Nonlinear Fuzzy Stability of a Functional Equation Related to a Characterization of Inner Product Spaces via Fixed Point Technique

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1 Filoma 29:5 (2015), DOI /FI W Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp:// Nonlinear Fuzzy Sabiliy of a Funcional Equaion Relaed o a Characerizaion of Inner Produc Spaces via Fixed Poin Technique Zhihua Wang a, Prasanna K. Sahoo b a School of Science, Hubei Universiy of Technology, Wuhan, Hubei , P.R. China b Deparmen of Mahemaics, Universiy of ouisville, ouisville, KY 40292, USA Absrac. Using he fixed poin mehod, we prove some resuls concerning he sabiliy of he funcional equaion f (x i 1 j=1 x j ) = f (x i ) f ( 1 x i ) where f is defined on a vecor space and aking values in a fuzzy Banach space, which is said o be a funcional equaion relaed o a characerizaion of inner produc spaces. 1. Inroducion Sabiliy problem of funcional equaions was firs posed by Ulam in [35] which was answered by Hyers in [14] for addiive mappings. Hyers resul, using unbounded Cauchy differen, was generalized for addiive mappings in [1] and for linear mappings in [31]. Since hen here have been several new resuls on sabiliy of various classes of funcional equaions in he Hyers-Ulam sense or Hyers-Ulam-Rassias sense in normed spaces (see [10, 15, 16, 34] and references cied herein). Sabiliy problems of funcional equaions on arbirary groups and on non-abelian groups were reaed in [6 9]. In [20 23], various sabiliy resuls concerning Cauchy, Jensen, quadraic and cubic funcional equaions were invesigaed in fuzzy normed spaces. Furhermore some sabiliy resuls concerning addiive, quadraic, Cauchy-Jensen, mixed ype cubic and quaric funcional equaions were invesigaed (cf. [5, 12, 19, 24, 33]) in he seing of non-archimedean fuzzy normed spaces, non-archimedean -fuzzy normed spaces and generalized fuzzy normed spaces, respecively. I was shown by Rassias [32] ha a normed space (X, ) is an inner produc space if and only if for any finie se of vecors x 1,..., x n X, and a fixed ineger n 2 n x i 1 n n x j 2 = x i 2 n 1 n x i 2. (1) n n j= Mahemaics Subjec Classificaion. Primary 39B82 ; Secondary 39B72, 03F55 Keywords. Fixed poin mehod; Funcional equaions relaed o inner produc space; Fuzzy Banach spaces; Hyers-Ulam-Rassias sabiliy. Received: 8 November 2013; Acceped: 17 July 2014 Communicaed by Dragan S. Djordjevic Research suppored by NSFC # , BSQD12077 and 2014CFB189 addresses: mawzh2000@126.com (Zhihua Wang), corresponding auhor (Zhihua Wang), sahoo@louisville.edu (Prasanna K. Sahoo)

2 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), Employing he above ideniy, Najai and Rassias [26] obained he funcional equaion n f (x i 1 n n x j ) = j=1 n f (x i ) n f ( 1 n n x i ). (2) I is easy o see ha he funcion f (x) = ax 2 + bx is a soluion of he funcional equaion (2). In [17, 28], he auhors inroduced he following funcional equaion f (x i 1 j=1 x j ) = f (x i ) f ( 1 x i ), (3) which is said o be a funcional equaion relaed o a characerizaion of inner produc spaces. They obained he general soluion of equaion (3) and proved he Hyers-Ulam-Rassias sabiliy of his equaion. For noaional simpliciy, we will denoe he funcional equaion (3) as D f (x 1,..., x ) = 0, where D f is given by D f (x 1,..., x ) := f (x i 1 j=1 x j ) f (x i ) + f ( 1 x i ). (4) There are many ineresing new resuls concerning funcional equaions relaed o inner produc spaces have been obained by Park e al. [27] as well as for fuzzy sabiliy of funcional equaions relaed o inner produc spaces [11, 29]. The main purpose of his paper is o esablish a fuzzy version of he Hyers-Ulam-Rassias sabiliy for he funcional equaion (3) in fuzzy Banach spaces by using he fixed poin mehod. This paper is differen from he earlier papers [11, 29] in he sense ha he bound used in his paper for D f (x 1,..., x ) is quie differen from he bound used in [11] (or see [29]). Specializing he funcion ϕ(x 1,..., x ) which is as a par of he bound, we obain several resuls similar o resuls known for sabiliy of he equaion (3) in Banach spaces. 2. Preliminaries In his secion, some definiion and preliminary resuls are given which will be used in his paper. Following [2, 20, 21], we give he following noion of a fuzzy norm. Definiion 2.1. e X be a real vecor space. A funcion N : X R [0, 1] (he so-called fuzzy subse) is said o be a fuzzy norm on X if for all x, y X and all s, R: (N1) N(x, c) = 0 for c 0; (N2) x = 0 if and only if N(x, c) = 1 for all c > 0; (N3) N(cx, ) = N(x, c ) if c 0; (N4) N(x + y, s + ) min{n(x, s), N(y, )}; (N5) N(x, ) is a non-decreasing funcion on R and lim N(x, ) = 1; (N6) for x 0, N(x, ) is coninuous on R. In his case (X, N) is called a fuzzy normed vecor space. Example 2.2. (cf. [25]). e (X, ) be a normed vecor space and α, β > 0. Then { α α+β x, > 0, x X, N(x, ) = 0, 0, x X is a fuzzy norm on X.

3 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), Example 2.3. (cf. [25]). e (X, ) be a normed vecor space and β > α > 0. Then 0, α x, N(x, ) = +(β α) x, α x < β x, 1, > β x is a fuzzy norm on X. Definiion 2.4. (cf. [2, 20, 21]). e (X, N) be a fuzzy normed vecor space. A sequence {x n } in X is said o be convergen if here exiss x X such ha lim N(x n x, ) = 1( > 0). In ha case, x is called he limi of he sequence n {x n } and we denoe by N lim x n = x. n Definiion 2.5. (cf. [2, 20, 21]). e (X, N) be a fuzzy normed vecor space. A sequence {x n } in X is called Cauchy if for each ε > 0 and δ > 0, here exiss n 0 N such ha N(x m x n, δ) > 1 ε (m, n n 0 ). If 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. Definiion 2.6. e E be a se. A funcion d : E E [0, ] is called a generalized meric on E if d saisfies (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) = d(y, x), x, y E; (3) d(x, z) d(x, y) + d(y, z), x, y, z E. The nex resul is due o Diaz and Margolis [4]. emma 2.7. (cf. [4] or [30]). e (E, d) be a complee generalized meric space and J : E E be a sricly conracive mapping wih ipschiz consan < 1. Then for each fixed elemen x E, eiher d(j n x, J n+1 x) = n 0, or d(j n x, J n+1 x) < n n 0, for some naural number n 0. Moreover, if he second alernaive holds hen: (i) The sequence {J n x} is convergen o a fixed poin y of J; (ii) y is he unique fixed poin of J in he se E := {y E d(j n 0 x, y) < + } and d(y, y ) 1 1 d(y, Jy), x, y E. 3. Fuzzy Sabiliy of Funcional Equaions Before proceeding o he proof of he main resuls in his secion, we shall need he following wo lemmas. emma 3.1. (cf. [28]). e V and W be real vecor spaces. If an odd mapping f : V W saisfies (3), hen he mapping f : V W is addiive, ha is, f is a soluion of f (x + y) = f (x) + f (y) for all x, y V. emma 3.2. (cf. [17]). e V and W be real vecor spaces. If an even mapping f : V W saisfies (3), hen he mapping f : V W is quadraic, ha is, f is a soluion of f (x + y) + f (x y) = 2 f (x) + 2 f (y) for all x, y V. In his secion, we assume ha X is a vecor space and (Υ, N) is a fuzzy Banach space. We will esablish he following sabiliy resuls for funcional equaions (3) in fuzzy Banach spaces by using he fixed poin mehod, which is said o be a funcional equaion relaed o inner producs space. For given mapping f : X Υ, le D f : X Υ be a mapping as defined in (4) for all x 1,..., x X.

4 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), Theorem 3.3. e f : X Υ be a mapping saisfying f (0) = 0 for which here exiss a funcion ϕ : X [0, ) such ha N(D f (x 1,..., x ), ) + ϕ(x 1,..., x ) for all x 1,..., x X. If here exis a consan 0 < < 1 such ha ϕ(x 1,..., x ) 2 ϕ(2x 1,..., 2x ) (6) for all x 1,..., x X, hen here exiss a unique addiive mapping A : X Υ saisfying (3) such ha (5) N( f (x) f ( x) A(x), ) for all x X and > 0, where n(1 ) n(1 ) + Φ(x) (7) Φ(x) = ϕ(2x,..., 2x, 0,..., 0) + ϕ( 2x,..., 2x, 0,..., 0). (8) } {{ }} {{ }} {{ }} {{ } Proof. Seing x 1 = = x n = x and x n+1 = = x = 0 in (5), we obain N(3n f ( x 2 ) + n f ( x 2 ) n f (x), ) + ϕ(x,..., x, 0,..., 0) } {{ }} {{ } for all x X and all > 0. Replacing x by x in (9), we ge N(3n f ( x 2 ) + n f (x 2 ) n f ( x), ) + ϕ( x,..., x, 0,..., 0) } {{ }} {{ } for all x X and all > 0. Thus N(( f ( x 2 ) f ( x )) n( f (x) f ( x)), 2) 2 min{n(3n f ( x 2 ) + n f ( x ) n f (x), ), N(3n f ( x 2 2 ) + n f (x ) n f ( x), )} 2 + ϕ(x,..., x, 0,..., 0) + ϕ( x,..., x, 0,..., 0) } {{ }} {{ }} {{ }} {{ } for all x X and all > 0. eing (x) = f (x) f ( x) and (9) (10) (11) Φ(x) = ϕ(2x,..., 2x, 0,..., 0) + ϕ( 2x,..., 2x, 0,..., 0) } {{ }} {{ }} {{ }} {{ } for all x X, we ge N( ( x ) n (x), 2) 2 + Φ( x 2 ) (12) for all x X and all > 0. Therefore N( (2x) 2 (x), 2 n ) + Φ(x) (13)

5 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), for all x X and all > 0. e E 1 be he se of all funcions q 1 : X Υ. e us inroduce a generalized meric on E 1 as follows: { } d 1 (q 1, h 1 ) := inf λ [0, ] N(q 1(x) h 1 (x), λ), x X, > 0. + Φ(x) I is easy o prove ha (E 1, d 1 ) is a complee generalized meric space [3, 13, 18]. Now we consider he funcion J 1 : E 1 E 1 defined by J 1 q 1 (x) := 2q 1 ( x 2 ), for all q 1 E 1 and x X. (14) e q 1, h 1 E 1 and le λ [0, ] be an arbirary consan wih d 1 (q 1, h 1 ) λ. From he definiion of d 1, we have N(q 1 (x) h 1 (x), λ) + Φ(x) for all x X and all > 0. Hence N(J 1 q 1 (x) J 1 h 1 (x), λ) = N(2q 1 ( x 2 ) 2h 1( x ), λ) 2 for all x X and all > 0. Thus = N(q 1 ( x 2 ) h 1( x 2 ), 2 λ) = Φ( x 2 ) + Φ(x) Φ(x) (15) d 1 (J 1 q 1, J 1 h 1 ) d 1 (q 1, h 1 ) (16) for all q 1, h 1 E 1. I follows from (13) ha N( (x) 2 ( x 2 ), 2 n 2 ) Φ( x 2 ) = + Φ(x) Φ(x) for all x X and all > 0. So, we have d 1 (, J 1 ) n. Therefore according o emma 2.7, he sequence J k 1 converges o a fixed poin A of J 1, ha is, A : X Υ, N lim n 2 k ( x 2 k ) = A(x) and A(2x) = 2A(x) for all x X. Also A is he unique fixed poin J 1 in he se E 1 = {q 1 E 1 : d 1 (, q 1 ) < } and d 1 (, A) 1 1 d 1(, J 1 ) n(1 ), ha is, inequaliy (7) holds rue for all x X and all > 0. I follows from (5) ha N(2 k D ( x 1 2 k,..., x 2 k ), 2k ) + ϕ( x 1 2 k,..., x 2 k ) + ϕ( x 1 2 k,..., x 2 k )

6 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), for all x 1,..., x X, all > 0 and all k N. By (5), we ge N(2 k D ( x 1 2,..., x k 2 ), ) 2 k k + k (ϕ( x 1,..., x ) + ϕ( x 1,..., x )) 2 k 2 k 2 k 2 k 2 k 2 k for all x 1,..., x X, all > 0 and all k N. Since lim k 2 k 2 k + k 2 k (ϕ( x 1 2 k,..., x 2 k )+ϕ( x 1 2 k,..., x N(D A (x 1,..., x ), ) = 1 2 k )) = 1 for all x 1,..., x X and all > 0, we have for all x 1,..., x X and all > 0. Since every fuzzy Banach space Υ is a real vecor space, by emma 3.1, he mapping A : X Υ is addiive. Finally i remains o prove he uniqueness of A. e T : X Υ is anoher addiive mapping saisfying (3) and (7). Since d 1 (, T) n(1 ) and T is addiive, we ge T E 1 and J 1 T(x) = 2T( x 2 ) = T(x) for all x X, i.e., T is a fixed poin of J 1. Since A is he unique fixed poin of J 1 in, hen T = A. This complees he proof of he heorem. E 1 Corollary 3.4. e p > 1 and θ be non-negaive real numbers. e X be a normed vecor space wih norm. e f : X Υ be a mapping saisfying f (0) = 0 and N(D f (x 1,..., x ), ) + θ( x 1 p + + x p ) for all x 1,..., x X and all > 0. Then here exiss a unique addiive mapping A : X Υ saisfying (3) such ha N( f (x) f ( x) A(x), ) for all x X and all > 0. Proof. The proof follows from Theorem 3.3 by aking ϕ(x 1,..., x ) = θ( x 1 p + + x p ) (17) (2 p 2) (2 p 2) + 2 p+2 θ x p (18) for all x 1,..., x X. Then we can choose = 2 1 p and we ge he desired resul. Corollary 3.5. e f : X Υ be an odd mapping for which here exiss a funcion ϕ : X [0, ) saisfying (5) and (6). Then here exiss a unique addiive mapping A : X Υ saisfying (3) such ha N(2 f (x) A(x), ) n(1 ) n(1 ) + Φ(x) for all x X and > 0, where Φ(x) is defined in (8). Theorem 3.6. e f : X Υ be a mapping saisfying f (0) = 0 for which here exiss a funcion φ : X [0, ) such ha N(D f (x 1,..., x ), ) (20) + φ(x 1,..., x ) for all x 1,..., x X. If here exis a consan 0 < < 1 such ha φ(2x 1,..., 2x ) 2φ(x 1,..., x ) (21) for all x 1,..., x X, hen here exiss a unique addiive mapping A : X Υ saisfying (3) such ha N( f (x) f ( x) A(x), ) for all x X and > 0, where n(1 ) n(1 ) + Ψ(x) Ψ(x) = φ(2x,..., 2x, 0,..., 0) + φ( 2x,..., 2x, 0,..., 0). (23) } {{ }} {{ }} {{ }} {{ } (19) (22)

7 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), Proof. Using he same mehod as in he proof of Theorem 3.3, we have N( (2x) 2 (x), 2 n ) + Ψ(x) (24) for all x X and all > 0, where Ψ(x) = φ(2x,..., 2x, 0,..., 0) + φ( 2x,..., 2x, 0,..., 0). } {{ }} {{ }} {{ }} {{ } We inroduce he definiions for E 1 and d 1 as in he proof of Theorem 3.3 (by replacing Φ by Ψ) such ha (E 1, d 1 ) becomes a complee generalized meric space. Now we consider he funcion J 1 : E 1 E 1 defined by J 1 q 1 (x) := 1 2 q 1(2x), for all q 1 E 1 and x X. (25) e q 1, h 1 E 1 and le λ [0, ] be an arbirary consan wih d 1 (q 1, h 1 ) λ. From he definiion of d 1, we have N(q 1 (x) h 1 (x), λ) + Ψ(x) for all x X and all > 0. Hence N(J 1 q 1 (x) J 1 h 1 (x), λ) for all x X and all > 0. So = N( 1 2 q 1(2x) 1 2 h 1(2x), λ) = N(q 1 (2x) h 1 (2x), 2λ) Ψ(2x) Ψ(x) = + Ψ(x) (26) d 1 (J 1 q 1, J 1 h 1 ) d 1 (q 1, h 1 ) (27) for all q 1, h 1 E 1. I follows from (24) ha N( (x) 1 2 (2x), 1 n ) + Ψ(x) for all x X and all > 0. So, we have d 1 (, J 1 ) 1 n. Therefore according o emma 2.7, he sequence J k 1 converges o a fixed poin A of J 1, ha is, A : X Υ, 1 N lim n 2 k (2k x) = A(x) and A(2x) = 2A(x) for all x X. Also A is he unique fixed poin J 1 in he se E 1 = {q 1 E 1 : d 1 (, q 1 ) < } and d 1 (, A) 1 1 d 1 1(, J 1 ) n(1 ), ha is, inequaliy (22) holds rue for all x X and all > 0. The res of he proof is similar o he proof of Theorem 3.3 and we omi he deails. This complees he proof of he heorem.

8 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), Corollary 3.7. e 0 < p < 1 and θ be non-negaive real numbers. e X be a normed vecor space wih norm. e f : X Υ be a mapping saisfying f (0) = 0 and (17). Then here exiss a unique addiive mapping A : X Υ saisfying (3) such ha N( f (x) f ( x) A(x), ) for all x X and all > 0. Proof. The proof follows from Theorem 3.6 by aking φ(x 1,..., x ) = θ( x 1 p + + x p ) (2 2 p ) (2 2 p ) + 2 p+2 θ x p (28) for all x 1,..., x X. Then we can choose = 2 p 1 and we ge he desired resul. Corollary 3.8. e f : X Υ be an odd mapping for which here exiss a funcion ϕ : X [0, ) saisfying (20) and (21). Then here exiss a unique addiive mapping A : X Υ saisfying (3) such ha N(2 f (x) A(x), ) n(1 ) n(1 ) + Ψ(x) for all x X and > 0, where Ψ(x) is defined in (23). Theorem 3.9. e f : X Υ be a mapping saisfying f (0) = 0 for which here exiss a funcion ϕ : X [0, ) such ha N(D f (x 1,..., x ), ) (30) + ϕ(x 1,..., x ) for all x 1,..., x X. If here exis a consan 0 < < 1 such ha ϕ(x 1,..., x ) 4 ϕ(2x 1,..., 2x ) (31) (29) for all x 1,..., x X, hen here exiss a unique quadraic mapping Q : X Υ saisfying (3) such ha N( f (x) + f ( x) Q(x), ) n(2 2) n(2 2) + Φ(x) for all x X and > 0, where Φ(x) is defined in (8). Proof. Seing x 1 = = x n = x and x n+1 = = x = 0 in (30), we obain N(3n f ( x 2 ) + n f ( x 2 ) n f (x), ) + ϕ(x,..., x, 0,..., 0) } {{ }} {{ } for all x X and all > 0. Replacing x by x in (33), we ge N(3n f ( x 2 ) + n f (x 2 ) n f ( x), ) + ϕ( x,..., x, 0,..., 0) } {{ }} {{ } for all x X and all > 0. Thus N(4n( f ( x 2 ) + f ( x )) n( f (x) + f ( x)), 2) 2 min{n(3n f ( x 2 ) + n f ( x ) n f (x), ), N(3n f ( x 2 2 ) + n f (x ) n f ( x), )} 2 + ϕ(x,..., x, 0,..., 0) + ϕ( x,..., x, 0,..., 0) } {{ }} {{ }} {{ }} {{ } (32) (33) (34) (35)

9 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), for all x X and all > 0. eing (x) = f (x) + f ( x) and Φ(x) = ϕ(2x,..., 2x, 0,..., 0) + ϕ( 2x,..., 2x, 0,..., 0) } {{ }} {{ }} {{ }} {{ } for all x X, we ge N(4n ( x ) n (x), 2) 2 + Φ( x 2 ) (36) for all x X and all > 0. So N( (2x) 4 (x), 2 n ) + Φ(x) (37) for all x X and all > 0. e E 2 be he se of all funcions q 2 : X Υ and inroduce a generalized meric on E 2 as follows: { } d 2 (q 2, h 2 ) := inf µ [0, ] N(q 2(x) h 2 (x), µ), x X, > 0. + Φ(x) So (E 2, d 2 ) is a complee generalized meric space. e J 2 : E 2 E 2 defined by J 2 q 2 (x) := 4q 2 ( x 2 ), for all q 2 E 2 and x X. (38) e q 2, h 2 E 2 and le µ [0, ] be an arbirary consan wih d(q 2, h 2 ) µ. From he definiion of d 2, we have N(q 2 (x) h 2 (x), µ) + Φ(x) for all x X and all > 0. Hence N(J 2 q 2 (x) J 2 h 2 (x), µ) = N(4q 2 ( x 2 ) 4h 2( x ), µ) 2 = N(q 2 ( x 2 ) h 2( x 2 ), 4 µ) = Φ( x 2 ) + Φ(x) Φ(x) (39) for all x X and all > 0. Therefore d 2 (J 2 q 2, J 2 h 2 ) d 2 (q 2, h 2 ) (40) for all q, h E. I follows from (37) ha N( (x) 4 ( x 2 ), 2 n 2 ) Φ( x 2 ) = + Φ(x) Φ(x)

10 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), for all x X and all > 0. So, we have d 2 (, J 2 ). Therefore according o emma 2.7, he sequence J k 2 converges o a fixed poin Q of J 2, ha is, Q : X Υ, N lim n 4 k ( x 2 k ) = Q(x) and Q(2x) = 4Q(x) for all x X. Also Q is he unique fixed poin J 2 in he se E 2 = {q 2 E 2 : d 2 (, q 2 ) < } and d 2 (, Q) 1 1 d 2(, J 2 ) (1 ), ha is, inequaliy (32) holds rue for all x X and all > 0. I follows from (30) ha N(4 k D ( x 1 2 k,..., x 2 k ), 4k ) + ϕ( x 1 2 k,..., x 2 k ) + ϕ( x 1 2 k,..., x 2 k ) for all x 1,..., x X, all > 0 and all k N. By (30), we ge N(4 k D ( x 1 2,..., x k 2 ), ) 4 k k + k (ϕ( x 1,..., x ) + ϕ( x 1,..., x )) 4 k 4 k 2 k 2 k 2 k 2 k for all x 1,..., x X, all > 0 and all k N. Since lim k 4 k 4 k + k 4 k (ϕ( x 1 2 k,..., x 2 k )+ϕ( x 1 2 k,..., x N(D Q (x 1,..., x ), ) = 1 2 k )) = 1 for all x 1,..., x X and all > 0, we have for all x 1,..., x X and all > 0. Since every fuzzy Banach space Υ is a real vecor space, by emma 3.2, he mapping Q : X Υ is quadraic. Finally i remains o prove he uniqueness of Q. e Q : X Υ is anoher quadraic mapping saisfying (3) and (32). Since d 2 (, Q ) n(1 ) and Q is quadraic, we ge Q E 2 and J 2Q (x) = 4Q ( x 2 ) = Q (x) for all x X, ha is, Q is a fixed poin of J 2. Since Q is he unique fixed poin of J 2 in E 2, hen Q = Q. This complees he proof of he heorem. Corollary e p > 2 and θ be non-negaive real numbers. e X be a normed vecor space wih norm. e f : X Υ be a mapping saisfying f (0) = 0 and (17). Then here exiss a unique quadraic mapping Q : X Υ saisfying (3) such ha N( f (x) + f ( x) Q(x), ) for all x X and all > 0. Proof. The proof follows from Theorem 3.6 by aking ϕ(x 1,..., x ) = θ( x 1 p + + x p ) for all x 1,..., x X and choosing = 2 2 p. (2 p 4) (2 p 4) p θ x p (41) Corollary e f : X Υ be an even mapping for which here exiss a funcion ϕ : X [0, ) saisfying f (0) = 0, (30) and (31). Then here exiss a unique quadraic mapping Q : X Υ saisfying (3) such ha N(2 f (x) Q(x), ) n(2 2) n(2 2) + Φ(x) (42) for all x X and > 0, where Φ(x) is defined in (8).

11 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), Theorem e f : X Υ be a mapping saisfying f (0) = 0 for which here exiss a funcion φ : X [0, ) such ha N(D f (x 1,..., x ), ) + φ(x 1,..., x ) for all x 1,..., x X. If here exis a consan 0 < < 1 such ha φ(2x 1,..., 2x ) 4φ(x 1,..., x ) (44) for all x 1,..., x X, hen here exiss a unique quadraic mapping Q : X Υ saisfying (3) such ha (43) N( f (x) + f ( x) Q(x), ) n(2 2) n(2 2) + Ψ(x) for all x X and > 0, where where Ψ(x) is defined in (23). (45) Proof. Using he same mehod as in he proof of Theorem 3.9, we have N( (2x) 4 (x), 2 n ) for all x X and all > 0, where + Ψ(x) (46) Ψ(x) = φ(2x,..., 2x, 0,..., 0) + φ( 2x,..., 2x, 0,..., 0). } {{ }} {{ }} {{ }} {{ } We inroduce he definiions for E 2 and d 2 as in he proof of Theorem 3.9 (by replacing Φ by Ψ) such ha (E 2, d 2 ) becomes a complee generalized meric space. Now we consider he funcion J 2 : E 2 E 2 defined by J 2 q 2 (x) := 1 4 q 2(2x), for all q 2 E 2 and x X. (47) e q 2, h 2 E 2 and le µ [0, ] be an arbirary consan wih d 2 (q 2, h 2 ) µ. From he definiion of d 2, we have N(q 2 (x) h 2 (x), µ) + Ψ(x) for all x X and all > 0. Hence N(J 2 q 2 (x) J 2 h 2 (x), µ) for all x X and all > 0. Therefore = N( 1 4 q 2(2x) 1 4 h 2(2x), µ) = N(q 2 (2x) h 2 (2x), 4µ) Ψ(2x) Ψ(x) = + Ψ(x) (48) d 2 (J 2 q 2, J 2 h 2 ) d 2 (q 2, h 2 ) (49) for all q 2, h 2 E 2. I follows from (46) ha N( (x) 1 4 (2x), 1 ) + Ψ(x)

12 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), for all x X and all > 0. So, we have d 2 (, J 2 ) 1. Therefore according o emma 2.7, he sequence J k 2 converges o a fixed poin Q of J 2, ha is, 1 Q : X Υ, N lim n 4 k (2k x) = Q(x) and Q(2x) = 4Q(x) for all x X. Also Q is he unique fixed poin J 2 in he se E 2 = {q 2 E 2 : d 2 (, q 2 ) < } and d 2 (, Q) 1 1 d 1 2(, J 2 ) (1 ), ha is, inequaliy (45) holds rue for all x X and all > 0. The res of he proof is similar o he proof of Theorem 3.9 and we omi he deails. This complees he proof of he heorem. Corollary e 0 < p < 2 and θ be non-negaive real numbers. e X be a normed vecor space wih norm. e f : X Υ be a mapping saisfying f (0) = 0 and (17). Then here exiss a unique quadraic mapping Q : X Υ saisfying (3) such ha N( f (x) + f ( x) Q(x), ) for all x X and all > 0. Proof. The proof follows from Theorem 3.9 by aking φ(x 1,..., x ) = θ( x 1 p + + x p ) (4 2 p ) (4 2 p ) p θ x p (50) for all x 1,..., x X. Nex choosing = 2 p 2, we ge he desired resul. Corollary e f : X Υ be an even mapping for which here exiss a funcion ϕ : X [0, ) saisfying f (0) = 0, (43) and (44). Then here exiss a unique quadraic mapping Q : X Υ saisfying (3) such ha N(2 f (x) Q(x), ) n(2 2) n(2 2) + Ψ(x) for all x X and > 0, where Φ(x) is defined in (23). Combining Theorems 3.3 and 3.9, we obain he following resul. Theorem e f : X Υ be a mapping saisfying f (0) = 0 for which here exiss a funcion ϕ : X [0, ) saisfying (5) and (31). Then here exiss a unique addiive mapping A : X Υ saisfying (3) and a unique quadraic mapping Q : X Υ saisfying (3) such ha N(2 f (x) A(x) Q(x), ) n(1 ) n(1 ) + 3Φ(x) for all x X and > 0, where Φ(x) is defined in (8). Proof. By Theorems 3.3 and 3.9, we obain n(1 ) N( f (x) f ( x) A(x), ) n(1 ) + Φ(x), n(2 2) N( f (x) + f ( x) Q(x), ) n(2 2) + Φ(x) for all x X and > 0. Thus N(2 f (x) A(x) Q(x), 2) min{n( f (x) f ( x) A(x), ), N( f (x) + f ( x) Q(x), } n(2 2) n(2 2) + 3Φ(x) for all x X and > 0, and we ge he desired resul. (51) (52)

13 Zhihua Wang and Prasanna K. Sahoo / Filoma 29:5 (2015), Corollary e p > 2 and θ be non-negaive real numbers. e X be a normed vecor space wih norm. e f : X Υ be a mapping saisfying f (0) = 0 and (17). Then here exiss a unique addiive mapping A : X Υ saisfying (3) and a unique quadraic mapping Q : X Υ saisfying (3) such ha N(2 f (x) A(x) Q(x), ) for all x X and all > 0. (2 p 4) (2 p 4) p θ x p (53) Proof. Define ϕ(x 1,..., x ) = θ( x 1 p + + x p ), = 2 2 p, and apply Theorem 3.15 o ge he desired resul. Similarly, combining Theorems 3.6 and 3.12, we obain he following resul. Theorem e f : X Υ be a mapping saisfying f (0) = 0 for which here exiss a funcion ϕ : X [0, ) saisfying (20) and (21). Then here exiss a unique addiive mapping A : X Υ saisfying (3) and a unique quadraic mapping Q : X Υ saisfying (3) such ha N(2 f (x) A(x) Q(x), ) n(1 ) n(1 ) + 3Ψ(x) for all x X and > 0, where Ψ(x) is defined in (23). (54) Proof. Similar o he proof of Theorem 3.15, he resul follows from Theorems 3.6 and Corollary e 0 < p < 1 and θ be non-negaive real numbers. e X be a normed vecor space wih norm. e f : X Υ be a mapping saisfying f (0) = 0 and (17). Then here exiss a unique addiive mapping A : X Υ saisfying (3) and a unique quadraic mapping Q : X Υ saisfying (3) such ha N(2 f (x) A(x) Q(x), ) for all x X and all > 0. (2 2 p ) (2 2 p ) p θ x p (55) Proof. Define ϕ(x 1,..., x ) = θ( x 1 p + + x p ), = 2 p 1, and apply Theorem 3.17 o ge he desired resul. Acknowledgemens: The work was done while he firs auhor sudied a Universiy of ouisville as a Posdocoral Fellow from Hubei Universiy of Technology during References [1] T. Aoki, On he sabiliy of he linear ransformaion in Banach spaces, J. Mah. Soc. Japan 2(1950), [2] T. Bag, S.K. Samana, Finie dimensional fuzzy normed linear spaces, J. Fuzzy Mah. 11(2003), [3]. Cǎdariu and V. Radu, On he sabiliy of he Cauchy funcional equaion: A fixed poin approach, Grazer Mah. Ber. 346(2004), [4] J. B. Diaz and B. Margolis, A fixed poin heorem of he alernaive for conracions on a generalized complee meric space. Bull. Amer. Mah. Soc. 74(1968), [5] A. Ebadian, N. Ghobadipour, M. B. Savadkouhi and M. E. Gordji, Sabiliy of a mixed ype cubic and quaric funcional equaion in non-archimedean -fuzzy normed spaces, Thai J. Mah. 9(2011), [6] V. A. Faiziev, The sabiliy of a funcional equaion on groups, Russian Mah. Surveys, 48(1993), [7] V. A. Faiziev, Th. M. Rassias and P.K. Sahoo, The space of (ψ, γ)-addiive mappings on semigroups, Trans. Amer. Mah. Soc. 354(2002), [8] V. A. Faiziev and P. K. Sahoo, On (ψ, γ)-sabiliy of a Cauchy equaion on some noncommuaive groups, Pub. Mah. Debreceen, 75(2009), [9] V. A. Faiziev and P. K. Sahoo, Sabiliy of a funcional equaion of Whiehead on semigroups, Ann. Func. Anal. 3(2012), [10] P. Găvrua, A generalizaion of he Hyers-Ulam-Rassias sabiliy of approximaely addiive mappings, J. Mah. Anal. Appl. 184(1994), [11] M. E. Gordji and H. Khodaei, The fixed poin mehod for fuzzy approximaion of a funcional equaion associaed wih inner produc spaces, Discree Dynamics in Naure and Sociey Volume 2010, Aricle ID , 15 pages.

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