HYERS-ULAM STABILITY OF THE GENERALIZED
QUADRATIC FUNCTIONAL EQUATION IN
AMENABLE SEMIGROUP
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
BOUIKHALENE BELAID
Department of Mathematics, Faculty of Sciences,
University of Ibn Tofail, Kenitra, Morocco.
EMail: bbouikhalene@yahoo.fr
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ELQORACHI ELHOUCIEN AND REDOUANI AHMED
Laboratory LAMA, Harmonic Analysis and Functional Equations Team
Department of Mathematics
Faculty of Sciences, University Ibn Zohr
Agadir, Morocco
EMail: elqorachi@hotamail.com and redouani_ahmed@yahoo.fr
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Received:
6 March, 2007
Accepted:
26 April, 2007
Communicated by:
Th.M. Rassias
2000 AMS Sub. Class.:
39B82, 39B52.
Key words:
Hyers-Ulam stability, Quadratic functional
Amenable semigroup, Morphism of semigroup.
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equation,
Abstract:
In this paper we derive the Hyers-Ulam stability of the quadratic functional equation
f (xy) + f (xσ(y)) = 2f (x) + 2f (y), x, y ∈ G,
respectively the functional equation
f (xy) + g(xσ(y)) = f (x) + g(y), x, y ∈ G,
where G is an amenable semigroup, σ is a morphism of G such that σ ◦ σ = I,
respectively where G is an amenable semigroup and σ is an homomorphism of
G such that σ ◦ σ = I.
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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Contents
1
Introduction
4
2
Stability of Equation (1.4) in Amenable Semigroups
6
3
Stability of Equation (1.5) in Amenable Semigroups
22
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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1.
Introduction
In 1940, Ulam [21] raised a question concerning the stability problem of group homomorphisms:
Given a group G1 , a metric group (G2 , d), a number ε > 0 and a mapping
f : G1 −→ G2 which satisfies the inequality d(f (xy), f (x)f (y)) < ε for
all x, y ∈ G1 , does there exist a homomorphism h : G1 −→ G2 and a
constant k > 0, depending only on G1 and G2 such that d(f (x), h(x)) ≤
kε for all x in G1 ?
The case of approximately additive mappings was solved by D. H. Hyers [8]
under the assumption that G1 and G2 are Banach spaces.
In 1978, Th. M. Rassias [16] gave a remarkable generalization of the Hyers’s result which allows the Cauchy difference to be unbounded. Since then, several mathematicians have been attracted to the results of Hyers and Rassias and investigated a
number of stability problems of different functional equations. See for example the
monographs of the following references [5, 6, 9, 10, 11, 16].
The quadratic functional equation
(1.1)
f (x + y) + f (x − y) = 2f (x) + 2f (y), x, y ∈ G
has been much studied. It was generalized by Stetkær [19] to the more general
equation
(1.2)
f (x + y) + f (x + σ(y)) = 2f (x) + 2f (y), x, y ∈ G,
where σ is an automorphism of the abelian group G such that σ 2 = I, (I denotes the
identity).
A stability result for the quadratic functional equation (1.1) was derived by Skof
[18], Cholewa [3] and by Czerwik [4].
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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Recently, Bouikhalene, Elqorachi and Rassias stated the stability theorem of
equation (1.2), see [1] and [2].
Székelyhidi [20] extended the Hyers’s result to amenable semigroups. He replaced the original proof given by Hyers by a new one based on the use of invariant
means.
In [22] Yang obtained the stability of the quadratic functional equation
(1.3)
f (xy) + f (xy −1 ) = 2f (x) + 2f (y), x, y ∈ G,
in amenable groups.
The purpose of the present paper is a joint treatment of the functional equations
(1.1) and (1.3) and their generalization, where the unifying object is a morphism like
the one introduced in (1.2).
New features of the paper:
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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1. In comparison with [20] and [22], we work with a general morphism σ.
2. In contrast to [1] and [2] we here allow the (semi)group G to be non-abelian.
In Section 2, we obtain the stability of the quadratic functional equation
(1.4)
f (xy) + f (xσ(y)) = 2f (x) + 2f (y), x, y ∈ G,
where σ is a morphism of G such that σ ◦ σ = I. The result of this section can be
compared with the ones of Yang [22] because we formulate them in the same way
by using some ideas from [22].
In Section 3, we obtain the stability of the generalized quadratic functional equation
(1.5)
f (xy) + g(xσ(y)) = f (x) + g(y), x, y ∈ G,
where σ is an automorphism of G such that σ ◦ σ = I.
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2.
Stability of Equation (1.4) in Amenable Semigroups
In this section we investigate the Hyers-Ulam stability of the quadratic functional
equation
(2.1)
f (xy) + f (xσ(y)) = 2f (x) + 2f (y),
x, y ∈ G,
where G is an amenable semigroup with unit element e and σ : G −→ G is a
morphism of G, i.e. σ is an antiautomorphism: σ(xy) = σ(y)σ(x) for all x, y ∈ G
or σ is an automorphism: σ(xy) = σ(x)σ(y) for all x, y ∈ G. Furthermore, we
assume that σ satisfies (σ ◦ σ)(x) = x, for all x ∈ G.
We recall that a semigroup G is said to be amenable if there exists an invariant
mean on the space of the bounded complex functions defined on G. We refer to [7]
for the definition and properties of invariant means.
Throughout this paper, as in [5], we use the following definition.
Definition 2.1. Let G be a semigroup and B a Banach space. We say that the equation
(2.2)
f (xy) + f (xσ(y)) = 2f (x) + 2f (y),
x, y ∈ G
is stable for the pair (G, B) if for every function f : G −→ B such that
1
≤ δ, x, y ∈ G for some δ ≥ 0,
(2.3) [f
(xy)
+
f
(xσ(y))]
−
f
(x)
−
f
(y)
2
there exists a solution q of equation (2.2) and a constant γ ≥ 0 dependent only on δ
such that
(2.4)
kf (x) − q(x)k ≤ γ for all x ∈ G.
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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Proposition 2.1. Let σ be an antiautomorphism of the semigroup G such that σ◦σ =
I. Let B a Banach space. Suppose that f : G −→ B satisfies the inequality (2.3).
Then for every x ∈ G, the limit
"
#
n
X
n−k
n−k
k−1
n
2k−1 f ((x2 σ(x)2 )2 )
(2.5)
g(x) = lim 2−2n f (x2 ) +
n→+∞
k=1
exists. Moreover, g : G −→ C is a unique function satisfying
(2.6)
kf (x) − g(x)k ≤ δ, and g(x2 ) + g(xσ(x)) = 4g(x) for all x ∈ G.
Proof. Assume that f : G −→ C satisfies the inequality (2.3) and define by induction the sequence function f0 (x) = f (x) and fn (x) = 21 [fn−1 (x2 ) + fn−1 (xσ(x))]
for n ≥ 1. By direct computation, we obtain
"
#
n
X
n
n−k
n−k
k−1
fn (x) = 2−n f (x2 ) +
2k−1 f ((x2 σ(x)2 )2 )
k=1
for all n ≥ 1.
By letting x = y, in (2.3) we get
1
2
(2.7)
2 [f (x ) + f (xσ(x))] − 2f (x) ≤ δ,
kf1 (x) − 2f0 (x)k ≤ δ for all x ∈ G.
In the following, we prove by induction the inequalities
(2.9)
vol. 8, iss. 2, art. 47, 2007
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so
(2.8)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
kfn (x) − 2fn−1 (x)k ≤ δ
(2.10)
kfn (x) − 2n f (x)k ≤ (2n − 1)δ
for all n ∈ N and x ∈ G. It is clear that (2.8) is (2.9) for n = 1. The inductive step
must now be demonstrated to hold true for the integer n + 1, that is
kfn+1 (x) − 2fn (x)k
1
1
2
2
=
2 [fn (x ) + fn (xσ(x))] − 2 2 [fn−1 (x ) + fn−1 (xσ(x))]
1
1
2
2
≤ [kfn (x ) − 2fn−1 (x )k] + [f
(xσ(x))
−
2f
(xσ(x))]
n
n−1
2
2
1
≤ [δ + δ] = δ.
2
This proves that (2.9) is true for any natural number n.
Now, by using the inequality
(2.11)
(2.12)
kfn (x) − 2n f (x)k ≤ kfn (x) − 2fn−1 (x)k + 2kfn−1 (x) − 2n−1 f (x)k
we check that (2.10) holds true for any n ∈ N.
Let us define
"
#
n
X
fn (x)
−2n
2n
k−1
2n−k
2n−k 2k−1
(2.13)
gn (x) =
=2
f (x ) +
2 f ((x
σ(x)
)
)
2n
k=1
for any positive integer n and x ∈ G.
Now, by using (2.11) and (2.13), we get
(2.14)
kgn+1 (x) − gn (x)k ≤
δ
2n+1
.
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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It easily follows that {gn (x)} is a Cauchy sequence for all x ∈ G. Since B is
complete, we can define g(x) = limn−→+∞ gn (x) for any x ∈ G. From (2.10),
one can verify that g satisfies the first assertion of (2.6). Now, we will show that
g satisfies the second assertion of (2.6). By induction one proves that the sequence
fn (x) satisfies
1
2
[fn (x ) + fn (xσ(x))] − fn (x) − fn (x) ≤ δ
(2.15)
2
for all n ∈ N.
For n = 1, we have
1
2
(2.16) [f1 (x ) + f1 (xσ(x))] − f1 (x) − f1 (x)
2
h i
1 1
22
2
2
2
2
=
f
x
+
f
(x
σ(x)
)
+
f
((xσ(x))
)
+
f
((xσ(x))
)
2 2
1
1
2
2
− [f (x ) + f (xσ(x))] − [f (x ) + f (xσ(x))] 2
2
h i
1 1
22
2
2
2
≤
f x
+ f (x σ(x) ) − 2f (x )
2 2
1 1
2
2
f ((xσ(x)) ) + f ((xσ(x)) ) − 2f (xσ(x)) +
2 2
1
≤ [δ + δ] = δ.
2
So (2.15) is true for n = 1. We assume then that (2.15) holds for n and we prove
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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that (2.15) is true for n + 1.
1
2
(2.17) 2 [fn+1 (x ) + fn+1 (xσ(x))] − fn+1 (x) − fn+1 (x)
h i
1 1
22
2
2
2
2
=
f
x
+
f
(x
σ(x)
)
+
f
((xσ(x))
)
+
f
((xσ(x))
)
n
n
n
2 2 n
1
1
2
2
− [fn (x ) + fn (xσ(x))] − fn (x ) + fn (xσ(x)) 2
2
h i
1 1
22
2
2
2
≤
f
x
+
f
(x
σ(x)
)
−
2f
(x
)
n
n
n
2 2
1 1
2
2
+
[fn ((xσ(x)) ) + fn ((xσ(x)) ) − 2fn (xσ(x))] 2 2
1
≤ [δ + δ] = δ.
2
Consequently, the sequence gn (x) satisfies
1
2
[gn (x ) + gn (xσ(x))] − gn (x) − gn (x) ≤ δ
(2.18)
2n
2
for all n ∈ N and then by letting n → +∞ we get the desired result.
Assume now that there exists another mapping h : G −→ B which satisfies
kf (x) − h(x)k ≤ δ and h(x2 ) + h(xσ(x)) = 4h(x) for all x ∈ G.
First, we will prove by mathematical induction that
(2.19)
kfn (x) − 2n h(x)k ≤ δ for all x ∈ G.
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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For n = 1, we have
1
1
2
2
(2.20)
kf1 (x) − 2h(x)k = [f
(x
)
+
f
(xσ(x))]
−
[h(x
)
+
h(xσ(x))]
2
2
1
≤ [kf (x2 ) − h(x2 )k + kf (xσ(x)) − h(xσ(x))k]
2
1
≤ [δ + δ] = δ.
2
Suppose (2.19) is true for n and we will prove it for n + 1. Hence, we have
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
kfn+1 (x) − 2n+1 h(x)k
1
1
2
n
2
=
[f
(x
)
+
f
(xσ(x))]
−
2
[h(x
)
+
h(xσ(x))]
n
n
2
2
1
≤ [kfn (x2 ) − 2n h(x2 )k + kfn (xσ(x)) − 2n h(xσ(x))k]
2
1
≤ [δ + δ] = δ.
2
This proves that (2.19) is true for all n ∈ N. From (2.19), we obtain fn2(x)
−
h(x)
≤
n
(2.21)
δ
,
2n
so by letting n → +∞ and by using the definition of g we get g = h. This completes the proof of the theorem.
By using the precedent proof we easily obtain the following result.
Proposition 2.2. Let σ be an automorphism of the semigroup G such that σ ◦ σ = I.
Let B a Banach space. Suppose that f : G −→ B satisfies the inequality (2.3). Then
for every x ∈ G, the limit
g(x) = lim 2−n fn (x)
n→+∞
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exists, and g is a unique function satisfying
kf (x) − g(x)k ≤ δ, and g(x2 ) + g(xσ(x)) = 4g(x) for all x ∈ G,
where the sequence of functions fn is defined on G by the formulas f0 (x) = f (x)
and fn (x) = 12 [fn−1 (x2 ) + fn−1 (xσ(x))] for n ≥ 1.
The following result shows that in this context the only property of B (Definition
2.1) involved is the completeness. For the proof, we refer to the one used by Yang
in [22].
Theorem 2.3. Let σ be a morphism of the semigroup G such that σ ◦σ = I. Suppose
that equation (2.2) is stable for the pair (G, C) (resp. (G, R)) Then for every complex
(resp. real) Banach space B, (2.2) is stable for the pair (G, B).
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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The main result of the present section is the following
Theorem 2.4. Let σ be an antiautomorphism of the amenable semigroup G such
that σ ◦ σ = I. Then equation (2.2) is stable for the pair (G, C).
First, we prove the following useful lemma
Lemma 2.5. Let σ be an antiautomorphism of the semigroup G such that σ ◦ σ = I.
Let B be a Banach space. Suppose that f : G −→ B satisfies the inequality
1
(f (xy) + f (xσ(y))) − f (x) − f (y) ≤ δ, for some δ ≥ 0.
(2.22)
2
Then for every x ∈ G, the limit
(2.23)
n
o
n
n−1
n−1
q(x) = lim 2−2n f (x2 ) + (2n − 1)f (x2 σ(x)2 )
n→+∞
exists. Moreover, the mapping q satisfies the inequality
(2.24)
k f (x) − q(x) k≤ 7δ for all x ∈ G.
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Proof. Assume that f : G −→ B satisfies the inequality (2.22). We will prove by
induction that
1
n
n
n−1
n−1
(2.25) f (x) − 2n {f (2 x) + (2 − 1)f (2 x + 2 σ(x))}
2
7
3
19
≤2
+
−
δ
2 22n−1 2n+1
for some positive integer n. By letting y = x, in (2.22) we get
(2.26)
kf (x2 ) + (2 − 1)f (xσ(x)) − 22 f (x)k ≤ 2δ,
so
(2.27)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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1
1
2
f (x) − {f (x ) + (2 − 1)f (xσ(x))} ≤ δ 1 −
for all x ∈ G.
22
2
This proves (2.25) for n = 1. The inductive step must now be demonstrated to hold
true for the integer n + 1, that is
n
o
1
n+1
n
n
2
n+1
2
2
f (x) −
f
(x
)
+
(2
−
1)f
(x
σ(x)
)
22(n+1)
1 n+1
n
n
n ≤ 2(n+1) f (x2 ) + f x2 σ(x)2 − 4f (x2 )
2
n−1
n−1
1 n−1
n−1
n−1
n−1 + 2(n+1) 2(2n − 1)f x2 σ(x)2 x2 σ(x)2
− 4(2n − 1)f x2 σ(x)2
2
n−1
1 n
n−1
+ 2(n+1) 4f (x2 ) + 4(2n − 1)f x2 σ(x)2
− 22(n+1) f (x)
2
n−1
2(2n − 1) 2n
2n
2
2n−1 2n−1
2n−1 + 2(n+1) f x σ(x)
−f x
σ(x)
x
σ(x)
2
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7
(2n − 1)2δ
3
19
≤ 2(n+1) +
+
+
−
2δ
2
22(n+1)
2 22n−1 2n+1
n−1
2(2n − 1) 2n
2n
2
2n−1 2n−1
2n−1 + 2(n+1) f (x σ(x) ) − f x
σ(x)
x
σ(x)
.
2
2δ
To complete the proof of the induction assumption (2.25), we need the following
inequalities. Let x = y = e in (2.22) to get k2f (e)k ≤ 2δ. Putting x = e in (2.22),
gives
k f (y) + f (σ(y)) − 2f (e) − 2f (y)k ≤ 2δ.
vol. 8, iss. 2, art. 47, 2007
Consequently,
(2.28)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
kf (y) − f (σ(y))k ≤ 4δ.
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By interchanging x by y in (2.22), we obtain
(2.29)
Contents
kf (yx) + f (yσ(x)) − 2f (x) − 2f (y)k ≤ 2δ.
By using (2.22), (2.28), (2.29) and the triangle inequality, we deduce that
(2.30)
Now, from (2.22), we obtain
n−1
n−1
n
n−1
n−1
n−1
n−1 (2.31) 2f x2 σ(x)2 x2
− 2f x2 σ(x)2
− 2f σ(x)2 x2
≤ 2δ.
(2.32)
f x2n σ(x)2n − f σ(x)2n x2n ≤ 8δ
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kf (xy) − f (yx)k ≤ 8δ.
Since
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and
(2.33)
2n−1
2n 2n−1
2n
2n σ(x) x
− f (x σ(x) ) ≤ 8δ
f x
then
(2.34)
n−1
n−1 n−1
n
n−1
− 4f x2 σ(x)2
≤ 18δ
2f x2 σ(x)2 x2
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
and
(2.35)
n−1
n
n−1 n
f (x2 σ(x)2 ) − 2f x2 σ(x)2
n−1
n
n−1
n
n ≤ f x2 σ(x)2 x2
− f (x2 σ(x)2 )
n−1
2n−1
2n 2n−1
2
2n−1 + f x
σ(x) x
−f x
σ(x)
≤ 8δ +
18δ
= 17δ.
2
Finally, we deduce that
n−1
2n
2n
2
2n−1 2n−1
2n−1 (2.36)
f
(x
σ(x)
)
−
f
x
σ(x)
x
σ(x)
n−1 n
n
n−1
≤ f (x2 σ(x)2 ) − 2f x2 σ(x)2
n−1
n−1
n−1
n−1 n−1
n−1
− f x2 σ(x)2 x2 σ(x)2
+ 2f x2 σ(x)2
≤ 17δ + δ = 18δ
vol. 8, iss. 2, art. 47, 2007
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and consequently,
n n+1 o
2
n+1
2n
2n
f (x) − 1
f x
+ (2
− 1)f x σ(x)
22(n+1)
2δ
2(2n − 1)δ
7
3
19
2(2n − 1)
≤ 2(n+1) +
+
+
−
2δ
+
18δ
2
22(n+1)
2 22n−1 2n+1
22(n+1)
3
19
7
=2
+
−
δ.
2 22n+1 2n+2
This proves the validity of the inequality (2.25).
Let us define
o
1 n
2n
n
2n−1
2n−1
σ(x)
)
(2.37)
qn (x) = 2n f (x ) + (2 − 1)f (x
2
for any positive integer n and x ∈ G.
Then {qn (x)} is a Cauchy sequence for every x ∈ G. In fact by using (2.22),
(2.36) and (2.37), we get
kqn+1 (x) − qn (x)k
n+1 1 2
2n
2n
2n ≤ 2(n+1) f x
+ f (x σ(x) ) − 4f (x )
2
n−1
1 n
2n
2n
n
2
2n−1 + 2(n+1) 2(2 − 1)f x σ(x)
− 4(2 − 1)f x
σ(x)
2
n−1
1 δ
n−1
n−1
n−1
≤ 2(n+1) + 2(n+1) 2(2n − 1)f x2 σ(x)2 x2 σ(x)2
2
2
n−1
n−1
− 4(2n − 1)f x2 σ(x)2
n−1
2(2n − 1) 2n
2n
2
2n−1 2n−1
2n−1 + 2(n+1) f (x σ(x) ) − f x
σ(x)
x
σ(x)
2
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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2(2n − 1)δ 36δ(2n − 1)
40δ
+
≤
.
22(n+1)
22(n+1)
22(n+1)
2n
It easily follows that {qn (x)} is a Cauchy sequence for all x ∈ G. Since B is
complete, we can define q(x) = limn−→+∞ qn (x) for any x ∈ G and, in view of
(2.25) one can verify that q satisfies the inequality (2.24). This completes the proof
of Lemma 2.5.
≤
2δ
+
Proof of Theorem 2.4. We follow the ideas and the computations used in [22]. By
using (2.30) one derives the functional inequality
vol. 8, iss. 2, art. 47, 2007
|f ((xy)n ) − f ((yx)n )| ≤ 8δ.
(2.38)
In addition, from (2.3), (2.38) and the triangle inequality we deduce that
2n
2n
2n
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
Title Page
2n
(2.39) |f ((xy) (σ(xy)) ) − f ((yx) (σ(yx)) )|
n
n
n
n
n
n
Contents
n
≤ |f ((xy)2 (σ(xy))2 ) + f ((xy)2 (xy)2 ) − 4f ((xy)2 )|
n
n
n
+ | − f ((yx)2 (yx)2 ) − f ((yx)2 (σ(yx))2 ) + 4f ((yx)2 )|
n
n n n
n
n
+ f ((xy)2 (xy)2 ) − f ((yx)2 (yx)2 ) + 4 f ((xy)2 ) − f ((yx)2 )
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≤ 2δ + 2δ + 8δ + 32δ = 44δ.
From Lemma 2.5, for every x ∈ G, the limit
n
n−1
o
n
n−1
(2.40)
q(x) = lim 2−2n f (x2 ) + (2n − 1)f x2 σ(x)2
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n→+∞
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exists and
(2.41)
|f (x) − q(x)| ≤ 7δ.
Furthermore, in view of (2.38) – (2.39) q satisfies the relation
(2.42)
q(xy) = q(yx),
x, y ∈ G
and by (2.41) – (2.22) q satisfies the inequality equation
|q(xy) + q(xσ(y)) − 2q(x) − 2q(y)| ≤ 44δ
(2.43)
for all x, y ∈ G.
Consequently, for any fixed y ∈ G the function
x 7−→ q(xy) + q(xσ(y)) − 2q(x)
is bounded. Since G is amenable, there exists an invariant mean mx on the space
of bounded, complex-functions on G. With the help of mx we define the following
function on G
ψ(y) = mx {qy + qσ(y) − 2q}
(2.44)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
Title Page
for all y ∈ G, where qy (z) = q(zy), z ∈ G.
Furthermore, by using (2.29) and (2.44), we get
(2.45)
ψ(zy)+ψ(σ(z)y)
= mx {qzy + qσ(y)σ(z) − 2q} + mx {qσ(z)y + qσ(y)z − 2q}
= mx {zy q + qσ(y)σ(z) − 2q} + mx {σ(z)y q + qσ(y)z − 2q}
= mx {zy q +σ(z)y q − 2(y q)} + mx {qσ(y)σ(z) + qσ(y)z − 2qσ(y) }
+ mx {2qy + 2qσ(y) − 4q}
= 2ψ(z) + 2ψ(y).
So, Q(y) =
(2.46)
ψ(y)
2
satisfies equation (2.2) and the following inequality
1
|Q(y) − q(y)| = |mx {qy + qσ(y) − 2q − 2q(y)}|
2
1
≤ sup |{q(xy) + q(xσ(y)) − 2q(x) − 2q(y)}|
x∈G 2
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44
δ = 22δ.
2
Consequently, there exists a mapping Q which satisfies the functional equation (2.2)
and the inequality |f (y) − Q(y)| ≤ 29δ. This completes the proof of the theorem.
≤
Theorem 2.6. Let σ be an automorphism of the amenable semigroup G such that
σ ◦ σ = I. Then equation (2.2) is stable for the pair (G, C).
Proof. From inequality (2.3), we deduce that for any fixed y ∈ G the function x 7−→
f (xy) + f (xσ(y)) − 2f (x) is bounded. Since G is amenable, then we can define
φ(y) = mx {fy + fσ(y) − 2f }
(2.47)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
Title Page
for all y ∈ G.
We have
(2.48)
Contents
φ(yz)+φ(yσ(z))
= mx {fyz + fσ(y)σ(z) − 2f } + mx {fyσ(z) + fσ(y)z − 2f }
= mx {fyz + fyσ(z) − 2fy } + mx {fσ(y)σ(z) + fσ(y)z − 2fσ(y) }
+ 2mx {fy + fσ(y) − 2f }
= φ(z) + φ(σ(z)) + 2φ(y) = 2φ(z) + 2φ(y),
so φ is a solution of equation (2.2). Moreover, we have
1
φ(y)
f (y) −
= |mx {fy + fσ(y) − 2f − 2f (y)}|
(2.49)
2 2
≤
1
sup |f (xy) + f (xσ(y)) − 2f (x) − 2f (y)| ≤ δ.
2 x∈G
This completes the proof of theorem.
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By using Theorem 2.4 and the proof of Proposition 2.1 we get the following
corollaries.
In the first corollary, the estimate improves the ones obtained in the proof of
Theorem 2.4.
Corollary 2.7. Let σ be an antiautomorphism of the amenable semigroup G such
that σ ◦ σ = I. Let B a Banach space. Suppose that f : G −→ B satisfies the
inequality (2.3). Then for every x ∈ G, the limit
#
"
n
n−k
X
−2n
2n
k−1
2
2n−k 2k−1
(2.50)
Q(x) = lim 2
f (x ) +
2 f (x
σ(x)
)
n→+∞
vol. 8, iss. 2, art. 47, 2007
k=1
exists. Moreover, Q is the unique solution of equation (1.4) satisfying
(2.51)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
Title Page
kf (x) − Q(x)k ≤ δ for all x ∈ G.
Corollary 2.8. Let σ be a morphism of the amenable semigroup G such that σ ◦ σ =
I. Then for every Banach space B, equation (2.2) is stable for the pair (G, B).
Corollary 2.9 ([20]). Let σ = I. Let G be an amenable semigroup. Then for every
Banach space B, equation
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(2.52)
f (xy) = f (x) + f (y), x, y ∈ G
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is stable for the pair (G, B).
Corollary 2.10 ([22]). Let σ(x) = x−1 . Let G be an amenable group. Then for
every Banach space B, equation
(2.53)
f (xy) + f (xy −1 ) = 2f (x) + 2f (y), x, y ∈ G
is stable for the pair (G, B).
Close
Corollary 2.11 ([2]). Let σ be an automorphism of the vector space G such that
σ ◦ σ = I. Then for every Banach space B, the equation
(2.54)
f (x + y) + f (x + σ(y)) = 2f (x) + 2f (y), x, y ∈ G
is stable for the pair (G, B).
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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3.
Stability of Equation (1.5) in Amenable Semigroups
In this section we investigate the Hyers-Ulam stability of the functional equation
(3.1)
f (xy) + g(xσ(y)) = f (x) + g(y),
x, y ∈ G,
where G is an amenable semigroup with element unity e and σ : G −→ G is an
automorphism of G such that σ ◦ σ = I.
The stability of equation (3.1) was studied by several authors in the case where
G is an abelian group and σ = −I. For more information, see for example [12].
First we establish some results which will be instrumental in proving our main
results.
In the following lemma, we will present a Hyers-Ulam stability result for Jensen’s
functional equation:
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
Title Page
Contents
(3.2)
f (xy) + f (xσ(y)) = 2f (x), x, y ∈ G.
Lemma 3.1. Let G be an amenable semigroup. Let σ be an homomorphism of G
such that σ ◦ σ = I and let f : G −→ C be a function. Assume that there exists
δ ≥ 0 such that
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|f (xy) + f (xσ(y)) − 2f (x)| ≤ δ
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for all x, y ∈ G. Then, there exists a solution J : G −→ C of Jensen’s functional
equation (3.2) such that
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(3.3)
Close
(3.4)
|f (x) − J(x) − f (e)| ≤ δ
for all x ∈ G.
Proof. Let us denote by f e (x) =
f (x)−f (σ(x))
the odd part of f .
2
f (x)+f (σ(x))
2
the even part of f and by f o (x) =
By replacing x by σ(x) and y by σ(y) in (3.3), we get
(3.5)
|f (σ(x)σ(y)) + f (σ(x)y) − 2f (σ(x))| ≤ δ.
Now, if we add (subtract) the argument of the inequality (3.3) to (from) inequality
(3.5), we deduce that the functions f e and f o satisfy the following inequalities
(3.6)
|f e (xy) + f e (xσ(y)) − 2f e (x)| ≤ δ
(3.7)
|f o (xy) + f o (xσ(y)) − 2f o (x)| ≤ δ
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
for all x, y ∈ G.
By putting x = e in (3.6), we obtain
(3.8)
δ
|f e (y) − f (e)| ≤ .
2
The inequality (3.7) can be written as follows
(3.9)
|f o (yx) − f o (σ(y)x) − 2f o (y)| ≤ δ.
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This implies that for fixed y ∈ G, the function x 7−→ f o (yx)−f o (σ(y)x) is bounded.
Since G is amenable, let mx be an invariant mean on the space of complex bounded
functions on G and define the mapping:
(3.10)
ψ(y) = mx {y f o −σ(y) f o } for all y ∈ G.
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Consequently from (3.10), we obtain that ψ satisfies the Jensen’s functional equation
(3.11)
ψ(yz)+ψ(yσ(z))
= mx {yz f o −σ(y)σ(z) f o } + mx {yσ(z) f o −σ(y)z f o }
= mx {yz f o −σ(y)z f o } + mx {yσ(z) f o −σ(y)σ(z) f o }
= mx {z [y f o −σ(y) f o ]} + mx {σ(z) [y f o −σ(y) f o ]}
= 2ψ(y).
The function J(y) =
following inequality
(3.12)
ψ(y)
2
satisfies the Jensen’s functional equation (3.2) and the
|J(y) − f o (y)| ≤
1
δ
sup |f o (yx) − f o (σ(y)x) − 2f o (y)| ≤ .
2 x∈G
2
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
Title Page
Contents
Finally, we obtain
(3.13)
|f (y) − J(y) − f (e)| = |f e (y) + f o (y) − J(y) − f (e)|
≤ |f e (y) − f (e)| + |f o (y) − J(y)| ≤ δ.
By using the proof of the preceding lemma, we get the stability of the Jensen
function equation
f (yx) + f (σ(y)x) = 2f (x),
x, y ∈ G.
Lemma 3.2. Let G be an amenable semigroup. Let σ be a homomorphism of G such
that σ ◦ σ = I and let f : G −→ C be a function. Assume that there exists δ ≥ 0
such that
(3.15)
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This completes the proof of Lemma 3.1.
(3.14)
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|f (yx) + f (σ(y)x) − 2f (x)| ≤ δ
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for all x, y ∈ G. Then, there exists a solution J : G −→ C of Jensen’s functional
equation (3.14) such that
|f (x) − J(x) − f (e)| ≤ δ
(3.16)
for all x ∈ G. More precisely, J is given by the formula
o
J(y) = mx {fyo − fσ(y)
} for all y ∈ G.
(3.17)
In the following lemma, we obtain a partial stability theorem for the Pexider’s
functional equation
vol. 8, iss. 2, art. 47, 2007
f1 (xy) + f2 (xσ(y)) = f3 (x) + f4 (y), x, y ∈ G
(3.18)
that includes the functional equation (1.5) and the Drygas’s functional equation:
(3.19)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
Title Page
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f (xy) + f (xσ(y)) = 2f (x) + f (y) + f (σ(y)), x, y ∈ G
as special cases.
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Lemma 3.3. Let G be an amenable semigroup. Let σ be an automorphism of G such
that σ ◦ σ = I. If the functions f1 , f2 , f3 , f4 : G −→ C satisfy the inequality
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|f1 (xy) + f2 (xσ(y)) − f3 (x) − f4 (y)| ≤ δ
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for all x, y ∈ G, then there exists a unique function q : G −→ C, a solution of
equation (1.4). Also, there exists a solution J1 , (resp. J2 ): G −→ C of Jensen’s
functional equation (3.14), (resp. (3.2)) such that,
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(3.20)
(3.21)
|f3 (x) − J2 (x) − q(x) − f3 (e)| ≤ 16δ,
(3.22)
|f4 (x) − J1 (x) − q(x) − f4 (e)| ≤ 16δ,
Close
(3.23)
(3.24)
and
e
1
1
e
f1 (x) + f2 (x) − q(x) − f1 (e) − f2 (e) ≤ 6δ,
2
2
|(f1e − f2e )(xy) − (f1e − f2e )(xσ(y))| ≤ 12δ,
o
1
1
f1 (x) − J1 (x) − J2 (x) ≤ 10δ
2
2
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
o
1
1
f2 (x) − J2 (x) + J1 (x) ≤ 10δ
2
2
Title Page
for all x, y ∈ G.
Contents
Proof. In the present proof, we follow the computations used in the papers [1], [12],
and [23].
For any function f : G −→ C, we define F (x) = f (x) − f (e).
By putting x = y = e in (3.20), we get
(3.25)
|f1 (e) + f2 (e) − f3 (e) − f4 (e)| ≤ δ.
Consequently, if we subtract the inequality (3.20) from the new inequality (3.25), we
obtain
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(3.26)
|F1 (xy) + F2 (xσ(y)) − F3 (x) − F4 (y)| ≤ 2δ.
Now, by replacing x by σ(x) and y by σ(y) in (3.26) and if we add (subtract) the
inequality obtained to (3.26), we deduce that
(3.27)
|F1e (xy) + F2e (xσ(y)) − F3e (x) − F4e (y)| ≤ 2δ,
and
(3.28)
|F1o (xy) + F2o (xσ(y)) − F3o (x) − F4o (y)| ≤ 2δ
for all x, y ∈ G. Hence, if we replace y by e, and x by e respectively in (3.27), we
get
|F1e (x) + F2e (x) − F3e (x)| ≤ 2δ
(3.29)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
and
vol. 8, iss. 2, art. 47, 2007
|F1e (y) + F2e (y) − F4e (y)| ≤ 2δ.
(3.30)
So, in view of (3.27), (3.29) and (3.30), we obtain
(3.31)
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|F1e (xy) + F2e (xσ(y)) − (F1e + F2e )(x) − (F1e + F2e )(y)|
≤ |F1e (xy) + F2e (xσ(y)) − F3e (x) − F4e (y)|
+ |F1e (x) + F2e (x) − F3e (x)| + |F1e (y) + F2e (y) − F4e (y)|
≤ 6δ.
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Page 27 of 36
By replacing y by σ(y) in (3.31), we get the following
(3.32)
|F1e (xσ(y))
+
F2e (xy)
−
(F1e
+
F2e )(x)
−
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(F1e
+
F2e )(y)|
≤ 6δ.
If we add (subtract) the inequality (3.31) to (3.32), we get
(3.33) |(F1e +F2e )(xy)+(F1e +F2e )(xσ(y))−2(F1e +F2e )(x)−2(F1e +F2e )(y)| ≤ 12δ,
(3.34)
|(F1e − F2e )(xy) − (F1e − F2e )(xσ(y))| ≤ 12δ
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for all x, y ∈ E1 . Hence, in view of Theorem 2.6, there exists a unique function q, a
solution of equation (1.4) such that
(3.35)
|(F1e + F2e )(x) − q(x)| ≤ 6δ for all x ∈ G.
Consequently, from (3.29), (3.30) and (3.35), we deduce that
(3.36)
|F3e (x) − q(x)| ≤ 8δ
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
and
vol. 8, iss. 2, art. 47, 2007
(3.37)
|F4e (x) − q(x)| ≤ 8δ
for all x ∈ G.
On the other hand, from (3.28) we get
(3.38)
|F3o (x)
−
F1o (x)
Title Page
Contents
−
F2o (x)|
≤ 2δ
and
(3.39)
|F4o (x) − F1o (x) + F2o (x)| ≤ 2δ,
for all x ∈ G. Hence, we obtain
(3.40)
|2F1o (x) − F3o (x) − F4o (x)| ≤ 4δ
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and
(3.41)
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|2F2o (x) − F3o (x) + F4o (x)| ≤ 4δ
for all x ∈ G and consequently, we have
(3.42)
|F3o (xy) + F3o (xσ(y)) − 2F3o (x)|
≤ |F3o (xy) − F1o (xy) − F2o (xy)|
+ |F3o (xσ(y)) − F1o (xσ(y)) − F2o (xσ(y))|
+ |F1o (xy) + F2o (xσ(y)) − F3o (x) − F4o (y)|
+ |F1o (xσ(y)) + F2o (xy) − F3o (x) − F4o (σ(y))|
≤ 8δ
vol. 8, iss. 2, art. 47, 2007
and
(3.43)
|F4o (yx) + F4o (σ(y)x) − 2F4o (x)|
≤ |F4o (yx) − F1o (yx) + F2o (yx)|
+ |F4o (σ(y)x) − F1o (σ(y)x) + F2o (σ(y)x)|
+ |F1o (yx) + F2o (yσ(x)) − F3o (y) − F4o (x)|
+ |F1o (σ(y)x) + F2o (σ(y)σ(x)) − F3o (σ(y)) − F4o (x)|
≤ 8δ
for all x, y ∈ G.
Now, from Lemma 3.1 and Lemma 3.2 there exist two solutions of Jensen’s functional equation (3.14) and (3.2), J1 , J2 : G −→ C such that
(3.44)
|F4o (x) − J1 (x)| ≤ 8δ.
and
(3.45)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
|F3o (x) − J2 (x)| ≤ 8δ
for all x ∈ G. Now, by small computations, we obtain the rest of the proof.
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By using the previous lemmas, we may deduce our main result.
Theorem 3.4. Let G be an amenable semigroup. Let σ be an automorphism of G
such that σ ◦ σ = I. If the functions f, g : G −→ C satisfy the inequality
(3.46)
|f (xy) + g(xσ(y)) − f (x) − g(y)| ≤ δ,
for all x, y ∈ G, then there exists a unique function q : G −→ C, a solution of
equation (1.4). Also, there exist two solutions J1 , (resp. J2 ): G −→ C of Jensen’s
functional equation (3.14), (resp. (3.2)) such that
(3.47)
|f (x) − J2 (x) − q(x) − f (e)| ≤ 16δ
and
(3.48)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
Title Page
|g(x) − J1 (x) − q(x) − g(e)| ≤ 16δ,
for all x ∈ G.
The stability of the Drygas’s functional equation (3.19) is a consequence of the
preceding theorem.
Theorem 3.5. Let G be an amenable semigroup. Let σ be an automorphism of G
such that σ ◦ σ = I. Let the function f : G −→ C satisfy the inequality
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|f (xy) + f (xσ(y)) − 2f (x) − f (y) − f (σ(y))| ≤ δ,
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for all x, y ∈ G. Then there exists a unique function q : G −→ C, a solution of
equation (1.4), and a solution J : G −→ C of Jensen’s functional equation (3.2)
such that
Close
(3.49)
(3.50)
for all x ∈ G.
|f (x) − J(x) − q(x) − f (e)| ≤ 16δ
Corollary 3.6. Let G be an amenable semigroup with a unity element. Let σ be an
automorphism of G such that σ ◦ σ = I. If the functions f1 , f2 , f3 , f4 : G −→ C
satisfy the functional equation
(3.51)
f1 (xy) + f2 (xσ(y)) = f3 (x) + f4 (y)
for all x, y ∈ G, then there exists a quadratic function q : G −→ C. There also
exists a function ν : G −→ C, a solution of
(3.52)
ν(xy) = ν(xσ(y)), x, y ∈ G.
vol. 8, iss. 2, art. 47, 2007
In addition, there exist α, β, γ, δ ∈ C and two solutions J1 , (resp. J2 ) of Jensen’s
equation (3.14) (resp. (3.2)) such that
(3.53)
(3.54)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
1
1
1
1
f1 (x) = J1 (x) + J2 (x) + ν(x) + q(x) + α,
2
2
2
2
1
1
1
1
f2 (x) = − J1 (x) + J2 (x) − ν(x) + q(x) + β,
2
2
2
2
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(3.55)
f3 (x) = J2 (x) + q(x) + γ
and
(3.56)
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f4 (x) = J1 (x) + q(x) + δ
for all x ∈ G.
From Lemma 3.3, we can deduce the results obtained in [12].
Corollary 3.7. Let G be a vector space. If the functions f1 , f2 , f3 , f4 : G −→ C
satisfy the inequality
|f1 (x + y) + f2 (x − y) − f3 (x) − f4 (y)| ≤ δ
(3.57)
for all x, y ∈ G, then there exists a unique function q : G −→ C, a solution of
equation (1.2). Also, α ∈ C exists, and there are exactly two additive functions a1 ,
a2 : G −→ C such that
1
1
1
(3.58)
f1 (x) − 2 a1 (x) − 2 a2 (x) − 2 q(x) − f1 (0) − α ≤ 19δ,
(3.59)
1
1
1
f2 (x) + a1 (x) − a2 (x) − q(x) − f2 (0) + α ≤ 19δ,
2
2
2
(3.60)
|f3 (x) − a2 (x) − q(x) − f3 (0)| ≤ 16δ
and
(3.61)
vol. 8, iss. 2, art. 47, 2007
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|f4 (x) − a1 (x) − q(x) − f4 (0)| ≤ 16δ
for all x ∈ G.
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The following corollary follows from Lemma 3.3. This result is well known in
the commutative case, see for example [15].
Corollary 3.8. Let G be an amenable semigroup. If the functions f1 , f2 , f3 : G −→
C satisfy the inequality
(3.62)
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
|f1 (xy) − f2 (x) − f3 (y)| ≤ δ
Close
for all x, y ∈ G, then there exists a unique additive function a : G −→ C such that
(3.63)
|f1 (x) − a(x) − f1 (e)| ≤ 38δ,
(3.64)
|f2 (x) − a(x) − f2 (e)| ≤ 16δ
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
and
(3.65)
|f3 (x) − a(x) − f3 (e)| ≤ 16δ
vol. 8, iss. 2, art. 47, 2007
for all x ∈ G.
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References
[1] B. BOUIKHALENE, E. ELQORACHI AND Th. M. RASSIAS, On the HyersUlam stability of approximately Pexider mappings, Math. Inequalities. and
Appl., (accepted for publication).
[2] B. BOUIKHALENE, E. ELQORACHI AND Th. M. RASSIAS, On the generalized Hyers-Ulam stability of the quadratic functional equation with a general
involution, Nonlinear Funct. Anal. Appl., (accepted for publication).
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
[3] P.W. CHOLEWA, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76–86.
vol. 8, iss. 2, art. 47, 2007
[4] S. CZERWIK, On the stability of the quadratic mapping in normed spaces, Abh.
Math. Sem. Univ. Hamburg, 62 (1992), 59–64.
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[5] G.L. FORTI, Hyers-Ulam stability of functional equations in several variables,
Aequationes Math., 50 (1995), 143–190.
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[6] Z. GAJDA, On stability of additive mappings, Internat. J. Math. Sci., 14 (1991),
431–434.
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[7] F.P. GREENLEAF, Invariant Means on Topological Groups, [Van Nostrand
Mathematical Studies V. 16], Van Nostrand, New York-Toronto-LondonMelbourne, 1969.
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[8] D.H. HYERS, On the stability of the linear functional equation, Proc. Nat.
Acad. Sci. U.S.A. 27 (1941), 222–224.
[9] D.H. HYERS AND Th.M. RASSIAS, Approximate homomorphisms, Aequationes Math., 44 (1992), 125–153.
[10] D.H. HYERS, G. I. ISAC AND Th.M. RASSIAS, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998.
[11] S.-M. JUNG, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press Inc., Palm Harbor, Florida, 2003.
[12] S.-M. JUNG, Hyers-Ulam-Rassias stability of Jensen’s equation and its application, Proc. Amer. Math. Soc. 126 (1998), 3137–3143.
[13] S.-M. JUNG AND P.K. SAHOO, Hyers-Ulam stability of the quadratic equation
of Pexider type, J. Korean Math. Soc., 38(3) (2001), 645–656.
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[16] Th.M. RASSIAS, On the stability of linear mapping in Banach spaces, Proc.
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[18] F. SKOF, Local properties and approximations of operators, Rend. Sem. Math.
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[19] H. STETKÆR, Functional equations on abelian groups with involution, Aequationes Math., 54 (1997), 144–172.
[20] L. SZÉKELYHIDI, Note on a stability theorem, Canad. Math. Bull., 25 (1982),
500–501.
Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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[21] S.M. ULAM, A Collection of Mathematical Problems, Interscience Publ. New
York, 1961. Problems in Modern Mathematics, Wiley, New York 1964.
[22] Contributions to the Theory of Functional Equations, PhD Thesis, University
of Waterloo, Waterloo, Ontario, Canada, 2006.
[23] D. YANG, D. YANG, Remarks on the stabilities of Drygas’ equation and
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Hyers-Ulam Stability
Bouikhalene Belaid, Elqorachi Elhoucien
and Redouani Ahmed
vol. 8, iss. 2, art. 47, 2007
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