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Stability of the Jensen equation in C*-algebras: a fixed point approach

  • 1,
  • 1,
  • 2 and
  • 3Email author
Advances in Difference Equations20122012:17

https://doi.org/10.1186/1687-1847-2012-17

  • Received: 1 January 2012
  • Accepted: 21 February 2012
  • Published:

Abstract

Using fixed point method, we prove the Hyers-Ulam stability of homomorphisms in C*-algebras and Lie C*-algebras and also of derivations on C*-algebras and Lie C*-algebras for the Jensen equation.

2010 Mathematics Subject Classification: 39B82; 47H10; 46L05; 39B52.

Keywords

  • Jensen equation
  • Hyers-Ulam stability
  • C*-algebras
  • fixed point method

1 Introduction

A classical question in the theory of functional equations is the following: "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 solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1]. In the next year, Hyers [2] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In [3], Rassias proved a generalization of the Hyers' theorem for additive mappings.

The result of Rassias has provided a lot of influence during the last three decades in the development of a generalization of the Hyers-Ulam stability concept. Furthermore, in 1994, a generalization of the Rassias' theorem was obtained by Gǎvruta [4] by replacing the bound ϵ(||x|| p + ||y|| p ) by a general control function φ(x, y).

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 [522].

Theorem 1.1. Let (X, d) be a complete generalized metric space and J: XX be a strictly contractive mapping with Lipschitz constant L < 1. Then, for all x X, either
d ( J n x , J n + 1 x ) =

for all nonnegative integers n or there exists a positive integer n 0 such that

(a) d(J n x, Jn+1x) < ∞ for all n0n0;

(b) the sequence {J n x} converges to a fixed point y* of J;

(c) y* is the unique fixed point of J in the set Y = { y X : d ( J n 0 x , y ) < } ;

(d) d ( y , y * ) 1 1 - L d ( y , J y ) for all y Y.

In [20], Park proved the Hyers-Ulam stability of the following functional equation:
2 f x + y 2 = f ( x ) + f ( y )
(1.1)

in fuzzy Banach spaces. In this article, using the fixed point alternative approach, we prove the Hyers-Ulam stability of homomorphisms in C*-algebras and Lie C*-algebras and also of derivations on C*-algebras and Lie C*-algebras for the Jensen Equation (1.1).

2 Stability of homomorphisms in C*-algebras

Throughout this section, assume that A is a C*-algebra with the norm ||.|| A and that B is a C*- algebra with the norm ||.|| B .

For a given mapping f: AB, we define
C μ f ( x , y ) : = 2 μ f x + y 2 - f ( μ x ) - f ( μ y )

for all μ T 1 : = { ν : ν = 1 } and all x, y A. Note that a -linear mapping H: AB is called a homomorphism in C*-algebras, if H satisfies H(xy) = H(x)H(y) and H(x*) = H(x)* for all x A. Throughout this section, we prove the Hyers-Ulam stability of homomorphisms in C*-algebras for the functional equation C μ f(x, y) = 0.

Theorem 2.1. Let f: AB be a mapping with f(0) = 0 for which there exists a function φ: A2 → [0, ∞) such that
C μ f ( x , y ) B φ ( x , y ) ,
(2.2)
f ( x y ) - f ( x ) f ( y ) B φ ( x , y ) ,
(2.3)
f ( x * ) - f ( x ) * B φ ( x , x )
(2.4)
for all μ T 1 and all x, y A. If there exists an L < 1 2 such that
φ ( x , y ) L φ ( 2 x , 2 y ) 2
(2.5)
for all x, y A, then there exists a unique C*-algebra homomorphism H: AB such that
f ( x ) - H ( x ) B φ ( x , 0 ) 1 - L .
(2.6)
Proof. It follows from (2.5) that
lim n 2 n φ x 2 n , y 2 n = lim n L n φ ( x , y ) = 0 .
Consider the set X := {g: AB;g(0) = 0} and the generalized metric d in X defined by
d ( f , g ) = inf { C + : g ( x ) - h ( x ) B C φ ( x , 0 ) , x A }
It is easy to show that (X, d) is complete. Now, we consider a linear mapping J : AA such that
J h ( x ) : = 2 h x 2
for all x A. By [[7], Theorem 3.1], d(Jg, Jh) ≤ Ld(g, h) for all g, h X. Letting μ = 1 and y = 0 in (2.2), we have
2 f x 2 - f ( x ) B φ ( x , 0 )
(2.7)
for all x A. It follows from (2.7) that d(f, Jf) ≤ 1. By Theorem 1.1, there exists a mapping H: AB satisfying the following:
  1. (1)
    H is a fixed point of J, that is,
    H x 2 = 1 2 H ( x )
    (2.8)
     
for all x A. The mapping H is a unique fixed point of J in the set Ω = {g X : d(f, g) < ∞}. This implies that H is a unique mapping satisfying (2.8) such that there exists C (0, ∞) satisfying ||f(x) - H(x)|| B (x, 0) for all x A.
  1. (2)
    d(J n f, H) → 0 as n → ∞. This implies the equality
    lim n 2 n f x 2 n = H ( x )
    (2.9)
     
for all x A.
  1. (3)
    d ( f , H ) d ( f , J f ) 1 - L , which implies the inequality d ( f , H ) 1 1 - L . This implies that the inequality (2.6) holds. It follows from (2.2) and (2.9) that
    2 H x + y 2 - H ( x ) - H ( y ) B = lim n 2 n + 1 f x + y 2 n + 1 - 2 n f x 2 n - 2 n f y 2 n B lim n 2 n φ x 2 n , y 2 n lim n L n φ ( x , y ) = 0
     

for all x, y A. So 2 H x + y 2 = H ( x ) + H ( y ) for all x, y X. Therefore, the mapping H: AB is Jensen additive.

Letting y = x in (2.2), we get μf(x) = f(μx) for all μ T 1 and all x A So, we get
μ H ( x ) - H ( μ x ) B = lim n 2 n μ f x 2 n - 2 n f μ x 2 n B = 0 .
So, μH(x) = H(μx) for all μ T 1 and all x A Thus one can show that the mapping H: AB is -linear. It follows from (2.3) that
H ( x y ) - H ( x ) H ( y ) B = lim n 4 n f x y 4 n - f x 2 n f y 2 n B lim n 4 n φ x 2 n , y 2 n lim n ( 2 L ) n φ ( x , y ) = 0
for all x A. Furthermore, By (2.4), we have
H ( x * ) - H ( x ) * B = lim n 2 n f x * 2 n - f x 2 n * B lim n 2 n φ x 2 n , y 2 n lim n L n φ ( x , y ) = 0

for all x A. Thus H: AB is a C*-algebra homomorphism satisfying (2.6), as desired.

Corollary 2.1. Let 0 < r < 1 and θ be nonnegative real numbers and f: AB be a mapping with f(0) = 0 such that
2 μ f x + y 2 - f ( μ x ) - f ( μ y ) B θ ( x A r + y A r ) , f ( x y ) - f ( x ) f ( y ) B θ ( x A r + y A r ) f ( x * ) - f ( x ) * B 2 θ x A r
(2.10)
for all μ T 1 and all x, y A. Then the limit H ( x ) = lim n 2 n f x 2 n exists for all x A and H: AB is a unique C*-algebra homomorphism such that
f ( x ) - H ( x ) B 2 θ x A r 2 - 2 r
(2.11)

for all x A.

Proof. The proof follows from Theorem 2.1, if we take φ ( x , y ) = θ ( x A r + y A r ) for all x, y A. In fact, if we choose L = 2r-1, then we get the desired result.

Theorem 2.2. Let f: AB be a mapping with f(0) = 0 for which there exists a function φ: A2 → [0, ∞) satisfying (2.2), (2.3), and (2.4). If there exists an L < 1 such that φ ( x , y ) 2 L φ x 2 , y 2 for all x, y A, then there exists a unique C*-algebra homomorphism H: AB such that
f ( x ) - H ( x ) B L φ ( x , 0 ) 1 - L .
(2.12)

for all x A.

Proof. We consider the linear mapping J: AA such that J g ( x ) = 1 2 g ( 2 x ) for all x A. It follows from (2.7) that
f ( x ) - 1 2 f ( 2 x ) φ ( 2 x , 0 ) 2 L φ ( x , 0 )
for all x X. Hence d(f, Jf) ≤ L. By Theorem 1.1, there exists a mapping H: AB satisfying the following:
  1. (1)
    H is a fixed point of J, that is,
    H ( 2 x ) = 2 H ( x )
    (2.13)
     
for all x A. The mapping H is a unique fixed point of J in the set Ω = {g X: d(f, g) < ∞}. This implies that H is a unique mapping satisfying (2.13) such that there exists C (0, ∞) satisfying ||f(x) - H(x)|| B (x, 0) for all x A.
  1. (2)

    d(J n f, H) → 0 as n → ∞. This implies the equality lim n f ( 2 n x ) 2 n = H ( x ) for all x A.

     
  2. (3)

    d ( f , H ) d ( f , J f ) 1 - L , which implies the inequality d ( f , H ) 1 1 - L . which implies that the inequality (2.12). The rest of the proof is similar to the proof of Theorem 2.1.

     
Corollary 2.2. Let r > 1 and θ be nonnegative real numbers and f: AB be a mapping satisfying f(0) = 0 and (2.10). Then the limit H ( x ) = lim n f ( 2 n x ) 2 n exists for all x A and H: AB is a unique C*-algebra homomorphism such that
f ( x ) - H ( x ) B 2 θ x A r 2 r - 2
(2.14)

for all x A.

Proof. The proof follows from Theorem 2.2 if we take φ ( x , y ) = θ ( x A r + y A r ) for all x, y A. In fact, if we choose L = 21-r, then we get the desired result.

3 Stability of derivations on C*-algebras

Throughout this section, assume that A is a C*-algebra with the norm ||.| A . Note that a -linear mapping δ: AA is called a derivation on A if δ satisfies δ(xy) = δ(x)y + (y) for all x, y A.

Throughout this section, using the fixed point alternative approach, We prove the Hyers-Ulam stability of derivations on C*-algebras for the functional equation (1.1).

Theorem 3.1. Let f: AA be a mapping with f(0) = 0 for which there exists a function φ: A2 → [0, ∞) such that
2 μ f x + y 2 - f ( μ x ) - f ( μ y ) A φ ( x , y )
(3.15)
f ( x y ) - f ( x ) y - x f ( y ) A φ ( x , y )
(3.16)
for all μ T 1 and all x, y A. If there exists an L < 1 2 such that φ ( x , y ) L φ ( 2 x , 2 y ) 2 for all x, y A, then there exists a unique derivation δ: AA such that
f ( x ) - δ ( x ) A φ ( x , 0 ) 1 - L .
(3.17)
Proof. By the same reasoning as in the proof of Theorem 2.1, there exists a unique -linear mapping δ: AA satisfying (3.17). The mapping δ: AA is given by
δ ( x ) = lim n 2 n f x 2 n
for all x A. It follows from (3.2) that
δ ( x y ) - δ ( x ) y - x δ ( y ) B = lim n 4 n f x y 4 n - f x 2 n y 2 n - x 2 n f y 2 n B lim n 4 n φ x 2 n , y 2 n lim n ( 2 L ) n φ ( x , y ) = 0
for all x, y A. So
δ ( x y ) - δ ( x ) y - x δ ( y )

for all x, y A. Thus δ: AA is a derivation satisfying (3.17).

Corollary 3.1. Let 0 < r < 1 and θ be nonnegative real numbers and f: AA be a mapping with f(0) = 0 such that
2 μ f x + y 2 - f ( μ x ) - f ( μ y ) A θ ( x A r + y A r ) ,
(3.18)
f ( x y ) - f ( x ) y - x f ( y ) A θ ( x A r + y A r )
(3.19)
for all μ T 1 and all x, y A. Then the limit H ( x ) = lim n 2 n f x 2 n exists for all x A and δ: AA is a unique derivation such that
f ( x ) - δ ( x ) 2 θ x A r 2 - 2 r
(3.20)

for all x A.

Proof. The proof follows from Theorem 3.1 if we take φ ( x , y ) = θ ( x A r + y A r ) for all x, y A. In fact, if we choose L = 2r-1, then we get the desired result.

Theorem 3.2. Let f: AA be a mapping with f(0) = 0 for which there exists a function φ: A2 → [0, ∞) satisfying (3.15) and (3.2). If there exists an L < 1 such that φ ( x , y ) 2 L φ x 2 , y 2 for all x, y A, then there exists a unique derivation δ: AA such that
f ( x ) - δ ( x ) A L φ ( x , 0 ) 1 - L .
(3.21)

Proof. The proof is similar to the proofs of Theorems 2.2 and 3.1.

Corollary 3.2. Let r > 1 and θ be nonnegative real numbers and f: AA be a mapping satisfying f(0) = 0, (3.4) and (3.5). Then the limit H ( x ) = lim n f ( 2 n x ) 2 n exists for all x A and δ: AA is a unique derivation such that
f ( x ) - δ ( x ) A 2 θ x A r 2 r - 2
(3.22)

for all x A.

Proof. The proof follows from Theorem 3.2 if we take φ ( x , y ) = θ ( x A r + y A r ) for all x, y A. In fact, if we choose L = 21-r, then we get the desired result.

4 Stability of homomorphisms in Lie C*-algebras

A C*-algebra C , endowed with the Lie product [ x , y ] : = x y - y x 2 on C , is called a Lie C*-algebra (see, [1719]).

Definition 4.1. Let A and B be Lie C*-algebras, A -linear mapping H: AB is called a Lie C*-algebra homomorphism if H ( [ x , y ] ) = [ H ( x ) , H ( y ) ] = H ( x ) H ( y ) - H ( y ) H ( x ) 2 for all x, y A.

Throughout this section, assume that A is a Lie C*-algebra with the norm ||.|| A and B is a Lie C*-algebra with the norm ||.|| B .

We prove the Hyers-Ulam stability of homomorphisms in Lie C*-algebras for the functional Equation (1.1).

Theorem 4.1. Let f: AB be a mapping with f(0) = 0 for which there exists a function φ: A2 → [0, ∞) satisfying (2.2) such that
f ( [ x , y ] ) - [ f ( x ) , f ( y ) ] B φ ( x , y )
(4.23)

for all x, y A. If there exists an L < 1 2 such that φ ( x , y ) L 2 φ ( 2 x , 2 y ) for all x, y A, then there exists a unique Lie C*-algebra homomorphism H: AB satisfying (2.6).

Proof. By the same reasoning as in the proof of Theorem 2.1, there exists a unique -linear mapping H: AB satisfying (2.6). The mapping H: AB is given by H ( x ) = lim n 2 n f x 2 n for all x A. It follows from (4.23) that
H ( [ x , y ] ) - [ H ( x ) , H ( y ) ] B = lim n 4 n f [ x , y ] 4 n - f x 2 n , f y 2 n B lim n 4 n φ x 2 n , y 2 n = 0

for all x, y A. So H([x, y]) = [H(x), H(y)] for all x, y A. Thus H: AB is a Lie C*-algebra homomorphism satisfying (2.6), as desired.

Corollary 4.1. Let 0 < r < 1 and θ be nonnegative real numbers, and let f: AB be a mapping satisfying f(0) = 0 such that
C μ f ( x , y ) B θ ( x A r + y A r ) ,
(4.24)
f ( [ x , y ] ) - [ f ( x ) , f ( y ) ] B θ ( x A r + y A r )
(4.25)

for all μ T 1 and all x, y A. Then there exists a unique Lie C*-algebra homomorphism H: AB satisfying (2.11).

Proof. The proof follows from Theorem 4.1 by taking φ ( x , y ) = θ ( x A r + y A r ) for all x, y A. Then L = 2r-1and we get the desired result.

Theorem 4.2. Let f: AB be a mapping with f(0) = 0 for which there exists a function φ: A2 → [0, ∞) satisfying (2.2) and (4.23). If there exists an L < 1 such that φ ( x , y ) 2 L φ x 2 , y 2 for all x, y A, then there exists a unique Lie C*-algebra homomorphism H: AB satisfying (2.12).

Corollary 4.2. Let r > 1 and θ be nonnegative real numbers, and let f: AB be a mapping satisfying f(0) = 0, (4.2) and (4.3). Then there exists a unique Lie C*-algebra homomorphism H: AB satisfying (2.14).

Proof. The proof follows from Theorem 4.2 by taking φ ( x , y ) = θ ( x A r + y A r ) for all x, y A. Then L = 21-rand we get the desired result.

5 Stability of Lie derivations on C*-algebras

Definition 5.1. Lat A be a Lie C*-algebras, A -linear mapping δ: AA is called a Lie derivation if δ([x, y]) = [δ(x),y] + [x, δ(y)] for all x, y A.

Throughout this section, assume that A is a Lie C*-algebra with the norm ||.|| A . In this section, we prove the Hyers-Ulam stability of derivations on Lie C*-algebras for the functional Equation (1.1).

Theorem 5.1. Let f: AB be a mapping with f(0) = 0 for which there exists a function φ: A2 → [0, ∞) satisfying (3.15) such that
f ( [ x , y ] ) - [ f ( x ) , y ] - [ x , f ( y ) ] B φ ( x , y )
(5.26)

for all x, y A. If there exists an L < 1 2 such that φ ( x , y ) L 2 φ ( 2 x , 2 y ) for all x, y A, then there exists a unique Lie derivation δ: AA satisfying (3.17).

Proof. By the same reasoning as in the proof of Theorem 2.1, there exists a unique -linear mapping δ: AA satisfying (3.17). The mapping δ: AA is given by δ ( x ) = lim n 2 n f x 2 n for all x A. It follows from (5.26) that
δ ( [ x , y ] ) - [ δ ( x ) , y ] - [ x , δ ( y ) ] A = lim n 4 n f [ x , y ] 4 n - f x 2 n , y 2 n - x 2 n , f y 2 n A lim n 4 n φ x 2 n , y 2 n lim n ( 2 L ) n φ ( x , y ) = 0

for all x, y A. So δ([x, y]) = [δ(x),y] + [x, δ(y)] for all x, y A. Thus δ: AA is a Lie derivation satisfying (3.17), as desired.

Corollary 5.1. Let 0 < r < 1 and θ be nonnegative real numbers, and let f: AB be a mapping satisfying f(0) = 0 and (3.4) such that
f ( [ x , y ] ) - [ f ( x ) , y ] - [ x , f ( y ) ] B θ ( x A r + y A r )
(5.27)

for all x, y A. Then there exists a unique Lie derivation δ: AA satisfying (3.20).

Proof. The proof follows from Theorem 5.1 by taking φ ( x , y ) = θ ( x A r + y A r ) for all x, y A. Then L = 2r-1and we get the desired result.

Theorem 5.2. Let f: AB be a mapping with f(0) = 0 for which there exists a function φ: A2 → [0, ∞) satisfying (3.15) and (5.26). If there exists an L < 1 such that φ ( x , y ) 2 L φ x 2 , y 2 for all x, y A, then there exists a unique Lie derivation δ: AA satisfying (3.21).

Corollary 5.2. Let r > 1 and θ be nonnegative real numbers, and let f: AB be a mapping satisfying f(0) = 0, (3.4) and (5.27). Then there exists a unique Lie derivation δ: AA satisfying (3.22).

Proof. The proof follows from Theorem 5.2 by taking φ ( x , y ) = θ ( x A r + y A r ) for all x, y A. Then L = 21-rand we get the desired result.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, College of Sciences, Yasouj University, Yasouj, 75914-353, Iran
(2)
Department of Mathematics, Islamic Azad University, Firoozabad Branch, Firoozabad, Iran
(3)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Republic of Korea

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© Azadi Kenary et al; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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