On approximate cubic homomorphisms

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Mar 5, 2009 - f(xy) = f(x)f(y), . f(2x + y) + f(2x − y)=2f(x + y)+2f(x − y) + 12f(x), ... functional equations have been extensively investigated by a number of au-.
arXiv:0903.1066v1 [math.CA] 5 Mar 2009

ON APPROXIMATE CUBIC HOMOMORPHISMS M. ESHAGHI GORDJI AND M. BAVAND SAVADKOUHI Abstract. In this paper, we investigate the generalized Hyers–Ulam– Rassias stability of the system of functional equations    f (xy) = f (x)f (y),   

f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x),

on Banach algebras. Indeed we establish the superstability of above system by suitable control functions.

1. Introduction A definition of stability in the case of homomorphisms between metric groups was suggested by a problem by S. M. Ulam [21] in 1940. Let (G1 , .) be a group and let (G2 , ∗) be a metric group with the metric d(., .). Given ǫ > 0, does there exist a δ > 0 such that if a mapping h : G1 −→ G2 satisfies the inequality d(h(x.y), h(x) ∗ h(y)) < δ for all x, y ∈ G1 , then there exists a homomorphism H : G1 −→ G2 with d(h(x), H(x)) < ǫ for all x ∈ G1 ? In this case, the equation of homomorphism h(x.y) = h(x) ∗ h(y) is called stable. In the other hand we are looking for situations when the homomorphisms are stable, i.e., if a mapping is an approximate homomorphism, then there exists an exact homomorphism near it. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, D. H. Hyers [8] gave a positive answer to the question of Ulam for Banach spaces. Let f : E1 −→ E2 be a mapping between Banach spaces such that kf (x + y) − f (x) − f (y)k ≤ δ 2000 Mathematics Subject Classification. Primary 39B52, Secondary 39B82, 46H25. Key words and phrases. Cubic functional equation; Homomorphism; Hyer-UlamRassias stability. 1

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M. ESHAGHI GORDJI AND M. BAVAND SAVADKOUHI

for all x, y ∈ E1 and for some δ ≥ 0. Then there exists a unique additive mapping T : E1 −→ E2 satisfying kf (x) − T (x)k ≤ δ for all x ∈ E1 . Moreover, if f (tx) is continuous in t for each fixed x ∈ E1 , then the mapping T is linear. Th. M. Rassias [20] succeeded in extending the result of Hyers’ Theorem by weakening the condition for the Cauchy difference controlled by (kxkp + kykp), p ∈ [0, 1) to be unbounded. This condition has been assumed further till now, through the complete Hyers direct method, in order to prove linearity for generalized Hyers-Ulam stability problem forms. A number of mathematicians were attracted to the pertinent stability results of Th. M. Rassias [20], and stimulated to investigate the stability problems of functional equations. The stability phenomenon that was introduced and proved by Th. M. Rassias is called Hyers–Ulam–Rassias stability. And then the stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [5-7], [9], [12-13] and [16-18]). D.G. Bourgin [4] is the first mathematician dealing with stability of (ring) homomorphism f (xy) = f (x)f (y). The topic of approximate homomorphisms was studied by a number of mathematicians, see [2,3,10,14,15,19] and references therein. Jun and Kim [11] introduced the following functional equation f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x)

(1.1)

and they established the general solution and generalized Hyers–Ulam– Rassias stability problem for this functional equation. It is easy to see that the function f (x) = cx3 is a solution of the functional equation (1.1). Thus, it is natural that (1.1) is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic function. Let R be a ring. Then a mapping f : R −→ R is called a cubic homomorphism if f is a cubic function satisfying f (ab) = f (a)f (b),

(1.2)

for all a, b ∈ R. For instance, let R be commutative, then the mapping f : R −→ R defined by f (a) = a3 (a ∈ R), is a cubic homomorphism. It

ON APPROXIMATE CUBIC HOMOMORPHISMS

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is easy to see that a cubic homomorphism is a ring homomorphism if and only if it is zero function. In this paper we study the stability of cubic homomorphisms on Banach algebras.

2. Main results In the following we suppose that A is a normed algebra, B is a Banach algebra and f is a mapping from A into B, and ϕ, ϕ1 , ϕ2 are maps from A × A into R+ . Also, we put 0p = 0 for p ≤ 0. Theorem 2.1. Let kf (xy) − f (x)f (y)k ≤ ϕ1 (x, y),

(2.1)

and kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ ϕ2 (x, y), (2.2) for all x, y ∈ A. Assume that the series ∞ X ϕ2 (2i x, 2i y) Ψ(x, y) = 23i i=0 converges and that ϕ1 (2n x, 2n y) lim = 0, n−→∞ 26n for all x, y ∈ A. Then there exists a unique cubic homomorphism T : A −→ A such that 1 kT (x) − f (x)k ≤ Ψ(x, 0), (2.3) 16 for all x ∈ A. Proof. Setting y = 0 in (2.2) yields k2f (2x) − 24 f (x)k ≤ ϕ2 (x, 0) ,

(2.4)

and then dividing by 24 in (2.4) to obtain ϕ2 (x, 0) f (2x) − f (x)k ≤ , 3 2 2.23 for all x ∈ A. Now by induction we have k

(2.5)

n−1

f (2n x) 1 X ϕ2 (2i x, 0) k 3n − f (x)k ≤ . 2 2.23 i=0 23i

(2.6)

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M. ESHAGHI GORDJI AND M. BAVAND SAVADKOUHI n

x) In order to show that the functions Tn (x) = f (2 is a convergent sequence, 23n we use the Cauchy convergence criterion. Indeed, replace x by 2m x and divide by 23m in (2.6), where m is an arbitrary positive integer. We find that

k

n−1 n+m−1 1 X ϕ2 (2i+m x, 0) 1 X ϕ2 (2i x, 0) f (2n+m x) f (2m x) − k ≤ = 23(n+m) 23m 2.23 i=0 23(i+m) 2.23 i=m 23i

for all positive integers m,n. Hence by the Cauchy criterion the limit T (x) = limn→∞ Tn (x) exists for each x ∈ A. By taking the limit as n −→ ∞ in P ϕ2 (2i x,0) 1 = 16 Ψ(x, 0) and (2.3) (2.6), we see that kT (x) − f (x)k ≤ 2.21 3 ∞ i=0 23i n n holds for all x ∈ A. If we replace x by 2 x and y by 2 y respectively in (2.2) and divide by 23n , we see that f (2.(2n x) + 2n y) f (2.(2n x) − 2n y) f (2n x + 2n y) f (2n x − 2n y) k + −2 −2 23n 23n 23n 23n n n n ϕ2 (2 x, 2 y) f (2 x) . − 12 3n k ≤ 2 23n Taking the limit as n −→ ∞, we find that T satisfies (1.1) (see Theorem 3.1 of [11]). On the other hand we have f (2n xy) f (2n x) f (2n y) − lim . lim k n→∞ n→∞ 23n 23n n→∞ 23n f (2n x2n y) f (2n y)f (2ny) = lim k − k n→∞ 26n 26n ϕ1 (2n x, 2n y) ≤ lim =0. n→∞ 26n

kT (xy) − T (x).T (y)k = k lim

for all x, y ∈ A. We find that T satisfies (1.2). To prove the uniqueness property of T , let T´ : A → A be a functions satisfies T´ (2x+y)+ T´ (2x−y) = 1 2T´(x + y) + 2T´(x − y) + 12T´(x) and kT´ (x) − f (x)k ≤ 16 Ψ(x, 0) . Since T, T´ are cubic, then we have T (2n x) = 23n T (x), T´ (2n x) = 23n T´ (x)

ON APPROXIMATE CUBIC HOMOMORPHISMS

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for all x ∈ A, hence, 1 kT (x) − T´(x)k = 3n kT (2n x) − T´ (2n x)k 2 1 ≤ 3n (kT (2n x) − f (2n x)k + kT´ (2n x) − f (2n x)k) 2 1 1 1 Ψ(2n x, 0)) ≤ 3n ( 3 Ψ(2n x, 0) + 3 2 2.2 2.2 ∞ 1 X 1 1 n ϕ2 (2i+n x, 0) = 3(n+1) Ψ(2 x, 0) = 3(n+1) 3i 2 2 2 i=0 ∞ ∞ 1 X 1 1 X 1 i+n = 3 ϕ2 (2 x, 0) = 3 ϕ2 (2i x, 0) . 3(i+n) 3i 2 i=0 2 2 i=n 2

By taking n → ∞ we get, T (x) = T´ (x).



Corollary 2.2. Let θ1 and θ2 be nonnegative real numbers, and let p ∈ (−∞, 3). Suppose that kf (xy) − f (x)f (y)k ≤ θ1 , kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ θ2 (kxkp + kykp) , for all x, y ∈ A. Then there exists a unique cubic homomorphism T : A −→ A such that 1 θ2 kxkp , kT (x) − f (x)k ≤ 16 1 − 2p−3 for all x, y ∈ A. Proof. In Theorem 2.1, let ϕ1 (x, y) = θ1 and ϕ2 (x, y) = θ2 (kxkp + kykp ) for all x, y ∈ A.  Corollary 2.3. Let θ1 and θ2 be nonnegative real numbers. Suppose that kf (xy) − f (x)f (y)k ≤ θ1 , kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ θ2 , for all x, y ∈ A. Then there exists a unique cubic homomorphism T : A −→ A such that θ2 kT (x) − f (x)k ≤ , 14 for all x ∈ A. Proof. It follows from Corollary 2.2.



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Corollary 2.4. Let p ∈ (−∞, 3) and let θ be a positive real number. Suppose that ϕ(2n x, 2n y) lim =0, n→∞ 26n for all x, y ∈ A. Moreover, Suppose that kf (xy) − f (x)f (y)k ≤ ϕ(x, y), and that kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ θkykp, (2.7) for all x, y ∈ A. Then f is a cubic homomorphism. Proof. Letting x = y = 0 in (2.7), we get that f (0) = 0. So by y = 0, in (2.7), we get f (2x) = 23 f (x) for all x ∈ A. By using induction we have f (2n x) = 23n f (x),

(2.8)

for all x ∈ A and n ∈ N. On the other hand by Theorem 2.1, the mapping T : A → A defined by f (2n x) , T (x) = lim n→∞ 23n is a cubic homomorphism. Therefore it follows from (2.8) that f = T. Hence it is a cubic homomorphism.  Corollary 2.5. Let p, q, θ ≥ 0 and p + q < 3. Let ϕ(2n x, 2n y) =0, n→∞ 26n for all x, y ∈ A. Moreover, Suppose that lim

kf (xy) − f (x)f (y)k ≤ ϕ(x, y), and that kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ θkxkq kykp, for all x, y ∈ A. Then f is a cubic homomorphism. Proof. If q = 0, then by Corollary 2.4 we get the result. If q 6= 0, it follows from Theorem 2.1, by putting ϕ1 (x, y) = ϕ(x, y) and ϕ2 (x, y) = θ(kxkp kykp ) for all x, y ∈ A. 

ON APPROXIMATE CUBIC HOMOMORPHISMS

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Corollary 2.6. Let p ∈ (−∞, 3) and θ be a positive real number. Let θ2np kykp = 0, n→∞ 26n for all x, y ∈ A. Moreover, suppose that lim

kf (xy) − f (x)f (y)k ≤ θkykp , and kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ θkykp , for all x, y ∈ A. Then f is a cubic homomorphism. Proof. Let ϕ(x, y) = θkykp . Then by Corollary 2.4, we get the result.



Theorem 2.7. Let kf (xy) − f (x)f (y)k ≤ ϕ1 (x, y) , and kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ ϕ2 (x, y) , (2.9) for all x, y ∈ A. Assume that the series Ψ(x, y) =

∞ X i=1

x y 23i ϕ2 ( i , i ) 2 2

converges and that

x y , ) = 0, n−→∞ 2n 2n for all x, y ∈ A. Then there exists a unique cubic homomorphism T : A −→ A such that 1 kT (x) − f (x)k ≤ Ψ(x, 0) , (2.10) 16 for all x ∈ A. lim 26n ϕ1 (

Proof. Setting y = 0 in (2.9) yields k2f (2x) − 2.23 f (x)k ≤ ϕ2 (x, 0) . Replacing x by

x 2

(2.11)

in (2.11) to get 1 x x kf (x) − 23 f ( )k ≤ ϕ2 ( , 0) , 2 2 2

(2.12)

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M. ESHAGHI GORDJI AND M. BAVAND SAVADKOUHI

for all x ∈ A. By (2.12) we use iterative methods and induction on n to prove our next relation n x 1 X 3i x 3n 2 ϕ2 ( i , 0) . (2.13) kf (x) − 2 f ( n )k ≤ 3 2 2.2 i=1 2 In order to show that the functions Tn (x) = 23n f ( 2xn ) is a convergent sequence, replace x by 2xm in (2.13), and then multiplying by 23m , where m is an arbitrary positive integer. We find that n x x 1 X 3(i+m) x 3m 3(n+m) k2 f ( m ) − 2 f ( n+m )k ≤ 2 ϕ2 ( i+m , 0) 3 2 2 2.2 i=1 2 n+m 1 X 3i x = 2 ϕ2 ( i , 0) 3 2.2 i=1+m 2

for all positive integers. Hence by the Cauchy criterion the limit T (x) = limn−→∞ Tn (x) exists for each x ∈ A. By taking the limit as n −→ ∞ P x 1 3i in (2.13), we see that kT (x) − f (x)k ≤ 2.21 3 ∞ i=1 2 ϕ2 ( 2i , 0) = 16 Ψ(x, 0) and (2.10) holds for all x ∈ A. The rest of proof is similar to the proof of Theorem 2.1.  Corollary 2.8. Let p > 3 and θ be a positive real number. Let x y lim 26n ϕ( n , n ) = 0 , n→∞ 2 2 for all x, y ∈ A. Moreover, Suppose that kf (xy) − f (x)f (y)k ≤ ϕ(x, y) , and kf (2x + y) + f (2x − y) − 2f (x + y) − 2f (x − y) − 12f (x)k ≤ θkykp , (2.14) for all x, y ∈ A. Then f is a cubic homomorphism. Proof. Letting x = y = 0 in (2.14), we get that f (0) = 0. So by y = 0, in (2.14), we get f (2x) = 23 f (x) for all x ∈ A. By using induction we have x f (x) = 23n f ( n ), (2.15) 2 for all x ∈ A and n ∈ N. On the other hand by Theorem 2.8, the mapping T : A → A defined by x T (x) = lim 23n f ( n ), n→∞ 2

ON APPROXIMATE CUBIC HOMOMORPHISMS

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is a cubic homomorphism. Therefore it follows from (2.15) that f = T. Hence f a cubic homomorphism. 

Example. Let 

  A :=   then A is a Banach algebra and the following norm:  0 a1  0 0  k  0 0 0 0

0 0 0 0

R 0 0 0

R R 0 0

R R R 0



  , 

equipped with the usual matrix-like operations  a2 a3 6 X a4 a5   |ai | (ai ∈ R). k = 0 a6  i=1 0 0

Let 

  a :=  

0 0 0 0

0 0 0 0

1 0 0 0

2 1 0 0

    

and we define f : A → A by f (x) = x3 + a, and ϕ1 (x, y) := kf (xy) − f (x)f (y)k = kak = 4 , ϕ2 (x, y) := kf (2x+y)+f (2x−y)−2f (x+y)−2f (x−y)−12f (x)k = 14kak = 56 , for all x, y ∈ A. Then we have ∞ X ϕ2 (2k x, 2k y) k=0

23k

∞ X 56 = = 64 , 23k k=0

and ϕ1 (2n x, 2n y) =0. n→∞ 26n lim

Thus the limit T (x) = limn→∞

f (2n x) 23n

= x3 exists. Also,

T (xy) = (xy)3 = x3 y 3 = T (x)T (y) .

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M. ESHAGHI GORDJI AND M. BAVAND SAVADKOUHI

Furthermore, T (2x + y) + T (2x − y) = (2x + y)3 + (2x − y)3 = 16x3 + 12xy 2 = 2T (x + y) + 2T (x − y) + 12T (x) . Hence T is cubic homomorphism. Also from this example it is clear that the superstability of the system of functional equations    f (xy) = f (x)f (y),   

f (2x + y) + f (2x − y) = 2f (x + y) + 2f (x − y) + 12f (x),

with the control functions in Corollaries 2.4, 2.5 and 2.6 does not hold. Acknowledgement. The authors would like to thank the referees for their valuable suggestions. Also, the second author would like to thank the office of gifted students at Semnan University for its financial support. References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan. 2(1950), 64-66. [2] R. Badora, On approximate ring homomorphisms,J. Math. Anal. Appl. 276, 589-597 (2002). [3] J. Baker, J. Lawrence and F. Zorzitto, The stability of the equation f (x + y) = f (x)f (y),Proc. Amer. Math. Soc. 74 (1979), no. 2, 242-246. [4] D. G. Borugin, Class of transformations and bordering transformations, Bull. Amer. Math. Soc. 27 (1951) 223-237. [5] V. A. Faizev, Th. M. Rassias and P. K. Sahoo, The space of (ψ, γ)-additive mappings on semigroups,Transactions of the Amer. Math. Soc. 354(11)(2002),4455-4472. [6] G. L. Forti, An existence and stability Theorem for a class of functional equations,Stochastica, 4 (1980) 23-30. [7] G. L. Forti, Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations, J. Math. Anal. Appl, 295 (2004), 127-133. [8] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941) 222-224. [9] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables,Birkhauser, Boston, Basel, Berlin, 1998. [10] D. H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequationes Math, 44 (1992), 125-153.

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[11] K. W. Jun and H. M. Kim, The generalized Hyers-Ulam-Russias stability of a cubic functional equation, J. Math. Anal. Appl. 274, (2002), no. 2, 267–278. [12] G. Isac and Th. M. Rassias, On the Hyers–Ulam stability of ?-additive mappings, J. Approx. Theory 72(1993), 131-137. [13] L. Maligranda, A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions- a question of priority,Aequat. Math. 75 (2008) 289-296. [14] C. Park, Hyers–Ulam–Rassias stability of homomorphisms in quasi-Banach algebras,Bull. Sci. Math. 132 (2008), no. 2, 87-96. [15] Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappings,J. Math. Anal. Appl. 246 (2000), no. 2, 352-378. [16] Th. M. Rassias and J. Tabor, Stability of mappings of Hyers-Ulam Type,Hadronic Press Inc. Florida, 1994. [17] Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158(1991),106-113. [18] Th. M. Rassias, On the stability of functional equations originated by a problem of Ulam,Mathematica, 44(67)(1)(2002),39-75. [19] Th. M. Rassias, On the stability of functional equations and a problem of Ulam,Acta Applicandae Math. 62(1)(2000),23-130. [20] Th. M. Rassias, On the stability of the linear mapping in Banach spaces,Proc. Amer. Math. Soc. 72 (1978) 297-300. [21] S. M. Ulam, A collection of Mathematical problems, Interscience Publ, New York, 1960. Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran E-mail address: [email protected] & [email protected]