Int. J. Adv. Appl. Math. and Mech. 3(4 (06 8 (ISSN: 347-59 Journal homepage: www.ijaamm.com IJAAMM International Journal of Advances in Applied Mathematics and Mechanics A fixed point approach to orthogonal stability of an Additive - Cubic functional equation Research Article R. Murali, M. Deboral, A. Antony Raj Department of Mathematics, Sacred Heart College, Tirupattur - 635 60, Tamil Nadu, India Received 09 March 06; accepted (in revised version 05 April 06 Abstract: Using fixed point method, we prove the Hyers-Ulam stability of the orthogonally additive-cubic functional equation f (x + y + f (x y f (4x = f (x + y + f (x y 8f (x + 0f (x f ( x for all x, y with x y. MSC: 39B5 39B55 39B8 46H5 47H0 Keywords: Hyers-Ulam stability Additive and cubic functional equations Fixed point method Orthogonality space 06 The Author(s. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/3.0/.. Introduction A basic equation in the theory of functional equations is as follows: when is it true that a function, which approximately satisfies a functional equation, must be close to an exact solution of the equation? If the problem accepts a unique solution, then we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [] and affirmatively solved by Hyers []. The result of Hyers was generalized by Aoki [3] for approximatively additive mappings and by Rassias [4] for approximate linear mappings by allowing the Cauchy difference operator D f (x, y = f (x + y f (x f (y to be controlled by ɛ( x p + y p. In 994, a generalization of Rassias s theorem was obtained by Gavruta [5], who replaced ɛ( x p + y p by a general control function φ(x, y. In addition, Rassias et al. [6], [7] generalized the Hyers stability result by introducting two weaker conditions controlled by a norms respectively. Recently, several further interesting discussions, modifications, extensions and generalizations of the original problem of Ulam have been proposed see [8], [9], [0]. Let us recall the orthogonality in the sense of Ratz see []. Let X be a real vector space with dim X and is a binary relation on X with the following properties: (a totality of for zero: x 0,0 x for all x X ; (b independence: if x, y X {0}, then x, y are linearly independent; (c homogeneity: if x, y X, x y, then αx βy for all α,β X ; (d the Thalesian property: Let P be a - dimensional subspace of X. If x P and λ R +, then there exists y 0 P such that x y 0 and x + y 0 λx y 0. The pair (X, is called an orthogonality space (in the sense of Ratz. By an orthogonality normed space, we mean an orthogonality space equipped with a norm. Some examples of special interest are Corresponding author. E-mail address: dksflow@hotmail.com (R. Murali
A fixed point approach to orthogonal stability of an Additive - Cubic functional equation (i The trivial orthogonality on a vector space X defined by (a, and for non-zero elements x, y X, x y if and only if x, y are linearly independent, (ii The ordinary orthogonality on an inner product space (X,(.,. given by x y if and only if (x, y = 0, (iii The Birkhoff-James orthogonality on a normed space (X,. defined by x y if and only if x + y x for all λ R. The relation is called symmetric if x y implies that y x for all x, y X. Clearly conditions (i and (i i are symmetric but (i i i is not. It is remarkable to note, however, that a real normed space of dimension greater than or equal to 3 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. The orthogonal Cauchy functional equation f (x + y = f (x + f (y, x y in which is an abstract orthogonally was first investigated by S. Gudder and D.Strawther []. The orthogonally quadratic equation f (x + y + f (x y = f (x + f (y, x y, was first investigated by Vajzovic [3] when X is a Hilbert space, Y is the scalar field, f is continuous and means the Hilbert space orthogonality. Later, Drljevic [4], Fochi [5], Moslehian [6], Szabo [7], Moslehian and Th. M. Rassias [8], [9] and Paganoni and Ratz [0] have investigated the orthogonal stability of functional equations. Ashish and Renu chugh [] proved the Hyers-Ulam-Rassias stability of the orthogonally cubic and quartic functional equation in the sense of Ratz orthogonality. Let X be a set. A function d : X X [0, ] is called a generalized metric on X if d satisfies. d(x, y = 0 if and only if x = y;. d(x, y = d(y, x for all x, y X ; 3. d(x, z d(x, y + d(y, z for all x, y, z X. We recall a fundamental result in fixed point theory. Theorem. ([]. Let (X,d be a complete generalized metric space and let J : X X be a strictly contractive mapping with Lipschitz constant α <. Then for each given element x X, either d(j n x, J n+ x = for all nonnegative integers n or there exists a positive integer n 0 such that (i d(j n x, J n+ x < for all n n 0 ; (ii The sequence { J n x } is convergent to a fixed point y of J; (iii y is the unique fixed point of T in the set Y = { y X : d(j n 0 x, y < } ; (iv d(y, y d(y, J y for all y Y.. α In 996, Isac and Rassias [3] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors. see [4], [5], [6]. Recently, Choonkil Park [7], proved the Hyers-Ulam stability of orthogonally additive-quadratic functional equation in orthogonality spaces by using the fixed point method. Sung jin Lee, Choonkil Park and Reza Saadati [8] proved the orthogonal stability of an additive quadratic functional equation by using fixed point method. This paper, we prove the Hyers-Ulam stability of the orthogonally additive-cubic functional equation f (x + y + f (x y f (4x = f (x + y + f (x y 8f (x + 0f (x f ( x. in orthogonality space of an odd mapping. Throughout this paper, assume that (X, is an orthogonality space and that (Y,. Y is a real Banach space.. Hyers-Ulam stability of the orthogonally Additive-Cubic functional equation : Fixed Point Method For a given mapping f : X Y, we define D f (x, y = f (x + y + f (x y f (4x f (x + y f (x y + 8f (x 0f (x + f ( x.
R. Murali et al. / Int. J. Adv. Appl. Math. and Mech. 3(4 (06 8 3 Theorem.. Let φ : X [0, be a function such that there exists an α < with φ(x, y αφ, y ( for all x, y X with x y. Let f : X Y be an odd mapping satisfying f (0 = 0 and D f (x, y Y φ(x, y ( for all x, y X with x y. Then there exists a unique orthogonally additive mapping A : X Y such that f (x 8f (x A(x Y α α φ,0 (3 Letting y = 0 in (, we get f (4x + 0f (x 6f (x Y φ(x,0 for all x X, since x 0. Now letting g (x = f (x 8 f (x in above equation, we get g (x g (x φ(x,0 g (x g (x φ(x,0 (4 g (x Y g (x φ(x,0 g (x Y g (x αφ,0 (5 Consider the set S := {h : X Y } and introduce the generalized metric on S. { m(g,h = inf µ R +, ( g (x h(x Y x } µφ,0, x X where, as usual infφ = +. It is easy to show that (S,m is complete. see [0] Now consider the linear mapping J : S S such that J g (x := g (x Let g,h S be given such that m(g,h = ɛ. Then ( g (x h(x Y x φ,0 Hence, J g (x Jh(x Y = g (x h(x αφ Y,0 So m(g,h = ɛ. implies that m(j g, Jh αɛ. This means that m(j g, Jh αm(g,h for all g,h S. It follows from (5 that m(g, J g α. By Theorem., there exists a mapping A : X Y satisfying the following: ( A is a fixed point of J, i.e., A(x = A(x (6 The mapping A is a unique fixed point of J in the set M = { g S : m(g,h < }.
4 A fixed point approach to orthogonal stability of an Additive - Cubic functional equation This implies that A is a unique mapping satisfying (6 such that there exists a µ (0, satisfying g (x A(x Y µφ(x,0 ( m(j n g, A 0 as n. This implies the equality lim n forall x X. (3 m(g, A n g (n x = A(x m(g, J g, which implies the inequality m(g, A α α. α This implies that the inequality (3 holds. It follows from ( and ( that D A(x, y Y = lim Dg ( n n n x, n y Y for all x, y X with x y. So D A(x, y = 0 lim n n φ(n x, n y n α n lim φ(x, y = 0 n n for all x, y X with x y. Hence A : X Y is an orthogonally additive mapping. Corollary.. Assume that (X, is an orhogonality normed space. Let θ be a positive real number and p a real number with 0 < p <. Let f : X Y be an odd mapping satisfying f(0= 0 and D f (x, y Y θ( x p + y p for all x, y X with x y. Then there exists a unique orhogonally additive mapping A : X Y such that f (x 8f (x A(x Y p θ( x p. result. Taking φ(x, y = θ( x p + y p for all x, y X with x y and choosing α = p in Theorem., we get the Theorem.. Let φ : X [0, be a function such that there exists an α < with φ(x, y α φ(x,y (7 for all x, y X with x y and satisfying (. Then there exists a unique orthogonally additive mapping A : X Y such that f (x 8f (x A(x Y α φ,0 Let (S,m be the generalized metric space defined in the proof of Theorem.. Now we consider the linear mapping J : S S such that J g (x = g It follows from (4 that m(g, J g. The rest of the proof is similiar to the proof of Theorem.. (8
R. Murali et al. / Int. J. Adv. Appl. Math. and Mech. 3(4 (06 8 5 Corollary.. Assume that (X, is an orhogonality normed space. Let θ be a positive real number and p a real number with p >. Let f : X Y be an odd mapping satisfying f(0= 0 and D f (x, y Y θ( x p + y p for all x, y X with x y. Then there exists a unique orhogonally additive mapping A : X Y such that f (x 8f (x A(x Y p θ( x p. result. Taking φ(x, y = θ( x p + y p for all x, y X with x y and choosing α = p in Theorem., we get the Theorem.3. Let φ : X [0, be a function such that there exists an α < with φ(x, y 8αφ, y (9 or all x, y X with x y and satisfying (. Then there exists a unique orthogonally cubic mapping C : X Y such that f (x f (x C (x Y α α φ,0 (0 Letting y = 0 in (, we get f (4x + 0f (x 6f (x Y φ(x,0 for all x X, since x 0. Now, letting h(x = f (x f (x in above equation, We get h(x 8h(x φ(x,0. 8h(x h(x Y φ(x,0. ( h(x Y 8 h(x 8 φ(x,0 h(x Y 8 g (x 8 8αφ,0 ( Consider the set S := {h : X Y } and introduce the generalized metric on S : { m(g,h = inf µ R +, ( g (x h(x Y x } µφ,0, x X where, as usual infφ = +. It is easy to show that (S,m is complete [0]. Now consider the linear mapping J : S S such that J g (x := 8 g (x Let g,h S be given such that m(g,h = ɛ. Then, ( g (x h(x Y x φ,0
6 A fixed point approach to orthogonal stability of an Additive - Cubic functional equation Hence, J g (x Jh(x Y = 8 g (x 8 h(x αφ Y,0 m(g,h = ɛ implies that m(j g, Jh αɛ. This means that m(j g, Jh αm(g,h for all g,h S. It follows from ( that m(g, J g α. By Theorem., there exists a mapping C : X Y satisfying the following: ( C is a fixed point of J, i.e., C (x = 8C (x (3 The mapping C is a unique fixed point of J in the set M = { g S : m(g,h < }. This implies that C is a unique mapping satisfying (3 such that there exists a µ (0, satisfying h(x C (x Y µφ(x,0 ( m(j n h,c 0 as n. This implies the equality lim n 8 n h(n x = C (x forall x X. (3 m(h,c m(h, Jh, which implies the inequality α m(h,c α α. This implies that the inequality (0 hlods. It follows from (9 and ( that DC (x, y Y = lim Dg ( n n 8 n x, n y Y lim n 8 n φ(n x, n y 8 n α n lim φ(x, y = 0 n 8n for all x, y X with x y. So DC (x, y = 0 for all x, y X with x y. Hence C : X Y is an orthogonally cubic mapping. Corollary.3. Assume that (X, is an orhogonality normed space. Let θ be a positive real number and p a real number with 0 < p < 3. Let f : X Y be an odd mapping satisfying f(0= 0 and D f (x, y Y θ( x p + y p for all x, y X with x y. Then there exists a unique orhogonally cubic mapping C : X Y such that f (x f (x C (x Y 8 p θ( x p. result. Taking φ(x, y = θ( x p + y p for all x, y X with x y and choosing α = p 3 in Theorem.3, we get the
R. Murali et al. / Int. J. Adv. Appl. Math. and Mech. 3(4 (06 8 7 Theorem.4. Let φ : X [0, be a function such that there exists an α < with φ(x, y α φ(x,y (4 8 or all x, y X with x y and satisfying (. Then there exists a unique orthogonally cubic mapping C : X Y such that f (x f (x C (x Y α φ,0 (5 Let (S,m be the generalized metric space defined in the proof of Theorem.3. Now we consider the linear mapping J : S S such that J g (x = 8g It follows from ( that m(h, Jh. The rest of the proof is similiar to the proof of Theorem.3. Corollary.4. Assume that (X, is an orhogonality normed space. Let θ be a positive real number and p a real number with 0 < p < 3. Let f : X Y be an odd mapping satisfying f(0= 0 and D f (x, y Y θ( x p + y p for all x, y X with x y. Then there exists a unique orhogonally cubic mapping C : X Y such that f (x f (x C (x Y p 8 θ( x p. result. Taking φ(x, y = θ( x p + y p for all x, y X with x y and choosing α = 3 p in Theorem.4, we get the References [] S.M. Ulam,Problems in Modern Mathematics, Rend. Chap.VI, Wiley, New York, 960. [] D.H. Hyers, On the stability of the linear functional equation, Proceeding of National Academic Sciences, U.S.A., 7 (94-4. [3] T. Aoki, On the stability of the linear transformation in Banach spaces, Journal of Mathematical Society Japan, (95 64-66. [4] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 7 (978 97-300. [5] P.Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 84 (994 43-436. [6] J.M. Rassias, On some approximately quadratic mappings being exactly quadratic, J. Ind. Math. Soc 69 (00 55-60. [7] K. Ravi, J.M. Rassias, M. Arunkumar, Ulam stability for the orthogonal general Euler-Lagrange type functional equation, Internet.J.Math.Stat. 3(A08 (008 36-46. [8] J.R. Lee, J. Kim, C. Park, A Fixed point approach to the stability of an additive-quadratic-cubic-quartic functional equation, Fixed point theory and applications, Vol. 00, article ID 85780, 6 pages. [9] D. Mihet, The fixed points method for fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 60 (009 663-667. [0] D. Mihet, V. Radu, t On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (009 567-57. [] J. Ratz, On orthogonally additive mappings. Aequationes Math. 8 (985 35-49. [] S. Gudder and D. Strawther, Orthogonally additive and orthogonally increasing function on vector spacees, Pacific J. Math. 58 (995 47-436. [3] F. Vajzovic, Uber das Functional H mit der Eigenschaft (x, y = 0 H(x + y + H(x y = H(x + H(y. Glasnik Mat Ser III. (967 73-8. [4] F. Drljevic, On a functional which is quadratic on A-orthogonal vectors, Publ Inst Math (Beograd. 54 (986 63-7.
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