Hyers-Ulam-Rassias Stability of n-Apollonius Type Additive - Hikari

18 Pages·2011·0.15 MB·English

Int. Journal of Math. Analysis, Vol. 6, 2012, no. 3, 111 - 128
Hyers-Ulam-Rassias Stability of n-Apollonius Type
∗
Additive Mapping and Isomorphisms in C -Algebras
Fridoun Moradlou
Department of Mathematics
Sahand University of Technology, Tabriz, Iran
[email protected]
Choonkil Park
Department of Mathematics, Hanyang University
Seoul, 133–791, Republic of Korea
[email protected]
Jung Rye Lee
Department of Mathematics, Daejin University
Kyeonggi 487-711, Republic of Korea
[email protected]
Abstract. In this paper, we prove Hyers-Ulam-Rassias stability of the
following functional equation in Banach modules over a unital C∗-algebra:
(cid:3) (cid:4)
(cid:2)n (cid:2) (cid:2)n
1 1
f(z −x )+ f(x +x ) = nf z − x ,
i n i j n2 i
i=1 1≤i<j≤n i=1
which n is fixed integer and n ≥ 2. As an application, we show that every
almost linear bijection h : A → B of a unital C∗-algebra A onto a unital C∗-
(cid:5)(cid:5) (cid:6) (cid:6) (cid:5)(cid:5) (cid:6) (cid:6)
algebraB is a C∗-algebra isomorphism whenh n2−1 duy = h n2−1 du h(y)
n2 n2
for all unitaries u ∈ A, all y ∈ A, and all d ∈ Z.
112 F. Moradlou, C. Park and J. Lee
Mathematics Subject Classification: Primary 39B72, 47H10, 46L05,
46B03, 47Jxx
Keywords: Hyers-Ulam-Rassias stability, n-Apollonius type additive map-
ping, C∗-algebra homomorphism, generalized derivation
1. Introduction and preliminaries
The stability problem of functional equations originated from a question of
Ulam [36] concerning the stability of group homomorphisms: Let (G ,∗) be a
1
group and let (G ,(cid:6),d) be a metric group with the metric d(·,·). Given (cid:2) > 0,
2
does there exist a δ((cid:2)) > 0 such that if a mapping h : G → G satisfies the
1 2
inequality
d(h(x∗y),h(x)(cid:6)h(y)) < δ
for all x,y ∈ G , then there is a homomorphism H : G → G with
1 1 2
d(h(x),H(x)) < (cid:2)
for all x ∈ G ? If the answer is affirmative, we would say that the equation
1
of homomorphism H(x∗y) = H(x)(cid:6)H(y) is stable. 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. Thus the stability
question of functional equations is that how do the solutions of the inequality
differ from those of the given functional equation?
Hyers [9] gave a first affirmative answer to the question of Ulam for Banach
spaces. Let X and Y be Banach spaces. Assume that f : X → Y satisfies
(cid:7)f(x+y)−f(x)−f(y)(cid:7) ≤ ε
for all x,y ∈ X and some ε ≥ 0. Then there exists a unique additive mapping
T : X → Y such that
(cid:7)f(x)−T(x)(cid:7) ≤ ε
for all x ∈ X.
Th.M.Rassias[30]providedageneralizationofHyers’Theoremwhichallows
the Cauchy difference to be unbounded.
Theorem 1.1. (Th.M.Rassias). Let f : E → E(cid:4) be a mapping from a normed
vector space E into a Banach space E(cid:4) subject to the inequality
(1.1) (cid:7)f(x+y)−f(x)−f(y)(cid:7) ≤ (cid:2)((cid:7)x(cid:7)p +(cid:7)y(cid:7)p)
Stability of the n-Apollonius type additive mapping 113
for all x,y ∈ E, where (cid:2) and p are constants with (cid:2) > 0 and p < 1. Then the
limit
f(2nx)
L(x) = lim
n→∞ 2n
exists for all x ∈ E and L : E → E(cid:4) is the unique additive mapping which
satisfies
2(cid:2)
(cid:7)f(x)−L(x)(cid:7) ≤ (cid:7)x(cid:7)p
2−2p
for all x ∈ E. Also, if for each x ∈ E the mapping f(tx) is continuous in
t ∈ R, then L is R-linear.
Theabove inequality (1.1)hasprovided a lotof influence inthedevelopment
ofwhatisnowknownasaHyers-Ulam-Rassias stabilityoffunctionalequations.
Beginning around the year 1980 the topic of approximate homomorphisms, or
the stability of the equation of homomorphism, was studied by a number of
mathematicians. G˘avruta [8] generalized the Rassias’ result. 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 [2], [5], [7], [13]–[24], [31]–[33]).
J.M. Rassias [28] following the spirit of the innovative approach of Th.M.
Rassias [30] for the unbounded Cauchy difference proved a similar stability
theorem in which he replaced the factor (cid:7)x(cid:7)p+(cid:7)y(cid:7)p by (cid:7)x(cid:7)p·(cid:7)y(cid:7)q for p,q ∈ R
with p+q (cid:9)= 1 (see also [29] for a number of other new results).
Theorem 1.2. [27, 28, 29] Let X be a real normed linear space and Y a real
complete normed linear space. Assume that f : X → Y is an approximately
additive mapping for which there exist constants θ ≥ 0 and p ∈ R −{1} such
that f satisfies inequality
(cid:7)f(x+y)−f(x)−f(y)(cid:7) ≤ θ ·||x||p2 ·||y||2p
for all x,y ∈ X. Then there exists a unique additive mapping L : X → Y
satisfying
θ
(cid:7)f(x)−L(x)(cid:7) ≤ ||x||p
|2p −2|
for all x ∈ X. If, in addition, f : X → Y is a mapping such that the
transformation t → f(tx) is continuous in t ∈ R for each fixed x ∈ X, then L
is an R-linear mapping.
114 F. Moradlou, C. Park and J. Lee
The following functional equation
(1.2) Q(x+y)+Q(x−y) = 2Q(x)+2Q(y),
is called a quadratic functional equation, and every solution of equation (1.2)
is said to be a quadratic mapping. F. Skof [35] proved the Hyers-Ulam-Rassias
stability of the quadratic functional equation (1.2) for mappings f : E → E ,
1 2
where E is a normed space and E is a Banach space. In [6], S. Czerwik
1 2
proved the Hyers-Ulam-Rassias stability of the quadratic functional equation.
C. Borelli and G.L. Forti [4] generalized the stability result as follows: let G
be an abelian group, E a Banach space. Assume that a mapping f : G → E
satisfies the functional inequality
(cid:7)f(x+y)+f(x−y)−2f(x)−2f(y)(cid:7) ≤ ϕ(x,y)
for all x,y ∈ G, and ϕ : G×G → [0,∞) is a function such that
(cid:2)∞
1
φ(x,y) := ϕ(2ix,2iy) < ∞
4i+1
i=0
for all x,y ∈ G. Then there exists a unique quadratic mapping Q : G → E
with the properties
(cid:7)f(x)−Q(x)(cid:7) ≤ φ(x,x)
for all x ∈ G. Jun and Lee [11] proved the Hyers-Ulam-Rassias stability of the
Pexiderized quadratic equation
f(x+y)+g(x−y) = 2h(x)+2k(y)
for mappings f,g,h and k.
In an inner product space, the equality
(cid:7) (cid:7)
1 (cid:7) x+y(cid:7)2
(1.3) (cid:7)z −x(cid:7)2 +(cid:7)z −y(cid:7)2 = (cid:7)x−y(cid:7)2 +2(cid:7)z − (cid:7)
2 2
holds, andiscalledtheApollonius’ identity. Thefollowingfunctionalequation,
which was motivated by this equation,
(cid:8) (cid:9)
1 x+y
(1.4) Q(z −x)+Q(z −y) = Q(x−y)+2Q z − ,
2 2
is quadratic (see [25]). For this reason, the functional equation (1.4) is called
a quadratic functional equation of Apollonius type, and each solution of the
functional equation (1.4) is said to be a quadratic mapping of Apollonius type.
The quadratic functional equation and several other functional equations are
useful to characterize inner product spaces [1].
Stability of the n-Apollonius type additive mapping 115
In [25], C. Park and Th.M. Rassias introduced and investigated a functional
equation, which is called the generalized Apollonius type quadratic functional
equation. Recently in [26], C. Park and Th.M. Rassias introduced and inves-
tigated the following functional equation
(cid:8) (cid:9)
1 x+y
(1.5) f(z −x)+f(z −y) = − f(x+y)+2f z −
2 4
whichiscalledApollonius type additive functional equation, andwhosesolution
is called a Apollonius type additive mapping. In [23], C. Park introduced and
investigated a functional equation, which is called the generalized Apollonius-
Jensen type additive functional equation and whose solution of the functional
equation is said to be a generalized Apollonius-Jensen type additive mapping.
In this paper, for a fixed integer n ≥ 2, we introduce the new functional
equation, which is called the additive functional equation of n-Apollonius type
and whose solution is called an additive mapping of n-Apollonius type,
(cid:3) (cid:4)
(cid:2)n (cid:2) (cid:2)n
1 1
(1.6) f(z −x )+ f(x +x ) = nf z − x .
i n i j n2 i
i=1 1≤i<j≤n i=1
This paper is organized as follows: In Section 2, we investigate the Hyers-
Ulam-Rassias stability of additive functional equation of n-Apollonius type in
Banach modules over C∗-algebras.
In Section 3, we investigate C∗-algebra isomorphisims in unital C∗-algebras
associated with additive functional equation of n-Apollonius type.
2. Hyers-Ulam-Rassias Stability of n-Apollonius type additive
mapping in Banach modules over a C∗-algebra
Throughout this section, assume that A is a unital C∗-algebra with norm
| · | and unitary group U(A), and that X and Y are Banach modules over a
unital C∗-algebra A with norms (cid:7)·(cid:7) and (cid:7)·(cid:7) , respectively.
X Y
Lemma 2.1. A function f : X → Y satisfies (1.6) for all z,x ,··· ,x if and
1 n
only if the function f is additive.
Proof. Letting x = ··· = x = z = 0 in (1.6), we get that f(0) = 0. Let j and
1 n
k be fixed integers with 1 ≤ j < k ≤ n. Setting x = 0 for all 1 ≤ i ≤ n,i (cid:9)= j,k
i
116 F. Moradlou, C. Park and J. Lee
in (1.6), we have
(2.1)
1 n−2(cid:5) (cid:6)
f(z −x )+f(z −x )+(n−2)f(z) =− f(x +x )− f(x )+f(x )
j k j k j k
n n
(cid:5) (cid:6)
1
+nf z − (x +x )
n2 j k
for all x ,x ,z ∈ X. Replacing x by −x and x by x in (2.1), respectively,
j k j j k j
we get
n−2(cid:5) (cid:6)
(2.2) f(z +x )+f(z −x ) = − f(−x )+f(x ) +2f(z)
j j j j
n
for all x ,z ∈ X. Putting z = 0 in (2.2), we conclude that f(−x ) = −f(x )
j j j
for all x ∈ X. This means that f is an odd function. Letting x = z = 0 in
j k
(2.1) and using the oddness of f, we obtain that
1 1
2 2
(2.3) f( x ) = f(x ), f(n x ) = n f(x )
n2 j n2 j j j
for all x ∈ X. Letting z = 0 in (2.1), using the oddness of f and (2.3) we have
j
f(x +x ) = f(x )+f(x )
j k j j
for all x ,x ∈ X. Therefore, f : X → Y is a additive mapping.
j k
The converse is obviously true.
For a given mapping f : X → Y and a given u ∈ U(A) and a fixed integer
n ≥ 2, we define
(cid:2)n (cid:2)
1
D f(z,x ,··· ,x ) := f(uz −ux )+ f(ux +μux )
u 1 n i i j
n
i=1 1≤i<j≤n
(cid:3) (cid:4)
(cid:2)n
1
−nuf z − x
n2 i
i=1
for all z,x ,··· ,x ∈ X.
1 n
We investigate the Hyers-Ulam-Rassias stability of a type additive mapping
of n-Apollonius type in Banach modules over a C∗-algebra.
Stability of the n-Apollonius type additive mapping 117
Theorem 2.2. Let f : X → Y be a mapping satisfying f(0) = 0 for which
there exists a function ϕ : An+1 → [0,∞) such that
(2.4)
(cid:3)
(cid:2)∞ (cid:8) n2 (cid:9)j (cid:8)n2 −1(cid:9)j (cid:8)n2 −1(cid:9)j
ϕ(cid:10)(z,x) = ϕ z,0,··· ,0, x,··· ,0) < ∞,
i n2 −1 n2 (cid:11) n(cid:12)2(cid:13) (cid:14)
j=0
i th
(2.5)
(cid:3) (cid:4)
(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)
n2 j n2 −1 j n2 −1 j n2 −1 j
lim ϕ z, x ,··· , x = 0,
j→∞ n2 −1 n2 n2 1 n2 n
(2.6)
(cid:7)D f(z,x ,··· ,x )(cid:7) ≤ ϕ(z,x ,··· ,x )
μ 1 n Y 1 n
for all u ∈ U(A) and all x,z,x ,··· ,x ∈ X. Then there exists a unique
1 n
A-Linear additive mapping L : X → Y of n-Apollonius type such that
n
(2.7) (cid:7)f(x)−L(x)(cid:7) ≤ ϕ(cid:10)(x,x)
Y n2 −1 i
for all x ∈ X.
Proof. Let u = 1 ∈ U(A). Letting z = x = x and for each 1 ≤ j ≤ n with
i
j (cid:9)= i, x = 0 in (2.6), we get
j
(cid:7) (cid:7)
(cid:7)n2 −1 (cid:5)n2 −1 (cid:6)(cid:7)
(2.8) (cid:7) f(x)−nf x (cid:7) ≤ ϕ(x,0,··· ,0,(cid:11)(cid:12)x(cid:13)(cid:14),0,··· ,0)
n n2 Y
ith
for all x ∈ X. For convenience, set
ϕi(z,x) = ϕ(z,0,··· ,0,(cid:11)(cid:12)x(cid:13)(cid:14),0,··· ,0)
ith
n2 −1
for all x,z ∈ X and all 1 ≤ i ≤ n. Letting α = , we get
n
(cid:7) (cid:7)
(cid:5) (cid:6)
(cid:7) α (cid:7)
(2.9) (cid:7)αf(x)−nf x (cid:7) ≤ ϕ (x,x)
i
n Y
for all x ∈ X. It follows from (2.9) that
(cid:7) (cid:7)
(cid:5) (cid:6)
(cid:7) n α (cid:7) 1
(2.10) (cid:7)f(x)− f x (cid:7) ≤ ϕ (x,x)
i
α n Y α
118 F. Moradlou, C. Park and J. Lee
(cid:8) (cid:9)
α k
for all x ∈ X. If we replace x in (2.10) by x and multiply both sides of
(cid:8) (cid:9) n
n k
(2.10) by , then we have
α
(cid:7)(cid:8) (cid:9) (cid:3)(cid:8) (cid:9) (cid:4) (cid:8) (cid:9) (cid:3)(cid:8) (cid:9) (cid:4)(cid:7)
(cid:7) n k α k n k+1 α k+1 (cid:7)
(cid:7) f x − f x (cid:7)
α n α n Y
(2.11) (cid:8) (cid:9) (cid:3)(cid:8) (cid:9) (cid:8) (cid:9) (cid:4)
1 n k α k α k
≤ ϕ x, x
i
α α n n
for all x ∈ X and all non-negative integers k. So we obtain that for all non-
negative integers m,l with l > m
(cid:3) (cid:4) (cid:3) (cid:4)
(cid:7)(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:7)
(cid:7) n m α m n l+1 α l+1 (cid:7)
(cid:7) f x − f x (cid:7)
α n α n Y
(cid:3) (cid:4) (cid:3) (cid:4)
(cid:2)l (cid:7)(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:7)
(cid:7) n i α i n i+1 α i+1 (cid:7)
≤ (cid:7) f x − f x (cid:7)
α n α n Y
i=m
(cid:3) (cid:4)
(cid:2)l (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)
1 n i α i α i
(2.12) ≤ ϕ x, x
i
α α n n
i=m
(cid:15)
(cid:8) (cid:9)
n d
for all x ∈ X. So it follows from (2.4) and (2.12) that the sequence
(cid:16) α
(cid:8) (cid:9)
(cid:5) (cid:6)
α d
f x is Cauchy for all x ∈ X, and thus converges by the completeness
n
of Y. Thus we can define a mapping L : X → Y by
(cid:8) (cid:9) (cid:8) (cid:9)
(cid:5) (cid:6)
n d α d
L(x) = lim f x
d→∞ α n
for all x ∈ X. Letting m = 0 in (2.12), we obtain
(cid:7) (cid:8) (cid:9) (cid:3)(cid:8) (cid:9) (cid:4)(cid:7)
(cid:7) n l+1 α l+1 (cid:7)
(cid:7)f(x)− f x (cid:7)
α n Y
(2.13) (cid:2)l (cid:8) (cid:9) (cid:3)(cid:8) (cid:9) (cid:8) (cid:9) (cid:4)
1 n j α j α j
≤ ϕ x, x
i
α α n n
j=0
for all x ∈ X and all l ∈ N. Taking the limit as l → ∞ in (2.13), we obtain
the inequality (2.7).
It follows from (2.4) and (2.6) that
(cid:7) (cid:3) (cid:4)(cid:7)
(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)
(cid:7) (cid:7)
(cid:7)D1L(z,x1,··· ,xn)(cid:7)Y = lim n d(cid:7)(cid:7)D1f α dz, α dx1,··· , α dxn (cid:7)(cid:7)
d→∞ α n n n
(cid:3) (cid:4)
(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)
n d α d α d α d
≤ lim ϕ z, x ,··· , x = 0.
1 n
d→∞ α n n n
Stability of the n-Apollonius type additive mapping 119
Therefore, the mapping L : X → Y satisfies the equation (1.6) and hence L is
a generalized additive mapping of n-Apollonius type.
To prove the uniqueness, let L(cid:4) be another generalized additive mapping of
n-Apollonius type satisfying (2.7). Then we have, for any positive integer k
(cid:15)(cid:7) (cid:3) (cid:4) (cid:3) (cid:4)(cid:7)
(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)
(cid:7) (cid:7)
(cid:7)L(x)−L(cid:4)(x)(cid:7) ≤ n k (cid:7)(cid:7)L α kx −f α kx (cid:7)(cid:7)
α n n
(cid:7) (cid:3) (cid:4) (cid:3) (cid:4)(cid:7)(cid:16)
(cid:8) (cid:9) (cid:8) (cid:9)
(cid:7) (cid:7)
+(cid:7)(cid:7)f α kx −L(cid:4) α kx (cid:7)(cid:7)
n n
(cid:3) (cid:4)
(cid:2)∞ (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)
2n n k+j α k+j α k+j
≤ ϕ x, x ,
n2 −1 α i n n
j=0
which tends to zero as k → ∞. So we conclude that L(x) = L(cid:4)(x) for all
x ∈ X.
By the assumption, for each u ∈ U(A), we get
(cid:7) (cid:3) (cid:4)(cid:7)
(cid:8) (cid:9) (cid:7) (cid:8) (cid:9) (cid:7)
n d(cid:7) α d (cid:7)
(cid:7)DuL(x,0(cid:11),·(cid:12)·(cid:13)· ,0(cid:14))(cid:7) = lim (cid:7)Duf x,0(cid:11),·(cid:12)·(cid:13)· ,0(cid:14) (cid:7)
d→∞ α n
ntimes ntimes
(cid:8) (cid:9) (cid:8) (cid:9)
n d α d
≤ lim ϕ( x,0,··· ,0) = 0
(cid:11) (cid:12)(cid:13) (cid:14)
d→∞ α n
ntimes
for all x ∈ X. Hence
nL(ux) = nuL(x)
for all u ∈ U(A) and all x ∈ X. So
L(ux) = uL(x)
for all u ∈ U(A) and all x ∈ X.
By the same reasoning as in the proofs of [19] and [22],
L(ax+by) = L(ax)+L(by) = aL(x)+bL(y)
for all a,b ∈ A(a,b (cid:9)= 0) and all x,y ∈ X. And L(0x) = 0 = 0L(x) for all
x ∈ X. SotheuniquegeneralizedadditivemappingL : X → Y ofn-Apollonius
type is an A-linear mapping.
Corollary 2.3. Let (cid:2) ≥ 0 and let p be a real number with p > 1. Assume that
a mapping f : X → Y satisfies the inequality
(cid:5) (cid:2)n (cid:6)
(2.14) (cid:7)D f(z,x ,··· ,x )(cid:7) ≤ (cid:2) (cid:7)z(cid:7)p + (cid:7)x (cid:7)p
u 1 n Y X j X
j=1
120 F. Moradlou, C. Park and J. Lee
for all u ∈ U(A) and all z,x ,··· ,x ∈ X. Then there exists a unique A-linear
1 n
additive mapping L : X → Y of n-Apollonius type such that
2n(n2 −1)1−p(cid:2)
(cid:7)f(x)−L(x)(cid:7) ≤ ||x||p
Y (n2 −1)(2−p) −(n2 −1)n2(1−p) X
for all x ∈ X.
Proof. It follows from (2.14) that f(0) = 0. Define
(cid:5) (cid:2)n (cid:6)
ϕ(z,x ,··· ,x ) := (cid:2) (cid:7)z(cid:7)p + ||x ||p ,
1 n X j X
j=1
and apply Theorem 2.2 to get the result.
Theorem 2.4. Let f : X → Y be a mapping satisfying f(0) = 0 for which
there exists a function ϕ : Xn+1 → [0,∞) satisfying (2.6) such that
(2.15)
(cid:3) (cid:4)
(cid:2)∞ (cid:8)n2 −1(cid:9)j (cid:8) n2 (cid:9)j (cid:8) n2 (cid:9)j
ϕ(cid:10)(z,x) = ϕ z,0,··· ,0, x,0,··· ,0 < ∞
i n2 n2 −1 n2 −1
(cid:11) (cid:12)(cid:13) (cid:14)
j=1
i th
(2.16)
(cid:3) (cid:4)
(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)
n2 −1 j n2 j n2 j n2 j
lim ϕ z, x ,··· , x = 0
j→∞ n2 n2 −1 n2 −1 1 n2 −1 n
for all x,z,x ,··· ,x ∈ X. Then there exists a unique A-linear additive map-
1 n
ping L : X → Y of n-Apollonius type such that
n
(2.17) (cid:7)f(x)−L(x)(cid:7) ≤ ϕ(cid:10)(x,x)
Y n2 −1 i
for all x ∈ X.
Proof. Let u = 1 ∈ U(A). It follows from (2.9) that
(cid:7) (cid:7) (cid:8) (cid:9)
(cid:5) (cid:6)
(cid:7) α n (cid:7) 1 n n
(2.18) (cid:7)f(x)− f x (cid:7) ≤ ϕ x, x
i
n α Y n α α
(cid:8) (cid:9)
n2 −1 n k
for all x ∈ X, where α = . If we replace x in (2.18) by x and
n (cid:8) (cid:9) α
α k
multiply both sides of (2.18) by , then we have
n
(cid:3) (cid:4) (cid:3) (cid:4)
(cid:7)(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9) (cid:7)
(cid:7) α k n k α k+1 n k+1 (cid:7)
(cid:7) f x − f x (cid:7)
n α n α Y
(2.19) (cid:8) (cid:9) (cid:3)(cid:8) (cid:9) (cid:8) (cid:9) (cid:4)
1 α k n k+1 n k+1
≤ ϕ x, x
i
n n α α