Stability Results in Intuitionistic Fuzzy Normed Spaces for a Cubic Functional Equation

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1 Appl. Mah. Inf. Sci. 7 No Applied Mahemaics & Informaion Sciences An Inernaional Journal hp://dx.doi.org/ /amis/ Sabiliy Resuls in Inuiionisic Fuzzy Normed Spaces for a Cubic Funcional Equaion M. Mursaleen Khursheed J. Ansari Deparmen of Mahemaics Aligarh Mus Universiy Aligarh 000 India Received: 18 Dec. 01 Revised: 3 Apr. 013 Acceped: 7 Apr. 013 Published online: 1 Sep. 013 Absrac: In his paper we deermine some sabiliy resuls concerning he cubic funcional equaion fx+y+fx y = +1 fx+y+fx y]+ 4 1fy where is a fixed ineger in he seing of inuiionisic fuzzy normed spaces IFNS. Furher we sudy he inuiionisic fuzzy coninuiy hrough he exisence of a cerain soluion of a fuzzy sabiliy problem for approximaely cubic funcional equaion. Keywords: Inuiionisic fuzzy normed spaces Cubic funcional equaion Hyers-Ulam sabiliy. 1. Inroducion preinaries Someime in modeling applied problems here may be a degree of uncerainy in he parameers used in he model or some measuremens may be imprecise. Due o such feaures we are emped o consider he sudy of funcional equaions in he fuzzy seing. The noion of fuzzy ses was firs inroduced by Zadeh 31] in 1965 which is a powerful h se for modeling uncerainy vagueness in various problems arising in he field of science engineering. For he las four decades fuzzy heory has become very acive area of research a lo of developmens have been made in he heory of fuzzy ses o find he fuzzy analogues of he classical se heory. The noion of inuiionisic fuzzy norm see ] is also useful one o deal wih he inexacness vagueness arising in modeling. There are many siuaions where he norm of a vecor is no possible o find he concep of inuiionisic fuzzy norm seems o be more suiable in such cases ha is we can deal wih such siuaions by modeling he inexacness hrough he inuiionisic fuzzy norm. In 1940 S.M. Ulam 30] raised he following quesion. Under wha condiions does here exis an addiive mapping near an approximaely addiion mapping? The case of approximaely addiive funcions was solved by D.H. Hyers 3] under cerain assumpion. In 1978 a generalized version of he heorem of Hyers for approximaely linear mapping was given by Th.M. Rassias 6]. A number of mahemaicians were araced by he resul of Th.M. Rassias. The sabiliy concep ha was inroduced invesigaed by Rassias is called he Hyers-Ulam-Rassias sabiliy. During he las decades he sabiliy problems of several funcional equaions have been exensively invesigaed by a number of auhors c.f ] references herein. Recenly Bae Lee Par ] esablished some sabiliy resuls for he funcional equaion fx+y+fx y = +1 fx+y+fx y]+ 4 1fy where is a fixed ineger in he seing of non- Archimedean L-fuzzy normed spaces. In his paper we deermine some sabiliy resuls concerning he above cubic funcional equaion in he seing of inuiionisic fuzzy normed spaces IFNS. We also sudy he inuiionisic fuzzy coninuiy hrough he exisence of a cerain soluion of a fuzzy sabiliy problem for approximaely cubic funcional equaion. Corresponding auhor mursaleenm@gmail.com c 013 NSP Naural Sciences Publishing Cor.

2 1678 M. Mursaleen K. J. Ansari : Sabiliy Resuls in Inuiionisic Fuzzy... In his secion we recall some noaions basic definiions used in his paper. Definiion 1.. A binary operaion : 0 1] 0 1] 01] is said o be a coninuous -norm if i saisfies he following condiions: a is associaive commuaive b is coninuous c a 1 = a for alla 01] d a b c d whenever a c b d for each abcd 01]. Definiion 1.3. A binary operaion : 01] 01] 01] is said o be a coninuous -conorm if i saisfies he following condiions: a is associaive commuaive b is coninuous c a 0 = a for all a 01] d a b c d whenever a c b d for each abcd 01]. Using he noions of coninuous-norm -conorm Saadai Par 9] inroduced he concep of inuiionisic fuzzy normed space as follows: Definiion 1.4. The five-uple X is said o be an inuiionisic fuzzy normed spaces for shor IFNS ifx is a vecor space is a coninuous-norm is a coninuous -conorm are fuzzy ses on X 0 saisfying he following condiions. For every x y X s > 0 ix+x 1iix > 0iiix = 1 for each α 0vx ys x+y+svix : 0 01] is coninuous vii x = 1 if only if x = 0 iv αx = x α 0 x = 0viiix < 1ixx = 0 if only if x = 0 x αx = x α for each α 0 xi x ys x + y + s xii x : 0 0 1] is coninuousxiii x = 0 x = 1. 0 In his case is called an inuiionisic fuzzy norm. Example 1.1. LeX. be a normed spacea b = ab a b = min{a + b1} for all ab 01]. For all x X every > 0 consider { x = + x if > 0 0 if 0; { x x = + x if > 0 0 if 0. Then X is an IFNS. The conceps of convergence Cauchy sequences in an inuiionisic fuzzy normed space are sudied in 9]. Le X be an IFNS. Then a sequence x = x is said o be inuiionisic fuzzy convergen o L X if x L = 1 x L = 0 for all IF > 0. In his case we wriex L as. LeX be an IFNS. Thenx = x is said o be inuiionisic fuzzy Cauchy sequence ifx +p x = 1 x +p x = 0 for all > 0 p = 1. Le X be an IFNS. Then X is said o be complee if every inuiionisic fuzzy Cauchy sequence in X is inuiionisic fuzzy convergen inx.. Inuiionisic fuzzy sabiliy The funcional equaion fx+y+fx y = +1 fx+y+fx y]+ 4 1fy.1 where is a fixed ineger is called he cubic funcional equaion since he funcion fx = cx 3 is is soluion. Every soluion of he cubic funcional equaion is said o be a cubic mapping. We begin wih a generalized Hyers-Ulam-Rassias ype heorem in IFNS for he cubic funcional equaion. Theorem.1. Le X be a linear space le Z be an IFNS. Leϕ : X X Z be a funcion such ha for someα > 3 ϕ x 0 ϕx0α n ϕ x 0 ϕx0α. n 3n ϕ x n y n 3n ϕ x y n =1 n = 0 for all xy X > 0. Le Y be an inuiionisic fuzzy Banach space le f : X Y be a ϕ-approximaely cubic mapping in he sense ha fx+y+fx y +1 fx+y ] +fx y 4 1fy ϕxy fx+y+fx y +1 fx+y ] +fx y 4 1fy ϕxy.3 for all > 0 all xy X. Then here exiss a unique cubic mapping g : X Y such ha gx fx ϕx0 α 3 c 013 NSP Naural Sciences Publishing Cor.

3 Appl. Mah. Inf. Sci. 7 No / gx fx ϕx0 α 3.4 for all x X all > 0. Proof. Pu y=0 in.3. Then for all x X > 0 which implies ha fx 3 fx ϕx0 3 f x fx ϕ x 0 ϕx0α 3 f x fx ϕ x 0 ϕx0α. Replacing x by x/ n in.5 we ge 3n+1 x f n+1 3n f x n3n ϕ x n0α 3n+1 x f ϕx0α n+1 n+1 3n f x ϕ x n0α Replacing by /α n+1 we obain n3n ϕx0α n+1. 3n+1 x f 3n f x n+1 3n n α n+1 ϕx0 3n+1 x f 3n f x n+1 3n n ϕx0. α n I follows from 3n f x fx = n n 1 3j+1 f x 3j f x j+1.6 ha j 3n f x fx n n 1 n 1 n 1 3j 3j+1 f x j+1 3j f x j 3j ϕx0 3n f x fx n 3j+1 f x ϕx0 n 1 3j j+1 3j f x j 3j.7 for all x X > 0 n > 0 where n 1 a j = a 1 a... a n n 1 b j = b 1 b... b n. By replacing x wihx/ m in.7 we ge 3n+m f x n+m 3m f x n 1 m ϕ x m0 ϕx0 3n+m x f n+m 3m f x n 1 m ϕ x m0 ϕx0 Thus 3n+m x f n+m 3m f x 3n+m f ϕx0 x n+m 3m f x ϕx0 n+m 1 m n+m 1 m for allx X > 0 m 0 n 0. Hence 3n+m x f 3m f x n+m m ϕx0 n+m 1 3j α j+1 3n+m x f 3m f x n+m m ϕx0 n+m 1 3j 3j+m α j+m+1 3j+m α j+m+1 3j 3j.8 for allx X > 0 m 0 n 0. Since α > 3 3 α < he Cauchy crierion for convergence in IFNS shows ha 3n f x is n a Cauchy sequence in Y. Since Y is complee his sequence converges o some poin gx Y. Fix x X pu m = 0 in.8 o obain 3n f x n fx ϕx0 3n f x n fx ϕx0 n 1 n 1 3j 3j c 013 NSP Naural Sciences Publishing Cor.

4 1680 M. Mursaleen K. J. Ansari : Sabiliy Resuls in Inuiionisic Fuzzy... for all > 0 n > 0. Thus we obain gx fx gx 3n f x n/ 3n f x n fx/ ϕx0 gx fx gx 3n f x 3n f x n fx/ ϕx0 n 1 n/ n 1 3j 3j for large n. Taing he i as n using he definiion of IFNS we ge gx fx ϕx0 α 3 gx fx ϕx0 α 3 for all x X > 0. Replace x y by x/ n y/ n respecively in.3 we have 3n f x+y n + 3n f x y n +1 3n f x+y n +3n f x y ] n + 3n 4 1f y n ϕ x y n n 3n 3n f x+y n 3n f x+y n +3n f x y + 3n 4 1f y n + 3n f x y n +1 ] n ϕ x y n n 3n for all xy X all > 0. Since n 3n ϕ x y n n = 1 n 3n ϕ x y n n = 0 for all xy X all > 0. We observe ha g fulfills.1. Therefore g is a cubic mapping. To Prove he uniqueness of he cubic mapping g assume ha here exiss a cubic mapping h : X Y which saisfies.4. For fix x X clearly 3n g x n = gx 3n h x = hx for all n. I follows from.4 ha n gx hx = 3n g x n 3n h x n 3n g x n 3n f x n 3n f x n 3n h x n ϕ x α 3 n0 3n ϕx0 αn α 3 3n similarly gx hx ϕx0 αn α 3 3n. α Since n α 3 n = as α > 3 we ge 3n Therefore n ϕx0 αn α 3 3n = 1 n ϕx0 αn α 3 3n = 0. gx hx = 1 gx hx = 0 for all > 0. Hence gx = hx. This complees he proof. In he following heorem we consider 0 < α < 3. Theorem.. Le X be a linear space le Z be an IFNS. Leϕ : X X Z be a funcion such ha for some0 < α < 3 ϕx0 αϕx0 ϕx0 αϕx0 n ϕ n x n y 3n = 1 n ϕ n x n y 3n = 0 for all xy in X > 0. Le Y be an inuiionisic fuzzy Banach space le f : X Y be a ϕ-approximaely cubic mapping in he sense ha fx+y+fx y +1 fx+y ] +fx y 4 1fy ϕxy fx+y+fx y +1 fx+y ] +fx y 4 1fy ϕxy c 013 NSP Naural Sciences Publishing Cor.

5 Appl. Mah. Inf. Sci. 7 No / for all > 0 all xy X. Then here exiss a unique cubic mapping g : X Y such ha gx fx ϕx0 3 α gx fx ϕx0 3 α for all x X all > 0. Proof. The echniques are similar o ha of Theorem.1. Hence we presen a sech of proof. Puy = 0 in.3 we ge fx 3 fx ϕx0 fx 3 fx ϕx0 for all x X > 0. Therefore f n+1 x 3 f n x ϕx0 α n f n+1 x 3 f n x ϕx0 α n for all x X > 0. For each x Xn 0m 0 > 0 we can deduce f n+m x fm x 3n+m 3m ϕx0 n+m 1 α j 3j+1.9 f n+m x fm x 3n+m 3m ϕx0 n+m 1 α j 3j+1 for all x X > 0 m 0 n 0. Thus f n x 3 n is a Cauchy sequence in inuiionisic fuzzy Banach space. Therefore here is a funcion g : X Y defined by f gx = n x n..9 wihm = 0 implies 3n gx fx ϕx0 3 α gx fx ϕx0 3 α for all x X all > 0. This complees he proof. Example.3. Le X be a Hilber space Z be a normed space. Denoe by he inuiionisic fuzzy norms given as in Example 1.1 on X Z respecively. Le ϕ : X X Z be defined by ϕxy = 1 x y + 4 y z where z is a fixed uni vecor in Z. Define f : X X by fx = x x + x x for some uni vecor x X. Then fx+y+fx y +1 = ] +fx y 4 1fy + 1x y 4 y + 1 x y + 4 y fx+y + 1 x y + 4 y = ϕxy ] fx+y+fx y +1 fx+y+fx y Also = 4 1fy ϕxy. ϕx0 = + ϕx0 + 1 x = ϕx0 ϕx0 = ϕx0. Thus n ϕ n x n y 3n 3n = n 3n + n 1x y + y ] = 1 n ϕ n x n y 3n 5n 1x y + y ] = n 3n + n 1x y + y ] = 0. Hence condiions of Theorem. forα = are fulfilled. Therefore here is a unique cubic mapping g : X Y such ha gx fx ϕx0 gx fx ϕx0. This complees he proof. c 013 NSP Naural Sciences Publishing Cor.

6 168 M. Mursaleen K. J. Ansari : Sabiliy Resuls in Inuiionisic Fuzzy Inuiionisic fuzzy coninuiy Recenly he inuiionisic fuzzy coninuiy is discussed in 0]. In his secion we esablish some ineresing resuls of coninuous approximaely cubic mappings. Definiion 3.1. Le f : R X be a funcion where is endowed wih he Euclidean opology X is an inuiionisic fuzzy normed space equipped wih inuiionisic fuzzy norm. Then f is called inuiionisic fuzzy coninuous a a poin s R if for all ǫ > 0 all 0 < α < 1 here exiss δ > 0 such ha for each s wih 0 < s s < δ fsx fs xǫ α fsx fs xǫ 1 α. Theorem 3.. Le X be a normed space Z be an IFNS. Le Y be an inuiionisic fuzzy Banach space f : X Y be a pq- approximaely cubic mapping in he sense ha for some pq somez Z ] fx+y+fx y +1 fx+y+fx y 4 1fy x p + y q z ] fx+y+fx y +1 fx+y+fx y 4 1fy x p + y q z for allxy X all > 0. Ifpq < 3 hen here exiss a unique cubic mapping g : X Y such ha gx fx x p z n3 n p gx fx x p z n3 n p 3.1 for all x X all > 0. Furhermore if for some x X all n N he mapping h : R Y defined by hs = f n sx is inuiionisic fuzzy coninuous. Then he mappings gsx fromroy is inuiionisic fuzzy coninuous. Proof. If we define ϕ : X X Z by ϕxy = x p + y q z. Exisence uniqueness of he cubic mapping g saisfying 3.1 are deduced from Theorem. see Example.3. Noe ha for each x X R n N we have gx fn x g = x 3n fn x 3n 3n = g n x f n x 3n np x p z 3n n 3 n p = x p z 3n n 3 n p np gx fn x x p z 3n 3n n 3 n p. np 3. Fixx X s R. Givenǫ > 0 0 < α < 1. From 3. i follows ha gsx fn sx 3n x p z 3n n 3 n p 1+ s p np gsx fn sx 3n x p z 3n n 3 n p 1+ s p np x p z 3n n 3 n p s p np x p z 3n n 3 n p s p np for all s s < 1 s R. Since here exissn N such ha sx gsx fn ǫ 3n 3 gsx fn sx ǫ 3n 3 n 3n n 3 n p 1+ s p np = α 1 α for all s s < 1 s R. By he inuiionisic fuzzy coninuiy of he mapping f n x here exiss δ < 1 such ha for each s wih0 < s s < δ we have f n sx s fn x ǫ α 3n 3n 3 f n sx s fn x ǫ 1 α. 3n 3n 3 I follows ha sx gsx gs xǫ gsx fn ǫ 3n 3 f n sx fn s x 3n ǫ f 3n 3 n s x gs 3n x ǫ 3 α gsx gs xǫ 1 α for each s wih 0 < s s < δ. Hence he mapping s gsx is inuiionisic fuzzy coninuous. This complees he proof. c 013 NSP Naural Sciences Publishing Cor.

7 Appl. Mah. Inf. Sci. 7 No / In he following heorem we prove a resul similar o Theorem 3. for he case pq > 3. Theorem 3.3. Le X be a normed space Z be an IFNS. Le Y be an inuiionisic fuzzy Banach space f : X Y be a pq- approximaely cubic mapping in he sense ha for some pq some z Z ] fx+y+fx y +1 fx+y+fx y 4 1fy x p + y q z ] fx+y+fx y +1 fx+y+fx y 4 1fy x p + y q z for all xy X all > 0. If pq > 3 here exiss a unique cubic mapping g : X Y such ha gx fx x p z np n 3 gx fx x p z np n for all x X all > 0. Furhermore if for some x X all n N he mapping h : R Y defined by hs = f n sx is inuiionisic fuzzy coninuous. Then he mappings gsx fromroy is inuiionisic fuzzy coninuous. Proof. If we define ϕ : X X Z by ϕxy = x p + y q z. Then ϕ x 0 = x p z p ϕ x 0 = x p z p for all x X > 0. Since p > 3 we have α = p > 3. By Theorem.1 here exiss a unique cubic mapping g which saisfies 3.3. Res of he proof can be done on he same lines as in Theorem 3.. This complees he proof. References 1] A. Aloaibi S.A. Mohiuddine On he sabiliy of a cubic funcional equaion in rom -normed spaces Adv. Diff. Equ ] J. H. Bae S. B. Lee W. G. Par Sabiliy resuls in non- Archimedean L-fuzzy normed spaces for a cubic funcional equaion Jour. Ineq. Appl ] D. H. Hyers On he sabiliy of he linear funcional equaion Proc. Nal. Acad. Sci ] D. H. Hyers G. Isac T. M. Rassias Sabiliy of Funcional Equaions in Several Variables Birhäuser Basel; ] K. W. Jun H. M. Kim The generalized Hyers-Ulam- Rassias sabiliy of a cubic funcional equaion J. Mah. Anal. Appl ] K. W. Jun H. M. Kim I. S. Chang On he Hyers-Ulam sabiliy of an Euler-Lagrange ype cubic funcional equaion J. Compu. Anal. Appl ] A. K. Mirmosafaee M. S. Moslehian Fuzzy approximaely cubic mappings Infor. Sci ] S. A. Mohiuddine Sabiliy of Jensen funcional equaion in inuiionisic fuzzy normed space Chaos Solions Fracals ] S. A. Mohiuddine M. A. Alghamdi Sabiliy of funcional equaion obained hrough a fixed-poin alernaive in inuiionisic fuzzy normed spaces Adv. Diff. Equ ] S. A. Mohiuddine A. Aloaibi Fuzzy sabiliy of a cubic funcional equaion via fixed poin echnique Adv. Diff. Equ ] S.A. Mohiuddine A. Aloaibi M. Obaid Sabiliy of various funcional equaions in non-archimedean inuiionisic fuzzy normed spaces Discree Dyn. Naure Soc ] S. A. Mohiuddine M. Cancan H. Şevli Inuiionisic fuzzy sabiliy of a Jensen funcional equaion via fixed poin echnique Mah. Compu. Model ] S. A. Mohiuddine Q. M. Danish Lohani On generalized saisical convergence in inuiionisic fuzzy normed space Chaos Solions Frac ] S. A. Mohiuddine H. Şevli Sabiliy of Pexiderized quadraic funcional equaion in inuiionisic fuzzy normed space J. Compu. Appl. Mah ] M. Mursaleen S. A. Mohiuddine On sabiliy of a cubic funcional equaion in inuiionisic fuzzy normed spaces Chaos Solions Fracals ] M. Mursaleen V. Karaaya S. A. Mohiuddine Schauder basis separabiliy approximaion propery in inuiionisic fuzzy normed space Absr. Appl. Anal ] M. Mursaleen S. A. Mohiuddine Saisical convergence of double sequences in inuiionisic fuzzy normed spaces Chaos Solions Fracals ] M. Mursaleen S. A. Mohiuddine On lacunary saisical convergence wih respec o he inuiionisic fuzzy normed space J. Compu. Appl. Mah ] M. Mursaleen S. A. Mohiuddine O.H.H. Edely On he ideal convergence of double sequences in inuiionisic fuzzy normed spaces Compu. Mah. Appl ] M. Mursaleen S. A. Mohiuddine Nonlinear operaors beween inuiionisic fuzzy normed spaces Fréche differeniaion Chaos Solions Fracals ] C. Par D. Y. Shin Funcional equaions in paranormed spaces Adv. Diff. Equ ] J. H. Par Inuiionisic fuzzy meric spaces Chaos Solions Fracals c 013 NSP Naural Sciences Publishing Cor.

8 1684 M. Mursaleen K. J. Ansari : Sabiliy Resuls in Inuiionisic Fuzzy... 3] K. H. Par Y. S. Jung Sabiliy for a cubic funcional equaion Bull. Korean Mah. Soc ] J. M. Rassias On approximaion of approximaely linear mappings by linear mapping J. Func. Anal ] J. M. Rassias On approximaion of approximaely linear mappings by linear mappings Bull. Sci. Mah ] T. M. Rassias On he sabiliy of he linear mapping in Banach spaces Proc. Amer. Mah. Soc ] T.M. Rassias On he sabiliy of funcional equaions a problem of Ulam Aca Appl. Mah ] K. Ravi J. M. Rassias P. Narasimman Sabiliy of cubic funcional equaion in fuzzy normed space Jour. Appl. Analy. Compu ] R. Saadai J. H. Par On he inuiionisic fuzzy opological spaces Chaos Solions Fracals ] S. M. Ulam Problems in Modern Mahemaics Science ed. John Wiley & Sons: New Yor; ] L. A. Zadeh Fuzzy ses Inform. Conrol M. Mursaleen is a full professor of mahemaics in he Deparmen of Mahemaics Aligarh Mus Universiy India. He has visied a number of counries e.g. USA UK Hungary Swizerl Turey Serbia Thail Libya Jordan Egyp UAE Saudi Arabia under various academic programmes. He is referee edior of many mahemaical journals of inernaional repue. His research ineress are in he areas of Sequence Spaces Summabiliy Approximaion Theory Funcional Equaions Fixed Poin Theory Measures of Noncompacness; has published abou 180 research papers in many repued scienific journals guided successfully 11 Ph.D. sudens. Khursheed J. Ansari received he Maser degree in Mahemaics from Banaras Hindu Universiy Varanasi India. Currenly he has joined he Ph. D. Program in he Deparmen of Mahemaics Aligarh Mus Universiy Aligarh India in Sepember 01 under he supervision of Prof. M. Mursaleen. His research ineress are in he sabiliy of funcional equaions waveles. c 013 NSP Naural Sciences Publishing Cor.