FIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES

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1 IJRRAS 6 () July 0 FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity of Techology-Khuzesta, Behbaha, Ia movahedia@bkatu.ac.i ABSTRACT I this pape, usig the fixed poit alteative appoach, we pove the Hyes Ulam-Rassias stability of the followig quadatic fuctioal equatio i vaious spaces. f ( x y) f ( x y) = f ( x y) f ( y) f ( x) Keywods: Fixed poit, Hyes-Ulam-Rassias stability,quadatic fuctioal equatio.. INTRODUCTION A classical questio i the theoy of fuctioal equatios is the followig: Whe is it tue that a fuctio which appoximately satisfies a fuctioal equatio D must be close to a exact solutio of D?. If the poblem accepts a solutio, we say that the equatio D is stable. The fist stability poblem coceig goup homomophisms was aised by Ulam [] i 90. I the ext yea D.H. Hyes [], gave a positive aswe to the above questio fo additive goups ude the assumptio that the goups ae Baach spaces. I 978, Th. M. Rassias [] poved a geealizatio of Hyes s theoem fo additive mappigs. The esult of Th. M. Rassias has iflueced the developmet of what is ow called the Hyes-Ulam-Rassias stability theoy fo fuctioal equatios. I 99, a geealizatio of Rassia s theoem was obtaied by G a vuta [] by eplacig the boud p p ( x y ) by a geeal cotol fuctio ( xy, ). The fuctioal equatio f ( x y) f ( x y) = f ( x) f ( y) is called a quadatic fuctioal equatio. I paticula, evey solutio of the quadatic fuctioal equatio is said to be a quadatic mappig. A geealized Hyes-Ulam stability poblem fo the quadatic fuctioal equatio was poved by Skof [5] fo mappigs f :, whee is a omed space ad is a Baach space. Cholewa [6] oticed that the theoem of Skof is still tue if the elevat domai is eplaced by a Abelia goup. Czewik [7] poved the geealized Hyes- Ulam stability of the quadatic fuctioal equatio. The stability poblems of seveal fuctioal equatios have bee extesively ivestigated by a umbe of authos, ad thee ae may iteestig esults coceig this poblem ([8],[9][][],[]). Defiitio. et be a set. A fuctio d : [0, ] is called a geealized metic o if d satisfies the followig coditios: ( a ) d( x, y ) = 0 if ad oly if x= y fo all xy, ; ( b ) d( x, y) = d( y, x ) fo all xy, ; () c d( x, z) d( x, y) d( y, z) fo all x, y, z. Note that the oly substatial diffeece of the geealized metic fom the metic is that the age of geealized metic icludes the ifiity. Theoem. [5] et (,d) be a complete geealized metic space ad : be a stictly cotactive mappig with ipschitz costat <. The, fo all x, eithe d( x, x) = () fo all oegative iteges o thee exists a positive itege 0 such that ( a ) d( x, x) < fo all 0 ;

2 IJRRAS 6 () July 0 Movahedia Fixed Poit ad Hyes-Ulam-Rassias Stability ( b ) the sequece { x} () c ( d ) coveges to a fixed poit y of ; 0 y is the uique fixed poit of i the set d( y, y ) d( y, y) fo all y. ={ y : d( x, y) < } ;. RESUTS AND DISCUSSION Theoem. Assume that : [0, ) be a fuctio such that, fo which 0 < < x y ( x, y), () ad f : be a mappig with f (0) = 0 satisfyig f ( x y) f ( x y) f ( x y) f ( x) f ( y) ( x, y) () fo all xy,. The thee exists a uique quadatic mappig Q : such that fo all xy,. ( xx, ) f ( x) Q( x) ( x,0) 9 9 () Poof. It follows fom () fo all, ( x, y) lim =0 xy. et coside the set :={ h: h(0) = 0} ad the mappig d defie o by ( xx, ) d( g, h) := if (0, ) : g( x) h( x) ( x,0), foall, x if. It is easy to show that (, d) whee = is a complete metic space (5). et coside the mappig :, g( x) = g( x) foall x. Fix a (0, ) ad take gh, such that d( g, h) <. By the defiitios of d ad, we have so by (), we have This implies that ( x, x) g x h x x x ( ) ( ) (,0) fo all ( xx, ) g( x) h( x) ( x,0) fo all x. d( g, h) < d( g, h) d( g, h) fo all gh., O the othe had, eplacig y by x i (), we obtai ( xx, ) f ( x) f ( x) f ( x). (6) fo all x. Also, eplacig y by 0 i (()), we have f ( x) f ( x) ( x,0). (7) fo all x. Combiig ((6)), ((7)) ad usig tiagula iequality, we get f ( x) f ( x) = f ( x) f ( x) f ( x) ( f ( x) f ( x)) (8) f ( x) f ( x) f ( x) f ( x) f ( x) (5)

3 IJRRAS 6 () July 0 Movahedia Fixed Poit ad Hyes-Ulam-Rassias Stability fo all x. Theefoe ( xx, ) ( x,0) fo all x. This meas that By Theoem., thee exists a mappig Q : () Q is a fixed poit of, i.e., ( xx, ) f ( x) f ( x) ( x,0), 9 d( f, f ) <. 9 satisfyig the followig: Q x = Q( x ) (0) fo all x. The mappig Q is a uique fixed poit of i the set M ={ g S : d( h, g) < }. implies that Q is a uique mappig satisfyig (0) such that thee exists a (0, ) satisfyig fo all x ; ( xx, ) f ( x) Q( x) Y ( x,0) () d( f, Q) 0 as. This implies the equality lim f x = Q( x) fo all x ; () d( f, Q) d( f, f ), which implies the iequality d( f, f ) d( f, Q). 9 9 xy by x, This implies that the iequalities ((0)) holds. Replacig, have This (9) () y i () ad applyig (()) ad (()), we Q( x y) Q( x y) Q( x y) Q( x) Q( y) () f (( x) y) f ( x ( y)) f ( x y) f ( x) f ( y) = lim ( x, y) lim = 0. So Q( x y) Q( x y) = Q( x y) Q( x) Q( y). Theefoe Q is a quadatic mappig, this completes the poof. Coollay. et, f (0) = 0 satisfyig be o-egative eal umbes such that < ad let f : be a mappig with f ( x y) f ( x y) f ( x y) f ( x) f ( y) ( x y ) () fo all xy,. The thee exists a uique quadatic mappig Q : such that fo all x. x f ( x) Q( x),

4 IJRRAS 6 () July 0 Movahedia Fixed Poit ad Hyes-Ulam-Rassias Stability Poof. The esult follow fom Theoem., whe (, ) = ( x y x y ) fo all xy, ad Theoem. Assume that : [0, ) be a fuctio such that, fo which 0 < < x (, ), y x y ad f : be a mappig with (0) = 0 Q : such that fo all x. Poof. Replacig x by x i (8), we obtai =. f satisfyig (()). The thee exists a uique quadatic mappig ( x, x) f ( x) Q( x) ( x,0) 9 9 (5) x x x x f ( x) f,,0 ( x, x) ( x,0) fo all x. et (, d) mappig : such that x h( x) := h fo all x. et gh, be such that d( g, h) =. The fo all x ad so be the geealized metic space defied i the poof of Theoem.. Coside a liea () (6) (7) ( xx, ) g( x) h( x) ( x,0) x x x x x g( x) h( x) = g h,,0 ( x, x). ( x,0) fo all x. Thus d( g, h) = implies that d( g, h). This meas that d( g, h) d( g, h) fo all gh., It follows fom ((6)) that d( f, f ) <. 9 By Theoem., thee exists a mappig Q : satisfyig the followig: (a) Q is a fixed poit of, that is, (8) x Q = Q( x) 9 fo all x. The mappig Q is a uique fixed poit of i the set ={ hs : d( g, h) < }. This implies that Q is a uique mappig satisfyig ((9)) such that thee exists (0, ) satisfyig (9)

5 IJRRAS 6 () July 0 Movahedia Fixed Poit ad Hyes-Ulam-Rassias Stability fo all x. ( xx, ) f ( x) Q( x) ( x,0) (b) d( f, Q) 0 as. This implies the equality x lim f ( ) = Q( x) fo all x. (c) d( f, f ) d( f, Q) with f, which implies the iequality d( f, Q). () 9 9 The est of the poof is simila to the poof of Theoem.. This completes the poof. be o-egative eal umbes such that > ad let f : f (0) = 0 satisfyig (()). The thee exists a uique quadatic mappig Q : Coollay. et, fo all x. x f ( x) Q( x), (0) be a mappig with such that Poof. The esult follow fom Theoem., whe (, ) = ( x y x y ) fo all xy, ad Theoem. Assume that : [0, ) be a fuctio such that, fo which 0 < < x y ( x, y), ad f : be a mappig with (0) = 0 Q : fo all xy,. such that Poof. By the same easoig as i the poof of Theoem., we obtai that =. f satisfyig (()). The thee exists a uique quadatic mappig () ( x,0) f ( x) Q ( x) () ( x, y) lim = 0. xy, () et coside the set S :={ h: h(0) = 0} ad the mappig d defie o S S by d( g, h):= if{ (0, ): g( x) h( x) ( x,0) foall x } (5) whee if =. It is easy to show that ( Sd, ) is a complete metic space. Replacig y by 0 i (()), we have f ( x) ( x,0) f( x), (6) fo all x. Coside a liea mappig : S S such that h( x) := hx fo all x. et g, h S be such that d( g, h) =. The g( x) h( x) ( x,0) fo all x ad so g( x) h( x) ( x,0) ( x,0) g( x) h( x) = = ( x,0) 5

6 IJRRAS 6 () July 0 Movahedia Fixed Poit ad Hyes-Ulam-Rassias Stability fo all x. Thus d( g, h) = implies that d( g, h). This meas that d( g, h) d( g, h) fo g h S. It follows fom ((6)) that d( f, f ) <. By Theoem., thee exists a mappig Q : satisfyig the followig: all, (a) Q is a fixed poit of, that is, fo all x. The mappig implies that f ( x) Q( x) ( x,0) fo all x. (b) (c) (7) Q x = Q ( x ) (8) Q is a uique fixed poit of i the set ={ h S : d( g, h) < }. This Q is a uique mappig satisfyig ((8)) such that thee exists (0, ) satisfyig f( x) d( f, Q ) 0 as. This implies the equality lim = Q ( x ) fo all x. d( f, f ) d( f, Q ) with f, which implies the iequality d( f, Q ). The est of the poof is simila to the poof of Theoem.. This completes the poof. be o-egative eal umbes such that < ad let f : f (0) = 0 satisfyig (()). The thee exists a uique quadatic mappig Q : Coollay. et, fo all x. x ( ) ( ), f x Q x be a mappig with such that Poof. The esult follow fom Theoem., whe (, ) = ( x y x y ) fo all xy, ad Similaly we have the followig ad we will omit the poofs. Theoem. Assume that : [0, ) be a fuctio such that, fo which 0 < < x (, ), y x y ad f : be a mappig with (0) = 0 Q : such that =. f satisfyig (()). The thee exists a uique quadatic mappig (9) fo all x. ( x,0) f ( x) Q ( x) (0) Coollay. et, be o-egative eal umbes such that > ad let f : f (0) = 0 satisfyig (()). The thee exists a uique quadatic mappig fo all x. x ( ) ( ), f x Q x Q : be a mappig with such that 6

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