Intuitionistic Fuzzy Stability of n-dimensional Cubic Functional Equation: Direct and Fixed Point Methods

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1 Ite. J. Fuzzy Mathematcal Achve Vol. 7 No ISSN: (P (ole Publhed o2 Jauay Iteatoal Joual of Itutotc Fuzzy Stablty of -Dmeoal Cubc Fuctoal Equato: Dect ad Fxed Pot Method T.Namachvayam ad M.Auuma 2 Depatmet of Mathematc Govemet At College Tuvaamala-60660TamlNadu Ida. e-mal: amach.va@edffmal.com Coepodg Autho 2 Depatmet of Mathematc Govemet At College Tuvaamala TamlNadu Ida. e-mal: aau2002@yahoo.co. Receved 4 Novembe 204; accepted 4 Decembe 204 Abtact. I th pape the autho vetgate geealzed Ulam-Hye tablty of a dmeoal cubc fuctoal equato h x h( 2 x = h( x + xl ( h( x = = = 6 = = < < l = < wth Itutotc fuzzy omed pace ug dect ad fxed pot method. Keywod: Cubc fuctoal equato geealzed Ulam - Hye tablty tutotc fuzzy omed pace fxed pot AMS Mathematc Subect Clafcato (200: 9B52 2B72 2B82. Itoducto The eeach of tablty poblem fo fuctoal equato led to the eowed Ulam poblem [4] ( 940 coceg the tablty of goup homomophm whch wa ft elucdated by Hye [0] 94. Th tablty poblem wa moe wdepead by qute a lot of ceato [ ]. Othe petet eeach wo elated to vaou fuctoal equato ug dect ad fxed pot method wee dcued (ee[ ]. Recetly Muthy et al. [2] toduced ad vetgate the geeal oluto ad geealzed Ulam-Hye tablty of a ew fom of dmeoal cubc fuctoal equato h x h( 2 x = h( x + xl ( h( x (. = = = 6 = = < < l = < wth Felb type pace ug dect ad fxed pot method. I th pape the autho vetgate the geealzed Ulam-Hye tablty of the above dmeoal cubc fuctoal equato (. Itutotc fuzzy omed pace ug dect ad fxed pot method. I Secto we peet the oluto of the fuctoal equato (.. The geealzed Ulam-Hye tablty ug Baach pace gve Secto. I Secto 4 the bac otato ad pelmae about Itutotc Fuzzy Nomed Space

2 Itutotc Fuzzy Stablty of -Dmeoal Cubc Fuctoal Equato... peet. Alo the tablty of (. Itutotc Fuzzy Nomed Space ug dect ad fxed pot method ae dcued Secto 5 ad 6 epectvely. 2. Geeal oluto of the fuctoal equato (. I th ecto the autho dcu the geeal oluto of the fuctoal equato (. by codeg X ad Y ae eal vecto pace. Theoem 2.. [2] If f : X Y atfe the fuctoal equato (. fo all x x2 x K x X ad the thee ext a fucto B : X Y uch that f ( x = B( x x x fo all x X whee B ymmetc fo each fxed oe vaable ad addtve fo each fxed two vaable. Heeafte thoughout th pape we defe a mappg H : X Y by H ( x x2 x = h x h ( 2x = = = 6 = fo all x x2 x K x X. h( x x xl ( h( x x = < < l = <. Stablty eult Baach pace I th ecto the geealzed Ulam - Hye tablty of a -dmeoal cubc fuctoal equato (. povded. Thoughout th ecto aume X ad Y to be a omed pace ad a Baach pace epectvely. The poof of the followg theoem ad coollay mla le to that of Theoem 4. ad Coollay 4.2 of [2]. Hece the detal of the poof ae omtted. Theoem.. et = ±. et h : X Y be a mappg fo whch thee ext a fucto ξ : X [0 wth the codto lm ξ ( 2 2 lm ( 2 2 = 0 2 x x covege ad ξ 2 x x (. H x x x x ξ x x x x (.2 uch that the fuctoal equalty ( ( 2 2 fo all x x2 x x X. The thee ext a uque cubc mappg C : X Y atfyg the fuctoal equato (. ad ξ 2 x0 0 2 tme ( 2( h( x C( x l = (. 2 l 2 2 fo all = = 2 x X. The mappg C ( x defed by f (2 x C( x = lm 2 (.4 fo all x X. Coollay.. et h : X Y be a fucto ad thee ext eal umbe λ ad uch that 2

3 T.Namachvayam ad M.Auuma λ (.5 H ( x x2 x x λ x < o > ; = λ x + x < o > ; = = fo all x x2 x x X. The thee ext a uque cubc fucto C : X Y uch that λ 7l λ x f ( x C( x l 8 2 λ x l 8 2 fo all x X. (.6.. Stablty eult: tutotc fuzzy omed pace I th ecto we gve ome bac defto ad otato about tutotc fuzzy metc pace toduced by J.H. Pa [24] ad R. Saadat ad J.H. Pa [0]. Defto... et µ ad ν be membehp ad omembehp degee of a tutotc fuzzy et fom X ( 0 + to [0] uch that µ x ( t + ν x ( t fo all x X ad all 0 X P M ad to be a tutotc fuzzy omed pace t >. The tple ( (befly IFN-pace f X a vecto pace M a cotuou t epeetable ad Pµ ν a mappg X ( 0 + atfyg the followg codto: fo all x y X ad t > 0 ( IFN P ( x0 = 0 ; ( IFN 2 P ( x t = f ad oly f x = 0; µ ν µ ν t ( IFN Pµ ν ( α x t = Pµ ν x fo all α 0;( IFN 4 Pµ ν ( x + y t + M ( Pµ ν ( x t Pµ ν ( y. α I th cae Pµ ν called a tutotc fuzzy om. Hee Pµ. ν ( x t = ( µ x( t ν x( t. Example... et( X. be a omed pace. et T( a b = ( a b m ( a2 + b2 fo all a = ( a a2 b = ( b b2 ad µ ν be membehp ad o-membehp degee of a tutotc fuzzy et defed by t x + Pµ. ν ( x t = ( µ x ( t ν x ( t = t R. The ( X P T a IFN-pace. t+ x t+ x Defto..2. A equece { } f fo ay ε > 0ad t > 0 thee ext 0 m 0 whee N the tadad egato. x a IFN-pace ( X Pµ ν T called a Cauchy equece N uch that P ( x x t > ( N ( ε ε. m

4 Itutotc Fuzzy Stablty of -Dmeoal Cubc Fuctoal Equato... Defto... The equece { } x ad to be coveget to a pot x X. (deoted by P x x f Pµ. ν ( x x t a fo evey t > 0. Defto..4. A IFN-pace ( X Pµ ν T ad to be complete f evey Cauchy equece X coveget to a pot x X.Fo futhe detal about IFN pace oe ca ee ([ ]. Thoughout th ecto let u code X ( Z Pµ ν M ad ( Y P µ ν M ae lea pace Itutotc fuzzy omed pace ad Complete Itutotc fuzzy omed pace. Theoem... et κ { } be fxed ad let ξ : X Z be a mappg uch that fo ome b wth ( b fo all 0 < 2 < x X ad all > 0 0 κ κ κ P µ ν ( ξ ( x P ( b ( x ξ κ κ κ b > ad lmpµ ν ( ξ ( 2 x 2 x 2 = (.. (..2 fo all x x2 x x X ad all > 0. Suppoe that a fucto h : X equalty Pµ ν ( H ( x x2 x x P µ ν ( ξ ( x x2 x x 4 Y atfe the (.. fo all x x2 x x X ad all > 0. The the lmt κ f (2 x Pµ ν C( x a 0 κ > (..4 2 ext fo all x X ad the mappg C : X Y a uque cubc mappg atfyg (. ad l 8 b Pµ ν ( h( x C( x P ( x0 0 ξ (..5 8 whee l = fo all x X ad all > 0. = 2 Poof: Ft aume κ =. Replacg ( x x2 x x by ( x00 0 (4. we ave Pµ ν h x h x Pµ ν ξ x = 2 = 6 fo all x X ad all > 0. Ug ( IFN the above equato we get ( (2 ( ( 0 0 h(2 x Pµ ν h( x P ( ( x0 0 ξ (..6 2 l whee l = fo all x X ad all > 0. Replacg x by 2 x (..6 we = 2 obta + h(2 x Pµ ν h(2 x P ( ( 2 x0 0 ξ (..7 2 l fo all x X ad all > 0. Ug (.. ( IFN (..7 we ave + h(2 x Pµ ν h(2 x P ( x0 0 ξ 2 l b (..8

5 T.Namachvayam ad M.Auuma fo all x X ad all > 0. It eay to vefy fom (..8 that + h(2 x h(2 x Pµ ν ( ( P x ξ ( l b hold fo all x X ad all > 0. Replacg by b (..9 we get + h(2 x h(2 x b Pµ ν ( ( ( P x ξ ( l fo all x X ad all > 0. It eay to ee that (2 + h x h(2 x h(2 x h( x = (.. ( + 2 =0 2 2 fo all x X. Fom equato (..0 ad (.. we have + h(2 x b h(2 x h(2 x b Pµ ν h( x 0 M P = ( + 2 =0 2 l = l { ( ξ ( 0 0 } ( ξ ( 0 0 M 0 P x P = x (..2 m fo all x X ad all > 0. Replacg x by 2 x (..2 ad ug (.. ( IFN we obta + m m (2 (2 h x h x b Pµ ν ( ( ( m m P x m ξ m ( =0 2 l b m fo all x X ad all > 0 ad all m 0. Replacg by b (.. we get + m m (2 (2 m + h x h x b Pµ ν ( ( ( m m P x 0 0 ξ = m 2 l fo all x X ad all > 0 ad all m 0. It follow fom (..4 that + m m m+ h(2 x h(2 x b Pµ ν ( ( m m P x ξ = m 2 l fo all (..4 (..5 x X ad all > 0 ad all m 0. Sce 0 < b < 2 ad ( b 2 < th mple h(2 x a Cauchy equece ( Y P M. Sce ( Y P M a complete IFN 2 pace th equece covege to ome pot C( x Y. So oe ca defe the mappg f (2 x C : X Y by Pµ ν C( x a 0 > 2 fo all x X. ettg m = 0 (..5 we get h(2 x Pµ ν h( x P ( x0 0 ξ 2 b =0 2 l fo all x X ad all > 0. ettg (..6 we ave =0 (..5a (..6 5

6 Itutotc Fuzzy Stablty of -Dmeoal Cubc Fuctoal Equato... l(8 b Pµ ν ( h( x C( x P ( x0 0 ξ fo all x X ad all > 0. To pove 8 C atfe the (. eplacg ( x x by (2 x 2 x ad dvdg by 2 (..2 we obta P H ( 2 x 2 x P ( ( 2 x 2 x 2 ξ 2 (..7 fo all x x X ad all > Pµ ν C x C ( 2x C ( x + xl + ( C ( x = = = 6 = = < < l = < M Pµ ν C x h 2 x = = 2 = = 5 P µ ν = 6 = = = C ( x + h( x Pµ ν C ( x + xl + ( 2 ( l h x + x + x = < < l 2 = < < l 5 Pµ ν + ( C ( x + ( ( 2 ( h x + x = < 2 = < Pµ ν h 2 x 2 2 h x 2 = = = 6 = ( h( 2 ( x + xl + ( h( 2 ( x = < < l = < 5 (..8 fo all x x X ad all > 0. Ug (..5a (..7 (..2 ad ( IFN 2 (..8 we ave C x C ( 2x = C ( x + xl ( C ( x = = = 6 = = < < l = < fo all x x X. Hece C atfe the cubc fuctoal equato (.. I ode to pove C( x uque let C '( x be aothe cubc fuctoal mappg atfyg (..4 ad (..5. Hece C(2 x h(2 x C '(2 x h(2 x Pµ ν ( C( x C '( x M P P l(8 b 2 l(8 b P ( 2 x0 0 P ( x0 0 ξ ξ b fo all x X ad all > 0. Sce 2 l(8 b lm = 8b 2 l(8 b lm µ ν ξ ( ( '( = 8b hece C( x = C '( x. Theefoe C( x uque. we obta P ( x0 0 =. Thu P C x C x fo all x X ad all > 0 6

7 T.Namachvayam ad M.Auuma Fo κ = we ca pove the eult by a mla method. Th complete the poof of the theoem. Fom Theoem.. we obta the followg coollay coceg the tablty fo the fuctoal equato (.. Coollay... Suppoe that a fucto h : X Y atfe the equalty ( λ P µ ν P µ ν x P ( H ( x x λ = (..20 P µ ν λ x = = fo all all x x X ad all > 0 whee λ ae cotat wth λ > 0. The thee ext a uque cubc mappg C : X Y uch that P µ ν ( λ l l 8 2 Pµ ν ( h( x C( x P x λ (4.2fo all x X ad all > l P µ ν λx 8.2. Stablty eult: fxed pot method I th ecto the autho peet the geealzed Ulam - Hye tablty of the fuctoal equato (. IFN - pace ug fxed pot method. Now we wll ecall the fudametal eult fxed pot theoy. Theoem.2.. [2] (The alteatve of fxed pot Suppoe that fo a complete geealzed metc pace ( X d ad a tctly cotactve mappg T : X X wth pchtz cotat. The fo each gve elemet x X ethe ( B + d( T x T x = 0 o (B 2 thee ext a atual umbe uch that: 0 + ( d( T x T x < fo all ; 0 ( The equece ( T x coveget to a fxed pot y of T ( (v d( y y d( y Ty fo all y Y. Fo to pove the tablty eult we defe the followg: 2 f = 0 a = f = 2 y the uque fxed pot of T the et 0 Y = { y X : d( T x y < }; 7 a a cotat uch that ad Α the et uch that Α { g g X Y g } = : (0 = 0.

8 Itutotc Fuzzy Stablty of -Dmeoal Cubc Fuctoal Equato... Theoem.2.2. et h : X Y be a mappg fo whch thee ext a fucto ξ : X Z lmp a x a x a = x x X > 0 (.2. wth the codto ( ξ ( ad atfyg the fuctoal equalty P H x x P x x x x X > 0 (.2.2 ( ( ( ( µ ν µ ν ξ If thee ext = ( uch that the fucto x x ψ ( x = ξ 0 0 ha the popety 2 ψ ( a x P = P µ ν ( ψ ( x x X > 0. (.2. a The thee ext uque cubc fucto C : X Y atfyg the fuctoal equato (. ad Pµ ν ( h( x C( x P ( x x X > 0. ψ l (.2.4 Poof: et d be a geeal metc o Α uch that { µ ν ( µ ν ( ψ > } d( g h = f K (0 P g( x h( x P ( x K x X 0. It eay to ee that ( Α d complete. Defe ϒ : Α Α by ϒ g( x = g( a x fo all a x X. By [2] we ee that ϒ tctly cotactve mappg o Α wth pchtz cotat. It follow fom (..6 that h(2 x Pµ ν h( x P ( ( x0 0 x X > 0. ξ ( l Replacg by l (.2.5 we ave h(2 x Pµ ν h( x P ( ( x0 0 x X > 0. ξ l ( Wth the help of (.2. whe = t follow fom (.2.6 that h(2 x Pµ ν h( x P ( ( x x X > 0. ψ l 2 d ϒh h = =. (.2.7 ( 0 Replacg x by x (.2.5 we obta 2 x x P h( x 2 h P l 0 0 x X > 0. ξ (.2.8 Wth the help of (.2. whe = 0 t follow fom (.2.8 that x Pµ ν h( x 2 h P ( ( x x X > 0 ψ 2 l d( h ϒh = =. (.2.9 The fom (.2.7 ad (.2.9 we ca coclude ( d h ϒh <. Now fom the fxed pot alteatve both cae t follow that thee ext a fxed pot C of ϒ Α uch that 8

9 ( Pµ ν (. a T.Namachvayam ad M.Auuma h a x C x x X (.2.0 Replacg ( x x by ( a x a x (.2.2 we ave ( ( ( Pµ ν H a x a x P a x a x a x x X > 0. ξ a By poceedg the ame pocedue the Theoem.. we ca pove the fucto C : X Y cubc ad t atfe the fuctoal equato (.. Sce C uque fxed Β = h Α d( h C < uch that pot of ϒ the et { } ( ( P h( x C( x P ( x K x X > 0. (.2. µ ν µ ν ψ Aga ug the fxed pot alteatve we obta d ( h C d ( h ϒh d ( h C. Hece we have Pµ ν ( h( x C( x P ( x x X > 0. ψ l (.2.2 Th complete the poof of the theoem. Fom Theoem.2.2 we obta the followg coollay coceg the tablty fo the fuctoal equato (.. Coollay.2.. Suppoe that a fucto h : X Y atfe the equalty P µ ν ( ε ; P µ ν x P ( H ( x x ε (.2. = P µ ν ε x + x ; = = fo all x x X ad all > 0 whee ε ae cotat wth ε > 0. The thee ext a uque cubc mappg C : X Y uch that 8 P µ ν ε l 7 2 (.2.4 Pµ ν ( h( x C( x P x ε l P µ ν ε x l 8 2 fo all x X ad all > 0. Poof: The poof follow by eplacg = 2 fo = 0 ad = 2 fo = = fo > = ad = fo < = = fo > = ad = fo < =

10 Itutotc Fuzzy Stablty of -Dmeoal Cubc Fuctoal Equato... Theoem.2.2 we deed ou eult. REFERENCES. J.Aczel ad J.Dhombe Fuctoal Equato Seveal Vaable Cambdge Uv Pe T.Ao O the tablty of the lea tafomato Baach pace J. Math. Soc. Japa 2 ( M.Auuma S.Muthy ad G.Gaapathy Stablty of geealzed -dmeoal cubc fuctoal equato fuzzy omed pace Iteatoal Joual of pue ad Appled Mathematc 77 (2 ( K.T.Ataaov Itutotc fuzzy et Fuzzy Set Syt. 20 ( H.Y.Chu ad D.S.Kag O the tablty of a dmeoal cubc fuctoal equato J. Math. Aal. Appl. 25 ( ( S.Czew Fuctoal Equato ad Iequalte Seveal Vaable Wold Scetfc Rve Edge NJ G.Dechve ad E.E.Kee O the elatohp betwee ome exteo of fuzzy et theoy Fuzzy Set ad Sytem 2 ( T.Gate R.Stelage ad R.Wae Compacte fuzzy topologcal pace J. Math. Aal. Appl. 62 ( P.Gavuta A geealzato of the Hye-Ulam-Raa tablty of appoxmately addtve mappg J. Math. Aal. Appl. 84 ( D.H.Hye O the tablty of the lea fuctoal equato Poc.Nat. Acad.Sc. U.S.A. 27 ( D.H.Hye G.Iac ad T.M.Raa Stablty of fuctoal equato eveal vaablebhaue Bael U.Hoehle Fuzzy eal umbe a Deded cut wth epect to a multple-valued logc Fuzzy Set Syt. 24 ( S.B.Hoe D.O'Rega ad R.Saadat Some eult o tutotc fuzzy pace Iaa J. Fuzzy Syt. 4 ( S.M.Jug Hye-Ulam-Raa Stablty of Fuctoal Equato Mathematcal Aaly Hadoc Pe Palm Habo P.Kaappa Fuctoal Equato ad Iequalte wth Applcato Spge Moogaph Mathematc O.Kaleva ad S.Seala O fuzzy metc pace Fuzzy Set Syt. 2 ( O.Kaleva The completo of fuzzy metc pace J. Math. Aal. Appl. 09 ( O.Kaleva A commet o the completo of fuzzy metc pace Fuzzy Set Syt. 59(6 ( H.A.Keay S.Y.Jag ad C.Pa Fxed pot appoach to the Hye-Ulam tablty of a fuctoal equato vaou omed pace Fxed Pot Theoy ad Applcato do:0.86/ R.owe Fuzzy Set Theoy (Ch. 5 : Fuzzy eal umbe Kluwe Dodecht B.Magol J.B.Daz A fxed pot theoem of the alteatve fo cotacto o a geealzed complete metc pace Bull. Ame. Math. Soc (

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