Novi Sad J. Mah. Vol. 46, No. 1, 2016, 15-25 STABILITY OF PEXIDERIZED QUADRATIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN FUZZY NORMED SPASES N. Eghbali 1 Absrac. We deermine some sabiliy resuls concerning he pexiderized quadraic funcional equaion in non-archimedean fuzzy normed spaces. Our resul can be regarded as a generalizaion of he sabiliy phenomenon in he framework of L-fuzzy normed spaces. AMS Mahemaics Subjec Classificaion (2010): 00A11; Primary 46S40; Secondary 39B52; 39B82; 26E50; 46S50 Key words phrases: fuzzy normed space; non-archimedean fuzzy normed space; quadraic funcional equaion; pexiderized quadraic funcional equaion; sabiliy 1. Inroducion The sabiliy of funcional equaions is an ineresing area of research for mahemaicians, bu i can be also of imporance o persons who work ouside of he realm of pure mahemaics. I seems ha he sabiliy problem of funcional equaions had been firs raised by Ulam [14]. Moreover he approximaed ypes of mappings have been sudied exensively in several papers. (See for insance [10], [6], [3], [4]). Fuzzy noion inroduced firsly by Zadeh [15] ha has been widely involved in differen subjecs of mahemaics. Zadeh s definiion of a fuzzy se characerized by a funcion from a nonempy se X o [0, 1]. Goguen in [5] generalized he noion of a fuzzy subse of X o ha of an L-fuzzy subse, namely a funcion from X o a laice L. Laer, in 1984 Kasaras [7] defined a fuzzy norm on a linear space o consruc a fuzzy vecor opological srucure on he space. Wih [9] by modifying he definiion of a fuzzy normed space in [2], Mirmosafaee Moslehian in [8] inroduced a noion of a non-archimedean fuzzy normed space. Shekari e al. ([12]) considered he quadraic funcional equaion in L-fuzzy normed space. Also Saadai Park considered he equaion f(lx+y)+f(lx y) = 2l 2 f(x)+2f(y) proved he Hyers-Ulam-Rasssias sabiliy of his equaion in L-fuzzy normed spaces ([13]). Defining he class of approximae soluions of a given funcional equaion one can ask wheher every mapping from his class can be somehow 1 Deparmen of Mahemaics, Facualy of Mahemaical Sciences, Universiy of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran.
16 N. Eghbali approximaed by an exac soluion of he considered equaion in he non- Archimedean L-fuzzy normed spaces. To answer his quesion, we esablished a non-archimedean L-fuzzy Hyers-Ulam-Rassias sabiliy of he pexiderized quadraic funcional equaion f(x + y) + f(x y) = 2g(x) + 2h(y). 2. Preliminaries In his secion, we provide a collecion of definiions relaed resuls which are essenial used in he subsequen discussions. Definiion 2.1. Le X be a real linear space. A funcion N : X R [0, 1] is said o be a fuzzy norm on X if for all x, y X all, s R, (N1) N(x, c) = 0 for c 0; (N2) x = 0 if only if N(x, c) = 1 for all c > 0; ) if c 0; (N4) N(x + y, s + ) min{n(x, s), N(y, )}; (N5) N(x,.) is a non-decreasing funcion on R lim N(x, ) = 1; (N6) for x 0, N(x,.) is (upper semi) coninuous on R. The pair (X, N) is called a fuzzy normed linear space. (N3) N(cx, ) = N(x, c Example 2.2. Le (X,. ) be a normed linear space. We define funcion N by N(x, ) = 2 x 2 2 + x if > x N(x, ) = 0 if x. Then N defines a 2 fuzzy norm on X. Definiion 2.3. Le (X, N) be a fuzzy normed linear space {x n } be a sequence in X. Then {x n } is said o be convergen if here exiss x X such ha lim n N(x n x, ) = 1 for all > 0. In ha case, x is called he limi of he sequence {x n } we denoe i by N lim n x n = x. Definiion 2.4. A sequence {x n } in X is called Cauchy if for each ε > 0 each > 0 here exiss n 0 such ha for all n n 0 all p > 0, we have N(x n+p x n, ) > 1 ε. I is known ha every convergen sequence in a fuzzy normed space is Cauchy if each Cauchy sequence is convergen, hen he fuzzy norm is said o be complee furhermore he fuzzy normed space is called a fuzzy Banach space. Definiion 2.5. A binary operaion : [0, 1] [0, 1] [0, 1] is said o be a -norm if i saisfies he following condiions: ( 1) is associaive, ( 2) is commuaive, ( 3) a 1 = a for all a [0, 1] ( 4) a b c d whenever a c b d for each a, b, c, d [0, 1]. Definiion 2.6. ([5]) Le L = (L, L ) be a complee laice le U be a non-empy se called he universe. An L-fuzzy se in U is defined as a mapping A : U L. For each u in U, A(u) represens he degree (in L) o which u is an elemen of A.
Sabiliy of pexiderized quadraic funcional equaion... 17 Definiion 2.7. ([1]) A -norm on L is a mapping L : L 2 L saisfying he following condiions: (i) ( x L)(x L 1 L = x)(: boundary condiion); (ii) ( (x, y) L 2 )(x L y = y L x)(: commuaiviy); (iii) ( (x, y, z) L 3 )(x L (y L z)) = ((x L y) L z)(: associaiviy); (iv) ( (x, y, z, w) L 4 )(x L x y L y x L y L x L y )(: monooniciy). A -norm L on L is said o be coninuous if, for any x, y L any sequences {x n } {y n } which converges o x y, respecively, lim n (x n L y n ) = x L y. Definiion 2.8. The riple (V, P, L ) is said o be an L-fuzzy normed space if V is vecor space, L is a coninuous -norm on L P is an L-fuzzy se on V (0, ) saisfying he following condiions: for all x, y V, s (0, ); (a) P(x, ) > L 0 L ; (b) P(x, ) = 1 L if only if x = 0; ) for each α 0; (d) P(x, ) L P(y, s) L P(x + y, + s); (e) P(x, ) : (0, ) L is coninuous; (f) lim 0 P(x, ) = 0 L lim P(x, ) = 1 L. In his case, P is called an L-fuzzy norm. (c) P(αx, ) = P(x, α Definiion 2.9. A negaor on L is any decreasing mapping N : L L saisfying N (0 L ) = 1 L N (1 L ) = 0 L. Definiion 2.10. If N (N (x)) = x for all x L, hen N is called an involuive negaor. In his paper, he involuive negaor N is fixed. Definiion 2.11. A sequence (x n ) in an L-fuzzy normed space (V, P, L ) is called a Cauchy sequence if, for each ε L {0 L } > 0, here exiss n 0 N such ha, for all n, m n 0, P(x n x m, ) > L N (ε), where N is a negaor on L. A sequence (x n ) is said o be convergen o x V in he L-fuzzy normed space (V, P, L ), if P(x n x, ) 1 L, whenever n + for all > 0. An L-fuzzy normed space (V, P, L ) is said o be complee if only if every Cauchy sequence in V is convergen. Definiion 2.12. Le K be a field. A non-archimedean absolue value on K is a funcion. : K R such ha for any a, b K we have (1) a 0 equaliy holds if only if a = 0, (2) ab = a b, (3) a + b max{ a, b }. Noe ha n 1 for each ineger n. We always assume, in addiion, ha. is non-ivial, i.e., here exiss an a 0 K such ha a 0 0, 1.
18 N. Eghbali Definiion 2.13. A non-archimedean L-fuzzy normed space is a riple (V, P, L ), where V is a vecor space over non-archimedean field K, L is a coninuous -norm on L P is an L-fuzzy se on V (0, + ) such ha for all x, y V, s (0, ), saisfying he following condiions: (a) P(x, ) > L 0 L ; (b) P(x, ) = 1 L if only if x = 0; ) for all α 0; (d) P(x, ) L P(y, s) L P(x + y, max{, s}); (e) P(x,.) : (0, ) L is coninuous; (f) lim 0 P(x, ) = 0 L lim P(x, ) = 1 L. (c) P(αx, ) = P(x, α Example 2.14. Le (X,. ) be a non-archimedean normed linear space. Then he riple (X, P, min), where { 0, x ; P(x, ) = 1, > x. in a non-archimedean L-fuzzy normed space in which L = [0, 1]. 3. Sabiliy of pexiderized quadraic equaion in L-fuzzy normed spaces Le K be a non-archimedean field, X a vecor space over K (Y, P, L ) a non-archimedean L-fuzzy Banach space over K. In his secion we invesigae he pexiderized quadraic funcional equaion. We define an L-fuzzy approximaely pexiderized quadraic mapping. Le Ψ be an L-fuzzy se on X X [0, ) such ha Ψ(x, y,.) is nondecreasing, Ψ(cx, cx, ) L Ψ(x, x, c ), x X, c 0 lim Ψ(x, y, ) = 1 L, x, y X, > 0. Throughou his paper, we use he noaion n j=1 a j for he expression a 1 L a 2 L... L a n. Definiion 3.1. A mapping f : X Y is said o be Ψ-approximaely pexiderized quadraic if (3.1) P(f(x + y) + f(x y) 2g(x) 2h(y), ) L Ψ(x, y, ), x, y X, > 0. Proposiion 3.2. Le K be a non-archimedean field, X a vecor space over K (Y, P, L ) a non-archimedean L-fuzzy Banach space over K. Le f : X Y be a Ψ-approximaely pexiderized quadraic mapping. Suppose ha f, g
Sabiliy of pexiderized quadraic funcional equaion... 19 h are odd. If here exis an α R (α > 0), an ineger k, k 2 wih 2 k < α 2 0 such ha (3.2) Ψ(2 k x, 2 k y, ) L Ψ(x, y, α), x X, > 0, lim n j=n M(x, αj 2 kj ) = 1 L, x X, > 0, hen here exiss an addiive mapping T : X Y such ha (3.3) P(f(x) T (x), ) L where i=1 M(x, αi+1 ), x X, > 0, 2 ki P(g(x) + h(x) T (x), ) L 2 ki ), M(x, ) = Ψ(x, x, ) L Ψ(x, 0, ) L Ψ(x, 0, ) L Ψ(2x, 2x, ) L Ψ(2x, 0, ) L Ψ(0, 2x, ) L... L Ψ(2 k 1 x, 2 k 1 x, ) L Ψ(2 k 1 x, 0, ) L Ψ(0, 2 k 1 x, ), x X, > 0. Proof. By changing he roles of x y (3.1) we ge (3.4) P(f(x + y) f(x y) 2g(y) 2h(x), ) L Ψ(y, x, ). I follows from (3.1), (3.4) 2 1 ha (3.5) P(f(x + y) g(x) h(y) g(y) h(x), ) L P(f(x + y) g(x) h(y) g(y) h(x), 2 ) L P(f(x + y) + f(x y) 2g(x) 2h(y), ) L P(f(x + y) f(x y) 2g(y) 2h(x), ) L Ψ(x, y, ) L Ψ(y, x, ). If we pu y = 0 in (3.5), we ge (3.6) P(f(x) g(x) h(x), ) L Ψ(x, 0, ) L Ψ(0, x, ). Similarly by puing x = 0 in (3.5), we have (3.7) P(f(y) g(y) h(y), ) L Ψ(0, y, ) L Ψ(y, 0, ). From (3.5), (3.6) (3.7) we conclude ha (3.8) P(f(x+y) f(x) f(y), ) L P(f(x+y) g(x) h(y) g(y) h(x), ) L
20 N. Eghbali P(f(x) g(x) h(x), ) L P(f(y) h(y) g(y), ) L Ψ(x, y, ) L Ψ(y, x, ) L Ψ(x, 0, ) L Ψ(0, x, ) L Ψ(0, y, ) L Ψ(y, 0, ). We show, by inducion on j, ha, for all x X, > 0 j 1, (3.9) P(f(2 j x) 2 j f(x), ) L M j (x, ). If we pu x = y in (3.8), we ge (3.10) P(f(2x) 2f(x), ) L Ψ(x, x, ) L Ψ(x, 0, ) L Ψ(0, x, ). This proves (3.9) for j = 1. Le (3.9) holds for some j > 1. Replacing x by 2 j x in (3.10), we ge P(f(2 j+1 x) 2f(2 j x), ) L Ψ(2 j x, 2 j x, ) L Ψ(2 j x, 0, ) L Ψ(0, 2 j x, ). Since 2 1, i follows ha P(f(2 j+1 x) 2 j+1 f(x), ) L P(f(2 j+1 x) 2f(2 j x), ) L P(2f(2 j x) 2 j+1 f(x), ) = P(f(2 j+1 x) 2f(2 j x), ) L P(f(2 j x) 2 j f(x), / 2 ) L P(f(2 j+1 x) 2f(2 j x), ) L P(f(2 j x) 2 j f(x), ) L Ψ(2 j x, 2 j x, ) L Ψ(2 j x, 0, ) L Ψ(0, 2 j x, ) L M j (x, ) = M j+1 (x, ). Thus (3.9) holds for all j 1. In paricular, we have (3.11) P(f(2 k x) 2 k f(x), ) L M(x, ). Replacing x by 2 (kn+k) x in (3.11) using he inequaliy (3.2), we obain so P(f( x 2 kn ) 2 k f( x 2 kn+k ), ) L M( x 2 kn+k, ) L M(x, α n+1 ) P(2 kn f( x 2 kn ) 2 k(n+1) f( x 2 k(n+1) ), ) L M(x, αn+1 2 kn ). Hence i follows ha P(2 kn f( x ) 2 k(n+p) x f( ), ) 2 kn 2 k(n+p) L n+p j=n (P(2kj f( x ) 2 k(j+1) x n+p f( ), ) 2 kj 2 k(j+1) L j=n M(x, αj+1 2 kj ). Since lim n j=n M(x, αj+1 ) = 1 2 kj L for all x X > 0, {2 kn f( x )} 2 kn is a Cauchy sequence in he non-archimedean L-fuzzy Banach space (Y, P, L ). Hence we can define a mapping T : X Y such ha (3.12) lim n P(2 kn f( x 2 kn ) T (x), ) = 1 L. Nex, for all n 1, x X > 0, we have
Sabiliy of pexiderized quadraic funcional equaion... 21 so P(f(x) 2 kn f( x ), ) = P( n 1 2 kn i=0 2ki f( x ) 2 k(i+1) x f( ), ) 2 ki 2 k(i+1) L n 1 i=0 P(2ki f( x ) 2 k(i+1) x n 1 f( ), ) 2 ki 2 k(i+1) L i=0 M(x, αi+1 2 ki ) (3.13) P(f(x) T (x), ) L P(f(x) 2 kn f( x 2 kn ), ) L P(2 kn f( x 2 kn ) T (x), ) L n 1 i=0 M(x, αi+1 2 ki ) L P(2 kn f( x 2 kn ) T (x), ). Taking he limi as n in (3.13), we obain (3.14) P(f(x) T (x), ) L i=1 M(x, αi+1 2 ki ), which proves (3.3). As L is coninuous, from a well known resul in L-fuzzy normed space(see [11], Chaper 12), i follows ha lim n P(2 kn f(2 kn (x + y)) 2 kn f(2 kn x) 2 kn f(2 kn y), ) = P(T (x + y) T (x) T (y), ) for almos all > 0. On he oher h, replacing x, y by 2 kn x, 2 kn y in (3.8), we ge P(2 kn f(2 kn (x + y)) 2 kn f(2 kn x) 2 kn f(2 kn y), ) L Ψ(2 kn x, 2 kn y, 2 kn ) L Ψ(2 kn y, 2 kn x, 2 kn ) L Ψ(2 kn x, 0, 2 kn ) L Ψ(0, 2 kn x, 2 kn ) L Ψ(0, 2 kn y, 2 kn ) L Ψ(2 kn y, 0, 2 kn ) L Ψ(x, y, αn 2 kn ) L Ψ(y, x, αn 2 kn ) L Ψ(x, 0, αn 2 kn ) L Ψ(0, x, αn 2 kn ) L Ψ(0, y, αn 2 kn ) L Ψ(y, 0, αn 2 kn ). Since All erms of he righ h side of above inequaliy end o 1 as n, i follows from (3.6) (3.14) ha P(g(x) + h(x) T (x), ) L P(f(x) T (x), ) L P(g(x) + h(x) f(x), ) L 2 ki ) L Ψ(x, 0, ) L Ψ(0, x, ) L 2 ki ). For he uniqueness of T, le T : X Y be anoher addiive mapping such ha P(T (x) f(x), ) L M(x, ). Then for all x, y X > 0, we have P(T (x) T (x), ) L P(T (x) 2 kn f( x 2 kn ), ) L P(2 kn f( x 2 kn ) T (x), ). Therefore from (3.12), we have T = T.
22 N. Eghbali Proposiion 3.3. Le K be a non-archimedean field, X a vecor space over K (Y, P, L ) a non-archimedean L-fuzzy Banach space over K. Le f : X Y be a Ψ-approximaely pexiderized quadraic mapping. Suppose ha f, g h are even f(0) = g(0) = h(0) = 0. If here exis an α R (α > 0), an ineger k, k 2 wih 2 k < α 2 0 such ha Ψ(2 k x, 2 k y, ) L Ψ(x, y, α), x X, > 0, lim n j=n M(x, αj 2 kj ) = 1 L, x X, > 0, hen here exiss a unique quadraic mapping Q : X Y such ha (3.15) P(f(x) Q(x), ) L, M(x, αi+1 ) x X, > 0, 2 ki i=1 P(Q(x) g(x), ) L 2 ki ), where P(Q(x) h(x), ) L 2 ki ), M(x, ) = Ψ(x, x, ) L Ψ(x, 0, ) L Ψ(x, 0, ) L Ψ(2x, 2x, ) L Ψ(2x, 0, ) L Ψ(0, 2x, ) L... L Ψ(2 k 1 x, 2 k 1 x, ) L Ψ(2 k 1 x, 0, ) L Ψ(0, 2 k 1 x, ), x X, > 0. Proof. Pu y = x in (3.1). Then for all x X > 0, Pu x = 0 in (3.1), we ge P(f(2x) 2g(x) 2h(x), ) L Ψ(x, x, ). (3.16) P(2f(y) 2h(y), ) L Ψ(0, y, ), for all x X > 0. For y = 0, (3.1) becomes (3.17) P(2f(x) 2g(x), ) L Ψ(x, 0, ). Combining (3.1), (3.16) (3.17) we ge (3.18) P(f(x + y) + f(x y) 2f(x) 2f(y), ) L P(f(x + y) + f(x y) 2g(x) 2h(y), ) L P(2f(y) 2h(y), ) L P(2f(x) 2g(x), ) L Ψ(x, y, ) L Ψ(0, y, ) L Ψ(x, 0, ).
Sabiliy of pexiderized quadraic funcional equaion... 23 We show, by inducion on j, ha, for all x X, > 0 j 1, (3.19) P(f(2 j x) 4 j f(x), ) L M j (x, ). Similar he proof of Proposiion (3.2) we can obain he resuls. Here, by (3.15) (3.17) we ge P(Q(x) g(x), ) L P(Q(x) f(x), ) L P(f(x) g(x), ) L i=1 M(x, αi+1 ) 2 ki L Ψ(x, 0, / 2 ) L ) 2 ki L Ψ(x, 0, ) = ). 2 ki A similar inequaliy holds for h. Theorem 3.4. Le K be a non-archimedean field, X a vecor space over K (Y, P, L ) a non-archimedean L-fuzzy Banach space over K. Le f : X Y be a Ψ-approximaely quadraic mapping (i.e. P(f(x + y) + f(x y) 2f(x) 2f(y), ) L Ψ(x, y, )). Suppose ha f(0) = 0. If here exis an α R (α > 0), an ineger k, k 2 wih 2 k < α 2 0 such ha (3.20) Ψ(2 k x, 2 k y, ) L Ψ(x, y, α), x X, > 0, lim n j=n M(x, αj 2 kj ) = 1 L, x X, > 0. Then here are unique mappings T Q from X o Y such ha T is addiive, Q is quadraic (3.21) P(f(x) T (x) Q(x), ) L where i=1 M(x, αi+1 ) x X, > 0, 2 ki M(x, ) = Ψ(x, x, ) L Ψ(x, 0, ) L Ψ(x, 0, ) L Ψ(2x, 2x, ) L Ψ(2x, 0, ) L Ψ(0, 2x, ) L... L Ψ(2 k 1 x, 2 k 1 x, ) L Ψ(2 k 1 x, 0, ) L Ψ(0, 2 k 1 x, ), x X, > 0. Proof. Passing o he odd par f o even par f e of f we deduce from (3.1) ha P(f o (x + y) + f o (x y) 2f o (x) 2f o (y), ) L Ψ(x, y, ) P(f e (x + y) + f e (x y) 2f e (x) 2f e (y), ) L Ψ(x, y, ). Using he proofs of Proposiions (3.2) (3.3) we ge unique addiive mapping T unique quadraic mapping Q saisfying
24 N. Eghbali P(f o (x) T (x), ) L 2 ki ), x X, > 0, P(f e (x) Q(x), ) L 2 ki ) x X, > 0. Therefore P(f(x) T (x) Q(x), ) L P(f o (x) T (x), ) L P(f e (x) Q(x), ) L 2 ki ). Example 3.5. Le (X,. ) be a non-archimedean Banach space. Denoe T M (a, b) = (min{a 1, b 1 }, max{a 2, b 2 }) for all a = (a 1, a 2 ), b = (b 1, b 2 ) L le P µ,ν be he fuzzy se on X (0, ) defined as follows: P µ,ν (x, ) = ( + x, x + x ) for all R +. Then (X, P µ,ν, T M ) is a complee non-archimedean fuzzy normed space. Define Ψ(x, y, ) = ( +1, 1 +1 ). I is easy o see ha (3.20) holds for α = 1. Also, since we have M(x, ) = ( 1+, 1 1+ ), lim n j=n M(x, αj 2 kj ) = lim n (lim m j=n M(x, 2 kj )) = lim n lim m (, 2k n + 2 k n + 2 k ) = (1, 0) = 1 L for all x X, > 0. Le f : X Y be a Ψ-approximaely quadraic mapping. Thus all he condiions of Theorem (3.4) hold so here exiss a unique quadraic mapping Q : X Y such ha References P µ,ν (f(x) Q(x), ) L ( + 2 k, 2 k + 2 k ). [1] Aanassov, K. T., Inuiuinisic fuzzy meric spaces. Fuzzy Ses Sysems 20 (1986), 87 96. [2] Bag, T., Samana, S. K., Fuzzy bounded linear operaors. Fuzzy Ses Sysems 151 (2005), 513 547. [3] Z. Gajda, On sabiliy of addiive mappings. Inerma. J. Mah. Sci. 14 (1991), 431 434. [4] P. Gavrua, P., A generalizaion of he Hyers-Ulam-Rassias sabiliy of approximaely addiive mappings. J. Mah. Anal. Appl. 184 (1994), 431 436. [5] J. A. Goguen, J. A., L-fuzzy ses. J. Mah. Anal. Appl. 18 (1967), 145 174.
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