Using the fixed point method, we prove the HyersUlam stability of lin ear mappings in Banach modules over a unital
C
*algebra and in nonArchimedean Banach modules over a unital
C
*algebra associated with the orthogonally Cauchy Jensen additive functional equation.
1. INTRODUCTION AND PRELIMINARIES
Assume that X is a real inner product space and
f
: X → ℝ is a solution of the orthogonal Cauchy functional equation
f
(
x
+
y
) =
f
(
x
) +
f
(
y
), <
x
,
y
>= 0. By the Pythagorean theorem
f
(
x
) = ∥
x
∥
^{2}
is a solution of the conditional equation. Of course, this function does not satisfy the additivity equation everywhere. Thus orthogonal Cauchy equation is not equivalent to the classic Cauchy equation on the whole inner product space.
G. Pinsker
[40]
characterized orthogonally additive functionals on an inner prod uct space when the orthogonality is the ordinary one in such spaces. K. Sundaresan
[50]
generalized this result to arbitrary Banach spaces equipped with the Birkho® James orthogonality. The orthogonal Cauchy functional equation
in which ⊥ is an abstract orthogonality relation, was first investigated by S. Gudder and D. Strawther
[18]
. They defined ⊥ by a system consisting of five axioms and described the general semicontinuous realvalued solution of conditional Cauchy functional equation. In 1985, J. Rätz
[47
] introduced a new definition of orthogonal ity by using more restrictive axioms than of S. Gudder and D. Strawther. Moreover, he investigated the structure of orthogonally additive mappings. J. Rätz and Gy. Szabó
[48]
investigated the problem in a rather more general framework.
Let us recall the orthogonality in the sense of J. Rätz; cf.
[47]
.
Suppose
X
is a real vector space (algebraic module) with dim
X
≥ 2 and ⊥ is a binary relation on
X
with the following properties:
(
O
_{1}
) totality of ⊥ for zero:
x
⊥ 0, 0 ⊥
x
for all
x
∈
X
;
(
O
_{2}
) independence: if
x
,
y
∈
X
 {0},
x
⊥
y
, then
x
,
y
are linearly independent;
(
O
_{3}
) homogeneity: if
x
,
y
∈
X
,
x
⊥
y
, then
αx
⊥
βy
for all
α
,
β
∈ ℝ;
(
O
_{4}
) the Thalesian property: if
P
is a 2dimensional subspace of
X
,
x
∈
P
and
λ
∈ ℝ
_{+}
, which is the set of nonnegative real numbers, 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 (module). By an orthogonality normed space (normed module) we mean an orthogonality space (module) having a normed (normed module) structure.
Assume that if
A
is a
C
*algebra and
X
is a module over
A
and if
x
,
y
∈
X
,
x
⊥
y
, then
ax
⊥
by
for all
a
,
b
∈
A
.
Some interesting examples are
(i) The trivial orthogonality on a vector space
X
defined by (
O
_{1}
), and for nonzero 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 BirkhoffJames orthogonality on a normed space (
X
, ∥.∥) defined by
x
⊥
y
if and only if ∥
x
+
λy
∥ ≥∥
x
∥ for all
λ
∈ ℝ.
The relation ⊥ is called symmetric if
x
⊥
y
implies that
y
⊥
x
for all
x
,
y
∈
X
. Clearly examples (i) and (ii) are symmetric but example (iii) is not. It is remarkable to note, however, that a real normed space of dimension greater than 2 is an inner product space if and only if the BirkhoffJames orthogonality is symmetric. There are several orthogonality notions on a real normed space such as BirkhoffJames, Boussouis, Singer, Carlsson, unitaryBoussouis, Roberts, Phythagorean, isosceles and Diminnie (see
[1]
−
[3]
,
[7
,
14
,
23
,
24
,
36]
).
The stability problem of functional equations was originated from the following question of Ulam
[52]
:
Under what condition does there is an additive mapping near an approximately additive mapping?
In 1941, Hyers
[20]
gave a partial affrmative answer to the question of Ulam in the context of Banach spaces. In 1978, Th.M. Rassias
[42]
extended the theorem of Hyers by considering the unbounded Cauchy difference ∥
f
(
x
+
y
) −
f
(
x
) −
f
(
y
)∥ ≤
ε
(∥
x
∥
^{p }
+ ∥
y
∥
^{p}
), (
ε
> 0,
p
∈ [0, 1)). The result of Rassias has provided a lot of infiuence in the development of what we now call
generalized HyersUlam stability
or
HyersUlam stability
of functional equations. During the last decades several stability problems of functional equations have been investigated in the spirit of HyersUlamRassias. The reader is referred to
[11
,
21
,
25
,
46]
and references therein for detailed information on stability of functional equations.
R. Ger and J. Sikorska
[17]
investigated the orthogonal stability of the Cauchy functional equation
f
(
x
+
y
) =
f
(
x
) +
f
(
y
), namely, they showed that if
f
is a mapping from an orthogonality space
X
into a real Banach space
Y
and ∥
f
(
x
+
y
) −
f
(
x
) −
f
(
y
)∥ ≤
ε
for all
x
,
y
∈
X
with
x
⊥
y
and some
ε
> 0, then there exists exactly one orthogonally additive mapping
g
:
X
→
Y
such that
for all
x
∈
X
.
The first author treating the stability of the quadratic equation was F. Skof
[49]
by proving that if
f
is a mapping from a normed space
X
into a Banach space
Y
satisfying ∥
f
(
x
+
y
)+
f
(
x
−
y
)−2
f
(
x
)−2
f
(
y
)∥≤
ε
for some
ε
> 0, then there is a unique quadratic mapping
g
:
X
→
Y
such that
. P.W. Cholewa
[8]
extended the Skof’s theorem by replacing
X
by an abelian group
G
. The Skof’s result was later generalized by S. Czerwik
[9]
in the spirit of HyersUlamRassias. The stability problem of functional equations has been extensively investigated by some mathematicians (see
[10
,
39]
,
[43]
–
[45]
).
The orthogonally quadratic equation
was first investigated by F. Vajzović
[53]
when
X
is a Hilbert space,
Y
is the scalar field,
f
is continuous and ⊥ means the Hilbert space orthogonality. Later, H. Drljević
[15]
, M. Fochi
[16]
, M.S. Moslehian
[31
,
32]
and Gy. Szabó
[51]
generalized this result.
In 1897, Hensel
[19]
introduced a normed space which does not have the Archimedean property. It turned out that nonArchimedean spaces have many nice applications (see
[12
,
27
,
28
,
35]
).
Definition 1.1.
By a
nonArchimedean field
we mean a field 𝕂 equipped with a function (valuation) ｜ · ｜ : 𝕂 → [0,∞) such that for all
r
,
s
∈ 𝕂, the following conditions hold:
(1) ｜
r
｜ = 0 if and only if
r
= 0;
(2) ｜
rs
｜ = ｜
r
∥
s
｜;
(3) ｜
r
+
s
｜ ≤ max{｜
r
｜, ｜
s
｜}.
Definition 1.2
(
[34]
)
.
Let
X
be a vector space over a scalar field 𝕂 with a non Archimedean nontrivial valuation ｜ · ｜ . A function ∥·∥ :
X
→
R
is a
non Archimedean norm
(valuation) if it satisfies the following conditions:
(1) ∥
x
∥ = 0 if and only if
x
= 0;
(2) ∥
rx
∥ = ｜
r
⦀
x
∥ (
r
∈ 𝕂,
x
∈
X
);
(3) The strong triangle inequality (ultrametric); namely,
∥
x
+
y
∥ ≤ max{∥
x
∥, ∥
y
∥},
x
,
y
∈
X
.
Then (
X
, ∥.∥) is called a nonArchimedean space.
Assume that if
A
is a
C
*algebra and
X
is a module over
A
, which is a non Archimedean space, and if
x
,
y
∈
X
,
x
⊥
y
, then
ax
⊥
by
for all
a
,
b
∈
A
. Then (
X
, ∥.∥) is called an
orthogonality nonArchimedean module
.
Due to the fact that
Definition 1.3.
A sequence {
x_{n}
}is
Cauchy
if and only if {
x
_{n+1}
−
x_{n}
} converges to zero in a nonArchimedean space. By a complete nonArchimedean space we mean one in which every Cauchy sequence is convergent.
Let
X
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
,
y
) ≤
d
(
x
,
y
) +
d
(
y
,
z
) for all
x
,
y
,
z
∈
X
.
We recall a fundamental result in fixed point theory.
Theorem 1.4
(
[4
,
13]
)
.
Let (X, d) be a complete generalized metric space and let
J
:
X
→
X
be a strictly contractive mapping with Lipschitz constant
α
< 1.
Then for each given element x
∈
X
,
either
for all nonnegative integers n or there exists a positive integer n_{0} such that
(1)
, ∀
n
≥
n
_{0}
;
(2)
the sequence
{
J
^{n}
x
}
converges to a fixed point y
*
of J
;
(3)
y
*
is the unique fixed point of J in the set Y
= {
y
∈
X
｜
d
(
J
^{n0}
x
,
y
) < ∞};
(4)
for all
y
∈
Y
.
In 1996, G. Isac and Th.M. Rassias
[22]
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
[5
,
6
,
30
,
37
,
38
,
41]
).
This paper is organized as follows: In Section 2, we prove the HyersUlam stability of the orthogonally CauchyJensen additive functional equation in Banach modules over a unital
C
*algebra. In Section 3, we prove the HyersUlam stability of the orthogonally CauchyJensen additive functional equation in nonArchimedean Banach modules over a unital
C
*algebra.
2. STABILITY OF THE ORTHOGONALLY CAUCHYJENSEN ADDITIVE FUNCTIONAL EQUATION IN BANACH MODULES OVER AC*ALGEBRA
Throughout this section, assume that
A
is a unital
C
*algebra with unit
e
and unitary group
U
(
A
) := {
u
∈
A
｜
u
*
u
=
uu
* =
e
}, (
X
,⊥) is an orthogonality normed module over
A
and (
Y
, ∥.∥
_{Y}
) is a Banach module over
A
.
In this section, applying some ideas from
[17
,
21]
, we deal with the stability problem for the orthogonally CauchyJensen additive functional equation
for all
x, y, z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
.
Theorem 2.1.
Let
φ
:
X
^{3}
→ [0,∞)
be a function such that there exists an
α
< 1
with
for all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
.
Let
f
:
X
→
Y
be a mapping satisfying f(0) = 0 and
for all u
∈
U
(
A
)
and all
x
,
y
,
z
∈
X with x
⊥
y
,
x
⊥
z and y
⊥
z
.
If for each x
∈
X the mapping f
(
tx
)
is continuous in t
∈ ℝ,
then there exists a unique orthogonally CauchyJensen additive and Alinear mapping L
:
X
→
Y such that
for all x
∈
X.
Proof
. Putting
y
=
z
= 0 and
u
=
e
in (2.2), we get
for all
x
∈
X
, since
x
⊥ 0. So
for all
x
∈
X
.
Consider the set
and introduce the generalized metric on
S
:
where, as usual, inf 𝜙 = +∞. It is easy to show that (
S
,
d
) is complete (see
[29]
).
Now we consider the linear mapping
J
:
S
→
S
such that
for all
x
∈
X
.
Let
g
,
h
∈
S
be given such that
d
(
g
,
h
) =
ε
. Then
for all
x
∈
X
. Hence
for all
x
∈
X
. So
d
(
g
,
h
) =
ε
implies that
d
(
Jg
,
Jh
) ≤α
ε
. This means that
for all
g
,
h
∈
S
.
It follows from (2.5) that
d
(
f
,
Jf
)≤
α
.
By Theorem 1.4, there exists a mapping
L
:
X
→
Y
satisfying the following:
(1)
L
is a fixed point of
J
, i.e.,
for all
x
∈
X
. The mapping
L
is a unique fixed point of
J
in the set
This implies that
L
is a unique mapping satisfying (2.6) such that there exists a
μ
∈(0,∞) satisfying
for all
x
∈
X
;
(2)
d
(
J
^{n}
f
,
L
) → 0 as
n
→∞. This implies the equality
for all
x
∈
X
;
(3)
, which implies the inequality
This implies that the inequalities (2.3) holds.
Let
u
=
e
in (2.2). It follows from (2.1) and (2.2) that
for all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
. So
for all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
. Hence
L
:
X
→
Y
is an orthogonally CauchyJensen additive mapping.
Let
y
=
z
= 0 in (2.2). It follows from (2.1) and (2.2) that
for all
x
∈
X
. So
for all
x
∈
X
. Hence
for all
u
∈
U
. (
A
) and all
x
∈
X
.
By the same reasoning as in the proof of [
42
, Theorem], we can show that
L
:
X
→
Y
is ℝlinear, since the mapping
f(tx)
is continuous in
t
∈ ℝ for each
x
∈
X
and
L
:
X
→
Y
is additive.
Since
L
is ℝlinear and each
a
∈
A
is a finite linear combination of unitary elements (see [
26
, Theorem 4.1.7]), i.e.,
, it follows from (2.7) that
for all
x
∈
X
. It is obvious that
. Thus
L
:
X
→
Y
is a unique orthogonally CauchyJensen additive and
A
linear mapping satisfying (2.3).
Corollary 2.2.
Let 𝜽 be a positive real number and p a real number with
0 <
p
< 1.
Let f : X
→
Y be a mapping satisfying
for all
u
∈
U
(
A
)
and all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
.
If for each
x
∈
X
the mapping f
(
tx
)
is continuous in t
∈ ℝ,
then there exists a unique orthogonally CauchyJensen additive and Alinear mapping
L
:
X
→
Y
such that
for all
x
∈
X
.
Proof
. The proof follows from Theorem 2.1 by taking
φ
(
x
,
y
) =
θ
(∥
x
∥
^{p}
+∥
y
∥
^{p}
+∥
z
∥
^{p}
) for all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
. Then we can choose
α
= 2
^{p−1}
and we get the desired result.
Theorem 2.3.
Let
f
:
X
→
Y
be a mapping satisfying
(2.2)
and f
(0) = 0
for which there exists a function
φ
:
X
^{3}
→ [0,∞)
such that
for all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
.
If for each
x
∈
X
the mapping f
(
tx
)
is continuous in t
∈ ℝ,
then there exists a unique orthogonally CauchyJensen additive and Alinear mapping
L
:
X
→
Y
such that
for all
x
∈
X
.
Proof.
Let (
S, d
) be the generalized metric space defined in the proof of Theorem 2.1.
Now we consider the linear mapping
J
:
S
→
S
such that
for all
x
∈
X
.
It follows from (2.4) that
d
(
f
,
Jf
) ≤1.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 2.4.
Let θ be a positive real number and p a real number with p
> 1.
Let
f
:
X
→
Y be a mapping satisfying
(2.8).
If for each x
∈
X the mapping f
(
tx
)
is continuous in t
∈ ℝ,
then there exists a unique orthogonally CauchyJensen additive and Alinear mapping
L
:
X
→
Y
such that
for all
x
∈
X
.
Proof.
The proof follows from Theorem 2.3 by taking
for all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
. Then we can choose
α
= 2
^{1−p}
and we get the desired result.
3. STABILITY OF THE ORTHOGONALLY CAUCHYJENSEN ADDITIVE FUNCTIONAL EQUATION IN NONARCHIMEDEAN BANACH MODULES OVER AC*ALGEBRA
Throughout this section, assume that
A
is a unital
C
*algebra with unit
e
and unitary group
U
(
A
) := {
u
∈
A
｜
u
*
u
=
uu
* =
e
}, (
X
,⊥) is an orthogonality nonArchimedean normed module over
A
and (
Y
, ∥.∥
_{Y}
) is a nonArchimedean Banach module over
A
. Assume that ｜2｜ ≠ 1.
In this section, applying some ideas from
[17
,
21]
, we deal with the stability problem for the orthogonally CauchyJensen additive functional equation.
Theorem 3.1.
Let
φ
:
X
^{3}
→ [0,∞)
be a function such that there exists an
α
< 1
with
for all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
.
Let
f
:
X
→
Y
be a mapping satisfying
f
(0) = 0
and
(2.2).
If for each
x
∈
X
the mapping f
(
tx
)
is continuous in t
∈ ℝ,
then there exists a unique orthogonally CauchyJensen additive and Alinear mapping
L
:
X
→
Y
such that
for all x
∈
X
.
Proof.
It follows from (2.4) that
for all
x
∈
X
.
Let (
S, d
) be the generalized metric space defined in the proof of Theorem 2.1.
Now we consider the linear mapping
J
:
S
→
S
such that
for all
x
∈
X
.
It follows from (3.3) that
d
(
f
,
Jf
) ≤
α
.
By Theorem 1.4, there exists a mapping
L
:
X
→
Y
satisfying the following:
(1)
d
(
J^{n}
f
,
L
) → 0 as
n
→ ∞. This implies the equality
for all
x
∈
X
;
(2)
, which implies the inequality
This implies that the inequality (3.2) holds.
It follows from (3.1) and (2.2) that
for all
u
∈
U
(
A
) and all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
. So
for all
u
∈
U
(
A
) and all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 3.2.
Let θ be a positive real number and p a real number with p
> 1.
Let
f
:
X
→
Y
be a mapping satisfying
(2.8).
If for each x
∈
X the mapping f
(
tx
)
is continuous in t
∈ℝ,
then there exists a unique orthogonally CauchyJensen additive and Alinear mapping
L
:
X
→
Y
such that
for all x
∈
X
.
Proof
. The proof follows from Theorem 3.1 by taking
for all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
. Then we can choose
α
=｜2｜
^{p−1}
and we get the desired result.
Theorem 3.3.
Let
f
:
X
→
Y
be a mapping satisfying
(2.2)
and f
(0) = 0
for which there exists a function
φ
:
X
^{3}
→ [0,∞)
such that
for all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
.
If for each x
∈
X
the mapping f
(
tx
)
is continuous in t
∈ ℝ,
then there exists a unique orthogonally CauchyJensen additive and Alinear mapping
L
:
X
→
Y
such that
for all x
∈
X
.
Proof
. Let (
S, d
) be the generalized metric space defined in the proof of Theorem 2.1.
Now we consider the linear mapping
J
:
S
→
S
such that
for all
x
∈
X
.
It follows from (2.4) that
d
(
f
,
Jf
) ≤ 1.
The rest of the proof is similar to the proofs of Theorems 2.1 and 3.1.
Corollary 3.4.
Let θ be a positive real number and p a real number with
0 <
p
< 1.
Let
f
:
X
→
Y
be a mapping satisfying
(2.8).
If for each x
∈
X the mapping f
(
tx
)
is continuous in t
∈ ℝ,
then there exists a unique orthogonally CauchyJensen additive and Alinear mapping
L
:
X
→
Y
such that
for all x
∈
X
.
Proof.
The proof follows from Theorem 3.3 by by taking
for all
x
,
y
,
z
∈
X
with
x
⊥
y
,
x
⊥
z
and
y
⊥
z
. Then we can choose
α
= ｜2｜
^{1−p}
and we get the desired result.
Acknowledgements
This research was supported by Hanshin University Research Grant.
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