On fuzzy normed algebras

Transcription

1 Available online a J. Nonlinear Sci. Appl. 9 (2016), Research Aricle On fuzzy normed algebras Tudor Bînzar a,, Flavius Paer a, Sorin Nădăban b a Deparmen of Mahemaics, Poliehnica Universiy of Timişoara, Regina Maria 1, RO Timişoara, Romania. b Deparmen of Mahemaics and Compuer Science, Aurel Vlaicu Universiy of Arad, Elena Drăgoi 2, RO-100, Arad, Romania. Communicaed by R. Saadai Absrac In his paper, a characerizaion for coninuous produc in a fuzzy normed algebra is esablished and i is proved ha any fuzzy normed algebra is wih coninuous produc. Anoher ype of coninuiy for he produc in a fuzzy normed algebras is inroduced and sudied. These conceps are illusraed by some examples. Also, he Caresian produc of fuzzy normed algebras is analyzed. c 2016 All righs reserved. Keywords: Fuzzy normed algebra, coninuous produc, fuzzy normed linear space MSC: 46S Inroducion Fuzzy logic and fuzzy ses inroduced by Zadeh in his famous paper [19] have quickly found heir applicabiliy in a wide variey of domains: conrol engineering, arificial inelligence, compuer science, roboics and many more. A he same ime, many mahemaicians have ried o ranslae he classical resuls of mahemaics in fuzzy conex. This approach is moivaed by he fac ha fuzzy heory has proved a useful ool o describe siuaions in which daa are imprecise. An imporan issue is finding a suiable definiion for he fuzzy norm. In he sudy of fuzzy opological vecor spaces, Kasaras [10] was he one who firs inroduced he noion of fuzzy norm. In 1992, Felbin [8] inroduced anoher concep of fuzzy norm by assigning a fuzzy real number o each elemen of he linear space. In 1994, Cheng and Mordenson [6] presened anoher idea of fuzzy norm on a linear space and in his siuaion he corresponding fuzzy meric is of Kramosil and Michálek ype. Following Cheng and Mordenson, in 200, Bag and Samana [] inroduced a new concep of fuzzy norm and hey sudied he properies of finie dimensional fuzzy normed linear space. In he paper [4], Bag and Samana inroduced Corresponding auhor addresses: udor.binzar@up.ro (Tudor Bînzar), flavius.paer@up.ro (Flavius Paer), snadaban@gmail.com (Sorin Nădăban) Received

2 T. Bînzar, F. Paer, S. Nădăban, J. Nonlinear Sci. Appl. 9 (2016), differen ypes of coninuiies and boundedness for linear operaors and esablished he principles of fuzzy funcional analysis. A comparaive sudy on fuzzy norms inroduced by Kasaras, by Felbin and by Bag and Samana was made in paper [5]. A new concep of fuzzy normed space was inroduced by Saadai and Vaezpour in [15]. Oher approaches for fuzzy normed linear spaces can be found in [1, 2, 9, 11, 12, 14, 17] ec. In his paper we will use he definiion of he fuzzy norm inroduced by Nădăban and Dziac [14]. From he noion of fuzzy normed linear space o he concep of fuzzy normed algebra i is one sep ha had o be done. Thus in paper [16], Sadeqi and Amiripour gave a definiion of fuzzy Banach algebra and esablished some resuls in his conex. We also noe ha Dinda e al. inroduced and sudied some imporan properies of inuiionisic fuzzy Banach algebra in he paper [7]. In his paper, we will use he concep of fuzzy normed algebra recenly inroduced by Mirmosafaee [11] in A characerizaion for coninuous produc in a fuzzy normed algebra is esablished and i is proved ha any fuzzy normed algebra is wih coninuous produc. Anoher ype of coninuiy for he produc in a fuzzy normed algebras is inroduced and sudied. These conceps are illusraed by some examples. Also, he Caresian produc of fuzzy normed algebras is analyzed. The resuls obained in his paper consiue a foundaion for he developmen of some specral properies in fuzzy normed algebras. 2. Preliminaries Definiion 2.1 ([18]). A binary operaion : [0, 1] [0, 1] [0, 1] is called riangular norm (-norm) if i saisfies he following condiion: 1. a b = b a, a, b [0, 1]; 2. a 1 = a, a [0, 1];. (a b) c = a (b c), a, b, c [0, 1]; 4. If a c and b d, wih a, b, c, d [0, 1], hen a b c d. Example 2.2. Three basic examples of coninuous -norms are,, L, which are defined by a b = min{a, b}, a b = ab (usual muliplicaion in [0, 1]) and a L b = max{a + b 1, 0} (he Lukasiewicz -norm). Definiion 2. ([1]). Le, be wo -norms. We say ha dominaes and we denoe if (x 1 x 2 ) (y 1 y 2 ) (x 1 y 1 ) (x 2 y 2 ) for all x 1, x 2, y 1, y 2 [0, 1]. Definiion 2.4 ([14]). Le X be a vecor space over a field K (where K is R or C) and be a coninuous -norm. A fuzzy se N in X [0, ) is called a fuzzy norm on X if i saisfies: (N1) N(x, 0) = 0, x X; (N2) [N(x, ) = 1, > 0] if and only if x = 0; ( ) (N) N(λx, ) = N x, λ, x X, 0, λ K ; (N4) N(x + y, + s) N(x, ) N(y, s), x, y X,, s 0; (N5) x X, N(x, ) is lef coninuous and lim N(x, ) = 1. The riple (X, N, ) will be called fuzzy normed linear space. Remark 2.5. N(x, ) is nondecreasing for all x X. Theorem 2.6 ([14]). Le (X, N, ) be a fuzzy normed linear space. For x X, r (0, 1), > 0 we define he open ball B(x, r, ) := {y X : N(x y, ) > 1 r}.

3 T. Bînzar, F. Paer, S. Nădăban, J. Nonlinear Sci. Appl. 9 (2016), Then 1. T N := {T X : x T iff > 0, r (0, 1) : B(x, r, ) T } is a opology on X; 2. If he -norm saisfies sup x (0,1) x x = 1, hen (X, T N ) is Hausdorff;. (X, T N ) is a merizable opological vecor space. Definiion 2.7 ([4]). Le (X, N 1, 1 ), (Y, N 2, 2 ) be fuzzy normed linear spaces. A mapping T : X Y is said o be fuzzy coninuous a x 0 X, if such ha ε > 0, α (0, 1), δ = δ(ε, α, x 0 ) > 0, β = β(ε, α, x 0 ) (0, 1) x X : N 1 (x x 0, δ) > β we have ha N 2 (T (x) T (x 0 ), ε) > α. If T is fuzzy coninuous a each poin of X, hen T is called fuzzy coninuous on X. Definiion 2.8 ([]). Le (X, N, ) be a fuzzy normed linear space and (x n ) be a sequence in X. 1. The sequence (x n ) is said o be convergen if here exiss x X such ha lim N(x n x, ) = 1, > 0. n In his case, x is called he limi of he sequence (x n ) and we denoe lim n x n = x or x n x. 2. The sequence (x n ) is called Cauchy sequence if lim N(x n+p x n, ) = 1, > 0, p N. n. (X, N, ) is said o be complee if any Cauchy sequence in X converges o a poin in X. A complee fuzzy normed linear space will be called a fuzzy Banach space.. Fuzzy normed algebras Definiion.1 ([11]). I is called fuzzy normed algebra he quadruple (X, N,, ) if we have (A1), are coninuous -norms; (A2) X is an algebra; (A) (X, N, ) is a fuzzy normed linear space; (A4) N(xy, s) N(x, ) N(y, s) x, y X,, s 0. If (X, N, ) is a fuzzy Banach space, hen (X, N,, ) will be called fuzzy Banach algebra. Example.2. Le (X, ) be a normed algebra,, be coninuous -norms and { 0, x N : X [0, ) [0, 1] defined by N(x, ) =. 1, > x Then (X, N,, ) is a fuzzy normed algebra. Proof. I is easy o check (N1)-(N) and (N5). We verify he condiion (N4). Le x, y X,, s [0, ). If x+y +s, hen x or s y (conrarily > x and s > y, hus +s > x + y x+y, conradicion). If x, hen N(x, ) = 0. If s y, hen N(y, s) = 0. Thus N(x, ) N(y, s) = 0. Therefore he inequaliy N(x + y, + s) N(x, ) N(y, s) holds. If x + y < + s, hen N(x + y, + s) = 1 and he inequaliy N(x + y, + s) N(x, ) N(y, s) holds. I remains o verify (A4). Le x, y X,, s [0, ). If xy s, hen x or s y (conrarily > x and s > y, hus s > x y xy, conradicion). If x, hen N(x, ) = 0. If s y, hen N(y, s) = 0. Thus N(x, ) N(y, s) = 0. Therefore he inequaliy N(xy, s) N(x, ) N(y, s) holds. If xy < s, hen N(xy, s) = 1 and he inequaliy N(xy, s) N(x, ) N(y, s) holds.

4 T. Bînzar, F. Paer, S. Nădăban, J. Nonlinear Sci. Appl. 9 (2016), Example.. Le (X, ) be a normed algebra and N : X [0, ) [0, 1] defined by, > 0 N(x, ) := + x. 0, = 0 Then (X, N,, ) is a fuzzy normed algebra. Proof. By [4], (X, N, ) is a fuzzy normed linear space. I remains o verify (A4), ha is N(xy, s) N(x, ) N(y, s), x, y X,, s [0, ). For = 0 or s = 0 he inequaliy is obvious. For 0 and s 0, he inequaliy is equivalen o s s + xy + x s s + y, namely s + y + s x + x y s + xy, which is evidenly rue. Example.4. Le (X, ) be a normed algebra and N : X [0, ) [0, 1] defined by, > 0 N(x, ) := + x. 0, = 0 Then (X, N,, ) is a fuzzy normed algebra. Proof. We will prove ha (X, N, ) is a fuzzy normed linear space. According o [4], condiions (N1)-(N) and (N5) are verified. I remains o prove (N4), ha is, N(x + y, + s) N(x, ) N(y, s), x, y X,, s 0. Indeed, for = 0 or s = 0 he inequaliy is obviously rue. For 0 and s 0, he inequaliy is equivalen o + s + s + x + y + x s s + y, namely ( + s)( + x )(s + y ) s( + s + x + y ), which is equivalen o s( x + y ) + s 2 x + 2 y + ( + s) x y s x + y. Because s( x + y ) s x + y and all he oher erms from he lef member are posiive, he inequaliy follows. Therefore (X, N, ) is a fuzzy normed linear space. Moreover, condiions (A1)-(A4) are saisfied similar o he proof from he previous example. I follows (X, N,, ) is a fuzzy normed algebra. Theorem.5. A fuzzy normed algebra (X, N,, ) is wih coninuous produc if and only if α (0, 1), β = β(α) (0, 1), M = M(α) > 0 such ha x, y X, s, > 0 : N(x, s) > β, N(y, ) > β N(xy, Ms) > α. Proof. ( ) Le α (0, 1) and V := {u X : N(u, 1) > α} be an open neighbourhood of zero. As X X (x, y) x y X is coninuous a (0, 0), here exis ɛ 1 = ɛ 1 (α) > 0, ɛ 2 = ɛ 2 (α) > 0, γ 1 = γ 1 (α) (0, 1), γ 2 = γ 2 (α) (0, 1) such ha u 1, u 2 X : N(u 1, ɛ 1 ) > γ 1, N(u 2, ɛ 2 ) > γ 2 we have ha N(u 1 u 2, 1) > α.

5 T. Bînzar, F. Paer, S. Nădăban, J. Nonlinear Sci. Appl. 9 (2016), Le β = max{γ 1, γ 2 } (0, 1), M = 1 ɛ 1 ɛ 2 > 0. Le x, y X, s, > 0 such ha N(x, s) > β, N(y, ) > β. Then N(x/s, 1) > β γ 1 and N(y/, 1) > β γ 2. Le u 1 = ɛ 1x s, u 2 = ɛ 2y. We noe ha N(u 1, ɛ 1 ) = N(u 1 /ɛ 1, 1) = N(x/s, 1) > γ 1, N(u 2, ɛ 2 ) = N(u 2 /ɛ 2, 1) = N(u/, 1) > γ 2. Hence N(u 1 u 2, 1) > α, i.e., N ( ( ) ɛ 1 x s ɛ2y, 1) s > α, namely N xy, ɛ 1 ɛ 2 > α. Thus N(xy, Ms) > α. ( ) Firs we will prove ha for each y 0 X, he mapping X x xy o X is coninuous. Le ɛ > 0, α (0, 1). Thus here exis β = β(α) (0, 1), M = M(α) > 0 such ha N(x, s) > β, N(y, ) > β N(xy, Ms) > α. As lim N(y 0, ) = 1, here exiss 0 > 0 such ha N(y 0, 0 ) > β. Le δ = δ(α, ɛ) = and β(α, ɛ) = β. Le x X such ha N(x, δ) > β. As N(y 0, 0 ) > β, we obain ha N(xy 0, M 0 δ) > α, namely N(xy 0, ɛ) > α. Similarly, we can esablish ha, for each x 0 X, he mapping X y x 0 y X is coninuous. Now, we will prove ha (X, N,, ) is wih coninuous produc. Le x n x 0, y n y 0. Thus x n y 0 x 0 y 0 and x 0 y n x 0 y 0. Hence lim N(x ny 0 x 0 y 0, s) = 1, lim N(x 0y n x 0 y 0, ) = 1 for all s, > 0. Therefore n n N(x n y n x 0 y 0, ) = N((x n x 0 )(y n y 0 ) + (x n x 0 )y 0 + x 0 (y n y 0 ), ) ( N (x n x 0 )(y n y 0 ), ) ( N (x n x 0 )y 0, ) N ( ( ) ( )) N x n x 0, N y n y 0, N 1. Hence x n y n x 0 y 0. Lemma.6. Any coninuous -norm saisfies: ɛ 0 M ( x 0 (y n y 0 ), ) γ (0, 1), α, β (0, 1) such ha α β = γ. ( (x n x 0 )y 0, ) ( N x 0 (y n y 0 ), ) Proof. Le γ (0, 1). Choose α > γ. Le g : [0, 1] [0, 1] defined by g(y) = α y. As is coninuous, we have ha g is coninuous. As g(0) = α 0 = 0 and g(1) = α 1 = α, for γ (0, α) here exiss β (0, 1) such ha g(β) = γ, namely α β = γ. Theorem.7. Any fuzzy normed algebra (X, N,, ) is wih coninuous produc. Proof. Le α (0, 1). Then here exiss ε > 0 such ha α + ε (0, 1). As is a coninuous -norm, by he previous lemma, we obain ha here exis β α, γ α (0, 1) such ha α + ε = β α γ α. We suppose ha β α γ α (he case β α γ α is similar). We choose M = M(α) = 1. Le x, y X, s, > 0 such ha N(x, s) > β α, N(y, ) > β α. Then N(xy, Ms) N(x, s) N(y, ) β α β α β α γ α = α + ε > α.

6 T. Bînzar, F. Paer, S. Nădăban, J. Nonlinear Sci. Appl. 9 (2016), Definiion.8. The fuzzy normed algebra (X, N,, ) is called wih muliplicaively coninuous produc if α (0, 1), x, y X, s, > 0 : N(x, s) > α, N(y, ) > α N(xy, s) α. Example.9 (Fuzzy normed algebra wih muliplicaively coninuous produc). The fuzzy normed algebra (X, N,, ) from Example.2 is wih muliplicaively coninuous produc. Proof. Indeed, le α (0, 1), x, y X, s, > 0 such ha N(x, s) > α, N(y, ) > α. Then N(x, s) = 1, N(y, ) = 1. Thus x < s, y <. Therefore xy x y < s. Hence N(xy, s) = 1 > α. Example.10 (Fuzzy normed algebra which is no wih muliplicaively coninuous produc). We consider he fuzzy normed algebra from Example., where X = R and he norm on X is he absolue value. Then (R, N,, ) is no wih muliplicaively coninuous produc. Proof. Indeed, for α = 1 5, x = 5 2 s, y = 5 2, s, > 0 we have ha N(x, s) = s N(y, ) > 1 5. Bu N(xy, s) = s coninuous produc. s+ xy = s s s = 4 29 < 1 5 s+ x = s = 2 s+ 5 2 s 7 > 1 5 and. Thus (R, N,, ) is no wih muliplicaively Proposiion.11. Le (X, N,, ) be a fuzzy normed algebra such ha α α α for all α (0, 1). Then (X, N,, ) is wih muliplicaively coninuous produc. Proof. Le α (0, 1), x, y X, s, > 0 such ha N(x, s) > α, N(y, ) > α. Then N(xy, s) N(x, s) N(y, ) α α α. Remark.12. The condiion α α α for all α (0, 1) from he previous proposiion is sufficien bu no necessary. Indeed, he algebra (X, N,, ) from Example.2 is wih muliplicaively coninuous produc, alhough = does no verify α α α for all α (0, 1). Proposiion.1. Le (X 1, N 1,, ) and (X 2, N 2,, ) be wo fuzzy normed algebras. If -norm dominaes boh and, hen ((X 1 X 2 ), N,, ) is a fuzzy normed algebra, where N((x 1, x 2 ), ) = N 1 (x 1, ) N 2 (x 2, ). Proof. According o [1], i remains o be proved ha: We have N((x 1 y 1, x 2 y 2 ), s) N((x 1, x 2 ), s) N((y 1, y 2 ), ), x 1, x 2 X 1, y 1, y 2 Y 2, s, (0, ). N((x 1 y 1, x 2 y 2 ), s) = N 1 (x 1 y 1, s) N 2 (x 2 y 2, s) [N 1 (x 1, s) N 1 (y 1, )] [N 2 (x 2, s) N 2 (y 2, )] [N 1 (x 1, s) N 2 (x 2, s)] [N 1 (y 1, ) N 2 (y 2, )] = N((x 1, x 2 ), s) N((y 1, y 2 ), ) x 1, x 2 X 1, y 1, y 2 Y 2, s, (0, ). Proposiion.14. Le be a -norm saisfying α α α for all α (0; 1) and le (X 1, N 1,, ) and (X 2, N 2,, ) be wo fuzzy normed algebras wih muliplicaively coninuous produc. If is a -norm ha dominaes boh and hen ((X 1 X 2 ), N,, ) is a fuzzy normed algebra wih muliplicaively coninuous produc.

7 T. Bînzar, F. Paer, S. Nădăban, J. Nonlinear Sci. Appl. 9 (2016), Proof. Le α (0, 1), (x 1, x 2 ) X 1 X 2, and (y 1, y 2 ) X 1 X 2, s, > 0 such ha N((x 1, x 2 ), s) > α and N((y 1, y 2 ), ) > α. Then we have successively: N((x 1 y 1, x 2 y 2 ), s) = N 1 (x 1 y 1, s) N 2 (x 2 y 2, s) [N 1 (x 1, s) N 1 (y 1, )] [N 2 (x 2, s) N 2 (y 2, )] [N 1 (x 1, s) N 2 (x 2, s)] [N 1 (y 1, ) N 2 (y 2, )] = N((x 1, x 2 ), s) N((y 1, y 2 ), ) α α α. Example.15. Le (X, N,, ) be a fuzzy normed algebra wih muliplicaively coninuous produc and le S X be a linear closed subalgebra of X. Then (S, N,, ) is a fuzzy normed algebra wih muliplicaively coninuous produc. Example.16 (Caresian produc of fuzzy normed algebras wih muliplicaively coninuous produc ha is no wih muliplicaively coninuous produc). Le (X, N,, ) be he fuzzy normed algebra from Example.2, where X = R. The fuzzy normed algebra (X X, N,, ), where N ((x 1, x 2 ), ) = N(x 1, ) N(x 2, ), (x 1, x 2 ) X X, > 0 is no wih muliplicaively coninuous produc. Proof. Taking ino accoun ha N ((x 1, x 2 ), ) = N(x 1, ) N(x 2, ) { 1, x1, = 0, for he res, = { 1, max{ x1, x 2 }, 0, for he res for α = 1 2, x 1 = x 2 = y 1 = y 2 = 1 2, s = =, we obain 5 { 1, x2, 0, for he res, N ((x 1, x 2 ), ) = 1 > 1 2, N ((y 1, y 2 ), s) = 1 > 1 2, N ((x 1 y 1, x 2 y 2 ), s) = 0 < 1 2. Therefore (X X, N,, ) is no wih muliplicaively coninuous produc. Proposiion.17. Le (X, N,, ) be a fuzzy normed algebra and le I X be a bilaeral closed ideal. Then (X/I, N,, ) is a fuzzy normed algebra, where N (ˆx, s) := inf N(x, s), ˆx X/I, s > 0. x ˆx Proof. Prove firs ha (X/I, N, ) is a fuzzy normed space. (N1) N (ˆx, 0) = inf x ˆx N(x, 0) = 0, ˆx X/I. (N2) One has o show ha [N (ˆx, ) = 1 > 0] iff ˆx = ˆ0. Indeed, [inf N(x, ) = 1 > 0] [N(x, ) = 1 > 0, x ˆx] x ˆx x = 0 x ˆx ˆx = ˆ0.

8 T. Bînzar, F. Paer, S. Nădăban, J. Nonlinear Sci. Appl. 9 (2016), (N) Le > 0, λ K, ˆx X/I. Then (N4) N (λˆx, ) = inf N(λx, ) = inf N x ˆx x ˆx N (ˆx + ŷ, + s) = N ( x + y, + s) = inf N(x, ) N(y, s) = x+y x+y ( x, ) ( = N ˆx, λ inf N(x + y, + s) x+y x+y inf x+y x+y N(x, ) inf N(x, ) inf N(y, s) = N (ˆx, ) N (ŷ, s) x ˆx y ŷ ). λ inf x+y x+y N(y, s) s, > 0, ˆx, ŷ X/I. (N5) Le x X and > 0 fixed. Consider an arbirary sequence ( n ) n 0 such ha n, n. Then, we have N(x, n ) N(x, ). Passing o infimum, i resuls inf N(x, n) inf N(x, ). Hence x ˆx x ˆx N (ˆx, n ) N (ˆx, ), ˆx X/I. Moreover, lim N (ˆx, ) = lim inf N(x, ) = inf lim N(x, ) = 1. I follows (X/I, N, ) is a fuzzy normed x ˆx x ˆx linear space. I is clear ha condiions (A1), (A2) are saisfied. To verify (A4), fix ˆx, ŷ X/I, s, > 0. Then N (ˆxŷ, s) = N ( xy, s) = Tha concludes he proof. inf N(xy, s) inf N(x, ) N(y, s) xy xy xy xy = inf N(x, ) inf N(y, s) inf N(x, ) inf N(y, s) = N (ˆx, ) N (ŷ, s). xy xy xy xy x x y ŷ Proposiion.18. If (X, N,, ) is a fuzzy normed algebra wih muliplicaively coninuous produc hen (X/I, N,, ) is a fuzzy normed algebra wih muliplicaively coninuous produc, where N (ˆx, s) := inf N(x, s), ˆx, ŷ X/I, s > 0. x ˆx Proof. Le α (0, 1), ˆx, ŷ X/I, s, > 0, N (ˆx, s) > α and N (ŷ, ) > α. I follows inf N(x, s) > α and x ˆx N(y, ) > α. This implies N(x, s) > α, N(y, ) > α for all x ˆx, y ŷ. Since (X, N,, ) is a fuzzy normed inf y ŷ algebra wih muliplicaively coninuous produc, i resuls N(xy, s) α, for all x ˆx, y ŷ, xy xy. Therefore inf N(xy, s) α. xy xy 4. Conclusion and fuure work In his paper, we iniiaed he sudy of fuzzy normed algebras. We have buil a ferile ground for sudying in he coming papers, some specral properies in fuzzy normed algebras. Also, fuzzy normed algebras will be used for applicaions in he heory of dynamical sysems, paricle physics, ec. References [1] C. Alegre, S. Romaguera, Characerizaions of fuzzy merizable opological vecor spaces and heir asymmeric generalizaion in erms of fuzzy (quasi-)norms, Fuzzy Ses and Sysems, 161 (2010), [2] R. Ameri, Fuzzy inner produc and fuzzy norm of hyperspaces, Iran. J. Fuzzy Sys., 11 (2014),

9 T. Bînzar, F. Paer, S. Nădăban, J. Nonlinear Sci. Appl. 9 (2016), [] T. Bag, S. K. Samana, Finie dimensional fuzzy normed linear spaces, J. Fuzzy Mah., 11 (200), , 2.8 [4] T. Bag, S. K. Samana, Fuzzy bounded linear operaors, Fuzzy Ses and Sysems, 151 (2005), , 2.7,, [5] T. Bag, S. K. Samana, A comparaive sudy of fuzzy norms on a linear space, Fuzzy Ses and Sysems, 159 (2008), [6] S. C. Cheng, J. N. Mordeson, Fuzzy linear operaor and fuzzy normed linear spaces, Bull. Calcua Mah. Soc., 86 (1994), [7] B. Dinda, T. K. Samana, U. K. Bera, Inuiionisic fuzzy Banach algebra, Bull. Mah. Anal. Appl., (2010), [8] C. Felbin, Finie-dimensional fuzzy normed linear space, Fuzzy Ses and Sysems, 48 (1992), [9] I. Goleţ, On generalized fuzzy normed spaces and coincidence poin heorems, Fuzzy Ses and Sysems, 161 (2010), [10] A. K. Kasaras, Fuzzy opological vecor spaces, II, Fuzzy Ses and Sysems, 12 (1984), [11] A. K. Mirmosafaee, Perurbaion of generalized derivaions in fuzzy Menger normed algebras, Fuzzy Ses and Sysems, 195 (2012), ,.1 [12] A. K. Mirmosafaee, M. Mirzavaziri, Uniquely remoal ses in c 0-sums and l -sums of fuzzy normed spaces, Iran. J. Fuzzy Sys., 9 (2012), [1] S. Nădăban, Fuzzy euclidean normed spaces for daa mining applicaions, In. J. Compu. Commun. Conrol, 10 (2014), , [14] S. Nădăban, I. Dziac, Aomic decomposiions of fuzzy normed linear spaces for wavele applicaions, Informaica (Vilnius), 25 (2014), , 2.4, 2.6 [15] R. Saadai, S. M. Vaezpour, Some resuls on fuzzy Banach spaces, J. Appl. Mah. Compu., 17 (2005), [16] I. Sadeqi, A. Amiripour, Fuzzy Banach algebra, Firs join congress on fuzzy and inelligen sysems, Ferdorwsi universiy of mashhad, Iran, (2007). 1 [17] I. Sadeqi, F. Moradlou, M. Salehi, On approximae Cauchy equaion in Felbin s ype fuzzy normed linear spaces, Iran. J. Fuzzy Sys., 10 (201), [18] B. Schweizer, A. Sklar, Saisical meric spaces, Pacific J. Mah., 10 (1960), [19] L. A. Zadeh, Fuzzy ses, Informaion and Conrol, 8 (1965),