Nonlinear L -Fuzzy stability of cubic functional equations

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1 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// RESEARCH Open Access Nonlinear L -Fuzzy sabiliy o cubic uncional equaions Ravi P Agarwal 1,2*, Yeol Je Cho 3, Reza Saadai 4 and Shenghua Wang 5 * Correspondence: agarwal@amuk. edu 1 Deparmen o Mahemaics, Texas A&M Universiy - Kingsville, Kingsville, TX 78363, USA Full lis o auhor inormaion is available a he end o he aricle Absrac We esablish some sabiliy resuls or he cubic uncional equaions 3 x +3y + 3x y = 15 x + y +15 x y +80 y, 2x + y + 2x y = 2 x + y +2 x y +12 x and 3x + y + 3x y = 3 x + y +3 x y +48 x in he seing o various L -uzzy normed spaces ha in urn generalize a Hyers-Ulam sabiliy resul in he ramework o classical normed spaces. Firs, we shall prove he sabiliy o cubic uncional equaions in he L -uzzy normed space under arbirary - norm which generalizes previous sudies. Then, we prove he sabiliy o cubic uncional equaions in he non-archimedean L -uzzy normed space. We hereore provide a link among dieren disciplines: uzzy se heory, laice heory, non- Archimedean spaces, and mahemaical analysis. Mahemaics Subjec Classiicaion 2000: Primary 54E40; Secondary 39B82, 46S50, 46S40. Keywords: sabiliy, cubic uncional equaion, uzzy normed space, -uzzy se 1. Inroducion The sudy o sabiliy problems or uncional equaions is relaed o a quesion o Ulam [1] concerning he sabiliy o group homomorphisms and i was airmaively answered or Banach spaces by Hyers [2]. Subsequenly, he resul o Hyers was generalized by Aoki [3] or addiive mappings and by Rassias [4] or linear mappings by considering an unbounded Cauchy dierence. The aricle [4] o Rassias has provided a lo o inluence in he developmen o wha we now call Hyers-Ulam-Rassias sabiliy o uncional equaions. For more inormaions on such problems, reer o he papers [5-15]. The uncional equaions 3 x +3y + 3x y = 15 x + y +15 x y +80 y, 1:1 2x + y + 2x y = 2 x + y +2 x y +12 x 1: Agarwal e al; licensee Springer. This is an Open Access aricle disribued under he erms o he Creaive Commons Aribuion License hp://creaivecommons.org/licenses/by/2.0, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied.

2 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 2 o 19 and 3x + y + 3x y = 3 x + y +3 x y +48 x 1:3 are called he cubic uncional equaions, sinceheuncionx =cx 3 is heir soluion. Every soluion o he cubic uncional equaions is said o be a cubic mapping. The sabiliy problem or he cubic uncional equaions was sudied by Jun and Kim [16] or mappings : X Y, wherex is a real normed space and Y is a Banach space. Laer a number o mahemaicians worked on he sabiliy o some ypes o cubic equaions [4,17-19]. Furhermore, Mirmosaaee and Moslehian [20], Mirmosaaee e al.[21], Alsina[22], Miheţ and Radu [23] and ohers [24-28] invesigaed he sabiliy in he seings o uzzy, probabilisic, and random normed spaces. 2. Preliminaries In his secion, we recall some deiniions and resuls which are needed o prove our main resuls. A riangular norm shorer -norm is a binary operaion on he uni inerval [0,1], i. e., a uncion T : [0,1] [0,1] [0,1] such ha or all a, b, c Î [0,1] he ollowing our axioms are saisied: i T a, b =T b, a : commuaiviy; ii T a, T b, c = T T a, b, c : associaiviy; iii T a, 1 = a : boundary condiion; iv T a, b T a, c whenever b c : monooniciy. Basic examples are he Lukasiewicz -norm T L, T L a, b =maxa + b-1, 0 a, b Î [0,1] and he -norms T P, T M, T D, where T P a, b :=ab, T M a, b := min{a, b}, T D a, b := { min a, b,imaxa, b =1; 0, oherwise. I T is a -norm hen x n is deined or every x Î [0,1] and n Î N {0} by 1, i n =0 T and T x n 1 T, x, i n 1. A -norm T is said o be o Hadžić-ype we denoe by T Î H i he amily is equiconinuous a x = 1 c. [29]. x n T n N Oher imporan riangular norms are see [30]: -he Sugeno-Weber amily { Tλ SW } λ [ 1, ] is deined by TSW 1 T SW λ x, y =max 0, x + y 1+λxy 1+λ = T D, T SW i lî-1,. -he Domby amily { Tλ D } λ [0, ], deined by T D, i l =0,T M, i l = and Tλ D x, y = x λ λ 1/λ x + 1 y y i l Î 0,. = T P and

3 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 3 o 19 -he Aczel-Alsina amily { Tλ AA } λ [0, ], deined by T D, i l =0,T M, i l = and Tλ AA log x λ + log y λ 1/λ x, y =e i l Î 0,. A -norm T can be exended by associaiviy in a unique way o an n-array operaion aking or x 1,...,x n Î [0,1] n he value T x 1,...,x n deined by T 0 i =1 x i =1,T n i=1 x i =T T n 1 i=1 x i, x n = T x1,..., x n. T can also be exended o a counable operaion aking or any sequence x n nîn in [0,1] he value T i=1 x i = lim T n i=1 x i. 2:1 The limi on he righ side o 2.1 exiss, since he sequence { T n i=1 x } i is nonincreasing and bounded rom below. Proposiion 2.1. [30] 1 For T T L he ollowing implicaion holds: lim T i=1 x n+i =1 1 x n <. n=1 n N 2 I T is o Hadžić-ype hen lim T i=1 x n+i = 1 or every sequence {x n } nîn in [0, 1] such ha lim x n = 1. 3 I T {Tλ AA} λ 0, {Tλ D} λ 0,, hen lim T i=1 x n+i =1 1 x n α <. n=1 4 I T {T sw λ } λ [ 1,, hen lim T i=1 x n+i =1 1 x n <. n=1 3. L-Fuzzy normed spaces The heory o uzzy ses was inroduced by Zadeh [31]. Aer he pioneering sudy o Zadeh, here has been a grea eor o obain uzzy analogs o classical heories. Among oher ields, a progressive developmen is made in he ield o uzzy opology [32-40,43-50]. One o he problems in L -uzzy opology is o obain an appropriae concep o L -uzzy meric spaces and L -uzzy normed spaces. Saadai and Park [40], respecively, inroduced and sudied a noion o inuiionisic uzzy meric normed spaces and hen Deschrijver e al. [41] generalized he concep o inuiionisic uzzy

4 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 4 o 19 meric normed spaces and sudied a noion o L -uzzy meric spaces and L -uzzy normed spaces also, see [41,42,51-55]. In his secion, we give some deiniions and relaed lemmas or our main resuls. In his secion, we give some deiniions and relaed lemmas which are needed laer. Deiniion 3.1 [43]. Le L = L, L be a complee laice and U be a non-empy se called universe. A L-uzzy se A on U is deined as a mapping A : U L. For any u Î U, A u represens he degree in L o which u saisies A. Lemma 3.2 [44]. Consider he se L* and operaion L deined by: L = {x 1, x 2 : x 1, x 2 [0, 1] 2 and x 1 + x 2 1}, x 1, x 2 L y 1, y 2 x 1 y 1 and x 2 y 2or all x 1, x 2, y 1, y 2 Î L*. Then L*, L* is a complee laice. Deiniion 3.3 [45]. An inuiionisic uzzy se A ζ,η on a universe U is an objec A ζ,η = {ζ A u, η A u : u U}, where, or all u Î U, ζ A u [0, 1] and η A u [0, 1] are called he membership degree and he non-membership degree, respecively, o u in A ζ,η and, urhermore, saisyζ A u + η A u 1. In Secion 2, we presened he classical deiniion o -norm, which can be easily exended o any laice L = L, L. Deine irs 0 L =inl and 1 L =supl. Deiniion 3.4. A riangular norm -norm on L is a mapping T : L 2 L saisying he ollowing condiions: i or any x L, T x, 1 L = x : boundary condiion; ii or any x, y L 2, T x, y = T y, x : commuaiviy; iii or any x, y, z L 3, T x, T y, z = T T x, y, z : associaiviy; iv or any x, x, y, y L 4, x L x and y L y T x, y L T x, y : monooniciy. A -norm can also be deined recursively as an n + 1-array operaion n Î N\ {0} by T 1 = T and T n x 1,..., x n+1 =T T n 1 x 1,..., x n, x n+1, n 2, x i L. The -norm M deined by { x i x L y Mx, y = y i y L x is a coninuous -norm. Deiniion 3.5. A -norm T on L* is said o be -represenable i here exis a -norm T and a -conorm S on [0,1] such ha T x, y = Tx 1, y 1, Sx 2, y 2, x = x 1, x 2, y = y 1, y 2 L. Deiniion 3.6. A negaion on L is any sricly decreasing mapping N 0 L =1 L saisying N 0 L =1 L and N 1 L =0 L.INNx = xor all x Î L, hen N is called an involuive negaion. In his aricle, le N : L L be a given mapping. The negaion N s on [0,1], deined as N s x =1-x or all x Î [0, 1] is called he sandard negaion on [0,1],.

5 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 5 o 19 Deiniion 3.7. The 3-uple V, P, T is said o be a L-uzzy normed space i V is a vecor space, T is a coninuous -norm on L and P is a L -uzzy se on V ]0, + [ saisying he ollowing condiions: or all x, y Î V and, s Î]0, + [, i 0 L < L Px, ; ii Px, =1 L i and only i x =0; iii Pαx, =Px, α or all a 0; iv T Px,, Py, s L Px + y, + s; v Px, :]0, [ L is coninuous; vi lim 0 Px, =0 L and lim Px, =1 L. In his case, P is called a L-uzzy norm. IP = P μ,ν is an inuiionisic uzzy se and he -norm T is -represenable, hen he 3-uple V, P μ,v, T is said o be an inuiionisic uzzy normed space. Deiniion Asequence{x n }inx is called a Cauchy sequence i, or any ε L\{0 L } and >0, here exiss a posiive ineger n 0 such ha N ε< L Px n+p x n,, n n 0, p > 0. 2 I every Cauchy sequence is convergen, hen he L -uzzy norm is said o be complee and he L -uzzy normed space is called a L-uzzy Banach space, wheren is an involuive negaion. 3 The sequence {x n } is said o be convergen o x Î V in he L -uzzy normed space P V, P, T denoed by x n x i Px n x, 1 L, whenever n + or all >0. Lemma 3.9 [46]. Le Pbe a L-uzzy norm on V. Then 1 For all Î V, Px, is nondecreasing wih respec o. 2 Px y, = Py x, or all x, y Î V and Î ]0, + [. Deiniion Le V, P, T be a L -uzzy normed space. For any Î ]0, + [, we deine he open ball Bx, r, wih cener x Î V and radius r L\{0 L,1 L } as Bx, r, = {y V : N r< L Px y, }. 4. Sabiliy resul in L -uzzy normed spaces In his secion, we sudy he sabiliy o uncional equaions in L -uzzy normed spaces. Theorem 4.1. Le X be a linear space and Y, P, T be a complee L-uzzy normed space. I : Yisamappingwih0 = 0 and Q is a L-uzzy se on X 2 0, wih he ollowing propery: P3 x +3y+ 3x y 15 x + y 15 x y 80 y, L Qx, y,, x, y X, > 0. 4:1 I T i=1 Q3n+i 1 x,0,3 3n+2i+1 = 1 L, x X, > 0,

6 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 6 o 19 and lim Q3n x,3 n y,3 3n =1 L, x, y X, > 0, hen here exiss a unique cubic mapping C : X Y such ha P x Cx, L T i=1 Q3i 1 x,0,3 2i+2, x X, > 0. 4:2 Proo. We brie he proo because i is similar as he random case [47,27]. Puing y = 0 in 4.1, we have 3x P 27 x, L Qx,0,3 3, x X, > 0. Thereore, i ollows ha 3 k+1 x P 3 3k+1 3k x 3 3k, 3 k+1 L Q3 k x,0,3 2k+1. k 1, > 0. By he riangle inequaliy, i ollows ha 3 n x P 27 n x, L Ti=1 n x,0,3 2i+2, x X, > 0. 4:3 { } In order o prove he convergence o he sequence 3 n x 27, we replace x wih 3 m x in n 4.3 o ind ha, or all m, n>0, 3 n+m x P 27 n+m 3m x 27 m, L Ti=1 n Q3i+m 1 x,0,3 2i+3m+2, x X, > 0. Since he righ-hand side o he inequaliy ends o 1 L as m ends o ininiy, he { } sequence 3 n x 3 is a Cauchy sequence. Thus, we may deine Cx = lim n x 3 3n 3 or all x 3n Î X. Replacingx, y wih 3 n x and 3 n y, respecively, in 4.1, i ollows ha C is a cubic mapping. To prove 4.2, ake he limi as n in 4.3. To prove he uniqueness o he cubic mapping C subjec o 4.2, le us assume ha here exiss anoher cubic mapping C which saisies 4.2. Obviously, we have C3 n x=3 3n Cx andc 3 n x= 3 3n C x or all x Î X and n Î N. Hence i ollows rom 4.2 ha P Cx C x, L P C3 n x C 3 n x, 3 3n L T P C3 n x 3 n x, 3 3n 1, P 3 n x C 3 n x, 2 3n 1 L T T i=1 Q3n+i 1 x,0,3 3n+2i+1, T i=1 Q3n+i 1 x,0,3 3n+2i+1 = T 1 L,1 L =1 L, x X, > 0, which proves he uniqueness o C. This complees he proo. Theorem 4.2. Le X be a linear space and Y, P, T be a complee L-uzzy normed space. I : X Yisamappingwih0 = 0 and Q is a L-uzzy se on X 2 0, wih he ollowing propery: P 2x + y + 2x y 2 x + y 2 x y 12 x, 4:4 L Qx, y,, x, y X, > 0.

7 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 7 o 19 I T i=1 Q2n+i 1 x,0,2 3n+2i+1 = 1 L, x X, > 0, and lim Q2n x,2 n y,2 3n =1 L, x, y X, > 0, hen here exiss a unique cubic mapping C : X Y such ha P x Cx, L T i=1 Q2i 1 x,0,2 2i+1, x X, > 0. 4:5 Proo. We omi he proo because i is similar as he las heorem and see [28]. Corollary 4.3. Le X, P, T be L-uzzy normed space and Y, P, T be a complee L-uzzy normed space. I : X Y is a mapping such ha and P 2x + y + 2x y 2 x + y 2 x y 12 x, L P x + y,, x, y X, > 0, i=1 P x,2 2n+i+2 = 1 L, x X, > 0, hen here exiss a unique cubic mapping C : X Y such ha P x Cx, L T i=1 P x,2 i+2, x X, > 0. Proo. See [28]. Now, we give an example o validae he main resul as ollows: Example 4.4 [28]. Le X, be a Banach space, X, P μ,ν, T M be an inuiionisic uzzy normed space in which T M a, b = min{a 1, b 1 },max{a 2, b 2 } and P μ,ν x, = + x, x + x, x X, > 0, also Y, P μ,ν, T M be a complee inuiionisic uzzy normed space. Deine a mapping : X Y by x =x 3 + x 0 or all x Î X, where x 0 is a uni vecor in X. A sraighorward compuaion shows ha P μ,ν 2x + y+ 2x y 2 x + y 2 x y 12 x, L P μ,ν x + y,, x, y X, > 0. Also, we have M,i=1 P μ,νx,2 2n+i+1 = lim lim T m m M,i=1 P μ,νx,2 2n+i+1 = lim lim P μ,νx,2 2n+2 m = lim P μ,ν x,2 2n+2 =1 L. Thereore, all he condiions o Theorem 4.2 hold and so here exiss a unique cubic mapping C : X Y such ha P μ,ν x Cx, L P μ,ν x,2 2, x X, > 0.

8 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 8 o Non-Archimedean L-uzzy normed spaces In 1897, Hensel [?] inroduced a ield wih a valuaion in which does no have he Archimedean propery. Deiniion 5.1. LeK be a ield. A non-archimedean absolue value on K is a uncion : K [0, + [such ha, or any a, b Î K, i a 0 and equaliy holds i and only i a =0; ii ab = a b ; iii a + b max { a, b } : he sric riangle inequaliy. Noe ha n 1 or each ineger n 1. We always assume, in addiion, ha is non-rivial, i.e., here exiss a 0 Î K such ha a 0 0, 1. Deiniion 5.2. A non-archimedean L-uzzy normed space is a riple V, P, T, where V is a vecor space, T is a coninuous -norm on L and P is a L -uzzy se on V ]0, + [ saisying he ollowing condiions: or all x, y Î V and, s Î ]0, + [, i 0 L < L Px, ; ii Px, =1 L i and only i x =0; iii Pαx, =Px, α or all a 0; vi T Px,, Py, s L Px + y,max{, s}; v Px, :]0, [ L is coninuous; vi lim 0 Px, =0 L and lim Px, =1 L. Example 5.3. LeX, be a non-archimedean normed linear space. Then he riple X, P,min, where { 0, i x ; Px, = 1, i > x, is a non-archimedean L -uzzy normed space in which L = [0,1]. Example 5.4. LeX, be a non-archimedean normed linear space. Denoe T M a, b = min{a 1, b 1 },max{a 2, b 2 } or all a =a 1, a 2, b =b 1, b 2 Î L* and P μ,ν be he inuiionisic uzzy se on X ]0, + [ deined as ollows: P μ,ν x, = + x, x + x, x X, R +. Then X, P μ,ν, T M is a non-archimedean inuiionisic uzzy normed space. 6. L-uzzy Hyers-Ulam-Rassias sabiliy or cubic uncional equaions in non- Archimedean L-uzzy normed space Le K be a non-archimedean ield, X beavecorspaceoverkand Y, P, T be a non-archimedean L -uzzy Banach space over K. Inhissecion,weinvesigaehe sabiliy o he cubic uncional equaion 1.1. Nex, we deine a L -uzzy approximaely cubic mapping. Le Ψ be a L -uzzy se on X X [0, such ha Ψ x, y, is nondecreasing, cx, cx, L x, x, c, x X, c = 0

9 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 9 o 19 and lim x, y, =1 L, x, y X, > 0. Deiniion 6.1. A mapping : X Y is said o be Ψ-approximaely cubic i P3 x +3y+ 3x y 15 x + y 15 x y 80 y, L x, y,, x, y X, > 0. 6:1 Here, we assume ha 3 0inK i.e., characerisic o K is no 3. Theorem 6.2. Le Kbe a non-archimedean ield, XbeavecorspaceoverKand Y, P, T be a non-archimedean L-uzzy Banach space over K: Le: X YbeaΨapproximaely cubic mapping. I here exis a a Î R a >0 andaninegerk, k 2 wih 3 k <a and 3 1 such ha and 3 k x,3 k y, L x, y, α, x, y X, > 0, 6:2 j=n x, M α j 3 kj =1 L, x X, > 0, hen here exiss a unique cubic mapping C : X Y such ha P x Cx, Ti=1 M x, αi+1 3 ki, x X, > 0, 6:3 where Mx, := T x, 0,, 3x, 0,,..., 3 k 1 x,0,, x X, > 0. Proo. Firs, we show, by inducion on j, ha, or all x Î X, >0 and j 1, P 3 j x 27 j x, L M j x, :=T x, 0,,..., 3 j 1 x,0,. 6:4 Puing y = 0 in 6.1, we obain P 3x 27 x, L x, 0,, x X, > 0. This proves 6.4 or j = 1. Le 6.4 hold or some j>1. Replacing y by 0 and x by 3 j x in 6.1, we ge P 3 j+1 x 27 3 j x, L 3 j x,0,, x X, > 0. Since 27 1, i ollows ha P 3 j+1 x 27 j+1 x, L T P 3 j+1 x 27 3 j x,, P8 3 j x 27 j+1 x, = T P 2 j+1 x 8 2 j x,, P 3 j x 27 j x, 27 L T P 3 j+1 x 27 3 j x,, P 3 j x 27 j x, L T 3 j x,0,, M j x, = M j+1 x,, x X, > 0.

10 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 10 o 19 Thus 6.4 holds or all j 1. In paricular, we have P 3 k x 27 k x, L Mx,, x X, > 0. 6:5 Replacing x by 3 -kn+k x in 6.5 and using he inequaliy 6.2, we obain x x x P 3 kn 27 k 3 kn+k, L M 3 kn+k, and so P 3 3k n L M x, L M x, x 3 k n 3 3k n+1 x 3 k n+1 α n+1 3 3k n α n+1 3 k n L Mx, α n+1 x X, > 0, n 0,, x X, > 0, n 0. Hence, i ollow ha P 3 3k n x 3 k n 3 3k n+p x 3 k n+p, L T n+p j=n P 3 3k j x 3 k j 3 3k j+p x 3 k j+p, L T n+p j=n x, M α j+1 3 k j, x X, > 0, n 0. Since lim Tj=n x, M α j+1 { 3 k j =1 L or all x Î X and >0, 3 3k n x 3 k }n N n is a Cauchy sequence in he non-archimedean L -uzzy Banach space Y, P, T. Hence we can deine a mapping C : X Y such ha lim P 3 3k n x 3 k n Cx, =1 L, x X, > 0. 6:6 Nex, or all n 1, x Î X and >0, we have P x 3 3k n x 3 k n, n 1 = P 3 3k i x i=0 3 k i 3 3k i+1 x 3 k i+1, L T n 1 i=0 P 3 3k i x 3 k i 3 3k i+1 x 3 k i+1, L T n 1 i=0 x, M α i+1 3 k i

11 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 11 o 19 and so P x Cx, L T P x 3 3k n x 3 k n L P T n 1 i=0 x, M α i+1 3 k i, P3 3k n Taking he limi as n in 6.7, we obain P x Cx, L Ti=1 x, M α i+1 3 k i,,, P 3 3k n x 3 k n. x 3 k n Cx, Cx, 6:7 which proves 6.3. As T is coninuous, rom a well known resul in L -uzzy probabilisic normed space see, [51, Chap. 12], i ollows ha lim P27k n 3 kn x +3y + 27 k n 3 kn 3x y 1527 k n 3 kn x + y 1527 k n 3 kn x y 8027 k n 3 kn y, = PCx +3y + C3x y 15Cx + y 15Cx y 80Cy,, > 0. On he oher hand, replacing x, y by 3 -kn x,3 -kn y in 6.1 and 6.2, we ge P27 k n 3 kn x +3y + 27 k n 3 kn 3x y 1527 k n 3 kn x + y 1527 k n 3 kn x y 8027 k n 3 kn y, L 3 kn x,3 kn y, 3 3k n L x, y, α n 3 k n, x, y X, > 0. α n Since lim x, y, 3 k n =1 L, we iner ha C is a cubic mapping. For he uniqueness o C, le C : X Y be anoher cubic mapping such ha PC x x, L Mx,, x X, > 0. Then we have, or all x, y Î X and >0, PCx C x, L T P Cx 3 3k n x 3 k n,, P 3 3k n x 3 k n C x,,. Thereore, rom 6.6, we conclude ha C = C. This complees he proo. Corollary 6.3. Le Kbe a non-archimedean ield, X be a vecor space over Kand Y, P, T be a non-archimedean L-uzzy Banach space over K under a -normt H. Le : X YbeaΨ-approximaely cubic mapping. I here exis a Î R a >0, 3 1 and an ineger k, k 3 wih 3 k <a such ha 3 k x,3 k y, L x, y, α, x, y X, > 0, hen here exiss a unique cubic mapping C : X Y such ha P x Cx, L Ti=1 x, M α i+1 3 ki, x X, > 0,

12 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 12 o 19 where Mx, := T x, 0,, 3x, 0,,..., 3 k 1 x,0,, x X, > 0. Proo. Since lim M x, α j 3 kj =1 L, x X, > 0, and T is o Hadžić ype, i ollows rom Proposiion 2.1 ha j=n x, M α j 3 kj =1 L, x X, > 0. Now, i we apply Theorem 6.2, we ge he conclusion. Now, we give an example o validae he main resul as ollows: Example 6.4. LeX, be a non-archimedean Banach space, X, P μ,ν, T M be non-archimedean L -uzzy normed space inuiionisic uzzy normed space in which P μ,ν x, = + x, x + x, x X, > 0, and Y, P μ,ν, T M be a complee non-archimedean L -uzzy normed space inuiionisic uzzy normed space see, Example 5.4. Deine x, y, = 1+, 1, x, y X, > I is easy o see ha 6.2 holds or a = 1. Also, since Mx, = 1+, 1, x X, > 0, 1+ we have M,j=n M x, α j 3 kj = lim lim = lim lim m T m m M,j=n M x, + 3 k n, 3 kj 2 k n + 3 k n = 1, 0 =1 L, x X, > 0. Le : X Y be a Ψ-approximaely cubic mapping. Thereore, all he condiions o Theorem 6.2 hold and so here exiss a unique cubic mapping C : X Y such ha P μ,ν x Cx, L + 3 k, 3 k + 3 k, x X, > 0. Deiniion 6.5. A mapping : X Y is said o be Ψ-approximaely cubic I i P 2x + y + 2x y 2 x + y 2 x y 12 x, 6:8 L x, y,, x, y X, > 0. In his secion, we assume ha 2 0inK i.e., he characerisic o K is no 2. Theorem 6.6. Le Kbe a non-archimedean ield, XbeavecorspaceoverKand Y, P, T be a non-archimedean L-uzzy Banach space overk. Le : X YbeaΨ-

13 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 13 o 19 approximaely cubic I mapping. I 2 1 and or some a Î R, a >0, and some ineger k, k 2 wih 2 k <a, and 2 k x,2 k y, L x, y, α, x, y X, > 0, 6:9 j=n x, M α j 2 kj =1 L, x X, > 0, hen here exiss a unique cubic mapping C : X Y such ha P x Cx, Ti=1 M x, αi+1 2 ki, x X, > 0, 6:10 where Mx, := T x, 0,, 2x, 0,,..., 2 k 1 x,0,, x X, > 0. Proo. We omi he proo because i is similar as he random case see, [28]. Corollary 6.7. Le Kbe a non-archimedean ield, X be a vecor space over Kand Y, P, T be a non-archimedean L -uzzy Banach space over Kunder a -norm T H. Le : X YbeaΨ-approximaely cubic I mapping. I here exis a a Î R a >0 and an ineger k, k 2 wih 2 k < a such ha 2 k x,2 k y, L x, y, α, x, y X, > 0, hen here exiss a unique cubic mapping C : X Y such ha P x Cx, L Ti=1 x, M α i+1 2 ki, x X, > 0, where Mx, := T x, 0,, 2x, 0,,..., 2 k 1 x,0,, x X, > 0. Proo. Since lim M x, α j 2 kj =1 L, x X, > 0, and T is o Hadžić ype, i ollows rom Proposiion 2.1 ha j=n x, M α j 2 kj =1 L, x X, > 0. Now, i we apply Theorem 6.2, we ge he conclusion. Now, we give an example o validae he main resul as ollows: Example 6.8. Le X, be a non-archimedean Banach space, X, P μ,ν, T M be non-archimedean L -uzzy normed space inuiionisic uzzy normed space in which P μ,ν x, = + x, x + x, x X, > 0, and Y, P μ,ν, T M be a complee non-archimedean L -uzzy normed space inuiionisic uzzy normed space see, Example 5.4. Deine

14 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 14 o 19 x, y, = 1+, 1, x, y X, > I is easy o see ha 6.9 holds or a = 1. Also, since Mx, = 1+, 1, x X, > 0, 1+ we have M,j=n M x, α j 2 kj = lim lim = lim lim m T m m M,j=n M x, + 2 k n, 2 kj 2 k n + 2 k n = 1, 0 =1 L, x X, > 0. Le : X Y be a Ψ-approximaely cubic I mapping. Thereore, all he condiions o Theorem 6.6 hold and so here exiss a unique cubic mapping C : X Y such ha P μ,ν x Cx, L + 2 k, 2 k + 2 k, x X, > 0. Deiniion 6.9. A mapping : X Y is said o be Ψ-approximaely cubic II i P 3x + y + 3x y 3 x + y 3 x y 48 x, L x, y,, x, y X, > 0. 6:11 Here, we assume ha 3 0inK i.e., he characerisic o K is no 3. Theorem Le Kbe a non-archimedean ield, X be a vecor space over Kand Y, P, Tbe a non-archimedean L-uzzy Banach space over K. Le : X YbeaΨapproximaely cubic II uncion. I 3 1 and, or some a Î R, a >0, and some ineger k, k 3, wih 3 k < a, 3 k x,3 k y, L x, y, α, x, y X, > 0, 6:12 and j=n x, M α j 3 kj =1 L, x X, > 0, 6:13 hen here exiss a unique cubic mapping C : X Y such ha P x C x, Ti=1 M x, αi+1 3 ki, 6:14 or all Î X and >0, where M x, := T x, 0,2, 3x, 0,2,..., 3 k 1 x,0,2, x X, > 0. Proo. Firs, we show, by inducion on j, ha, or all x Î X, >0 and j 1, P 3 j x 27 j x, L M j x, := T x, 0,2,..., 3 j 1 x,0,2. 6:15

15 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 15 o 19 Pu y = 0 in 6.11 o obain P 3x 27 x, L x, 0,2, x X, > 0. 6:16 This proves 6.15 or j = 1. Le 6.15 hold or some j>1. Replacing y by 0 and x by 3 j x in 6.16, we ge P 3 j+1 x 27 3 j x, L 3 j x,0,2, x X, > 0. Since 27 1, hen we have P 3 j+1 x 27 j+1 x, L T P 3 j+1 x 27 3 j x,, P 27 3 j x 27 j+1 x, = T P 3 j+1 x 27 3 j x,, P 3 j x 27 j x, L T P 3 j+1 x 27 3 j x,, P 3 j x 27 j x, L T 3 j x,0,2, M j x, = M j+1 x,, x X. 27 Thus 6.15 holds or all j 1. In paricular, i ollows ha P 3 k x 27 k x, L M x,, x X, > 0. 6:17 Replacing x by 3 -kn+k x in 6.17 and using inequaliy 6.12 we obain x x x P 3 kn 27 k 3 kn+k, L M 3 kn+k, L M x, α n+1, x X, > 0, n 0. Then we have P 3 3k n x 3 3k n 3 3k n+1 x, 3 3k n+1 α n+1 L M x, 3 3k n, x X, > 0, n 0. 6:18 6:19 Hence we have x x x P 3 kn 27 k 3 kn+k, L M 3 kn+k, L M x, α n+1, x X, > 0, n 0. Since lim T j=n M x, α j+1 3 3k j =1 L or all x Î X and >0, {k n k --n x} nîn is a Cauchy sequence in he non-archimedean L -uzzy Banach space Y, P, T. Hence we can deine a mapping C : X Y such ha x lim P 3 3k n 3 k n Cx, =1 L, x X, > 0. 6:20

16 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 16 o 19 Nex, or all n 1, x Î X and >0, P x 3 3k n x 3 k n, n 1 = P 3 3k i x i=0 3 k i 3 3k i+1 x 3 k i+1, L T n 1 i=0 P 3 3k i x 3 k i 3 3k i+1 x 3 k i+1, L T n 1 i=0 M x, αi+1 3 3k i. 6:21 Thereore, we have P x C x, L T P x 3 3k n x 3 k n,, P 3 3k n x 3 k n C x, L P T n 1 i=0 M x, αi+1 3 3k i, P 3 3k n x 3 k n C x,. By leing n in he above inequaliy, we obain P x C x, L Ti=1 x, M α i+1 3 3k i, x X, < 0. This proves Since T is coninuous, rom he well known resul in L -uzzy probabilisic normed space see, [51, Chap. 12], i ollows ha lim P 3 k n 3 kn 3x + y + 3 k n 3 kn 3x y 3 3 k n 3 kn x + y 3 3 k n 3 kn x y 48 3 k n 3 kn x, = P C 3x + y + C 3x y 3C x + y 3C x y 48C x,, > 0. On he oher hand, replace x, y by 3 -kn x,3 -kn y in 6.11 and 6.12 o ge P 3 k n 3 kn 3x + y + 3 k n 3 kn 3x y 3 3 k n 3 kn x + y 3 3 k n 3 kn x y 48 3 k n 3 kn y, L 3 kn x,3 kn y, 3 k n α n L x, y, 3 k n, x, y X, > 0. Since lim α x, y, n =1 3 k n L, we iner ha C is a cubic mapping. I C : X Y is anoher cubic mapping such ha P C x x, L M x, or all x Î X and >0, hen, or all n 1, x Î X and >0, P C x C x, L T PC x 3 3k n x 3 k n,, P3 3k n x 3 k n C x,,.

17 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 17 o 19 Thus, rom 6.20, we conclude ha C = C. This complees he proo. Corollary Le Kbe a non-archimedean ield, XbeavecorspaceoverKand Y, P, T.be a non-archimedean L-uzzy Banach space over K under a -norm T Î H. Le : X Ybea Ψ-approximaely cubic II uncion. I, or some a Î R, a >0 and an ineger k, k 3, wih 3 k < a, 3 k x,3 k y, L x, y, α, x X, > 0, hen here exiss a unique cubic mapping C : X Y such ha, or all Î Xand >0, P x C x, L Ti=1 M x, αi+1 3 ki, Where M x, := T x, 0,2, 3x, 0,,..., 3 k 1 x,0,2, x X, > 0. Proo. Since lim M x, α j 3 kj =1 L, x X, > 0, and T is o Hadžić ype, rom Proposiion 2.1, i ollows ha j=n x, M α j 3 kj =1 L, x X, > 0. Thus, i we apply Theorem 6.10, hen we can ge he conclusion. This complees he proo. 7. Conclusion We esablished he Hyers-Ulam-Rassias sabiliy o he cubic uncional equaions 1.1, 1.2, and 1.3 in various uzzy spaces. In Secion 4, we proved he sabiliy o uncional equaions 1.1, 1.2, and 1.3 in a L -uzzy normed space under arbirary - norm which is a generalizaion o [26]. In Secion 6, we proved he sabiliy o uncional equaions 1.1, 1.2, and 1.3 in a non-archimedean L -uzzy normed space. We hereore provided a link among hree various discipline: uzzy se heory, laice heory, and mahemaical analysis. Acknowledgmen This sudy was suppored by Basic Science Research Program hrough he Naional Research Foundaion o Korea NRF unded by he Minisry o Educaion, Science and Technology Gran Number: Auhor deails 1 Deparmen o Mahemaics, Texas A&M Universiy - Kingsville, Kingsville, TX 78363, USA 2 Deparmen o Mahemaics, Faculy o Science, King Abdulaziz Universiy, Jeddah 21589, Saudi Arabia 3 Deparmen o Mahemaics Educaion and he RINS, Gyeongsang Naional Universiy, Chinju , Korea 4 Deparmen o Mahemaics, Science and Research Branch, Islamic Azad Universiy, Tehran, I.R. Iran 5 Deparmen o Mahemaics and Physics, Norh China Elecric Power Universiy, Baoding , China Auhors conribuions All auhors read and approved he inal manuscrip. Compeing ineress The auhors declare ha hey have no compeing ineress.

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19 Agarwal e al. Journal o Inequaliies and Applicaions 2012, 2012:77 hp:// Page 19 o Agarwal, RP, Cho, YJ, Saadai, R: On random opological srucures. Absr Appl Anal, Ar ID , Krishna, SV, Sarma, KKM: Separaion o uzzy normed linear spaces. Fuzzy Ses Sys. 63, Xiao, JZ, Zhu, XH: Fuzzy normed spaces o operaors and is compleeness. Fuzzy Ses Sys. 133, Kasaras, AK: Fuzzy opological vecor spaces II. Fuzzy Ses Sys. 12, Schweizer, B, Sklar, A: Probabilisic Meric Spaces. Elsevier, Norh Holand, New York Bag, T, Samana, SK: Finie dimensional uzzy normed linear spaces. J Fuzzy Mah. 113: Biswas, R: Fuzzy inner produc spaces and uzzy norm uncions. In Sci. 53, Felbin, C: Finie dimensional uzzy normed linear space. Fuzzy Ses Sys. 48, Kramosil, I, Michalek, J: Fuzzy meric and saisical meric spaces. Kyberneica. 11, doi: / x Cie his aricle as: Agarwal e al.: Nonlinear -Fuzzy sabiliy o cubic uncional equaions. Journal o Inequaliies and Applicaions :77. Submi your manuscrip o a journal and benei rom: 7 Convenien online submission 7 Rigorous peer review 7 Immediae publicaion on accepance 7 Open access: aricles reely available online 7 High visibiliy wihin he ield 7 Reaining he copyrigh o your aricle Submi your nex manuscrip a 7 springeropen.com