Stability of General Cubic Mapping in Fuzzy Normed Spaces

An. Ş. Univ. Ovidius Consanţa Vol. 20, 202, 29 50 Sabiliy of General Cubic Mapping in Fuzzy ormed Spaces S. Javadi, J. M. Rassias Absrac We esablish some sabiliy resuls concerning he general cubic funcional equaion fx + ky kfx + y + kfx y fx ky = 2kk 2 fy for fixed k \} in he fuzzy normed spaces. More precisely, we show under some suiable condiions ha an approximaely cubic funcion can be approximaed by a cubic mapping in a fuzzy sense and we esablish ha he exisence of a soluion for any approximaely cubic mapping guaranees he compleeness of he fuzzy normed spaces. Inroducion and preliminary resuls In order o consruc a fuzzy srucure on a linear space, in 984, Kasaras [8] defined a fuzzy norm on a linear space o consruc a fuzzy vecor opological srucure on he space. A he same year Wu and Fang [8] also inroduced a noion of fuzzy normed space and gave he generalizaion of he Kolmogoroff normalized heorem for a fuzzy opological linear space. In [6], Biswas defined and sudied fuzzy inner produc spaces in a linear space. Since hen some mahemaicians have defined fuzzy norms on a linear space from various poins of view [5, 0, 20, 27, 29]. In 994, Cheng and Mordeson inroduced a definiion of fuzzy norm on a linear space in such a manner ha he corresponding induced fuzzy meric is of Kramosil and Michalek ype [9]. Key Words: Generalized Hyers-Ulam-Rassias sabiliy, Cubic funcional equaion, Fuzzy normed spaces. 200 Mahemaics Subjec Classificaion: Primary: 9B55; Secondary: 9B52. Received: May, 20. Acceped: Sepember, 20. 29

0 S. Javadi, J. M. Rassias In 200, Bag and Samana [] modified he definiion of Cheng and Mordeson [7] by removing a regular condiion. They also esablished a decomposiion heorem of a fuzzy norm ino a family of crisp norms and invesigaed some properies of fuzzy norms see [4]. The concep of sabiliy of a funcional equaion arises when one replaces a funcional equaion by an inequaliy which acs as a perurbaion of he equaion. In 940, Ulam [28] posed he firs sabiliy problem. In he nex year, Hyers [2] gave an affirmaive answer o he quesion of Ulam. Hyers s heorem was generalized by Aoki [] for addiive mappings and by Rassias [22] for linear mappings by considering an unbounded Cauchy difference. The concep of he generalized Hyers-Ulam sabiliy was originaed from Rassias s paper [22] for he sabiliy of funcional equaions. During he las decades several sabiliy problems for various funcional equaions have been invesigaed by many mahemaicians; we refer he reader o [9,, 6, 2, 24, 5]. Also, he ineresed reader should refer o [0,, 2, ] and [4] and he references herein. Following [], we give he noion of a fuzzy norm as follows. Le X be a real linear space. A funcion : X R [0, ] so-called fuzzy subse is said o be fuzzy norm on X if for all x, y X and all s, R, x, c = 0 for c 0; 2 x = 0 if and only if x, c = for all c > 0; cx, = x, c if c 0; 4 x + y, s + x, s, y, }; 5 x,. is a non-decreasing funcion of R and lim x, = ; 6 For x 0, x,. is upper semi coninuous on R. I follows from 2 and 4 ha if s <, hen x, x, s, 0, s} = x, s. Therefore he condiion x,. is a non-decreasing funcion of R in 5 can be omied. The pair X, is called a fuzzy normed linear space. One may regard x, as he ruh value of he saemen he norm of x is less han or equal o he real number. Example. Le X,. be a normed linear space. Then + x x, =, > 0, 0, 0 is a fuzzy norm on X. Example 2. Le X,. be a normed linear space. Then 0, < x, x, =, x is a fuzzy norm on X.

Sabiliy of General Cubic Mapping in Fuzzy ormed Spaces Le X, be a fuzzy normed linear space. Le x n } be a sequence in X. Then x n } is said o be convergen if here exiss x X such ha lim x n x, = n for all > 0. In ha case, x is called he limi of he sequence x n and we wrie lim x n = x. A sequence in a fuzzy normed space X, is called Cauchy if for each ε > 0 and each > 0 here exiss n 0 such ha for all n n 0 and all p > 0, x n+p x n, > ε. I is known ha every convergen sequence in a fuzzy normed space is Cauchy. If in a fuzzy normed space, each Cauchy sequence is convergen, hen he fuzzy norm is said o be complee and he fuzzy normed space is called a fuzzy Banach space. The funcional equaion fx + ky kfx + y + kfx y fx ky = 2kk 2 fy. for fixed k wih k \ is called he general cubic funcional equaion, since he funcion fx = x is is soluion. Every soluion of he general cubic funcional equaion is said o be cubic mapping. From., puing x = y = 0 yields f0 = 0. oe ha he lef hand side of. changes sign when y is replaced by y. Thus f is odd. Puing x = 0 and y = x in.. We conclude ha fkx = k fx. By inducion, we infer ha fk n x = k n fx for all posiive ineger n. The sabiliy problem for he cubic funcional equaion was proved by Jun and Kim [4] for mappings f : X Y, where X is a real normed space and Y is a Banach space. Laer a number of mahemaicians worked on he sabiliy of some ypes of he cubic equaion [2,, 5, 7, 25, 26]. ajai in [2] esablished he general soluion and he generalized Hyers-Ulam sabiliy for he equaion.. In he nex secion we proved he non-uniform version of he generalized Hyers-Ulam sabiliy of he general cubic funcional equaion. in fuzzy normed spaces. The uniform version is discussed in Secion. Finally, in secion 4, we show ha he exisence of a condiional cubic mapping for every approximaely cubic ype mapping implies ha our fuzzy normed space is complee. 2 Fuzzy generalized Hyers-Ulam heorem:non-uniform version Theorem 2.. Le k \ }, α [, + and α k. Le X be a linear space and le Z, be a fuzzy normed space. Suppose ha an even funcion

2 S. Javadi, J. M. Rassias φ : X X Z saisfies φk n x, k n y = α n φx, y, for all x, y X and for all n. Suppose ha Y, is a fuzzy Banach space. If a map f : X Y saisfies fx+ky kfx+y+kfx y fx ky 2kk 2 fy, φx, y, 2. for all x, y X and > 0, hen here exiss a unique cubic map C : X Y which saisfies. and inequaliy fx Cx, φ0, x, k α holds for all α < k, x X and > 0. Also, fx Cx, φ0, x, α k holds for all α > k, x X and > 0., φ0, x, kk2 k α α, φ0, x, kk2 α k α Proof. Case : α k. Replacing y := y in 2. and adding he resul o 2. yield fy + f y, φx, y, kk 2, φx, y, kk 2 }. 2.2 Since 2.2 and 2. hold for any x, le us fix x = 0 for convenience. Using 4, we obain 2fky 2k fy, φ0, y, min φ0, y, min φ0, y, I follows ha 2fky 2fy, k, fky + f ky,, fy + f y,, φ0, ky, kk2, φ0, y, k2, φ0, y, kk2 α φ0, y, k Replacing y by x in 2.4, by we have } k } }. 2., φ0, y, k4 k 2 }. α 2.4 fkx k fx, φ0, x, 2k, φ0, x, 2k4 k 2 }. α 2.5 } }

Sabiliy of General Cubic Mapping in Fuzzy ormed Spaces Replacing x by k n x in 2.5, we ge fk n+ x fkn x k n+ k n, k n φ0, k n x, 2k φ0, x, 2k α n I follows from and above inequaliy ha fk n x k n, φ0, k n x, 2k4 k 2 α, φ0, x, 2k4 k 2 }. α n+ fk n n x fk i+ x k n fx = fki x k i+ k i i=0 n fx, i=0 min φ0, x, 2k α i k i n i=0 fk i+ x, φ0, x, 2k4 k 2 α } fki x k i+ k i }., αi k i } We replace x by k m x o prove convergence of he sequence fkn x k n }. For m, n, fk n+m x fkm x k n+m k m n, i=0 φ0, k m x, 2k φ0, x, 2k α m α i k i+m Replacing by α m in las inequaliy o ge fk n+m x fkm x k n+m k m φ0, x, 2k, φ0, k m x, 2k4 k 2 α, φ0, x, 2k4 k 2 }. n+m, i=m For every n and m 0}, we pu a mn := α i k i α m+, φ0, x, 2k4 k 2 α n+m i=m α i k i. }. }

4 S. Javadi, J. M. Rassias Replacing by a mn in las inequaliy, we observe ha φ0, x, fk n+m x fkm x k n+m k m, 2k a mn, φ0, x, 2k4 k 2 αa mn }. Le > 0 and ε > 0 be given. Using he fac ha lim φ0, x, =, we can find some 0 such ha φ0, x, 2 > ε for every 2 >. The convergence of he series i=0 } αi k guaranees ha here exiss some m i 2k such ha min a mn, 2k4 k 2 αa mn >, for every m m and n. For every m m and n, we have fk n+m x fkm x k n+m k m, 2k φ0, x,, φ0, x, 2k4 k 2 } a mn αa mn ε, ε} = ε. 2.6 Hence fkn x k } is a Cauchy sequence in he fuzzy Banach space Y,, herefore his sequence converges o some poin Cx Y. n Fix x X and pu m = 0 in 2.6 o obain fk n x k n fx, φ0, x, 2k a 0n For every n, Cx fx,, φ0, x, 2k4 k 2 αa 0n Cx fkn x k n, fk n x, 2 k n fx, }. 2 }. The firs wo erms on he righ hand side of he above inequaliy end o as n. Therefore Cx fx, Cx fkn x k n, fk n x, 2 k n fx, } 2 k φ0, x,, φ0, x, k4 k 2 } a 0n αa 0n for n large enough. By las inequaliy, we have Cx fx, φ0, x, k α φ0, x, kk2 k α α, 2.7 }

Sabiliy of General Cubic Mapping in Fuzzy ormed Spaces 5 Replacing x, y by k n x and k n y, respecively in 2. o ge fk n x + ky k n kfkn x + y k n + kfkn x y k n fkn x ky φk n x, k n y, k n φx, y, kn α n for all x, y X and all > 0. Since α < k, by 5 lim φx, y, kn n α n =. k n 2kk2 fk n y k n We conclude ha C fulfills.. To prove he uniqueness of he cubic funcion C, assume ha here exiss cubic funcion C : X Y which saisfies 2.7. Fix x X. Obviously, Ck n x = k n Cx, C k n x = k n C x for all n. I follows from 2.7 ha Ck Cx C n x x, = k n C k n x k n, Ck n x k n fkn x k n, fk n x, 2 k n C k n x k n, } 2 φ0, k n x, k αk n, φ0, k n x, kk2 k αk n } 6 6α φ0, x, k αk n 6α n, φ0, x, kk2 k αk n } 6α n+. Since α < k, we obain lim φ0, x, k αk n n 6α n = lim 6α n+ n φ0, x, kk2 k αk n =. Therefore Cx C x, = for all > 0, whence Cx = C x. Case 2: α > k. We can sae he proof in same paern as we did in firs case. Replacing x, by x k and 2, respecively in 2. o ge fx k f x k, φ0, x k We replacing y and by x k and n k n 2,, φ0, x k, 2kk2 α }. in las inequaliy, respecively, we find

6 S. Javadi, J. M. Rassias ha k n f x k n x kn+ f k n+, x 2 φ0,, kn+ k n, φ0, φ0, x, 2αn+ k n, φ0, x, 2kk2 α n k n For each n, one can deduce k n f x k n fx, φ0, x, 2α b 0n x k n+, 2kk2 } k n α }., φ0, x, 2k2 b 0n } where b 0n = n i=0 ki α. I is easy o see ha k n f x i k } is a Cauchy sequence n in Y,. Therefore his sequence converges o some poin Cx Y in he Banach space Y. Moreover, C saisfies. and fx Cx, φ0, x, α k, φ0, x, kk2 α k α The proof for uniqueness of C for his case, proceeds similarly o ha in he previous case, hence i is omied. Corollary 2.2. Le X be a Banach space and ε > 0 be a real number. Suppose ha a funcion f : X X saisfies fx + ky kfx + y + kfx y fx ky 2kk 2 fy ε x 2p + y 2p + x p y p for all x, y X where 0 < p < 2 and k \ }. Then here exiss a unique cubic funcion C : X X which saisfying. and he inequaliy }. for all x X. Cx fx < ε x p k k 2p Proof. Define : X R [0, ] by + x x, =, > 0, 0, 0 I is easy o see ha X, is a fuzzy Banach space. Denoe by φ : X X R he map sending each x, y o ε x 2p + y 2p + x p y p. By assumpion fx+ky kfx+y+kfx y fx ky 2kk 2 fy, φx, y,,

Sabiliy of General Cubic Mapping in Fuzzy ormed Spaces 7 noe ha : R R [0, ] given by x, = + x, > 0, 0, 0 is a fuzzy norm on R. By Theorem 2., here exiss a unique cubic funcion C : X X saisfies. and inequaliy + fx Cx = fx Cx, min φ0, x, k k 2p, = min k k 2p k k 2p +ε x, p φ0, x, kk2 k k 2p k 2p } kk 2 k k 2p kk 2 k k 2p +ε x p } = k k 2p k k 2p +ε x p. for all x X and > 0. Consequenly fx Cx ε k k 2p x p. Corollary 2.. Le X be a Banach space and ε > 0 be a real number. Suppose ha a funcion f : X X saisfies fx + ky kfx + y + kfx y fx ky 2kk 2 fy ε x + y x 2 y 2 for all x, y X where k \ }. Then here exiss a unique cubic funcion C : X X which saisfying. and he inequaliy for all x X. Cx fx < x ε k k Proof. Consider he fuzzy norms defined by Corollary 2.2. We define for all x, y X. By assumpion φx, y = ε x + y x 2 y 2 fx+ky kfx+y+kfx y fx ky 2kk 2 fy, φx, y,,

8 S. Javadi, J. M. Rassias for all x, y X and > 0. Therefore here exiss a unique cubic funcion C : X X saisfies. and inequaliy + fx Cx = fx Cx, min φ0, x, k k, = min φ0, x, kk2 k k k k k k k+ x ε, kk 2 k k kk 2 k k+k x ε } } = k k k k+ x ε. for all x X and > 0. Consequenly Cx fx < x ε k k. Le X be a Banach space. Denoe by and he fuzzy norms obained as Corollary 2.2 on X and R, respecively. Le ϵ > 0 and le φ : X X R be defined by φx, y = ε for all x, y X. Le f : X X be a φ-approximaely cubic mapping in he sense ha fx + ky kfx + y + kfx y fx ky 2kk 2 fy < ε, hen here exiss a unique cubic funcion C : X X which saisfies fx Cx ε k for all x X. Le f be a mapping from X o Y. For each k, le Df k : X X Y be a mapping defined by Df k x, y = fx + ky fx ky kfx + y + kfx y 2kk 2 fy Proposiion 2.4. Le X be a linear space and le Y be a normed space. Le ε be a nonnegaive real number and le f be a mapping from X o Y. Suppose ha Df 2 x, y ε 2.8 for all x, y X. Then here exiss a sequence of nonnegaive real numbers ε k } k=0 such ha ε 0 = ε, ε = 4ε, ε 2 = 0ε,..., ε k = 2ε k + k + ε + ε k 2 k and Df k x, y ε k 2. 2.9

Sabiliy of General Cubic Mapping in Fuzzy ormed Spaces 9 Proof. Replacing x by x + y and x y in 2.8, respecively, we ge from 2.8 ha Df x, y 4ε = ε 2.0 for all x, y X. Replacing x by y + x and x y in 2.0, respecively, we ge from 2.8 ha Df 4 x, y 0ε = ε 2 2. for all x, y X. Replacing x by y + x and x y in 2., respecively, we ge from 2.8 and 2.0 Df 5 x, y 28ε = 2ε 2 + 4ε + ε = ε, for all x, y X. Therefore by using his mehod, by inducion we infer 2.9. From abou argumen we have he following Corollary. Corollary 2.5. Le X be a linear space and le Y be a Banach space. Le f be a mapping from X o Y and le ε be a nonnegaive real number. Suppose ha Df 2 x, y ε holds for all x, y X. Then for each posiive ineger k >, here exiss a unique cubic mapping C k : X Y such ha for all x X. C k x fx ε k 2 k Fuzzy Generalized Hyers-Ulam heorem: uniform version In his secion, we deal wih a fuzzy version of he generalized Hyers-Ulam sabiliy in which we have uniformly approximae cubic mapping. Theorem.. Le X be a linear space and Y, be a fuzzy Banach space. Le φ : X X [0, be a funcion such ha ϕx, y = k n φk n x, k n y <. n=0 for all x, y X. Le f : X Y be a uniformly approximaely cubic funcion respec o φ in he sense ha lim fx+ky kfx+y+kfx y fx ky 2kk2 fy, φx, y =.2

40 S. Javadi, J. M. Rassias fk uniformly on X X. Then T x := lim n x n k for each x X exiss n and defines a cubic mapping T : X Y such ha if for some δ > 0, α > 0, fx+ky kfx+y+kfx y fx ky 2kk 2 fy, δφx, y > α. for all x, y X, hen T x fx, δ k ϕ0, x > α for all x X Proof. Le ε > 0, by.2, we can find 0 > 0 such ha fx+ky kfx+y+kfx y fx ky 2kk 2 fy, φx, y ε.4 for all x, y X and all 0. By inducion on n, we shall show ha n fk n x k n fx, m=0 k n m φ0, k m x ε.5 for all 0, all x X and all posiive inegers n. Puing x = 0 and y = x in.4, we ge.5 for n =. Le.5 holds for some posiive inegers n. Then fk n+ x k n+ fx, n m=0 k n m φ0, k m x fk n+ x k fk n x, φ0, k n x, k fk n x k n+ fx, ε, ε} = ε n m=0 } k n m φ0, k m x This complees he inducion argumen. Le = 0 and pu n = p. Then by replacing x wih k n x in.5, we obain fk n+p x fkn x k n+p k n, 0 p m=0 k n+m+ φ0, k n+m x ε.6 for all inegers n 0, p > 0. The convergence of. and he equaion p m=0 k n+m+ φ0, k n+m x = n+p k m=n k m φ0, k m x,

Sabiliy of General Cubic Mapping in Fuzzy ormed Spaces 4 guaranees ha for given δ > 0 here exiss n 0 such ha n+p 0 k k m φ0, k m x < δ, m=n for all n n 0 and p > 0. I follows from.6 ha fk n+p x fkn x k n+p k n fk n+p x fkn x k n+p k n, δ, 0 p m=0 k n+m+ φ0, k n+m x ε.7 for each n n 0 and all p > 0. Hence fkn x k } is a Cauchy sequence in Y. n Since Y is a fuzzy Banach space, his sequence converges o some T x Y. fk Hence we can define a mapping T : X Y by T x := lim n x n namely. For each > 0 and x X, lim n k n, T x fkn x k, =. ow, n le x, y X. Fix > 0 and 0 < ε <. Since lim n k n φk n x, k n y = 0, here is some n > n 0 such ha 0 φk n x, k n y < kn 6 for all n n. Hence for each n n, we infer ha T x + ky kt x + y + kt x y T x ky 2kk 2 T y, T x + ky fkn x + ky k n,, T x + y fkn x + y 6 k n, T x y fkn x y k n, 6k T y fkn y k n, 2kk 2,, T x ky fkn x ky k n, 6 fk n x + ky kfk n x + y + kfk n x y fk n x ky 2kk 2 fk n y, kn 6 }. As n, we see ha he firs five erms on he righ-hand side of he above inequaliy end o and he sixh erm is greaer han D k fk n x, k n y, 0 φk n x, k n y, ha is, by.4, greaer han or equal o ε. Thus T x + ky kt x + y + kt x y T x ky 2kk 2 T y, ε, for all > 0 and 0 < ε <. I follows ha T x + ky kt x + y + kt x y T x ky 2kk 2 T y, =,,, 6k

42 S. Javadi, J. M. Rassias for all > 0 and by 2, we have T x + ky kt x + y + kt x y T x ky = 2kk 2 T y. To end he proof, le for some posiive δ and α,. holds. Le φ n x, y := n m=0 k m φk m x, k m y x, y X. Le x X. By a similar discussion as in he beginning of he proof we can obain from. n fk n x k n fx, δ m=0 k n m φ0, k m x α,.8 for all posiive inegers n. Le s > 0. We have fx T x, δφ n 0, x + s.9 fx fkn x fk n k n, δφ x n0, x, k n } T x, s fk n x k n Combining.7,.8 and he fac ha lim n obain ha fx T x, δφ n 0, x + s α..0 T x, s =, we for large enough n. By he upper semi coninuiy of he real funcion fx T x,., we obain ha fx T x, δ k ϕ0, x + s α. Taking he limi as s 0, we conclude ha fx T x, δ k ϕ0, x α. Corollary.2. Le X be a linear space and Y, be a fuzzy Banach space. Le φ : X X [0, be a funcion saisfying.. Le f : X Y be a uniformly approximaely cubic funcion wih respec o φ. Then here is a unique cubic mapping T : X Y such ha uniformly on X. lim fx T x, ϕ0, x =.

Sabiliy of General Cubic Mapping in Fuzzy ormed Spaces 4 Proof. The exisence of uniform limi.9 immediaely follows Theorem.. I remains o prove he uniqueness asserion. Le S be anoher cubic mapping saisfying.9. Fix c > 0. Given ε > 0, by.9 for T and S, we obain some 0 > 0 such ha fx T x, 2 ϕ0, x ε, fx Sx, 2 ϕ0, x ε, for all x X and all 0. Fix some x X and find some ineger n 0 such ha Since 0 m=n k n φ0, k m x = m=n k m φ0, k m x < c 2, n n 0. = k n k n k m n φ0, k m n k n x m=n k i φ0, k i k n x = k n ϕ0, kn x, we have fk n x Sx T x, c k n T x, c, Sx fkn x 2 k n, c } 2 } = min fk n x T k n x, kn 2 c, Sk n x fk n x, kn 2 c fk n x T k n x, k n 0 k n φ0, k m x, i=0 m=n Sk n x fk n x, k n 0 m=n } k n φ0, k m x = minfk n x T k n x, 0 ϕ0, k n x, Sk n x fk n x, 0 ϕk n x} ε I follows ha Sx T x, c = for all c > 0. Thus T x = Sx for all x X. 4 Fuzzy Compleeness We proved ha under suiable condiions including he compleeness of a space, for every approximaely cubic funcion here exiss a unique cubic mapping which is close o i. I is naural o ask wheher he converse of his resul

44 S. Javadi, J. M. Rassias holds. More precisely, under wha condiions involving approximaely cubic funcions, our fuzzy normed space is complee. The following resul gives a parial answer o his quesion. Definiion 4.. Le X, be a fuzzy normed space. A mapping f : 0} X is said o be approximaely cubic if for each α 0, here is some n α such ha fi + kj kfi + j + kfi j fi kj 2kk 2 fj, α, for all i 2j n α. Definiion 4.2. Le X, be a fuzzy normed space. A mapping f : 0} X is said o be a condiional cubic if for all i 2j. fi + kj kfi + j + kfi j fi kj = 2kk 2 fj, Theorem 4.. Le X, be a fuzzy normed space such ha for each approximaely cubic ype mapping f : 0} X, here is a condiional cubic mapping C : 0} X such ha lim n Cn fn, =. Then X, is a fuzzy Banach space. Proof. Le x n } be a Cauchy sequence in X,. sequence n l } of naural numbers There is an increasing x n x m, 0kl l for each n, m n l. Pu y l = x nl and define f : 0} X by fl = l y l. Le α 0, and find some n 0 such ha n 0 > α. Since fi + kj kfi + j + kfi j fi kj 2kk 2 fj =kj [y i+kj y i+j ] + kj [y i+kj y i j ] + k 2kj [y i+kj y j ] +k j [y i kj y j ] + k 2 ij 2 [y i+kj y i kj ] + kij 2 [y i+j y i j ] +ki 2 j[y i+kj y i+j ] + ki 2 j[y i kj y i j ] + i [y i+kj y i kj ] + ki [y i j y i+j ],

Sabiliy of General Cubic Mapping in Fuzzy ormed Spaces 45 for each i 2kj, hence fi + kj kfi + j + kfi j fi kj 2kk 2 fj, min y i+kj y i+j, 0kj, y i+kj y i j, 0kj, y i+kj y j, 0k 2kj, y i kj y j, 0k j, y i+kj y i kj, 0k 2 ij, y 2 i+j y i j, 0kij, 2 y i+kj y i+j, 0ki 2 j, y i kj y i j, 0ki 2 j, y i+kj y i kj, 0i, y i j y i+j, 0ki }. Le i > n 0. Then y i+kj y i+j, y i+kj y i j, 0kj y i+kj y i+j, 0kj y i+kj y i j, 0 k i + j α, 0 k i j α, y i+kj y j, 0k 2kj y i+kj y j, 0 k j α, y i kj y j, 0k j y i kj y j, 0 k j α, y i+kj y i+j, 0ki 2 y i+kj y i+j, j Since i kj kj and i i 2 kj, we infer ha y i+kj y i kj, y i+j y i j, 0k 2 ij 2 y i+kj y i kj, 0kij 2 y i+j y i j, 0 k i + j α. 0 k i kj α, 0 k i j α,

46 S. Javadi, J. M. Rassias y i kj y i j, y i+kj y i kj, Therefore y i j y i+j, 0ki 2 y i kj y i j, j 0i y i+kj y i kj, 0ki y i j y i+j, 0 k i j α, 0 k i kj α, 0 k i j α. fi + kj kfi + j + kfi j fi kj 2kk 2 fj, α. This means ha f is an approximaely cubic ype mapping. By our assumpion, here is a condiional cubic mapping C : 0} X such ha lim n Cn fn, =. In paricular, lim n Ck n fk n, =. This means ha lim n C y k n, k =. Hence he subsequence n y k n} of he cauchy sequence x n } converges o y = C. Therefore x n } also converges o y. References [] T. Aoki, On he sabiliy of he linear ransformaion in Banach spaces, J. Mah. Soc. Japan., 2 950, 64 66. [2] C. Baak and M. S. Moslehian, On he sabiliy of orhogonally cubic funcional equaions, Kyungpook. Mah. J., 47 2007, 69 76. [] T. Bag and S. K. Samana, Finie dimenional fuzzy normed linear spaces, J. Fuzzy Mah., 200, 687 705. [4] T. Bag and S. K. Samana, Fuzzy bounded linear operaors, Fuzzy Ses and Sysems, 5 2005, 5 547. [5] V. Balopoulos, A. G. Hazimichailidis and B. K. Papadopoulos, Disance and Similariy measures for fuzzy operaors, Inform. Sci., 77 2007, 26 248. [6] R. Biswas, Fuzzy inner produc spaces and fuzzy normed funcions, Inform. Sci., 5 99, 85 90. [7] S.C. Cheng and J.. Mordeson, Fuzzy linear operaors and fuzzy normed linear spaces, Bull. Calua Mah. Soc., 86 994, 429 46.

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Sabiliy of General Cubic Mapping in Fuzzy ormed Spaces 49 [5] T. Z. Xu, J. M. Rassias and W. X. Xu, On he sabiliy of a general mixed addiive-cubic funcional equaion in random normed spaces, Journal of Inequaliies and Applicaions, Vol. 200, Aricle ID 2847, 6 pages, 200. S. Javadi, Deparmen of Mahemaics, Semnan Universiy, P. O. Box 595-6, Semnan, Iran Email: s.javadi62@gmail.com J. M. Rassias, Pedagogical Deparmen aional and Capodisrian Universiy of Ahens 4, Agamemnonos Sr., Aghia Paraskevi, Aikis 542, Ahens, Greece. Email: jrassias@primedu.uoa.gr

50 S. Javadi, J. M. Rassias