# Stability of the Jensen-Type Functional Equation inC∗-Algebras: A Fixed Point Approach

Authors: Choonkil Park; John Michael Rassias
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms inC∗-algebras and Lie C∗-algebras and also of derivations on
C∗-algebras and Lie C∗-algebras for the Jensen-type functional equation
f((x+y)/2)+f((x−y)/2)=f(x).


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## Body

## 1. Introduction and Preliminaries

The stability problem  of functional equations
originated from a question  of Ulam [1] concerning the stability of group homomorphisms.
Hyers [2] gave a first
affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by
considering an unbounded Cauchy difference. The paper of Rassias
[4] has provided a
lot of influence in the development of what we call generalized Hyers-Ulam stability of
functional equations. A generalization of the Rassias theorem was
obtained by Găvruţa [5] by replacing the unbounded
Cauchy difference by a general control function in the spirit of Rassias'
approach. In 1982, Rassias [6] followed the innovative approach of the Rassias' theorem [4]
in which he replaced the factor ∥x∥p+∥y∥p by ∥x∥p⋅∥y∥q for p,q∈ℝ with p+q≠1. 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 [4, 7–27]).We recall a fundamental result in fixed point theory.LetX be a set. A
function d:X×X→[0,∞] is called a generalized metric on X if d satisfies(1)
d(x,y)=0 if and only if x=y;(2)
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.Theorem 1.1 (see [28, 29]).
Let(X,d) be a complete
generalized metric space and let J:X→X be a strictly
contractive mapping with Lipschitz constant L<1. Then for each given element x∈X, either(1.1)d(Jnx,Jn+1x)=∞for all nonnegative integers n or there exists
a positive integer n0 such that
(1)
d(Jnx,Jn+1x)<∞, for all n≥n0;(2)
the sequence{Jnx} converges to a
fixed point y* of J;(3)
y* is the unique
fixed point of J in the set Y={y∈X∣d(Jn0x,y)<∞};(4)
d(y,y*)≤(1/(1−L))d(y,Jy) for all y∈Y.By the using fixed point method, the stability
problems of several functional equations have been extensively investigated by
a number of authors (see [17, 30–33]).This paper is organized as follows: in Sections2 and
3, using the fixed point method, we prove the generalized Hyers-Ulam stability
of homomorphisms in C*-algebras and
of derivations on C*-algebras for
the Jensen-type functional equation.In Sections4 and 5, using the fixed point method, we
prove the generalized Hyers-Ulam stability of homomorphisms in Lie C*-algebras and
of derivations on Lie C*-algebras for
the Jensen-type functional equation.


## 2. Stability of Homomorphisms inC*-Algebras

Throughout this section, assume thatA is a C*-algebra with
norm ∥⋅∥A and that B is a C*-algebra with
norm ∥⋅∥B.For a given mappingf:A→B, we define(2.1)Dμf(x,y):=μf(x+y2)+μf(x−y2)−f(μx)for all μ∈𝕋1:={ν∈ℂ:|ν|=1} and all x,y∈A.Note that aℂ-linear mapping H:A→B is called a homomorphism in C*-algebras if H satisfies H(xy)=H(x)H(y) and H(x*)=H(x)* for all x,y∈A.We prove the generalized Hyers-Ulam stability of
homomorphisms inC*-algebras for
the functional equation Dμf(x,y)=0.Theorem 2.1.
Letf:A→B be a mapping
for which there exists a function φ:A2→[0,∞) such
that(2.2)∥Dμf(x,y)∥B≤φ(x,y),(2.3)∥f(xy)−f(x)f(y)∥B≤φ(x,y),(2.4)∥f(x*)−f(x)*∥B≤φ(x,x)for all μ∈𝕋1 and all x,y∈A. If there exists an L<1 such that φ(x,y)≤2Lφ(x/2,y/2) for all x,y∈A, then there exists a unique C*-algebra homomorphism H:A→B such
that(2.5)∥f(x)−H(x)∥B≤L1−Lφ(x,0)for all x∈A.Proof.
It
follows fromφ(x,y)≤2Lφ(x/2,y/2) that(2.6)limj→∞2−jφ(2jx,2jy)=0for all x,y∈A.
Consider the set(2.7)X:={g:A→B}and introduce the generalized metric on X:(2.8)d(g,h)=inf{C∈ℝ+:∥g(x)−h(x)∥B≤Cφ(x,0),∀x∈A}.It is easy to show that (X,d) is complete.
Now we consider the linear mappingJ:X→X such that(2.9)Jg(x):=12g(2x)for all x∈A.
By [28, Theorem 3.1],(2.10)d(Jg,Jh)≤Ld(g,h)for all g,h∈X.
Lettingμ=1 and y=0 in (2.2), we get(2.11)∥2f(x2)−f(x)∥B≤φ(x,0)for all x∈A. So(2.12)∥f(x)−12f(2x)∥B≤12φ(2x,0)≤Lφ(x,0)for all x∈A. Hence d(f,Jf)≤L.
By Theorem1.1, there exists a mapping H:A→B such that
(1)
H is a fixed
point of J, that is,(2.13)H(2x)=2H(x)for all x∈A. The mapping H is a unique
fixed point of J in the set(2.14)Y={g∈X:d(f,g)<∞}.This implies that H is a unique
mapping satisfying (2.13) such that there
exists C∈(0,∞) satisfying(2.15)∥H(x)−f(x)∥B≤Cφ(x,0)for all x∈A.(2)
d(Jnf,H)→0 as n→∞. This implies the equality(2.16)limn→∞f(2nx)2n=H(x)for all x∈A.(3)
d(f,H)≤(1/(1−L))d(f,Jf), which implies the inequality(2.17)d(f,H)≤L1−L.This implies that the
inequality (2.5) holds.
It follows from(2.2),(2.6), and (2.16) that(2.18)∥H(x+y2)+H(x−y2)−H(x)∥B=limn→∞12n∥f(2n−1(x+y))+f(2n−1(x−y))−f(2nx)∥B≤limn→∞12nφ(2nx,2ny)=0for all x,y∈A. So(2.19)H(x+y2)+H(x−y2)=H(x)for all x,y∈A. Letting z=(x+y)/2 and w=(x−y)/2 in (2.19), we get(2.20)H(z)+H(w)=H(z+w)for all z,w∈A. So the mapping H:A→B is Cauchy
additive, that is, H(z+w)=H(z)+H(w) for all z,w∈A.
Lettingy=x in (2.2), we get(2.21)μf(x)=f(μx)for all μ∈𝕋1 and all x∈A. By a similar method to above, we get(2.22)μH(x)=H(μx)for all μ∈𝕋1 and all x∈A. Thus one can show that the mapping H:A→B is ℂ-linear.
It follows from
(2.3) that(2.23)∥H(xy)−H(x)H(y)∥B=limn→∞14n∥f(4nxy)−f(2nx)f(2ny)∥B≤limn→∞14nφ(2nx,2ny)≤limn→∞12nφ(2nx,2ny)=0for all x,y∈A. So(2.24)H(xy)=H(x)H(y)for all x,y∈A.
It follows from
(2.4) that(2.25)∥H(x*)−H(x)*∥B=limn→∞12n∥f(2nx*)−f(2nx)*∥B≤limn→∞12nφ(2nx,2nx)=0for all x∈A. So(2.26)H(x*)=H(x)*for all x∈A.
ThusH:A→B is a C*-algebra
homomorphism satisfying (2.5), as desired.Corollary 2.2.
Let0<r<1 and θ be nonnegative
real numbers, and let f:A→B be a mapping
such that(2.27)∥Dμf(x,y)∥B≤θ(∥x∥Ar+∥y∥Ar),(2.28)∥f(xy)−f(x)f(y)∥B≤θ(∥x∥Ar+∥y∥Ar),(2.29)∥f(x*)−f(x)*∥B≤2θ∥x∥Arfor all μ∈𝕋1 and all x,y∈A. Then there exists a unique C*-algebra
homomorphism H:A→B such
that(2.30)∥f(x)−H(x)∥B≤2rθ2−2r∥x∥Arfor all x∈A.Proof.
The proof follows from Theorem2.1
by taking(2.31)φ(x,y):=θ(∥x∥Ar+∥y∥Ar)for all x,y∈A. Then L=2r−1 and we get the
desired result.Theorem 2.3.
Letf:A→B be a mapping
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)≤(1/2)Lφ(2x,2y) for all x,y∈A, then there exists a unique C*-algebra
homomorphism H:A→B such
that(2.32)∥f(x)−H(x)∥B≤L2−2Lφ(x,0)for all x∈A.Proof.
We
consider the linear mappingJ:X→X such that(2.33)Jg(x):=2g(x2)for all x∈A.
It follows from
(2.11) that(2.34)∥f(x)−2f(x2)∥B≤φ(x2,0)≤L2φ(x,0)for all x∈A. Hence d(f,Jf)≤L/2.
By Theorem1.1, there exists a mapping H:A→B such that
(1)
H is a fixed
point of J, that is,(2.35)H(2x)=2H(x)for all x∈A. The mapping H is a unique
fixed point of J in the set(2.36)Y={g∈X:d(f,g)<∞}.This implies that H is a unique
mapping satisfying (2.35) such that
there exists C∈(0,∞) satisfying(2.37)∥H(x)−f(x)∥B≤Cφ(x,0)for all x∈A.(2)
d(Jnf,H)→0 as n→∞. This implies the equality(2.38)limn→∞2nf(x2n)=H(x)for all x∈A.(3)
d(f,H)≤(1/(1−L))d(f,Jf), which implies the inequality(2.39)d(f,H)≤L2−2L,which implies that the inequality (2.32) holds.
The rest of the proof is similar to the proof of
Theorem2.1.Corollary 2.4.
Letr>2 and θ be nonnegative
real numbers, and let f:A→B be a mapping
satisfying (2.27), (2.28) and
(2.29). Then there exists a unique C*-algebra
homomorphism H:A→B such
that(2.40)∥f(x)−H(x)∥B≤θ2r−2∥x∥Arfor all x∈A.Proof.
The
proof follows from Theorem2.3 by taking(2.41)φ(x,y):=θ(∥x∥Ar+∥y∥Ar)for all x,y∈A. Then L=21−r and we get the
desired result.Theorem 2.5.
Letf:A→B be an odd
mapping for which there exists a function φ:A2→[0,∞) satisfying (2.2),
(2.3), (2.4) and (2.6). If there exists an L<1 such that φ(x,3x)≤2Lφ(x/2,3x/2) for all x∈A, then there exists a unique C*-algebra
homomorphism H:A→B such
that(2.42)∥f(x)−H(x)∥B≤12−2Lφ(x,3x)for all x∈A.Proof.
Consider
the set(2.43)X:={g:A→B}and introduce the generalized metric on X:(2.44)d(g,h)=inf{C∈ℝ+:∥g(x)−h(x)∥B≤Cφ(x,3x),∀x∈A}.It is easy to show that (X,d) is complete.
Now we consider the linear mappingJ:X→X such that(2.45)Jg(x):=12g(2x)for all x∈A.
By [28, Theorem 3.1],(2.46)d(Jg,Jh)≤Ld(g,h)for all g,h∈X.
Lettingμ=1 and relpacing y by 3x in (2.2), we get(2.47)∥f(2x)−2f(x)∥B≤φ(x,3x)for all x∈A. So(2.48)∥f(x)−12f(2x)∥B≤12φ(x,3x)for all x∈A. Hence d(f,Jf)≤1/2.
By Theorem1.1, there exists a mapping H:A→B such that
(1)
H is a fixed
point of J, that is,(2.49)H(2x)=2H(x)for all x∈A. The mapping H is a unique
fixed point of J in the set(2.50)Y={g∈X:d(f,g)<∞}.This implies that H is a unique
mapping satisfying (2.49) such that
there exists C∈(0,∞) satisfying(2.51)∥H(x)−f(x)∥B≤Cφ(x,3x)for all x∈A.(2)
d(Jnf,H)→0 as n→∞. This implies the equality(2.52)limn→∞f(2nx)2n=H(x)for all x∈A.(3)
d(f,H)≤(1/(1−L))d(f,Jf), which implies the inequality(2.53)d(f,H)≤12−2L.This implies that the inequality(2.42) holds.
The rest of the proof is similar to the proof of
Theorem2.1.Corollary 2.6.
Let0<r<1/2 and θ be nonnegative
real numbers, and let f:A→B be an odd
mapping such that(2.54)∥Dμf(x,y)∥B≤θ⋅∥x∥Ar⋅∥y∥Ar,∥f(xy)−f(x)f(y)∥B≤θ⋅∥x∥Ar⋅∥y∥Ar,∥f(x*)−f(x)*∥B≤θ∥x∥A2rfor all μ∈𝕋1 and all x,y∈A. Then there exists a unique C*-algebra
homomorphism H:A→B such
that(2.55)∥f(x)−H(x)∥B≤3rθ2−22r∥x∥A2rfor all x∈A.Proof.
The
proof follows from Theorem2.5 by taking(2.56)φ(x,y):=θ⋅∥x∥Ar⋅∥y∥Arfor all x,y∈A. Then L=22r−1 and we get the
desired result.Theorem 2.7.
Letf:A→B be an odd
mapping 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,3x)≤(1/2)Lφ(2x,6x) for all x∈A, then there exists a unique C*-algebra
homomorphism H:A→B such
that(2.57)∥f(x)−H(x)∥B≤L2−2Lφ(x,3x)for all x∈A.Proof.
We
consider the linear mappingJ:X→X such that(2.58)Jg(x):=2g(x2)for all x∈A.
It follows from
(2.47) that(2.59)∥f(x)−2f(x2)∥B≤φ(x2,3x2)≤L2φ(x,3x)for all x∈A. Hence d(f,Jf)≤L/2.
By Theorem1.1, there exists a mapping H:A→B such that
(1)
H is a fixed
point of J, that is,(2.60)H(2x)=2H(x)for all x∈A. The mapping H is a unique
fixed point of J in the set(2.61)Y={g∈X:d(f,g)<∞}.This implies that H is a unique
mapping satisfying (2.60) such that
there exists C∈(0,∞) satisfying(2.62)∥H(x)−f(x)∥B≤Cφ(x,3x)for all x∈A.(2)
d(Jnf,H)→0 as n→∞. This implies the equality(2.63)limn→∞2nf(x2n)=H(x)for all x∈A.(3)
d(f,H)≤(1/(1−L))d(f,Jf), which implies the inequality(2.64)d(f,H)≤L2−2L,which implies that the
inequality (2.57) holds.
The rest of the proof is similar to the proof of
Theorem2.1.Corollary 2.8.
Letr>1 and θ be nonnegative
real numbers, and let f:A→B be an odd
mapping satisfying (2.54). Then there exists a unique C*-algebra
homomorphism H:A→B such
that(2.65)∥f(x)−H(x)∥B≤θ22r−2∥x∥A2rfor all x∈A.Proof.
The
proof follows from Theorem2.7 by taking(2.66)φ(x,y):=θ⋅∥x∥Ar⋅∥y∥Arfor all x,y∈A. Then L=21−2r and we get the
desired result.


## 3. Stability of Derivations onC*-Algebras

Throughout this section, assume thatA is a C*-algebra with
norm ∥⋅∥A.Note that aℂ-linear mapping δ:A→A is called a derivation on A if δ satisfies δ(xy)=δ(x)y+xδ(y) for all x,y∈A.We prove the generalized Hyers-Ulam stability of
derivations onC*-algebras for
the functional equation Dμf(x,y)=0.Theorem 3.1.
Letf:A→A be a mapping
for which there exists a function φ:A2→[0,∞) such
that(3.1)∥Dμf(x,y)∥A≤φ(x,y),(3.2)∥f(xy)−f(x)y−xf(y)∥A≤φ(x,y)for all μ∈𝕋1 and all x,y∈A. If there exists an L<1 such that φ(x,y)≤2Lφ(x/2,y/2) for all x,y∈A. Then there exists a unique derivation δ:A→A such
that(3.3)∥f(x)−δ(x)∥A≤L1−Lφ(x,0)for all x∈A.Proof.
By the
same reasoning as the proof of Theorem2.1, there exists a unique involutive ℂ-linear mapping δ:A→A satisfying (3.3). The mapping δ:A→A is given
by(3.4)δ(x)=limn→∞f(2nx)2nfor all x∈A.
It follows from
(3.2) that(3.5)∥δ(xy)−δ(x)y−xδ(y)∥A=limn→∞14n∥f(4nxy)−f(2nx)⋅2ny−2nxf(2ny)∥A≤limn→∞14nφ(2nx,2ny)≤limn→∞12nφ(2nx,2ny)=0for all x,y∈A. So(3.6)δ(xy)=δ(x)y+xδ(y)for all x,y∈A. Thus δ:A→A is a derivation
satisfying (3.3).Corollary 3.2.
Let0<r<1 and θ be nonnegative
real numbers, and let f:A→A be a mapping
such that(3.7)∥Dμf(x,y)∥A≤θ(∥x∥Ar+∥y∥Ar),(3.8)∥f(xy)−f(x)y−xf(y)∥A≤θ(∥x∥Ar+∥y∥Ar)for all μ∈𝕋1 and all x,y∈A. Then there exists a unique derivation δ:A→A such
that(3.9)∥f(x)−δ(x)∥A≤2rθ2−2r∥x∥Arfor all x∈A.Proof.
The
proof follows from Theorem3.1 by taking(3.10)φ(x,y):=θ(∥x∥Ar+∥y∥Ar)for all x,y∈A. Then L=2r−1 and we get the
desired result.Theorem 3.3.
Letf:A→A be a mapping
for which there exists a function φ:A2→[0,∞) satisfying (3.1) and
(3.2). If there exists an L<1 such that φ(x,y)≤(1/2)Lφ(2x,2x) for all x,y∈A, then there exists a unique derivation δ:A→A such
that(3.11)∥f(x)−δ(x)∥A≤L2−2Lφ(x,0)for all x∈A.Proof.
The
proof is similar to the proofs of Theorems2.3 and 3.1.Corollary 3.4.
Letr>2 and θ be nonnegative real
numbers, and let f:A→A be a mapping
satisfying (3.7) and (3.8). Then there exists a unique derivation δ:A→A such
that(3.12)∥f(x)−δ(x)∥A≤θ2r−2∥x∥Arfor all x∈A.Proof.
The
proof follows from Theorem3.3 by taking(3.13)φ(x,y):=θ(∥x∥Ar+∥y∥Ar)for all x,y∈A. Then L=21−r and we get the desired result.Remark 3.5.
For inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems2.5 and
2.7 and Corollaries 2.6 and 2.8.


## 4. Stability of Homomorphisms in LieC*-Algebras

AC*-algebra 𝒞, endowed with the Lie product [x,y]:=(xy−yx)/2 on 𝒞, is called a LieC*-algebra (see [13–15]).Definition 4.1.
LetA  and  B  be Lie C*-algebras. A ℂ-linear mapping   H:A→B  is called
a Lie  C*-algebra
homomorphism if  H([x,y])=[H(x),H(y)]  for all   x,y∈A.Throughout this section, assume thatA is a Lie C*-algebra with
norm ∥⋅∥A and that B is a Lie C*-algebra with
norm ∥⋅∥B.We prove the generalized Hyers-Ulam stability of
homomorphisms in LieC*-algebras for
the functional equation Dμf(x,y)=0.Theorem 4.2.
Letf:A→B be a mapping
for which there exists a function φ:A2→[0,∞) satisfying (2.2) such that(4.1)∥f([x,y])−[f(x),f(y)]∥B≤φ(x,y)for all x,y∈A. If there exists an L<1 such that φ(x,y)≤2Lφ(x/2,y/2) for all x,y∈A, then there exists a unique Lie C*-algebra
homomorphism H:A→B satisfying (2.5).Proof.
By the same reasoning as the proof
of Theorem2.1, there exists a unique ℂ-linear mapping δ:A→A satisfying (2.5). The mapping H:A→B is given
by(4.2)H(x)=limn→∞f(2nx)2nfor all x∈A.
It follows from
(4.1) that(4.3)∥H([x,y])−[H(x),H(y)]∥B=limn→∞14n∥f(4n[x,y])−[f(2nx),f(2ny)]∥B≤limn→∞14nφ(2nx,2ny)≤limn→∞12nφ(2nx,2ny)=0for all x,y∈A. So(4.4)H([x,y])=[H(x),H(y)]for all x,y∈A.
ThusH:A→B is a Lie C*-algebra
homomorphism satisfying (2.5), as
desired.Corollary 4.3.
Let0<r<1 and θ be nonnegative
real numbers, and let f:A→B be a mapping
satisfying (2.27) such
that(4.5)∥f([x,y])−[f(x),f(y)]∥B≤θ(∥x∥Ar+∥y∥Ar)for all x,y∈A. Then there exists a unique Lie C*-algebra
homomorphism H:A→B satisfying (2.30).Proof.
The
proof follows from Theorem4.2 by taking(4.6)φ(x,y):=θ(∥x∥Ar+∥y∥Ar)for all x,y∈A. Then L=2r−1 and we get the
desired result.Theorem 4.4.
Letf:A→B be a mapping
for which there exists a function φ:A2→[0,∞) satisfying (2.2) and
(4.1). If there exists an L<1 such that φ(x,y)≤(1/2)Lφ(2x,2y) for all x,y∈A, then there exists a unique Lie C*-algebra
homomorphism H:A→B satisfying (2.32).Proof.
The proof is similar to the proofs
of Theorems2.3 and 4.2.Corollary 4.5.
Letr>2 and θ be nonnegative
real numbers, and let f:A→B be a mapping
satisfying (2.27) and (4.5). Then there exists a unique Lie C*-algebra
homomorphism H:A→B satisfying (2.40).Proof.
The
proof follows from Theorem4.4 by taking(4.7)φ(x,y):=θ(∥x∥Ar+∥y∥Ar)for all x,y∈A. Then L=21−r and we get the
desired result.Remark 4.6.
For inequalities controlled by the
product of powers of norms, one can obtain similar results to Theorems2.5 and 2.7 and Corollaries 2.6 and 2.8.


## 5. Stability of Lie Derivations onC*-Algebras

Definition 5.1.
LetA  be a Lie C*-algebra. A ℂ-linear mapping δ:A→A is called
a Lie derivation if δ([x,y])=[δ(x),y]+[x,δ(y)] for all x,y∈A.Throughout this section, assume thatA is a Lie C*-algebra with
norm ∥⋅∥A.We prove the generalized Hyers-Ulam stability of
derivations on LieC*-algebras for
the functional equation Dμf(x,y)=0.Theorem 5.2.
Letf:A→A be a mapping
for which there exists a function φ:A2→[0,∞) satisfying (3.1) such that(5.1)∥f([x,y])−[f(x),y]−[x,f(y)]∥A≤φ(x,y)for all x,y∈A. If there exists an L<1 such that φ(x,y)≤2Lφ(x/2,y/2) for all x,y∈A. Then there exists a unique Lie derivation δ:A→A satisfying (3.3).Proof.
By the same reasoning as the proof
of Theorem2.1, there exists a unique involutive ℂ-linear mapping δ:A→A satisfying (3.3). The mapping δ:A→A is given
by(5.2)δ(x)=limn→∞f(2nx)2nfor all x∈A.
It follows from
(5.1) that(5.3)∥δ([x,y])−[δ(x),y]−[x,δ(y)]∥A=limn→∞14n∥f(4n[x,y])−[f(2nx),2ny]−[2nx,f(2ny)]∥A≤limn→∞14nφ(2nx,2ny)≤limn→∞12nφ(2nx,2ny)=0for all x,y∈A. So(5.4)δ([x,y])=[δ(x),y]+[x,δ(y)]for all x,y∈A. Thus δ:A→A is a derivation
satisfying (3.3).Corollary 5.3.
Let0<r<1 and θ be nonnegative
real numbers, and let f:A→A be a mapping
satisfying (3.7) such
that(5.5)∥f([x,y])−[f(x),y]−[x,f(y)]∥A≤θ(∥x∥Ar+∥y∥Ar)for all x,y∈A. Then there exists a unique Lie derivation δ:A→A satisfying (3.9).Proof.
The
proof follows from Theorem5.2 by taking(5.6)φ(x,y):=θ(∥x∥Ar+∥y∥Ar)for all x,y∈A. Then L=2r−1 and we get the
desired result.Theorem 5.4.
Letf:A→A be a mapping
for which there exists a function φ:A2→[0,∞) satisfying (3.1) and
(5.1). If there exists an L<1 such that φ(x,y)≤(1/2)Lφ(2x,2y) for all x,y∈A, then there exists a unique Lie derivation δ:A→A satisfying (3.11).Proof.
The
proof is similar to the proofs of Theorems2.3 and 5.2.Corollary 5.5.
Letr>2 and θ be nonnegative
real numbers, and let f:A→A be a mapping
satisfying (3.7) and (5.5). Then there exists a unique Lie
derivation δ:A→A satisfying (3.12).Proof.
The
proof follows from Theorem5.4 by taking(5.7)φ(x,y):=θ(∥x∥Ar+∥y∥Ar)for all x,y∈A. Then L=21−r and we get the
desired result.Remark 5.6.
For inequalities controlled by the
product of powers of norms, one can obtain similar results to Theorems2.5 and 2.7 and Corollaries 2.6 and 2.8.



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*Source: 360432-2009-03-10.xml*