On the Stability of the n-dimensional Quadratic and Additive Functional Equation in Random Normed Spaces via Fixed Point Method

Transcription

1 In. Journal of Mah. Analysis, Vol. 7, 013, no. 49, HIKARI Ld, hp://d.doi.org/ /ijma On he Sabiliy of he n-dimensional Quadraic and Addiive Funcional Equaion in Random Normed Spaces via Fied Poin Mehod Sun Sook Jin Deparmen of Mahemaics Educaion Gongju Naional Universiy of Educaion Gongju , Republic of Korea Yang-Hi Lee Deparmen of Mahemaics Educaion Gongju Naional Universiy of Educaion Gongju , Republic of Korea Copyrigh c 013 Sun Sook Jin and Yang-Hi Lee. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac In his paper, we prove he sabiliy in random normed spaces via fied poin mehod for he funcional equaion n n f j +n ) f j ) f i + j )=0. j=1 j=1 1 i<j n Mahemaics Subjec Classificaion: 39B8, 46S50, 46S40 Keywords: Sabiliy, addiive mapping, random normed space, n-dimensional quadraic and addiive funcional equaion, fied poin heory

2 414 Sun Sook Jin and Yang-Hi Lee 1 Inroducion S. M. Ulam [] raised he sabiliy problem of group homomorphisms and D. H. Hyers [6] gave a parial soluion of Ulam s problem for he case of approimae addiive mappings. Hyers resul was generalized by T. Aoki [1] for addiive mappings and Th. M. Rassias [18] for linear mappings. Since hen, a number of mahemaicians have invesigaed he sabiliy problems of funcional equaions eensively, see [],[4],[5],[10]-[15]. Recall, almos all subsequen proofs in his very acive area have used Hyers mehod, called a direc mehod. Namely, he funcion F, which is he soluion of a funcional equaion, is eplicily consruced, saring from 1 he given funcion f, by he formulae F ) = lim n f n )orf ) = n lim n n f ). In 003, V. Radu [17] observed ha he eisence of he n soluion F of a funcional equaion and he esimaion of he difference wih he given funcion f can be obained from he fied poin alernaive. In 008, D. Mihe and V. Radu [16] applied his mehod o prove he sabiliy heorems of he Cauchy funcional equaion: f + y) f) fy) = 0 1) in random normed spaces. We call soluions of 1) by addiive mappings. Now we consider he following n-dimensional quadraic and addiive ype funcional equaion n n f j +n ) f j ) f i + j )=0. ) j=1 j=1 1 i<j n In 006, K.-W. Jun and H.-M. Kim [9] invesigaed he sabiliy of he funcional equaion ) in classical normed spaces by using he direc mehod. Recenly, he auhors proved he sabiliy of he funcional equaion ) in fuzzy spaces[8] and used he fied poin mehod[7] o prove he sabiliy for he funcional equaion ) in Banach spaces. I is easy o see ha he mappings f) =a + b is a soluion of he funcional equaion ). Every soluion of he n-dimensional quadraic and addiive funcional equaion is said o be a quadraic-addiive mapping. In his paper, using he fied poin mehod, we prove he sabiliy for he funcional equaion ) in random normed spaces. Preliminaries In his secion, we sae he usual erminology, noaions and convenions of he heory of random normed spaces, as in [0,1]. Firsly, he space of all

3 n-dimensional quadraic and addiive funcional equaion 415 probabiliy disribuion funcions is denoed by Δ + := F : R, } [0, 1] F is lef-coninuous and nondecreasing on R, where F 0) = 0 and F + ) =1}. And le he subse D + Δ + be he se D + := F Δ + l F + ) =1}, where l f) denoes he lef limi of he funcion f a he poin. The space Δ + is parially ordered by he usual poinwise ordering of funcions, ha is, F G if and only if F ) G) for all R. The maimal elemen for Δ + in his order is he disribuion funcion ε 0 : R 0} [0, ) given by ε 0 ) = 0, if 0, 1, if >0. Definiion.1 [0]) A mapping τ :[0, 1] [0, 1] [0, 1] is called a coninuous riangular norm briefly, a coninuous -norm) ifτ saisfies he following condiions: a) τ is commuaive and associaive; b) τ is coninuous; c) τa, 1) = a for all a [0, 1]; d) τa, b) τc, d) whenever a c and b d for all a, b, c, d [0, 1]. Typical eamples of coninuous -norms are τ P a, b) = ab, τ M a, b) = mina, b) and τ L a, b) = maa + b 1, 0). Definiion. [1]) A random normed space briefly, RN-space) is a riple X, Λ,τ), where X is a vecor space, τ is a coninuous -norm, and Λ is a mapping from X ino D + such ha he following condiions hold: RN1) Λ ) =ε 0 ) for all >0 if and only if =0, RN) Λ α ) =Λ / α ) for all in X, α 0 and all 0, RN3) Λ +y + s) τλ ), Λ y s)) for all, y X and all, s 0. If X, ) is a normed space, we can define a mapping Λ : X D + by Λ ) = + for all X and >0. Then X, Λ,τ M ) is a random normed space. This space is called he induced random normed space. Definiion.3 Le X, Λ,τ) be an RN-space.

4 416 Sun Sook Jin and Yang-Hi Lee i) A sequence n } in X is said o be convergen o a poin X if, for every >0 and ε>0, here eiss a posiive ineger N such ha Λ n ) > 1 ε whenever n N. ii) A sequence n } in X is called a Cauchy sequence if, for every >0 and ε>0, here eiss a posiive ineger N such ha Λ n m ) > 1 ε whenever n m N. iii) An RN-space X, Λ,τ) is said o be complee if and only if every Cauchy sequence in X is convergen o a poin in X. Theorem.4 [0]) If X, Λ,τ) is an RN-space and n } is a sequence such ha n, hen lim n Λ n ) =Λ ). 3 The sabiliy of he equaion ) for even n We recall he fundamenal resul in he fied poin heory. Theorem 3.1 [3] or [19]) Suppose ha a complee generalized meric space X, d), which means ha he meric d may assume infinie values, and a sricly conracive mapping J : X X wih he Lipschiz consan 0 <L<1 are given. Then, for each given elemen X, eiher dj n, J n+1 )=+, n N 0} or here eiss a nonnegaive ineger k such ha: 1) dj n, J n+1 ) < + for all n k; ) he sequence J n } is convergen o a fied poin y of J; 3) y is he unique fied poin of J in Y := y X, dj k, y) < + }; 4) dy, y ) 1/1 L))dy, Jy) for all y Y. Throughou his paper, le X be a real or comple) linear space and Y a Banach space. In his secion, le n be an even number greaer han 3. For a given mapping f : X Y, we use he following abbreviaions n n Df 1,,, n ) := f +n ) f j ) f i + j ), j j=1 n +1 n 1 }}}} ˆ :=,,,,, ) j=1 1 i<j n for all, 1,,, n X. Now we will esablish he sabiliy for he funcional equaion ) in random normed spaces via fied poin mehod for even n.

5 n-dimensional quadraic and addiive funcional equaion 417 Theorem 3. Le X be a real linear space, Z, Λ,τ M ) be an RN-space, Y,Λ,τ M ) be a complee RN-space, n be an even number greaer han 3, and ϕ :X\0}) n Z. Suppose ha ϕ saisfies one of he following condiions: i) Λ αϕ 1,,, ) n) Λ ϕ 1,,, n) ) for some 0 <α<, ii) Λ ϕ 1,,, ) n) Λ αϕ 1,,, n) ) for some 4 <α for all 1,,, n X\0} and >0. Iff : X Y is a mapping such ha Λ Df1,,, n)) Λ ϕ 1,,, n)) 3) for all 1,,, n X\0} and >0 wih f0) = 0, hen here eiss a unique mapping F : X Y such ha for all 1,,, n X\0} and DF 1,,, n )=0 M, n ) α)) if ϕ saisfies i), Λ f) F ) ) M, n )α 4)) if ϕ saisfies ii) 4) for all X\0} and >0, where M, ) :=τ M Λ ϕˆ)), Λ ϕ ) )}. Moreover if α<1 and Λ ϕ 1,,, n) is coninuous in 1,,, n under he condiion i), hen f F. Proof. We will prove he heorem in wo cases, ϕ saisfies he condiion i) or ii). Case 1. Assume ha ϕ saisfies he condiion i). Le S be he se of all funcions g : X Y wih g0) = 0 and inroduce a generalized meric on S by dg, h) = inf u R + Λ g) h) u) M, ) for all X\0} }. Consider he mapping J : S S defined by Jf) := f) f ) 4 + f)+f ) 8 hen we have J m f) = 1 4 m f m )+f m )) + m f m ) f m )) )

6 418 Sun Sook Jin and Yang-Hi Lee for all X. Le f,g S and le u [0, ] be an arbirary consan wih dg, f) u. From he definiion of d, RN), and RN3), for he given 0 <α< we have αu αu Λ Jg) Jf) = Λ3g) f)) g ) f ) 8 8 } 3αu αu Λ 3g) f)), Λ g ) f ) Λg) f) αu), Λ g ) f ) αu) } } Λ ϕ ) α), Λ α) ϕ ) M, ) for all X\0}, which implies ha djf,jg) α df,g). Tha is, J is a sricly conracive self-mapping of S wih he Lipschiz consan 0 < α < 1. By 3), we see ha Λ f) Jf) = Λ n ) n+)df ) n )Df ) 4nn ) n ) n +) Λ } n+)df ), Λ 4nn ) 4nn ) Df ) 4n 4n ΛDf ) ), Λ Df ) ) } Λ ϕ ) ), Λ ϕ ) )} for all X\0}. I means ha df,jf) 1 < by he definiion of n ) d. Therefore according o Theorem 3.1, he sequence J m f} converges o he unique fied poin F : X Y of J in he se T = g S df,g) < }, which is represened by F ) := lim f m )+f m ) m + fm ) f m ) 4 m m+1 for all X. Since df,f) α df,jf) n ) α) he inequaliy 4) holds. By RN3), we have Λ DF 1,,, n) ) Λ DJ m f 1,,, n), Λ F J 4) m n f) j=1, j) 4 } τ M Λ n )F J m f) i ) : i =1,,n, 4n ) }} τ M Λ J m f F ) i + j ) :1 i<j n 5) nn 1)

7 n-dimensional quadraic and addiive funcional equaion 419 for all 1,,, n X\0} and m N. The las hree erms on he righ hand side of he above inequaliy end o 1 as m by he definiion of F. Now consider ha Λ Df 4) m 1,, m n), Λ Df m 1,, m n), Λ DJ m f 1,, n) 4 m 16 4 m 16 } Λ Df m 1,, m n), Λ m Df 16 1,, m n) m 16 4 τ M Λ ) 4 m Df m 1,, m n), Λ Df 8 1,, m n) 8 m ) m )} Λ Df m 1,, m n), Λ Df 8 1,, m n) 8 4 τ M Λ ) 4 ϕ 1,,, n), Λ ) 8α m ϕ 1,,, n), 8α m Λ ) ϕ 1,,, n), Λ )} ϕ 1,,, n) 8α m 8α m which ends o 1 as m by RN3) and α > 1 for all 1,,, n X\0}. Therefore i follows from 5) ha Λ DF 1,,, n)) =1 for each 1,,, n X\0} and >0. By RN1), his means ha DF 1,,, n )=0 for all 1,,, n X\0}. Assume ha α<1 and Λ ϕ 1,,, n) is coninuous in 1,,, n.ifn, a 1,b 1,a,b,,a n, and b n are any fied inegers wih a 1,a,,a n 0, hen we have ), lim m Λ ϕ m a 1 +b 1 ) 1, m a +b ),, m a n+b n) n)) lim m Λ ϕa 1 + b 1 m ) 1,a + b m ),,a n+ bn )n) m α m lim m Λ ϕa 1 + b 1 m ) 1,a + b m ),,a n+ bn )n)n) m = Λ ϕa 1 1,a,,a N) n n) for all 1,,, n X\0} and >0. Since N is arbirary, we have lim m Λ ϕ m a 1 +b 1 ) 1, m a +b ),, m a n+b n) n)) lim N Λ ϕa 1 1,a,,a N)=1 n n) by RN3) for all 1,,, n X\0} and >0. From hese, we ge 3n ) 7n +6) Λ n 1)F ) f))

8 40 Sun Sook Jin and Yang-Hi Lee lim τ M Λ Df DF ) m m +1), m,, m )), Λ F f) n) m +1))), Λ n )f F ) m +1))n )), Λ n 1)n )F f) m )n 1)n )), } n 1)n ) =1 lim m τ M Λ n 1)n ) F f) m+1 ) Λ ϕ m +1), m,, m ) ),M n)m +1), n ) α)), M m +1), n ) α)),m m, n ) α)), M m+1, n ) α)) } for all X\0}. By RN1) and RN), his means ha f) =F ) for all X\0}. Togeher wih he fac f0) = 0 = F 0), we obain f F. Case. Assume ha ϕ saisfies he condiion ii). Le he se S, d) be as in he proof of Case 1. Now we consider he mapping J : S S defined by Jg) :=g g ) + g + g )) for all g S and V. Noice ha J m g) = g m 1 g )) + 4m m m g + g )) m m and J 0 g) =g) for all X. Le f,g S and le u [0, ] bean arbirary consan wih dg, f) u. From he definiion of d, RN), and RN3), we have 4u Λ Jg) Jf) α = 4u Λ 3g ) f ))+g ) f ) α for all X\0}, which implies ha Λ 3g ) f )) 3u } u α ), Λ g ) f ) α } u u = τ M Λ g ) f ) α, Λ g ) f ) α } Λ, Λ ϕ ) α ϕ ) α M, ) djf,jg) 4 α df,g).

9 n-dimensional quadraic and addiive funcional equaion 41 Tha is, J is a sricly conracive self-mapping of S wih he Lipschiz consan 0 < 4 < 1. Moreover, by 3), we see ha α Λ f) Jf) = Λ n )α n+4 Df n 4 )+ nn ) nn ) Df ) n )α n +4) Λ n+4, Df nn ) ) nn )α } n 4) Λ n 4 nn ) Df Λ ϕ ) α M, ) ) nn )α )} ), Λ ϕ ) α for all X\0}. I means ha df,jf) 1 < by he definiion of n )α d. Therefore according o Theorem 3.1, he sequence J m f} converges o he unique fied poin F : X Y of J in he se T = g S df,g) < }, which is represened by F ) := for all X. Since lim m m 1 f m ) f m )) + 4 m df,f) df,jf) α 1 n )α 4) f m ) + f m ))) he inequaliy 4) holds. Ne we have he inequaliy 5) for all 1,,, n X\0} and n N. The las hree erms on he righ hand side of he inequaliy 5) end o 1 as m by he definiion of F. Now consider ha Λ DJ m f 1,, n) 4) Λ m 1 Df 1 m,, n m ), Λ 16 m 1 Df 1, m,, n m ) 16 } Λ m 1 Df 1 m,, n m ), Λ 16 m 1 Df 1 m,, n m ) 16 α Λ m ) α ϕ 1,,, n), Λ m ) m+3 ϕ 1,,, n), m+3 α Λ m ) α ϕ 1,,, n), Λ m )} m+3 ϕ 1,,, n) m+3 which ends o 1 as m by RN3) for all 1,,, n X\0}. Therefore i follows from 5) ha Λ DF 1,,, n)) =1 for each 1,,, n X\0} and >0. By RN1), his means ha DF 1,,, n )=0

10 4 Sun Sook Jin and Yang-Hi Lee for all 1,,, n X\0}. I complees he proof of Theorem 3.. By he similar mehod used in he proof of Theorem 3., we ge he following heorem. Theorem 3.3 Le X be a linear space, Z, Λ,τ M ) be an RN-space, Y,Λ,τ M ) be a complee RN-space, n be an even number greaer han 3, and ϕ : X n Z. Assume ha ϕ saisfies one of he following condiions: i) Λ αϕ 1,,, ) n) Λ ϕ 1,,, n) ) for some 0 <α<, ii) Λ ϕ 1,,, ) n) Λ αϕ 1,,, n) ) for some 4 <α for all 1,,, n X and >0. Iff : X Y is a mapping saisfying 3) for all 1,,, n X and >0 wih f0) = 0, hen here eiss a unique quadraic-addiive mapping F : X Y saisfying 4) for all X and >0. Moreover if α<1 and Λ ϕ 1,,, is coninuous in n) 1,,, n X\0} under he condiion i), hen f is a quadraic-addiive mapping. Now we have he general Hyers-Ulam sabiliy of he quadraic-addiive funcional equaion ) in he framework of normed spaces. If X is a normed space, hen X, Λ,τ M ) is an induced random normed space. I leads us o ge he following resul. Corollary 3.4 Le X be a linear space, n be an even number greaer han 3, Y be a complee normed-space, and ϕ :X\0}) n [0, ). Suppose ha ϕ saisfies one of he following condiions: i) αϕ 1,,, n ) ϕ, y, z, w) for some 0 <α<, ii) ϕ 1,,, n ) αϕ 1,,, n ) for some 4 <α for all 1,,, n X\0}. Iff : X Y is a mapping such ha Df 1,,, n ) ϕ 1,,, n ) for all 1,,, n X\0} wih f0) = 0, hen here eiss a unique mapping F : X Y such ha DF 1,,, n )=0 for all 1,,, n X\0} and f) F ) Φ) n ) α) Φ) n )α 4) for all X\0}, where Φ) is defined by if ϕ saisfies i), if ϕ saisfies ii) Φ) = maϕ ),ϕ )). Moreover, if 0 <α<1 under he condiion i), hen f F.

11 n-dimensional quadraic and addiive funcional equaion 43 Now we have Hyers-Ulam-Rassias sabiliy resuls of he quadraic-addiive funcional equaion ) in he framework of normed spaces. Corollary 3.5 Le X be a normed space, n be an even number greaer han 3, p R\[1, ] and Y a complee normed-space. If f : X Y is a mapping such ha Df 1,,, n ) 1 p + p + + n p for all 1,,, n X\0} wih f0) = 0, hen here eiss a unique mapping F : X Y such ha DF 1,,, n )=0 for all 1,,, n X\0} and 0 if p<0, n f) F ) p if 0 p<1, n ) p ) n p if p> n ) p 4) 6) for all X\0}. Proof. If we denoe by ϕ 1,,, n )= 1 p + p + + n p, hen he induced random normed space X, Λ,τ M ) holds he condiions of Theorem 3. wih α = p. Corollary 3.6 Le X be a normed space, n be an even number greaer han 3, p R\[1, ] and Y a complee normed-space. If f : X Y is a mapping such ha n Df 1,,, n ) i p i=1, i 0 for all 1,,, n X wih f0) = 0, hen here eiss a unique quadraicaddiive mapping F : X Y saisfying 6) for all X\0}. Proof. If we denoe by ϕ 1,,, n )= n i=1,i 0 i p, hen he induced random normed space X, Λ,τ M ) holds he condiions of Theorem 3.3 wih α = p. 4 The sabiliy of he equaion ) for odd n In his secion, le n be an odd number greaer han 3. We will esablish he sabiliy for he funcional equaions ) in random normed spaces via fied poin mehod for odd n.

12 44 Sun Sook Jin and Yang-Hi Lee Theorem 4.1 Le X be a linear space, Z, Λ,τ M ) be an RN-space, Y,Λ,τ M ) be a complee RN-space, n be an odd number greaer han 3, and ϕ :X\0}) n Z. Assume ha ϕ saisfies one of he following condiions i) and ii) in Theorem 3.3. If f : X Y be a mapping saisfying 3) for all 1,,, n X\0} and >0 wih f0) = 0, hen here eiss a unique mapping F : X Y such ha DF 1,,, n )=0 for all 1,,, n X\0} and Λ f) F ) ) M, n 1) α) ) ) M, n 1)α 4) if ϕ saisfies i), if ϕ saisfies ii) 7) }}}} for all X and >0, where :=,,,,, ) and n+1 n 1 M, ) :=τ M Λ ϕ ) ), Λ ϕ ) )}. Moreover, if α<1 and Λ ϕ 1,,, n) is coninuous in 1,,, n under he condiion i), hen f F. Proof. We will prove he heorem in wo cases, ϕ saisfies he condiion i) or ii). Case 1. Assume ha ϕ saisfies he condiion i). Le S, J be as in Case 1 of he proof of Theorem 3.3. Now, we inroduce a generalized meric on S by dg, h) = inf u R + Λ g) h) u) M, ) for all X\0} }. Noice ha Λ f) Jf) n 1 = Λ Df )+Df ) n 1) Df ) Df ) + n 1) n n 1) Λ ndf ) n 1) ΛDf ) ), Λ Df ) ) } Λ ϕ ) ), Λ ϕ ) )} ) n 1 ), Λ n)df ) n 1) } n ) n 1) for all X\0}. I means ha df,jf) 1 < by he definiion of d. n 1 Using he similar mehod of Case 1 in he proof of Theorem 3.3, we can easily obain he resuls of his case.

13 n-dimensional quadraic and addiive funcional equaion 45 Case. Assume ha ϕ saisfies he condiion ii). Le he se S, d) be as in he proof of Case 1 and le J be as in Case of he proof of Theorem 3.3. Noice ha Λ f) Jf) n 1)α = Λ Df ) Df ) n 1)α τ M n 1) + Df )+Df ) n 1 Λ n+1)df ) n 1) n +1) n 1) α = τ M Λ Df ) α ), Λ Df ) α ) } Λ ϕ ) α ), Λ ϕ ) α ) } M, ) ), Λ n 3)Df ) n 1) n 3) n 1) α for all X\0}. I means ha df,jf) < by he definiion of n 1)α d. Using he similar mehod of Case in he proof of Theorem 3.3, we can easily obain he resul. By he similar mehod used in he proof of Theorem 4.1, we ge he following heorem. Theorem 4. Le X be a linear space, Z, Λ,τ M ) be an RN-space, Y,Λ,τ M ) be a complee RN-space, n be an odd number greaer han 3, and ϕ : X n Z. Assume ha ϕ saisfies one of he following condiions: i) Λ αϕ 1,,, ) n) Λ ϕ 1,,, n) ) for some 0 <α<, ii) Λ ϕ 1,,, ) n) Λ αϕ 1,,, n) ) for some 4 <α, for all 1,,, n X and >0. Iff : X Y is a mapping saisfying 3) for all 1,,, n X and >0 wih f0) = 0, hen here eiss a unique quadraic-addiive mapping F : X Y saisfying 7) for all X and >0. Moreover if α<1 and Λ ϕ 1,,, is coninuous in n) 1,,, n X\0} under he condiion i), hen f is a quadraic-addiive mapping. Now we have a general Hyers-Ulam sabiliy resuls of he quadraic-addiive funcional equaion ) in he framework of normed spaces for n is odd. Corollary 4.3 Le X be a linear space, n be an odd number greaer han 3, and Y be a complee normed-space and f : X Y be a mapping wih f0) = 0 for which here is ϕ :X\0}) n [0, ) such ha Df 1,,, n ) ϕ 1,,, n )

14 46 Sun Sook Jin and Yang-Hi Lee for all 1,,, n X\0}. Ifϕ saisfies one of he following condiions: i) αϕ 1,,, n ) ϕ, y, z, w) for some 0 <α<, ii) ϕ 1,,, n ) αϕ 1,,, n ) for some 4 <α for all 1,,, n X\0}, hen here eiss a unique mapping F : X Y such ha DF 1,,, n )=0 for all 1,,, n X\0} and f) F ) Φ) n 1) α) Φ) n 1)α 4) if ϕ saisfies i), if ϕ saisfies ii) for all X\0}, where Φ) is defined by Φ) = maϕ ),ϕ )). Moreover, if 0 <α<1 under he condiion i), hen f F. Now we have he Hyers-Ulam-Rassias sabiliy of he quadraic-addiive funcional equaion ) in he framework of normed spaces. Corollary 4.4 Le X be a normed space, n be an odd number greaer han 3, p R\[1, ], and Y a complee normed-space. If f : X Y is a mapping such ha Df 1,,, n ) 1 p + p + + n p for all 1,,, n X\0} wih f0) = 0, hen here eiss a unique mapping F : X Y such ha DF 1,,, n )=0 for all 1,,, n X\0} and 0 if p<0, n f) F ) p if 0 p<1, n 1) p ) n p if <p n 1) p 4) 8) for all X\0}. Proof. If we denoe by ϕ 1,,, n )= 1 p + p + + n p, hen he induced random normed space X, Λ,τ M ) holds he condiions of Theorem 4.1 wih α = p.

15 n-dimensional quadraic and addiive funcional equaion 47 Corollary 4.5 Le X be a normed space, n be an odd number greaer han 3, p R\[1, ] and Y a complee normed-space. If f : X Y is a mapping such ha n Df 1,,, n ) i p i=1, i 0 for all 1,,, n X wih f0) = 0, hen here eiss a unique quadraicaddiive mapping F : X Y saisfying 8) for all X\0}. Proof. If we denoe by ϕ 1,,, n )= n i=1,i 0 i p, hen he induced random normed space X, Λ,τ M ) holds he condiions of Theorem 4. wih α = p. References [1] T. Aoki, On he sabiliy of he linear ransformaion in Banach spaces, J. Mah. Soc. Japan, 1950), [] P. W. Cholewa, Remarks on he sabiliy of funcional equaions, Aeq. Mah., ), [3] J. B. Diaz and B. Margolis, A fied poin heorem of he alernaive for conracions on a generalized complee meric space, Bull. Amer. Mah. Soc., ), [4] Z. Gajda, On he sabiliy of addiive mappings, Inerna. J. Mah. and Mah. Sci., ), [5] P. Găvrua, A generalizaion of he Hyers-Ulam-Rassias sabiliy of approimaely addiive mappings, J. Mah. Anal. Appl., ), [6] D. H. Hyers, On he sabiliy of he linear funcional equaion, Proc. Nal. Acad. Sci. USA, ), -4. [7] S.-S. Jin and Y.-H. Lee, A fied poin approach o he sabiliy of he n-dimensional quadraic and addiive funcional equaion, In. J. Mah. Anal. Ruse), 7 013), [8] S.-S. Jin and Y.-H. Lee, Fuzzy sabiliy of an n-dimensional quadraic and addiive ype funcional equaion, In. J. Mah. Anal. Ruse), 7 013), [9] K. W. Jun and H.-M. Kim, On he sabiliy of an n-dimensional quadraic and addiive funcional equaion, Mah. Inequal. Appl., 9 006),

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