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Fixed points and stability of functional equations in fuzzy ternary Banach algebras

Abstract

By using Diaz and Margolis fixed point theorem, we establish the generalized Hyers-Ulam-Rassias stability of the ternary homomorphisms and ternary derivations between fuzzy ternary Banach algebras associated to the following (m,n)-Cauchy-Jensen additive functional equation:

1 i 1 < < i m n 1 k l n k l i j , j { 1 , , m } f ( j = 1 m x i j m + l = 1 n m x k l ) = ( n m + 1 ) n ( n m ) i = 1 n f( x i ).

MSC:39B52, 46S40, 26E50.

1 Introduction

A classical question in the theory of functional equations is the following:

When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of ?

If the problem admits a solution, we say that the equation is stable. Such a problem was formulated by Ulam [1] in 1940 and solved in the next year for the Cauchy functional equation by Hyers [2]. Since Hyers, many authors have studied the stability theory for functional equations. The result of Hyers was extended by Aoki [3] in 1950, by considering the unbounded Cauchy differences. Also, Hyers’ theorem was generalized by Rassias [4] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (TM Rassias)

Let f:E E be a mapping from a normed vector space E into a Banach space E subject to the following inequality:

f ( x + y ) f ( x ) f ( y ) ϵ ( x p + y p )

for all x,yE, where ϵ and p are constants with ϵ>0 and 0p<1. Then the limit L(x)= lim n f ( 2 n x ) 2 n exists for all xE, and L:E E is the unique additive mapping which satisfies

f ( x ) L ( x ) 2 ϵ 2 2 p x p

for all xE. Also, if for each xE, the function f(tx) is continuous in tR, then L is linear.

Găvruta [5] generalized the Rassias’ result. Beginning around the year 1980, the stability problems of several functional equations and approximate homomorphisms have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [629]).

Katsaras [30] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view (see [31, 32]). In particular, Bag and Samanta [33], following Cheng and Mordeson [34], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Karmosil and Michalek type [35]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [36].

Now, we consider a mapping f:XY satisfying the following functional equation, which is introduced by Rassias and Kim [37] (see also [38]):

1 i 1 < < i m n 1 k l n k l i j , j { 1 , , m } f ( j = 1 m x i j m + l = 1 n m x k l ) = ( n m + 1 ) n ( n m ) i = 1 n f( x i )
(1.1)

for all x 1 ,, x n X, where m,nN are fixed integers with n2 and 1mn. Especially, we observe that, in the case m=1, equation (1.1) yields the Cauchy additive equation

f ( l = 1 n x k l ) = l = 1 n f( x i ).

Also, we observe that, in the case m=n, equation (1.1) yields the Jensen additive equation

f ( j = 1 n x j n ) = 1 n l = 1 n f( x i ).

Therefore, equation (1.1) is a generalized form of the Cauchy-Jensen additive equation and thus every solution of equation (1.1) may be analogously called the general (m,n)-Cauchy-Jensen additive. For the case m=2, the authors have established new theorems about the Ulam-Hyers-Rassias stability in quasi-β-normed spaces [37].

Let X and Y be linear spaces. For each m with 1mn, a mapping f:XY satisfies equation (1.1) for all n2 if and only if f(x)f(0)=A(x) is Cauchy additive, where f(0)=0 if m<n. In particular, we have f((nm+1)x)=(nm+1)f(x) and f(mx)=mf(x) for all xX.

Definition 1.1 Let X be a real vector space. A function N:X×R[0,1] is called a fuzzy norm on X if for all x,yX and s,tR,

(N1) N(x,t)=0 for t0;

(N2) x=0 if and only if N(x,t)=1 for all t>0;

(N3) N(cx,t)=N(x, t | c | ) if c0;

(N4) N(x+y,s+t)min{N(x,s),N(y,t)};

(N5) N(x,) is a non-decreasing function of and lim t N(x,t)=1;

(N6) for any x0, N(x,) is continuous on .

Example 1.1 Let (X,) be a normed linear space and β>0. Then

N(x,t)={ t t + β x , t > 0 , x X , 0 , t 0 , x X

is a fuzzy norm on X.

Definition 1.2 Let (X,N) be a fuzzy normed vector space. A sequence { x n } in X is said to be convergent if there exists xX such that

lim n N( x n x,t)=1

for all t>0. In this case, x is called the limit of the sequence { x n } in X, which is denoted by N lim t x n =x.

Definition 1.3 Let (X,N) be a fuzzy normed vector space. A sequence { x n } in X is called a Cauchy sequence if for each ϵ>0 and each t>0, there exists n 0 N such that, for all n n 0 and p>0,

N( x n + p x n ,t)>1ϵ.

It is well known that every convergent sequence in a fuzzy normed vector space is a Cauchy sequence. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a mapping f:XY between fuzzy normed vector spaces X and Y is continuous at a point xX if for each sequence { x n } converging to x 0 X, the sequence {f( x n )} converges to f( x 0 ). If f:XY is continuous at each xX, then f:XY is said to be continuous on X (see [36]).

Ternary algebraic operations were considered in the nineteenth century by several mathematicians such as Cayley [39] who introduced the notion of cubic matrix which in turn was generalized by Kapranov, Gelfand and Zelevinskii in 1990 [40]. The comments on physical applications of ternary structures can be found in [4145].

Definition 1.4 Let X be a ternary algebra and (X,N) be a fuzzy normed space.

  1. (1)

    The fuzzy normed space (X,N) is called a ternary fuzzy normed algebra if

    N ( [ x y z ] , s t u ) N(x,s)N(y,t)N(z,u)

for all x,y,zX and s,t,u>0;

  1. (2)

    A complete ternary fuzzy normed algebra is called a ternary fuzzy Banach algebra.

Example 1.2 Let (X,) be a ternary normed (Banach) algebra. Let

N(x,t)={ t t + x , t > 0 , x X , 0 , t 0 , x X .

Then N(x,t) is a fuzzy norm on X and (X,N) is a ternary fuzzy normed (Banach) algebra.

Definition 1.5 Let (X,N) and (Y, N ) be two ternary fuzzy normed algebras.

  1. (1)

    A -linear mapping H:(X,N)(Y, N ) is called a ternary homomorphism if

    H ( [ x y z ] ) = [ H ( x ) H ( y ) H ( z ) ]

for all x,y,zX;

  1. (2)

    A -linear mapping D:(X,N)(X,N) is called a ternary fuzzy derivation if

    D ( [ x y z ] ) = [ D ( x ) y z ] + [ x D ( y ) z ] + [ x y D ( z ) ]

for all x,y,zX.

We apply the following theorem on weighted spaces (see [4649]).

Theorem 1.2 (The generalized fixed point theorem of Diaz and Margolis)

Let (X,d) be a complete metric space and T:XX be a contraction, i.e., there exists α[0,1) such that

d(Tx,Ty)αd(x,y)

for all x,yX. Then there exists a unique aX such that Ta=a. Moreover, a= lim n T n x and

d(a,x) 1 1 α d(x,Tx)

for all xX.

Throughout this paper, we suppose that X is a ternary fuzzy normed algebra and Y is a ternary fuzzy Banach algebra. Moreover, we assume that n 0 N is a positive integer and T 1 n o 1 :={ e i θ :0θ 2 π n o }. For the convenience, we use the following abbreviation for a given mapping f:XY:

2 Main results

In this section, by using the idea of Gavruta and Gavruta [14], we prove the generalized Hyers-Ulam-Rassias stability of ternary homomorphisms related to functional equation (1.1) on ternary fuzzy Banach algebras (see also [50]).

Theorem 2.1 Let n3 and φ: X n [0,) be a mapping such that there exists L< 1 ( n m + 1 ) n 2 such that

φ ( x 1 n m + 1 , , x n n m + 1 ) L φ ( x 1 , x 2 , , x n ) n m + 1

for all x 1 ,, x n X. Let f:XY with f(0)=0 be a mapping satisfying

N ( Δ f ( x 1 , , x n ) , t ) t t + φ ( x 1 , , x n )
(2.1)

and

N ( f ( [ a b c ] ) [ f ( a ) f ( b ) f ( c ) ] , t ) t t + φ ( a , b , c , 0 , , 0 )
(2.2)

for all μ T 1 n o 1 , x 1 ,, x n ,a,b,cX and t>0. Then there exists a unique ternary homomorphism H:XY such that

N ( f ( x ) H ( x ) , t ) ( n m + 1 ) ( n m ) ( 1 L ) t ( n m + 1 ) ( n m ) ( 1 L ) t + L φ ( x , , x )
(2.3)

for all xX and t>0.

Proof Letting μ=1 and putting x 1 = x 2 == x n =x in (2.1), we have

N ( ( n m ) f ( ( n m + 1 ) x ) ( n m ) ( n m + 1 ) f ( x ) , t ) t t + φ ( x , , x )
(2.4)

for all xX and t>0. Set S 0 :={h:XY:h(0)=0} and define a mapping d 0 : S 0 × S 0 [0,] by

d 0 (f,g)=inf { μ R + : N ( g ( x ) h ( x ) , μ t ) t t + φ ( x , , x ) , x X , t > 0 } ,

where inf=+. Also, put S:={h S 0 : d 0 (h,f)<}. Suppose that d is the restriction of d 0 on S×S. By using the same technique in the proof of Theorem 3.2 [50], we can show that (S,d) is a complete metric space. Now, we define a mapping J:SS by

Jg(x):=(nm+1)g ( x n m + 1 )

for all xX. It is easy to see that d(Jg,Jh)Ld(g,h) for all g,hS. This implies that

d(f,Jf) L ( n m + 1 ) ( n m ) .

Thus, by Banach’s fixed point theorem (Theorem 1.2), J has a unique fixed point H:XY in S satisfying

H ( x n m + 1 ) = H ( x ) n m + 1
(2.5)

for all xX. This implies that H is a unique mapping with (2.5) such that there exists μ(0,) satisfying

N ( f ( x ) H ( x ) , μ t ) t t + φ ( x , , x )

for all xX and t>0.

Moreover, we have d( J p f,H)0 as p, which implies

N- lim p f ( x ( n m + 1 ) p ) ( n m + 1 ) p =H(x)
(2.6)

for all xX. Thus it follows from (2.1) and (2.6) that

1 i 1 < < i m n 1 k l n k l i j , j { 1 , , m } H ( j = 1 m μ x i j m + l = 1 n m μ x k l ) = ( n m + 1 ) n ( n m ) i = 1 n μH( x i )

for all μ T 1 n o 1 and x 1 ,, x n X. This means that H:XY is additive. By using the same technique as in the proof of Theorem 2.1 [51], we can show that H is -linear. On the other hand, by (2.2), we have

N(α,β) t t + φ ( a ( n m + 1 ) p , b ( n m + 1 ) p , c ( n m + 1 ) p , 0 , 0 , , 0 )

for all a,b,cX and t>0, where

Then we have, as p+,

for all a,b,cX and t>0. So, it follows that

N ( H ( [ a b c ] ) [ H ( a ) H ( b ) H ( c ) ] , t ) =1

for all a,b,cX and t>0. Hence we have H([abc])=[H(a)H(b)H(c)] for all a,b,cX. This means that H is a ternary homomorphism. This completes the proof. □

Theorem 2.2 Let φ: X n [0,) be a mapping such that there exists L<1 with

φ( x 1 ,, x n )(nm+1)Lφ ( x 1 n m + 1 , , x n n m + 1 )

for all x 1 , x 2 ,, x n X. Let f:XY be a mapping with f(0)=0 satisfying (2.1). Then the limit H(x):=N- lim p f ( ( n m + 1 ) p x ) ( n m + 1 ) p exists for all xX and H:XY is defined as a ternary homomorphism such that

N ( f ( x ) H ( x ) , t ) ( n m + 1 ) ( n m ) ( 1 L ) t ( n m + 1 ) ( n m ) ( 1 L ) t + φ ( x , , x )
(2.7)

for all xX and t>0.

Proof Let (S,d) be the metric space defined as in the proof of Theorem 2.1. Consider the mapping T:SS defined by Tg(x):= g ( ( n m + 1 ) x ) n m + 1 for all xX. One can show that d(g,h)=ϵ implies that d(Tg,Th)Lϵ for all positive real numbers ϵ. This means that T is a contraction on (S,d). The mapping

H(x):=N- lim p f ( ( n m + 1 ) p x ) ( n m + 1 ) p

is the unique fixed point of T in S and H has the following property:

(nm+1)H(x)=H ( ( n m + 1 ) x )
(2.8)

for all xX. This implies that H is a unique mapping satisfying (2.8) such that there exists μ(0,) satisfying N(f(x)H(x),μt) t t + φ ( x , , x ) for all xX and t>0.

The rest of the proof is similar to the proof of Theorem 2.1. This completes the proof. □

Now, we investigate the Hyers-Ulam-Rassias stability of ternary derivations in ternary fuzzy Banach algebras.

Theorem 2.3 Let X be a fuzzy Banach algebra. Let φ: X n [0,) be a function such that there exists L< 1 ( n m + 1 ) n 2 with

φ ( x 1 n m + 1 , , x n n m + 1 ) L φ ( x 1 , x 2 , , x n ) n m + 1

for all x 1 ,, x n X. Let f:XX be a mapping with f(0)=0 satisfying (2.1) and

N ( f ( [ a b c ] ) [ f ( a ) b c ] [ a f ( b ) c ] [ a b f ( c ) ] , t ) t t + φ ( a , b , c , 0 , 0 , , 0 )
(2.9)

for all a,b,cX and t>0. Then D(x):=N- lim p f ( x ( n m + 1 ) p ) ( n m + 1 ) p exists for all xX and D:XX is defined as a unique ternary derivation such that

N ( f ( x ) D ( x ) , t ) ( n m + 1 ) ( n m ) ( 1 L ) t ( n m + 1 ) ( n m ) ( 1 L ) t + L φ ( x , , x )
(2.10)

for all xX and t>0.

Proof By the same reasoning as that in the proof of Theorem 2.1, the mapping D:XX is a unique -linear mapping which satisfies (2.10).

Now, we show that D is a ternary derivation. By (2.9), we have

(2.11)

for all a,b,cX and t>0. Then we have

for all a,b,cX and t>0. So, we have

N ( D ( [ a b c ] ) [ D ( a ) b c ] [ a D ( b ) c ] [ a b H ( c ) ] , t ) =1

for all a,b,cX and t>0. Hence we have D([abc])=[D(a)bc]+[aD(b)c]+[abD(c)] for all a,b,cX . This means that D is a ternary derivation. This completes the proof. □

Theorem 2.4 Let X be a fuzzy Banach algebra. Let φ: X n [0,) be a function such that there exists L<1 with

φ( x 1 ,, x n )(nm+1)Lφ ( x 1 n m + 1 , , x n n m + 1 )

for all x 1 , x 2 ,, x n X. Let f:XX be a mapping with f(0)=0 satisfying (2.1) and (2.9). Then the limit D(x):=N- lim p f ( ( n m + 1 ) p x ) ( n m + 1 ) p exists for all xX and D:XX is defined as a ternary derivation such that

N ( f ( x ) D ( x ) , t ) ( n m + 1 ) ( n m ) ( 1 L ) t ( n m + 1 ) ( n m ) ( 1 L ) t + φ ( x , , x )
(2.12)

for all xX and t>0.

References

  1. 1.

    Ulam SM: A Collection of the Mathematical Problems. Interscience, New York; 1960.

  2. 2.

    Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222

  3. 3.

    Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64–66. 10.2969/jmsj/00210064

  4. 4.

    Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1

  5. 5.

    Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431–436. 10.1006/jmaa.1994.1211

  6. 6.

    Bourgin DG: Classes of transformations and bordering transformations. Bull. Am. Math. Soc. 1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7

  7. 7.

    Cădariu L, Radu V: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 2003., 4(1): Article ID 4

  8. 8.

    Cădariu L, Radu V: Fixed points and the stability of quadratic functional equations. An. Univ. Timişoara, Ser. Mat. Inform 2003, 41: 25–48.

  9. 9.

    Czerwik S: The stability of the quadratic functional equation. In Stability of Mappings of Hyers-Ulam Type. Edited by: Rassias TM, Tabor J. Hadronic Press, Florida; 1994:81–91.

  10. 10.

    Eshaghi Gordji M, Rassias JM, Ghobadipour N:Generalized Hyers-Ulam stability of generalized(N,K)-derivations. Abstr. Appl. Anal. 2009., 2009: Article ID 437931

  11. 11.

    Eshaghi Gordji M, Ghaemi MB, Kaboli Gharetapeh S, Shams S, Ebadian A:On the stability of J -derivations. J. Geom. Phys. 2010, 60: 454–459. 10.1016/j.geomphys.2009.11.004

  12. 12.

    Eshaghi Gordji M, Najati A:Approximately J -homomorphisms: a fixed point approach. J. Geom. Phys. 2010, 60: 809–814. 10.1016/j.geomphys.2010.01.012

  13. 13.

    Rassias JM, Jun KW, Kim HM:Approximate(m,n)-Cauchy-Jensen additive mappings in C -algebras. Acta Math. Sin. 2011, 27(10):1907–1922. 10.1007/s10114-011-0179-4

  14. 14.

    Gǎvruta P, Gǎvruta L: A new method for the generalized Hyers-Ulam-Rassias stability. Int. J. Nonlinear Anal. Appl. 2010, 1: 11–18.

  15. 15.

    Khodaei H, Rassias TM: Approximately generalized additive functions in several variables. Int. J. Nonlinear Anal. Appl. 2010, 1: 22–41.

  16. 16.

    Cholewa PW: Remarks on the stability of functional equations. Aequ. Math. 1984, 27: 76–86. 10.1007/BF02192660

  17. 17.

    Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 1992, 62: 239–248.

  18. 18.

    Saadati R, Vaezpour M, Cho YJ: A note to paper ‘On the stability of cubic mappings and quartic mappings in random normed spaces’. J. Inequal. Appl. 2009., 2009: Article ID 214530

  19. 19.

    Saadati R, Zohdi MM, Vaezpour SM: Nonlinear L -random stability of an ACQ functional equation. J. Inequal. Appl. 2011., 2011: Article ID 194394

  20. 20.

    Azadi Kenary H, Rezaei H, Ghaffaripour A, Talebzadeh S, Park C, Lee JR: Fuzzy Hyers-Ulam stability of an additive functional equation. J. Inequal. Appl. 2011., 2011: Article ID 140

  21. 21.

    Park C, Lee JR, Rassias TM, Saadati R: Fuzzy -homomorphisms and fuzzy -derivations in induced fuzzy C -algebras. Math. Comput. Model. 2011, 54: 2027–2039. 10.1016/j.mcm.2011.05.012

  22. 22.

    Park C, Eshaghi Gordji M, Cho YJ: Stability and superstability of generalized quadratic ternary derivations on non-Archimedean ternary Banach algebras: a fixed point approach. Fixed Point Theory Appl. 2012., 2012: Article ID 97

  23. 23.

    Cho YJ, Kang SM, Sadaati R: Nonlinear random stability via fixed-point method. J. Appl. Math. 2012., 2012: Article ID 902931

  24. 24.

    Cho YJ, Park C, Saadati R: Functional inequalities in non-Archimedean Banach spaces. Appl. Math. Lett. 2010, 23: 1238–1242. 10.1016/j.aml.2010.06.005

  25. 25.

    Cho YJ, Saadati R, Vahidi J:Approximation of homomorphisms and derivations on non-Archimedean Lie C -algebras via fixed point method. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 373904

  26. 26.

    Saadati R, Cho YJ, Vahidi J: The stability of the quartic functional equation in various spaces. Comput. Math. Appl. 2010, 60: 1994–2002. 10.1016/j.camwa.2010.07.034

  27. 27.

    Saadati R, Park C: Non-Archimedean L -fuzzy normed spaces and stability of functional equations. Comput. Math. Appl. 2010, 60: 2488–2496. 10.1016/j.camwa.2010.08.055

  28. 28.

    Saadati R, Vaezpour SM, Park C: The stability of the cubic functional equation in various spaces. Math. Commun. 2011, 16: 131–145.

  29. 29.

    Rassias TM: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 2000, 251: 264–284. 10.1006/jmaa.2000.7046

  30. 30.

    Katsaras AK: Fuzzy topological vector spaces. Fuzzy Sets Syst. 1984, 12: 143–154. 10.1016/0165-0114(84)90034-4

  31. 31.

    Felbin C: Finite-dimensional fuzzy normed linear space. Fuzzy Sets Syst. 1992, 48: 239–248. 10.1016/0165-0114(92)90338-5

  32. 32.

    Park C: Fuzzy stability of a functional equation associated with inner product spaces. Fuzzy Sets Syst. 2009, 160: 1632–1642. 10.1016/j.fss.2008.11.027

  33. 33.

    Bag T, Samanta SK: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 2003, 11: 687–705.

  34. 34.

    Cheng SC, Mordeson JN: Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 1994, 86: 429–436.

  35. 35.

    Karmosil I, Michalek J: Fuzzy metric and statistical metric spaces. Kybernetica 1975, 11: 326–334.

  36. 36.

    Bag T, Samanta SK: Fuzzy bounded linear operators. Fuzzy Sets Syst. 2005, 151: 513–547. 10.1016/j.fss.2004.05.004

  37. 37.

    Rassias JM, Kim H: Generalized Hyers-Ulam stability for general additive functional equations in quasi β -normed spaces. J. Math. Anal. Appl. 2009, 356: 302–309. 10.1016/j.jmaa.2009.03.005

  38. 38.

    Azadi Kenary H: Non-Archimedean stability of Cauchy-Jensen type functional equation. J. Nonlinear Anal. Appl. 2011, 1: 1–10.

  39. 39.

    Cayley A: On the 34 concomitants of the ternary cubic. Am. J. Math. 1981, 4: 1–15.

  40. 40.

    Kapranov M, Gelfand IM, Zelevinskii A: Discrimininants, Resultants and Multidimensional Determinants. Birkhauser, Berlin; 1994.

  41. 41.

    Abramov V, Kerner R, Liivapuu O, Shitov S: Algebras with ternary law of composition and their realization by cubic matrices. J. Gen. Lie Theory Appl. 2009, 3: 77–94. 10.4303/jglta/S090201

  42. 42.

    Kerner R: Ternary and non-associative structures. Int. J. Geom. Methods Mod. Phys. 2008, 5: 1265–1294. 10.1142/S0219887808003326

  43. 43.

    Kerner R: The cubic chessboard. Geometry and physics. Class. Quantum Gravity 1997, 14: 203–225. 10.1088/0264-9381/14/1A/017

  44. 44.

    Sewell GL: Quantum Mechanics and Its Emergent Macrophysics. Princeton University Press, Princeton; 2002.

  45. 45.

    Zettl H: A characterization of ternary rings of operators. Adv. Math. 1983, 48: 117–143. 10.1016/0001-8708(83)90083-X

  46. 46.

    Diaz J, Margolis B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 1968, 74: 305–309. 10.1090/S0002-9904-1968-11933-0

  47. 47.

    Mihet D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 2008, 343: 567–572. 10.1016/j.jmaa.2008.01.100

  48. 48.

    Park C, Lee JR, Rassias TM, Saadati R: Fuzzy -homomorphisms and fuzzy -derivations in induced fuzzy C -algebras. Math. Comput. Model. 2011, 54: 2027–2039. 10.1016/j.mcm.2011.05.012

  49. 49.

    Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4: 91–96.

  50. 50.

    Eshaghi Gordji, M, Moradlou, F: Approximate Jordan derivations on Hilbert C -moduls. Fixed Point Theory (to appear)

  51. 51.

    Eshaghi Gordji M: Nearly involutions on Banach algebras; a fixed point approach. Fixed Point Theory 2011, 12: 341–348.

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Acknowledgements

The second author of this work was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2012-0008170).

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Asgari, G., Cho, Y., Lee, Y. et al. Fixed points and stability of functional equations in fuzzy ternary Banach algebras. J Inequal Appl 2013, 166 (2013). https://doi.org/10.1186/1029-242X-2013-166

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Keywords

  • Hyers-Ulam-Rassias stability
  • Diaz and Margolis contraction theorem
  • fuzzy ternary Banach algebra
  • ternary algebras
  • functional equations