m. Rassias, O the stabiit o the iea appig i Baach spaces,oc.ae.math. Soc., 7 978, []. S.M. Ua, obes i Mode Matheatics, Red. hap.vi,wie, New ok, 90. []. D.H.Hes, o the stabiit o the iea uctioa equatios, oc.nat.acad.sci., 9, -. []. h.m.rassias, o the stabiit o iea appig i Baach spaces, oc.ae.soc., 7978, [].. Aoki, O the stabiit o the iea tasoatio i Baach spaces, J.Math.Soc.Japa.950. DOI / age

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