On the stability of a Pexiderized functional equation in intuitionistic fuzzy Banach spaces

Transcription

1 Available a hp://pvamuedu/aam Appl Appl Mah ISSN: Vol 0 Issue December 05 pp Applicaions and Applied Mahemaics: An Inernaional Journal AAM On he sabiliy of a Pexiderized funcional equaion in inuiionisic fuzzy Banach spaces Nabin Chandra Kayal Praap Mondal and 3 T K Samana Deparmen of Mahemaics Moula Neaji Vidyalaya Moula Howrah 73 Wes Bengal India Corresponding auhor kayalnabin8@gmailcom Deparmen of Mahemaics Bijoy Krishna Girls College praapmondal@gmailcom 3 Deparmen of Mahemaics Uluberia College mumpu apas5@yahoocoin Received: July 6 0; Acceped: June 05 Absrac During he las few decades several researchers have been devoed o esablishing sabiliy of differen kinds of funcional equaions differenial equaions funcional differenial equaions fracional differenial equaions ec under differen sufficien condiions in differen spaces like Banach spaces Banach modules fuzzy Banach spaces ec In his paper we remain confined in he discussion of sabiliy of funcional equaions in inuiionisic fuzzy Banach spaces Ulam was he firs person who inroduced an open quesion concerning he sabiliy of a group homomorphism in an inernaional conference Thereafer several researchers have replied and are sill replying o his open quesion in differen conexs The objecive of he presen paper is o deermine he Hyers-Ulam-Rassias ype sabiliy concerning he Pexiderized funcional equaion in inuiionisic fuzzy Banach spaces Under a few sufficien condiions Hyers-Ulam-Rassias ype sabiliy of a Pexiderized funcional equaion has been esablished in inuiionisic fuzzy Banach spaces Keywords: -norm -conorm Inuiionisic fuzzy Banach space Pexiderized funcional equaion Hyers-Ulam-Rassias sabiliy MSC 00 No: 03E7 97I70 39B8 783

2 78 N C Kayal e al Inroducion The sabiliy problem of a funcional equaion was posed by Ulam 960 in 90 concerning he sabiliy of group homomorphisms and answered in he nex year by Hyers 9 for Cauchy funcional equaion in Banach spaces and hen generalized by T Aoki 950 and Th M Rassias 978 for addiive mappings and linear mappings by considering an unbounded Cauchy difference respecively In he spiri of Rassias s approach Gavrua 99 replaced he unbounded Cauchy difference by a general conrol funcion o generalize Rassias s heorem The Hyers-Ulam sabiliy heorem was generalized by F Skof 983 for he funcion f : X Y where X is a normed space and Y is a Banach space and hen he resul of Skof was exended by P W Cholewa 98 and S Czerwik 99 In his way several sabiliy problems for various funcional equaions have been invesigaed Recenly he fuzzy version of differen funcional equaions was discussed by A K Mirmosafaee and M S Moslehian 008 and C Park 009 The concep of inuiionisic fuzzy ses was inroduced by Aanassov 986 as a generalizaion of fuzzy ses One of he mos imporan problems in inuiionisic fuzzy opology is o obain an appropriae concep of inuiionisic fuzzy meric spaces and inuiionisic fuzzy normed spaces J H Park 00 Saadai and Park 006 and T K Samana and Iqbal 009 inroduced and sudied a few noions of inuiionisic fuzzy meric spaces and inuiionisic fuzzy normed spaces Several resuls for he Hyers-Ulam-Rassias sabiliy of many funcional equaions have been proved by several researchers like A K Mirmosafaee and M S Moslehian 008 C Park 009 Nabin e al 0 Nabin e al 0 Shakeri 009 Samana e al 0 and Samana e al 03 in fuzzy Banach spaces and inuiionisic fuzzy Banach spaces Our goal is o deermine some sabiliy resuls concerning he Pexiderized funcional equaion fx + y = gx + hy in inuiionisic fuzzy Banach spaces Preliminaries In his secion we recall some lemmas definiions and examples used in his paper Lemma Deschrijver and Kerre 003 Consider he se L and he order relaion L defined by L = {x x : x x [0 ] and x + x } x x L y y x y x y x x y y L Then L L is a complee laice We denoe is unis by 0 L = 0 and L = 0 Definiion Aanassov 986 An inuiionisic fuzzy se A ζ η in a universal se U is an objec A ζ η = { ζ A u η A u : u U } where ζ A u [ 0 ] and η A u [ 0 ] for all u U are called he membership degree and he non-membership degree respecively of u in A ζ η and furhermore saisfy ζ A u + η A u

3 AAM: Inern J Vol 0 Issue December Definiion Deschrijver e al 00 A riangular norm -norm on L is a mapping τ : L L saisfying he following condiions: a x L τx L = x boundary condiion; b x y L τx y = τy x commuaiviy; c x y z L 3 τx τy z = ττx y z associaiviy; d x x y y L x L x and y L y τx y L τx y monooniciy A -norm τ on L is said o be coninuous if for any x y L and any sequences {x n } and {y n } which converge o x and y respecively lim τx n y n = τx y n For example le a = a a b = b b L consider τa b = a b min{ a + b } and Ma b = min{ a b } max{ a b } Then τa b and Ma b are coninuous -norm Now we define a sequence τ n for all n and x i L recursively by τ = τ and τ n x x n + = τ τ n x x n x n + Definiion Deschrijver e al 00 A coninuous -norm τ on L is said o be coninuous -represenable if here exiss a coninuous -norm and a coninuous -conorm on [0 ] such ha for all x = x x y = y y L τx y = x y x y Definiion Deschrijver e al 00 A negaor on L is any decreasing mapping N : L L saisfying N0 L = L and N L = 0 L If NNx = x for all x L hen N is called an involuive negaor A negaor on [0 ] is a decreasing mapping N : [0 ] [0 ] saisfying N0 = and N = 0 N s denoes he sandard negaor on [0 ] defined by N s x = x for all x [0 ] Definiion Shakeri 009 Le L = L L The riple X P τ is said o be an L- fuzzy normed space if X is a vecor space τ is a coninuous -norm on L and P is an L-fuzzy se on X 0 + saisfying he following condiions for all x y X and s > 0 a P x > 0 L ; b P x = L if and only if x = 0; c P αx = P x for all α 0; α d P x + y + s L τp x P y s; e P x : 0 L is coninuous; f lim P x = 0 L and lim P x = L 0 In his case P is called an L-fuzzy norm briefly L -fuzzy norm

4 786 N C Kayal e al If P = P µ ν is an inuiionisic fuzzy se hen he riple X P µ ν τ is said o be an inuiionisic fuzzy normed space briefly IFN-space In his case P = P µ ν is called an inuiionisic fuzzy norm on X Noe ha if P is an L -fuzzy norm on X hen he following are saisfied: i P x is nondecreasing wih respec o for all x X ii P x y = P y x for all x y X and > 0 Example Le X be a normed space Le τa b = a b min{a + b } for all a = a a b = b b L and µ ν be membership and non-membership degree of an inuiionisic fuzzy se defined by P µ ν x = µ x ν x = + m x x + x for all R + in which m > Then X P µ ν τ is an IFN-space Here µx + νx = for x = 0 and µx + νx < for x 0 Le Ma b = min{a b } max{a b } for all a = a a b = b b L and µ ν be membership and non-membership degree of an inuiionisic fuzzy se defined by P µ ν x = µ x ν x = e x e x e x for all R + Then X P µ ν M is an IFN-space Definiion A sequence {x n } in an IFN-space X P µ ν τ is said o be convergen o a poin x X denoed by x n x if P µ ν x n x L as n for every > 0 A sequence {x n } in an IFN-space X P µ ν τ is said o be a Cauchy sequence if for any 0 < ɛ < and > 0 here exiss n 0 N such ha P µ ν x n x m > L N s ɛ ɛ for all n m n 0 where N s is he sandard negaor 3 An IFN-space X P µ ν τ is said o be complee if every Cauchy sequence in X P µ ν τ is convergen in X P µ ν τ A complee inuiionisic fuzzy normed space is called an inuiionisic fuzzy Banach space 3 Sabiliy Of The Funcional Equaion Throughou his secion X Y Z are assumed o be real vecor spaces Theorem Le Y P µ ν τ be a complee IFN-space and Z µ ν τ be an IFN-space Le φ : X Z be a mapping such ha 3x 3y L µ ν α φx y

5 AAM: Inern J Vol 0 Issue December for all x y X > 0 and for some 0 < α < 3 If f g h : X Y are mappings such ha P µ ν fx + y gx hy L x y for all x y X and > 0 hen here exiss a unique addiive mapping A : X Y such ha for all x X > 0 P µ ν fx Ax f0 L M x 3 α 3 and where f3 n x Ax as n 3 n g3 n x Ax h3 n x Ax as n 5 3 n 3 n x M x := τ 6 x x µ ν µ ν x φ x 3x 3x φ 3x µ ν 3x x x φ µ ν x φ x x Proof: Here for all x y X and > 0 we have x + y P µ ν f x L τ 3 y fx fy y x x x y y [ by ] 6 Le us now define F x = fx f0 for all x X Clearly F 0 = 0 and F saisfies 6 Puing y = x in 6 for he funcion F we ge for all x X > 0 x P µ ν F x F x L τ 3 x x x x x x x 7 Replacing x by x and y by 3x in 6 we ge for he funcion F

6 788 N C Kayal e al L τ 3 µ ν P µ ν F x F x F 3x x φ 3x x x 3x 3x 3x x 8 for all x X > 0 Now by using 7 and 8 we ge for all x X > 0 P µ ν F x F 3x L M x Clearly M 3x L ge M x α and lim M x = L Replacing x by 3 n x in 9 we P µ ν F 3 n x 3 n F 3 n + x 3 n + for all x X > 0 n N Clearly for all x X and n N F x F 3 n x 3 n = Now for all x X > 0 n N P µ ν F x F 3 n n x 3 n 3 r + Tha is n L M x 3 n + 0 α n F 3 r x F 3 r + x 3 r 3 r + L τ n P µ ν F x F 3x F 3x P µ ν F 3 x α = M x [ by 9 and 0 ] P µ ν F x F 3 n x 3 n F 3 n x P µ ν F 3 n x α n 3 n 3 n 3 n L M x n for all x X > 0 n N Replacing x by 3 m x in we ge F 3 m x P µ ν F 3 n + m x 3 m 3 n + m L M x 3 r + α m n 3 m 3 r +

7 AAM: Inern J Vol 0 Issue December for all x X > 0 m n N Since α m n 3 m M Thus from we see ha x α m n 3 m { } F 3 n x 3 n 3 r + 0 as m 3 r + L as m is a Cauchy sequence in Y P µ ν τ Since Y P µ ν τ is a complee IFN-space here exiss a mapping A : X Y such ha F 3 n x 3 n Ax as n This proves Le δ > 0 Now for all x X > 0 and n N P µ ν F x Ax + δ L τ M x n 3 r + P µ ν Taking he limi as n we ge by using P µ ν F x Ax + δ L F 3 n x 3 n Ax δ τ M x 3 r + L [ by ] for all x X > 0 Taking he limi as δ 0 we ge 3 From he definiion of A we ge for all x X n N A3 n x = 3 n Ax and A0 = 0 3 Now for all x X > 0 n N we have by using 3 P µ ν Ax Ax L τ 3 P µ ν Ax f3 n x P µ ν A3x f3 n + x 3 n 3 n P µ ν Ax f3 n x P 3 n µ ν f3 n x f3 n + x f3 n x 3 n By using 6 and we find ha las erm ends o L for all x X > 0 ie for all x X P µ ν Ax Ax = L [ by ] Now using for all x y X > 0 and n N as n This implies ha Ax = Ax

8 790 N C Kayal e al P µ ν Ax + y Ax Ay x + y L τ 3 P µ ν A 3 f n P µ ν Ay f3 n y 3 n P µ ν f x + y 3 n 3 n x + y P µ ν Ax f3 n x 3 n f3 n x f3 n y 3 n By using 6 and and aking he limi as n we ge for all x y X > 0 P µ ν Ax + y Ax Ay = L Therefore Ax + y = Ax + Ay for all x y X ie A is addiive To prove he uniqueness le us assume ha A : X Y is a mapping saisfying 3 and 3 Now using 3 and 3 we ge for all x X > 0 and n N P µ ν Ax A x A3 n x L τ P µ ν f3 n x + f0 3 n 3 n 3 n L τ M x 3 α 3 n α n M x 3 α 3 n α n P µ ν f3 n x 3 n A 3 n x 3 n f0 3 n Taking he limi as n we ge P µ ν Ax A x = L for all x X > 0 ie A x = Ax for all x X Thus A is unique Now using and we ge for all x X > 0 n N P µ ν g3 n x 3 n + h3 n x L L τ τ µ ν Ax 3 n φ3 n x 3 n x 3 n P µ ν Ax f3 n x x x 3 n P α n µ ν Ax f3 n x 3 n 3 n Taking limi as n we ge by for all x X > 0 g3 n x P µ ν + h3 n x Ax 3 n 3 n L as n Then by using we ge for all x X g3 n x 3 n + h3 n x 3 n Ax as n 5 Replacing x and y by 3 n + x and 3 n x respecively in we ge for all x X > 0 and n N P µ ν f3 n + x + 3 n x g3 n + x h3 n x L 3 n + x 3 n x 6

9 AAM: Inern J Vol 0 Issue December Again replacing x and y by 3 n x and 3 n + x respecively in we ge for all x X > 0 and n N P µ ν f3 n x + 3 n + x g3 n x h3 n + x L 3 n x 3 n + x 7 Now by using 6 and 7 we ge for all x X > 0 and n N g3 n + x h3 n + x P µ ν g3 n x h3 n x 3 n 3 n L τ 3 n + x 3 n x 3 n 3 n x 3 n + x 3 n L τ 3x x 3 n α n µ ν φx 3x 3 n α n Taking limi as n we ge for all x X > 0 g3 n + x h3 n + x P µ ν g3 n x h3 n x 3 n 3 n Corresponding o ɛ > 0 here exiss m N such ha g3 n + x h3 n + x P µ ν g3 n x h3 n x 3 n 3 n L as n L N s ɛ ɛ 8 for all x X > 0 n m For fixed x X and m N here exiss m N wih m m such ha g3 m x h3 m x P µ ν L 3 m N s ɛ ɛ 9 for all > 0 Now for n m : g3 n x h3 n x P µ ν 3 n L τ n m P µ ν g3 m x h3 m x 3 m g3 n x h3 n x L P µ ν 3 m n m + 3 m m P µ ν g3 m + x h3 m + x 3 m g3 m x h3 m x 3 m P µ ν g3 n x h3 n x 3 n g3 n x h3 n x 3 n L N s ɛ ɛ [ by 8 9 ] [ n m] n m + n m + 3 n m Thus g3 n x h3 n x 0 as n 0 3 n 3 n From and 0 we ge 5 This complees he proof of he heorem

10 79 N C Kayal e al Corollary Le ψ : [a R + be a mapping such ha i ψs ψψs ii ψ3 < 3 where a is a fixed real number saisfying 0 a 3 Le Y P µ ν M be a complee IFNspace and Z µ ν M be an IFN-space where M is given in Example Le z 0 Z If f g h : X Y are mappings such ha P µ ν fx + y gx hy L µ νψ x + ψ y z 0 for all x y X > 0 wih x y a hen here exiss a unique addiive mapping A : X Y such ha P µ ν fx Ax f0 { L µ ν ψ x z 0 min ψ + ψ 3 ψ } ψ 3 ψ3 3 8 and f3 n x Ax g3 n x Ax h3 n x Ax as n 3 n 3 n 3 n for all x X wih x a and > 0 Proof: Define φx y = ψ x + ψ y z 0 and ake α = ψ3 Clearly is saisfied and 0 < α < 3 Then { } M x L µ ν ψ x z 0 min ψ + ψ 3 ψ ψ 3 Corollary Le Y P µ ν M be a complee IFN-space and Z µ ν M be an IFN-space where M is given in Example 7 Le z 0 Z and p < If f g h : X Y are mappings such ha P µ ν fx + y gx hy L µ ν x p + y p z 0 for all x y X > 0 hen here exiss a unique addiive mapping A : X Y such ha P µ ν fx Ax f0 L µ ν x p z 0 p 3 3 p 3 p 6 and f3 n x Ax g3 n x Ax h3 n x 3 n 3 n 3 n Ax as n for all x X > 0

11 AAM: Inern J Vol 0 Issue December Proof: Define φx y = x p + y p z 0 and ake α = 3 p Clearly is saisfied and 0 < α < 3 Then M x = µ ν x p z 0 p 3 p 8 Theorem Le Y P µ ν τ be a complee IFN-space and Z µ ν τ be an IFN-space Le φ : X Z be a mapping such ha x 3 y L P 3 x y α for all x y X > 0 and for some α > 3 If f g h : X Y are mappings such ha P µ ν fx + y gx hy L x y for all x y X and > 0 hen here exiss a unique addiive mapping A : X Y such ha for all x X > 0 P µ ν fx Ax f0 L M x α 3 and 3 n f3 n x f0 Ax 3 n g3 n x f0 Ax 3 n h3 n x f0 Ax as n where M is given in Theorem Conclusion In his paper he Hyers-Ulam-Rassias sabiliy of fx + y = gx + hy he Pexiderized funcional equaion has been discussed in inuiionisic fuzzy Banach spaces Bu insead of considering he crisp mappings f g h if we consider he fuzzy mappings how he Hyers-Ulam- Rassias sabiliy of he corresponding Pexiderized funcional equaion can be esablished in inuiionisic fuzzy Banach spaces I is very imporan o invesigae he Hyers-Ulam-Rassias sabiliy of he fuzzy funcional equaions in inuiionisic fuzzy Banach spaces Acknowledgemens The auhors are graeful o he reviewers and also o Aliakbar Monazer Haghighi he Ediorin-Chief for heir valuable suggesions o recify his paper ino is presen form REFERENCES Aanassov K T 986 Inuiionisic fuzzy ses Fuzzy Ses and Sysems Vol 0 pp 87 96

12 79 N C Kayal e al Aoki T 950 On he Sabiliy of Linear Transformaion in Banach Spaces J Mah Soc Japan Vol pp 6 66 Cholewa P W 98 Remarks on he sabiliy of funcional equaions Aequaiones Mah Vol 7 pp Czerwik S 99 On he sabiliy of he quadraic mappings in normed spaces Abh Mah Sem Univ Hamburg Vol 6 pp 59 6 Deschrijver G Cornelis C and Kerre E E 00 On he represenaion of inuiionisic fuzzy -norms and -conorms IEEE Transacion on Fuzzy Sysems Vol pp 5 6 Deschrijver G and Kerre E E 003 On he relaionship beween some exensions of fuzzy se heory Fuzzy Ses and Sysems Vol 3 pp 7 35 Gavrua P 99 A generalizaion of he Hyers-Ulam-Rassias sabiliy of approximaely addiive mappings J Mah Anal appl Vol 8 pp 3 36 Hyers D H 9 On he sabiliy of he linear funcional equaion Proc Na Acad Sci USA Vol 7 pp Kayal N Chandra Mondal P and Samana T K 0 The Generalized Hyers-Ulam- Rassias Sabiliy of a Quadraic Funcional Equaion in Fuzzy Banach Spaces Journal of New Resuls in Science Vol No 5 pp Kayal N Chandra Mondal P and Samana T K 0 The Fuzzy Sabiliy of a Pexiderized Funcional Equaion Mahemaica Moravica Vol 8 No pp Mirmosafaee A K and Moslehian M S 008 Fuzzy versions of Hyers-Ulam-Rassias heorem Fuzzy Ses and Sysems Vol 59 pp Park C 009 Fuzzy sabiliy of a funcional equaion associaed wih inner produc space Fuzzy Ses and Sysems Vol 60 pp 63 6 Park J H 00 Inuiionisic fuzzy meric spaces Chaos Solions and Fracals Vol pp Rassias Th M 978 On he sabiliy of he linear mapping in Banach space Proc Amer Mahemaical Sociey Vol 7 No pp Saadai R and Park J H 006 On he inuiionisic fuzzy opological spaces Chaos Solions and Fracals Vol 7 pp 33 3 Samana T K and Jebril Iqbal H 009 Finie dimenional inuiionisic fuzzy normed linear space In J Open Problems Comp Mah Vol No pp Samana T K Mondal P and Kayal N Chandra 03 The generalized Hyers-Ulam- Rassias sabiliy of a quadraic funcional equaion in fuzzy Banach spaces Annals of Fuzzy Mahemaics and Informaics Vol 6 No pp 85 9 Samana T K Kayal N Chandra and Mondal P 0 The Sabiliy of a General Quadraic Funcional Equaion in Fuzzy Banach Space Journal of Hypersrucures Vol No pp 7 87 Shakeri S 009 Inuionisic fuzzy sabiliy of Jenson ype mapping J Non linear Sc Appl Vol No pp 05 Skof F 983 Propriea locali e approssimazione di opraori Rend Sem Ma Fis Milano Vol 53 pp 3 9 Ulam S M 960 Problems in Modern Mahemaics Chaper VI Science Ediions Wiley New York