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2、2BǤm:2012c4HÆa¬ÆØ©HÆa¬ÆØ©M5`²√£âÆØ©a.A/3/t0y/0¤ïĤJ.é©ïÄóÑ z<Ú8N,þ®3©¥®²(ª5².(²{Æ(Jòd<«ú.ö¶:FϵHÆa¬ÆØ©¦^
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4、,∗)ÚÝþ+(G2,·,d),Ù¥d(·,·)Ýþ.½ε>0,3δ>0¦XJf:G1→G2Né¤kx,y∈G1þkd(f(x∗y),f(x)·f(y))<δ.´Ä3Óg:G1→G2¦é¤kx∈G1,d(f(x),g(x))<ε?1941c,D.H.Hyers)ûBanachmþN½5¯K.3e5Acp,NõêÆ[é«ØÓ¼§½5?1XÚïÄ,~Xê§!g¼§!ng¼§±92¼§.1978c§Th.M.Rassias)û5N3Banachm½5¯K;1999cY.Lee
5、ÚK.JunïÄ2ÂJensen§½5.ù½5¤J3Å©Û!7KêÆÚ°êÆ+¥þk2A^.©©üÙ.31Ù¥,·ïÄÜÉܼ§x+y+zrf()=f(x)+f(y)+f(z),rÙ¥r∈(0,3){1},¿?ØÙ3C∗-êþ½5¯K.31Ù¥,·ïÄ#g!g!ngN¼§f(2x+y)+f(2x−y)=2f(2x)+2f(x+y)+2f(x−y)−4f(x)−f(y)−f(−y))9Ù½5¯K.·ÄkÑT§Ï),, ©O3n<âmÚCÄn<âm¥ïÄÙHyers-Ulam½5¯
6、K.'c:Hyers-Ulam½5;C∗-ê;N;gN;ngN;n<âm;CÄn<âm.iHÆa¬ÆØ©AbstractThestabilityproblemoffunctionalequationsoriginatedfromaquestionofUlamconcerningthestabilityofgrouphomomorphismsin1940:Giveagroup(G1,∗)andametricgroup(G2,·,d)withthemetricd(·,·).Give>0,doesthe
7、reexistsaδ>0suchthatiff:G1→G2satisfiesd(f(x∗y),f(x)·f(y))<δforallx,y∈G1,thenisthereahomomorphismg:G1→G2withd(f(x),g(x))<εforallx∈G1?In1941,D.H.HyerssolvedthestabilityproblemofadditivemappingonBanachspaces.Inthefollowingdecades,manymathematicianshavestudiedthe
8、stabilityofdifferentkindsoffunctionalequationssuchasexponentialequation,quadraticfunctionalequation,cubicfunctionalequation,generalizedadditiveequationandsoon.In1978,Th.M.RassiassolvedthestabilityproblemoflinearmappinginBanachspace.In1999,Y.LeeandK.Junstudied
9、thestabilityofgeneralizedJensenequation.Thesestabilityresultshaveapplicationsinsomerelatedfieldssuchasrandomanalysis,financialmathematicsandactuarialmathematics.Thisthesisconsistsoftwochapters.Inc