Journal of Inequalities in Pure and
Applied Mathematics
ON HYERS-ULAM STABILITY OF A SPECIAL CASE OF O’CONNOR’S
AND GAJDA’S FUNCTIONAL EQUATIONS
volume 6, issue 2, article 32,
2005.
1
BELAID BOUIKHALENE, 2 ELHOUCIEN ELQORACHI AND
2
AHMED REDOUANI
1 University
of Ibn Tofail
Faculty of Sciences
Department of Mathematics
Kenitra, Morocco
EMail: bbouikhalene@yahoo.fr
2 University
of Ibn Zohr
Faculty of Sciences
Department of Mathematics
Agadir, Morocco
EMail: elqorachi@hotmail.com
EMail: redouani_ahmed@yahoo.fr
Received 08 November, 2004;
accepted 01 March, 2005.
Communicated by: K. Nikodem
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
213-04
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Abstract
In this paper, we obtain the Hyers-Ulam stability for the following functional
equation
XZ
f(xkϕ(y)k−1 )dωK (k) = |Φ|a(x)a(y), x, y ∈ G,
ϕ∈Φ K
where G is a locally compact group, K is a compact subgroup of G, ωK is the
normalized Haar measure of K, Φ is a finite group of K-invariant morphisms
of G and f, a : G −→ C are continuous complex-valued functions such that f
satisfies the Kannappan type condition
Z Z
(*)
f(zkxk−1 hyh−1 )dωK (k)dωK (h)
K K
Z Z
=
f(zkyk−1 hxh−1 )dωK (k)dωK (h),
K K
for all x, y, z ∈ G.
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
Title Page
Contents
2000 Mathematics Subject Classification: 39B72.
Key words: Functional equations, Hyers-Ulam stability, Gelfand pairs.
Contents
1
2
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Generalized Stability Results of Cauchy’s and Wilson’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3
The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
References
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Page 2 of 38
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
1.
Introduction
Let G be a locally compact group. Let K be a compact subgroup of G. Let
ωK be the normalized Haar measure of K. A mapping ϕ : G → G is a
morphism of G if ϕ is a homeomorphism of G onto itself which is either
a group-homomorphism, (i.e. ϕ(xy) = ϕ(x)ϕ(y), x, y ∈ G), or a groupantihomomorphism, (i.e. ϕ(xy) = ϕ(y)ϕ(x), x, y ∈ G). We denote by
M or(G) the group of morphisms of G and Φ a finite subgroup of M or(G)
which is K-invariant (i.e. ϕ(K) ⊂ K, for all ϕ ∈ Φ). The number of elements of a finite group Φ will be designated by |Φ|. The Banach algebra of the
complex bounded measures on G is denoted by M (G), it is the topological dual
of C0 (G): Banach space of continuous functions vanishing at infinity. Finally
the Banach space of all complex measurable and essentially bounded functions
on G is denoted by L∞ (G) and C(G) designates the space of all continuous
complex valued functions on G.
The stability problem for functional equations are strongly related to the
question of S.M. Ulam concerning the stability of group homomorphisms [26],
[16]. During the last decades, the stability problems of several functional equations have been extensively investigated by a number of mathematicians ([16],
[2], [3], [22], [23], [24], [25], [20], ...). The main purpose of this paper is to generalize the Hyers-Ulam stability problem for the following functional equation
(1.1)
XZ
ϕ∈Φ
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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−1
f (xkϕ(y)k )dωK (k) = |Φ|a(x)a(y), x, y ∈ G,
K
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Page 3 of 38
where G is a locally compact group, and f, a ∈ C(G) with the assumption that
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
f satisfies the Kannappan type condition: (*)
Z Z
K
f (zkxk −1 hyh−1 )dωK (k)dωK (h)
K
Z Z
=
f (zkyk −1 hxh−1 )dωK (k)dωK (h),
K
K
for all x, y, z ∈ G.
In the case where G is a locally compact abelian group, O’Connor [19],
Gajda [14] and Stetkær [21] studied respectively the functional equation
(1.2)
f (x − y) =
n
X
ai (x)ai (y),
x, y ∈ G, n ∈ N,
i=1
(1.3)
f (x + y) + f (x − y) = 2
n
X
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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ai (x)ai (y),
x, y ∈ G, n ∈ N,
i=1
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and
Z
f (xh · y)dh = a(x)a(y),
(1.4)
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
x, y ∈ G̃,
H
where G̃ is a locally compact group and H is a compact subgroup of Aut(G̃).
In the case n = 1 equations (1.2) and (1.3) are special cases of (1.1). Moreover, taking G = G̃×s H the semi direct product of G̃ and H, and K = {e}×H,
we observe that equation (1.4) is also a special case of (1.1).
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
This equation may be considered as a common generalization of functional
equations
(1.5)
f (xy −1 ) = a(x)a(y),
(1.6)
f (xy) + f (xy −1 ) = 2a(x)a(y),
x, y ∈ G,
x, y ∈ G.
It is also a generalization of the equations
Z
(1.7)
f (xky −1 k −1 )dωK (k) = a(x)a(y),
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
x, y ∈ G,
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
K
Z
−1
Z
f (xky −1 k −1 )dωK (k)
f (xkyk )dωK (k) +
(1.8)
K
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K
= 2a(x)a(y),
Z
(1.9)
f (xky −1 )χ(k)dωK (k) = a(x)a(y),
x, y ∈ G,
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x, y ∈ G,
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Z
Z
f (xky)χ(k)dωK (k) +
(1.10)
K
f (xky −1 )χ(k)dωK (k)
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K
= 2a(x)a(y),
x, y ∈ G,
Page 5 of 38
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
Z
(1.11)
f (xky −1 )dωK (k) = a(x)a(y),
x, y ∈ G,
K
Z
Z
f (xky)dωK (k) +
(1.12)
K
f (xkϕ(y −1 ))dωK (k)
K
= 2a(x)a(y),
x, y ∈ G, ([10], [11]).
If G is a compact group, equation (1.1) may be considered as a generalization
of the equations
Z
(1.13)
f (xty −1 t−1 )dt = a(x)a(y),
x, y ∈ G,
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
G
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Z
(1.14)
f (xtyt−1 )dt +
G
(1.15)
Z
f (xty −1 t−1 )dt = 2a(x)a(y),
x, y ∈ G,
G
XZ
ϕ∈Φ
f (xtϕ(y −1 )t−1 )dt = |Φ|a(x)a(y),
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x, y ∈ G.
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Furthermore the following equations are also a particular case of (1.1).
X
(1.16)
f (xϕ(y −1 )) = |Φ|a(x)a(y),
x, y ∈ G,
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Page 6 of 38
ϕ∈Φ
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
(1.17)
XZ
ϕ∈Φ
(1.18)
XZ
ϕ∈Φ
f (xkϕ(y −1 ))dωK (k) = |Φ|a(x)a(y),
x, y ∈ G,
K
f (xkϕ(y −1 ))χ(k)dωK (k) = |Φ|a(x)a(y),
x, y ∈ G,
K
where χ is a character of K. For more information about the equations (1.1)
– (1.18) (see [1], [4], [6], [7], [11], [12], [14], [15], [19], [21]).
In the next section, we note some results for later use.
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
2.
Generalized Stability Results of Cauchy’s and
Wilson’s Equations
Let G, K and Φ be given as above. One can prove (see [4]) the following two
propositions.
Proposition 2.1. For an arbitrary fixed τ ∈ Φ, the mapping
Φ 3 ϕ 7→ ϕ ◦ τ ∈ Φ
is a bijection and for all x, y ∈ G, we have
XZ
XZ
−1
f (xkϕ(τ (y))k )dωK (k) =
f (xkψ(y)k −1 )dωK (k).
ϕ∈Φ
K
ψ∈Φ
K
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
Proposition 2.2. Let ϕ ∈ Φ and f ∈ C(G). Then
Title Page
i)
Z
Z
−1
f (xkϕ(yh)k −1 )dωK (k), x, y ∈ G, h ∈ K.
f (xkϕ(hy)k )dωK (k) =
K
K
ii) Moreover, if f satisfies the Kannappan type condition (∗), then we have
Z Z
f (zhϕ(ykxk −1 )h−1 )dωK (h)dωK (k)
K K
Z Z
=
f (zhϕ(xkyk −1 )h−1 )dωK (h)dωK (k),
K
K
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for all z, y, x ∈ G.
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
The next results extend the ones obtained in [4], [8], [9], [10] and [13].
Theorem 2.3. Let ε : G −→ R+ be a continuous function. Let f, g : G −→ C
be continuous functions such that f satisfies the Kannappan type condition (∗)
and
X Z
(2.1) f (xkϕ(y)k −1 )dωK (k) − |Φ|f (x)g(y) ≤ ε(y),
x, y ∈ G.
K
ϕ∈Φ
If f is unbounded, then g satisfies the functional equation
XZ
(2.2)
g(xkϕ(y)k −1 )dωK (k) = |Φ|g(x)g(y),
ϕ∈Φ
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
x, y ∈ G.
K
Proof. Let ε : G −→ R+ be a continuous function, and let f, g ∈ C(G) satisfying inequality (2.1). Let Φ = Φ+ ∪ Φ− , where Φ+ (resp. Φ− ) is a set of
group-homomorphisms (resp. of group-antihomomorphisms). By using Propositions 2.1, 2.2 and the fact that f satisfies the condition (*), for all x, y, z ∈ G,
we get
X Z
|Φ||f (z)| g(xkϕ(y)k −1 )dωK (k) − |Φ|g(x)g(y)
ϕ∈Φ K
X Z
−1
2
=
|Φ|f (z)g(xkϕ(y)k )dωK (k) − |Φ| f (z)g(x)g(y)
K
ϕ∈Φ
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
X Z X Z
≤
f (zhτ (xkϕ(y)k −1 )h−1 )dωK (k)dωK (h)
K
K
ϕ∈Φ
τ ∈Φ
XZ
−
|Φ|f (z)g(xkϕ(y)k −1 )dωK (k)
ϕ∈Φ K
X Z X Z
+
f (zhτ (xkϕ(y)k −1 )h−1 )dωK (k)dωK (h)
ϕ∈Φ K τ ∈Φ K
XZ
−|Φ|g(y)
f (zhτ (x)h−1 )dωK (h)
K
τ ∈Φ
X Z
+ |Φ||g(y)| f (zkτ (x)k −1 )dωK (k) − |Φ|f (z)g(x)
τ ∈Φ K
X Z X Z
=
f (zhτ (xkϕ(y)k −1 )h−1 )dωK (k)dωK (h)
ϕ∈Φ K τ ∈Φ K
XZ
−1
−
|Φ|f (z)g(xkϕ(y)k )dωK (k)
ϕ∈Φ K
X Z X Z
+
f (zhτ (x)kψ(y)k −1 )h−1 )dωK (k)dωK (h)
ψ∈Φ K τ ∈Φ+ K
XZ X Z
f (zhk −1 ψ(y)kτ (x)h−1 )dωK (k)dωK (h)
+
ψ∈Φ
K τ ∈Φ−
K
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
−|Φ|g(y)
−1
f (zhτ (x)h )dωK (h)
K
XZ
τ ∈Φ
X Z
+ |Φ||g(y)| f (zkτ (x)k −1 )dωK (k) − |Φ|f (z)g(x)
τ ∈Φ K
X Z X Z
=
f (zhτ (xkϕ(y)k −1 )h−1 )dωK (k)dωK (h)
ϕ∈Φ K τ ∈Φ K
XZ
−1
−
|Φ|f (z)g(xkϕ(y)k )dωK (k)
ϕ∈Φ K
X Z X Z
+
f (zhτ (x)h−1 kψ(y)k −1 )dωK (k)dωK (h)
ψ∈Φ K τ ∈Φ+ K
XZ X Z
+
f (zhτ (x)h−1 kψ(y)k −1 )dωK (k)dωK (h)
ψ∈Φ
K τ ∈Φ−
−|Φ|g(y)
K
−1
f (zhτ (x)h )dωK (h)
K
XZ
τ ∈Φ
X Z
−1
+ |Φ||g(y)| f (zkτ (x)k )dωK (k) − |Φ|f (z)g(x)
τ ∈Φ K
X Z X Z
=
f (zhτ (xkϕ(y)k −1 )h−1 )dωK (k)dωK (h)
K
K
ϕ∈Φ
τ ∈Φ
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
−1
−
|Φ|f (z)g(xkϕ(y)k )dωK (k)
ϕ∈Φ K
X Z X Z
+
f (zhτ (x)h−1 kψ(x)k −1 )dωK (k)dωK (h)
ψ∈Φ K τ ∈Φ K
XZ
−1
f (zhτ (x)h )dωK (h)
−|Φ|g(y)
K
τ ∈Φ
X Z
−1
+ |Φ||g(y)| f (zkτ (x)k )dωK (k) − |Φ|f (z)g(x)
τ ∈Φ K
X Z X Z
≤
f (zhτ (xkϕ(y)k −1 )h−1 )dωK (h)
ϕ∈Φ K τ ∈Φ K
−1 − |Φ|f (z)g(xkϕ(y)k )dωK (k)
X Z X Z
+
f (zhτ (x)h−1 kψ(y)k −1 ))dωK (h)
τ ∈Φ K ψ∈Φ K
−1
− |Φ|f (zhτ (x)h )g(y)dωK (h)
X Z
+ |Φ||g(y)| f (zhτ (x)h−1 )dωK (h) − |Φ|f (z)g(x)
τ ∈Φ K
Z
X
ε(xkϕ(y)k −1 )dωK (k) + |Φ|ε(y) + |Φ||g(y)|ε(x).
≤
XZ
ϕ∈Φ
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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K
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
Since f is unbounded, then it follows that g is a solution of (2.1). This ends the
proof of our theorem.
The next results extend the ones obtained in [8] and [13].
Theorem 2.4. Let ε : G −→ R+ . Let f, g : G −→ C be continuous functions
such that f satisfies the condition (∗) and
X Z
(2.3) f (xkϕ(y)k −1 )dωK (k) − |Φ|f (x)g(y) ≤ ε(y),
x, y ∈ G.
K
ϕ∈Φ
Suppose furthermore there exists x0 ∈ G such that |g(x0 )| > 1. Then there
exists exactly one solution F ∈ C(G) of the equation
XZ
(2.4)
F (xkϕ(y)k −1 )dωK (k) = |Φ|F (x)g(y),
x, y ∈ G,
ϕ∈Φ
Contents
such that F − f is bounded and one has
(2.5)
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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K
ε(x0 )
|F (x) − f (x)| ≤
,
|Φ|(|g(x0 )| − 1)
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
x ∈ G.
Proof. In the proof, we use the ideas and methods that are analogous to the ones
used in [8], [13] and [20]. Let β = |Φ|g(x0 ), for all x ∈ G, one has
X Z
−1
(2.6)
f (xkϕ(x0 )k )dωK (k) − βf (x) ≤ ε(x0 ),
x, y ∈ G.
K
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ϕ∈Φ
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
We define the following functions sequence
XZ
(2.7)
G1 (x) =
f (xkϕ(x0 )k −1 )dωK (k),
ϕ∈Φ
(2.8)
Gn+1 (x) =
XZ
ϕ∈Φ
x ∈ G,
K
Gn (xkϕ(x0 )k −1 )dωK (k),
x ∈ G and n ∈ N.
K
Next, we will prove the uniform convergence of the function sequence (β −n Gn )n≥1 ,
therefore we need to show by induction the following inequalities
(2.9)
(2.10)
|Gn+1 (x) − βGn (x)| ≤ |Φ|n ε(x0 ),
x ∈ G, n ≥ 1,
|Gn (x) − β n f (x)| ≤ ε(x0 )(|Φ|n−1 + |Φ|n−2 |β| + · · · + |β|n−1 ),
and
(2.11)
|β −(n+1) Gn+1 (x) − β −n Gn (x)| ≤ |β|−(n+1) |Φ|n ε(x0 ).
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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In view of (2.5) one has for all x ∈ G
Close
|G2 (x) − βG1 (x)|
Z
X Z
X
=
G1 (xkϕ(x0 )k −1 )dωK (k) − β
f (xkϕ(x0 )k −1 )dωK (k)
K
K
ϕ∈Φ
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Page 14 of 38
ϕ∈Φ
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
X Z X Z
=
f (xkϕ(x0 )k −1 hτ (x0 )h−1 )dωK (h)dωK (k)
K
K
ϕ∈Φ
τ ∈Φ
XZ
−β
f (xkτ (x0 )k −1 )dωK (k)
τ ∈Φ K
X Z X Z
≤
f (xkτ (x0 )k −1 hϕ(x0 )h−1 )dωK (h)
τ ∈Φ K ϕ∈Φ K
−1 − βf (xkτ (x0 )k )dωK (k)
≤ |Φ|ε(x0 ).
Assume (2.8) holds for n ≥ 1, then for n + 1, one has
X Z
|Gn+2 (x) − βGn+1 (x)| = Gn+1 (xkϕ(x0 )k −1 )dωK (k)
ϕ∈Φ K
XZ
−β
Gn (xkϕ(x0 )k −1 )dωK (k)
ϕ∈Φ K
XZ
≤
|Gn+1 (xkϕ(x0 )k −1 )dωK (k)
ϕ∈Φ
K
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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−1
− βGn (xkϕ(x0 )k )|dωK (k)
≤ |Φ|n+1 ε(x0 ).
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
In view of (2.5) we have for all x ∈ G
X Z
−1
|G1 (x) − βf (x)| = f (xkϕ(x0 )k )dωK (k) − βf (x)
K
ϕ∈Φ
≤ ε(x0 ).
Suppose (2.9) is true for n ≥ 1. For n + 1 one has
|Gn+1 (x) − β n+1 f (x)|
≤ |Gn+1 (x) − βGn (x)| + |β||Gn (x) − β n f (x)|
≤ |Φ|n ε(x0 ) + |β|ε(x0 )(|Φ|n−1 + |Φ|n−2 |β| + · · · + |β|n−1 )
= ε(x0 )(|Φ|n + |Φ|n−1 |β| + · · · + |β|n ).
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
For inequality (2.10), using (2.8), for all x ∈ G we get
|β −(n+1) Gn+1 (x) − β −n Gn (x)| = |β −(n+1) ||Gn+1 (x) − βGn (x)|
≤ |β|−(n+1) |Φ|n ε(x0 ).
So by using inequality (2.10) we deduce the uniform convergence of the sequence (β −n Gn )n≥1 . Let F be a continuous function defined by
F (x) =
lim β −n Gn (x),
n−→+∞
x ∈ G.
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Since
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β −(n+1) Gn+1 (x) = β −1
XZ
ϕ∈Φ
β −n Gn (xkϕ(x0 )k −1 )dωK (k),
Page 16 of 38
K
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
then one has
βF (x) =
XZ
ϕ∈Φ
F (xkϕ(x0 )k −1 )dωK (k),
x ∈ G.
K
In view of (2.9), one has for all x ∈ G
|β −n Gn (x) − f (x)| ≤ |β|−n ε(x0 )(|Φ|n−1 + |Φ|n−2 |β| + · · · + |β|n−1 ),
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
which proves that
|F (x) − f (x)| <
ε(x0 )
,
|Φ|(|g(x0 )| − 1)
Now we are going to show that F satisfies the equation
XZ
F (xkϕ(y)k −1 )dωK (k) = |Φ|F (x)g(y),
ϕ∈Φ
x ∈ G.
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
x, y ∈ G.
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K
Thus we need to show by induction the inequality
(2.12)
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X Z
−n
−1
−n
β Gn (xkϕ(y)k )dωK (k) − |Φ|β Gn (x)g(y)
K
ϕ∈Φ
ε(y)
≤
.
|g(x0 )|n
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
For n = 1, one has, by using the fact that f satisfies the condition (∗)
Z
1 X
G1 (xkϕ(y)k −1 )dωK (k) − |Φ|G1 (x)g(y)
β ϕ∈Φ K
Z XZ
1 X
f (xkϕ(y)k −1 hτ (x0 )h−1 )dωK (k)dωK (h)
= β ϕ∈Φ K τ ∈Φ K
XZ
−|Φ|g(y)
f (xkτ (x0 )k −1 )dωK (k)
τ ∈Φ K
Z XZ
1 X
= f (xkτ (x0 )k −1 hϕ(y)h−1 )dωK (k)dωK (h)
β τ ∈Φ K ϕ∈Φ K
XZ
−1
−|Φ|g(y)
f (xkτ (x0 )k )dωK (k)
τ ∈Φ K
Z X Z
1X
≤
f (xkτ (x0 )k −1 hϕ(y)h−1 )dωK (h)
β τ ∈Φ K ϕ∈Φ K
−1
− |Φ|g(y)f (xkτ (x0 )k )g(y)dωK (k)
|Φ|ε(y)
ε(y)
≤
=
.
|β|
|g(x0 )|
Assume (2.12) holds for some n ≥ 1. For n + 1, one has by using the fact that
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
f satisfies the condition (∗)
X Z
β −(n+1) Gn+1 (xkϕ(y)k −1 )dωK (k) − |Φ|β −(n+1) Gn+1 (x)g(y)
ϕ∈Φ K
Z XZ
1 X
β −n Gn (xkϕ(y)k −1 hτ (x0 )h−1 )dωK (h)dωK (k)
= β ϕ∈Φ K τ ∈Φ K
XZ
− |Φ|g(y)
β −n Gn (xkτ (x0 )k −1 )dωK (k)|
τ ∈Φ
K
Z X Z
1X
≤
β −n Gn (xkτ (x0 )k −1 hϕ(y)h−1 )dωK (h)
β τ ∈Φ K ϕ∈Φ K
− |Φ|g(y)β −n Gn (xkτ (x0 )k −1 )|dωK (k)|
ε(y)
|Φ|
ε(y)
≤
=
.
n
|Φ| | g(x0 ) || |g(x0 )|
|g(x0 )|n+1
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
3.
The Main Results
In the next proposition, we investigate the stability of the functional equation
(1.1).
Proposition 3.1. Let δ > 0. Let f, a ∈ C(G) such that
X Z
−1 −1
(3.1) f (xkϕ(y )k )dωK (k) − |Φ|a(x)a(y) ≤ δ,
K
x, y ∈ G.
ϕ∈Φ
Then
i) If f is bounded then a is bounded and one has,
s
δ
|a(x)| ≤ sup |f | +
,
x ∈ G,
|Φ|
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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s
δ
δ
|f (x)| ≤ |a(e)| sup |f | +
+
,
|Φ| |Φ|
x ∈ G.
ii) If f is unbounded then a(e) 6= 0. Furthermore there exists x0 ∈ G such
that |a(x0 )| > |a(e)|.
Proof. i) Let f be a continuous bounded solution of (3.1), then by taking x = y
in (3.1) we get
|Φ||a(x)|2 ≤ |Φ| sup |f | + δ,
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x ∈ G,
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
i.e.
s
|a(x)| ≤
sup |f | +
δ
,
|Φ|
x ∈ G.
For y = e in (3.1) we get
|f (x)| ≤ |a(x)||a(e)| +
i.e.
s
|f (x)| ≤ |a(e)| sup |f | +
δ
,
|Φ|
δ
δ
+
,
|Φ| |Φ|
x ∈ G.
δ
.
|Φ|
We will prove (ii) by contradiction. If a(e) = 0 then |f (x)| <
If |a(x)| ≤ |a(e)|, for all x ∈ G, then by taking y = e in (3.1) one has
δ
|f (x)| ≤ |a(e)| +
,
|Φ|
2
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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x ∈ G,
i.e. f is bounded, which is the desired contradiction.
The main results are the following theorems.
Theorem 3.2. Let δ > 0. Assume that f, a ∈ C(G) satisfy inequality (3.1) and
f fulfills (∗). Then
i) f, a are bounded
or
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Special Case of O’Connor’s and
Gajda’s Functional Equations
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
ii) f is unbounded and
XZ
(3.2) a(e)
ǎ(xkϕ(y)k −1 )dωK (k) = |Φ|ǎ(x)ǎ(y),
ϕ∈Φ
x, y ∈ G,
K
where ǎ(x) = a(x−1 ), for x ∈ G.
Proof. ii) Since f is unbounded then by Proposition 3.1 we have a(e) 6= 0. By
using the fact that f and a satisfy inequality (3.1) one has
XZ
f (xkϕ(y)k −1 )dωK (k) = |Φ|a(x)ǎ(y) + θ(x, y),
x, y ∈ G,
ϕ∈Φ
K
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
where |θ(x, y)| < δ. By taking y = e we get for all x ∈ G
|Φ|f (x) = |Φ|a(x)a(e) + θ(x, e),
so
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1
(f (x) − a(x)a(e)) =
θ(x, e),
|Φ|
then we get
X Z
−1
f (xkϕ(y)k )dωK (k) − |Φ|f (x)g(y) < ε(y),
K
ϕ∈Φ
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Gajda’s Functional Equations
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x, y ∈ G,
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where
g(y) =
ǎ(y)
,
a(e)
and
ε(y) = δ(1 + |g(y)|).
Page 22 of 38
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
In view of Theorem 2.3, we deduce that
XZ
a(e)
ǎ(xkϕ(y)k −1 )dωK (k) = |Φ|ǎ(x)ǎ(y),
ϕ∈Φ
x, y ∈ G.
K
The cases of f bounded follows from Proposition 3.1.
Theorem 3.3. Let δ > 0. Assume that f, a ∈ C(G) satisfy inequality (3.1) and
f fulfills (∗). Then either
s
δ
(3.3)
|a(x)| ≤ sup |f | +
,
x ∈ G,
|Φ|
s
|f (x)| ≤ |a(e)| sup |f | +
(3.4)
δ
δ
+
,
|Φ| |Φ|
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
x ∈ G,
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or there exist x0 ∈ G such that |a(x0 )| > |a(e)| and a unique continuous
function F : G −→ C such that
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a)
a(e)
XZ
ϕ∈Φ
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F (xkϕ(y)k −1 )dωK (k) = |Φ|F (x)ǎ(y),
x, y ∈ G,
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b) F − f is bounded and one has
δ(|a(e)| + |a(x0 )|)
|F (x) − f (x)| ≤
,
|Φ|(|a(x0 )| − |a(e)|)
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x ∈ G.
Page 23 of 38
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
Proof. If f is bounded, by using Theorem 3.2 and Proposition 3.1, we obtain
the first case of the theorem.
Now, let f be unbounded. Since a(e) 6= 0 it follows that
X Z
−1
f (xkϕ(y)k )dωK (k) − |Φ|f (x)g(y) < ε(y),
x, y ∈ G,
K
ϕ∈Φ
where
g(y) =
ǎ(y)
a(e)
, and ε(y) = δ(1 + |g(y)|).
Finally, by using Proposition 3.1 and Theorem 2.4 we get the rest of the proof.
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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4.
Applications
The following theorems are a particular case of Theorem 3.3.
If K ⊂ Z(G), then we have
Theorem 4.1. Let δ > 0. Let f, a be a complex-valued functions on G such that
f satisfies the Kannappan condition (see [18])
x, y ∈ G
f (zxy) = f (zyx),
(4.1)
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
and the functional inequality
X
(4.2)
f
(xϕ(y))
−
|Φ|a(x)a(y)
≤ δ,
x, y ∈ G.
ϕ∈Φ
Then either
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s
(4.3)
|a(x)| ≤
sup |f | +
s
(4.4)
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
|f (x)| ≤ |a(e)| sup |f | +
δ
,
|Φ|
x ∈ G,
δ
δ
+
,
|Φ| |Φ|
x ∈ G,
or there exist x0 ∈ G such that |a(x0 )| > |a(e)| and a unique function F :
G −→ C such that
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a)
a(e)
X
F (xϕ(y)) = |Φ|F (x)ǎ(y),
x, y ∈ G,
ϕ∈Φ
b) F − f is bounded and one has
|F (x) − f (x)| ≤
δ(|a(e)| + |a(x0 )|)
,
|Φ|(|a(x0 )| − |a(e)|)
x ∈ G.
If G is abelian then condition (4.1) holds. By taking Φ = {I} (resp. Φ =
{I, −I}), we get the following corollaries.
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Corollary 4.2. Let δ > 0. Let f, a be complex-valued functions on G such that
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
(4.5)
|f (x − y) − a(x)a(y)| ≤ δ,
x, y ∈ G.
Then either
(4.6)
(4.7)
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|a(x)| ≤
p
sup |f | + δ,
p
|f (x)| ≤ |a(e)| sup |f | + δ + δ,
x ∈ G,
x ∈ G,
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or there exist x0 ∈ G such that |a(x0 )| > |a(e)| and a unique function F :
G −→ C such that
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a)
a(e)F (x + y) = F (x)ǎ(y),
x, y ∈ G,
Page 26 of 38
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
b) F − f is bounded and one has
|F (x) − f (x)| ≤
δ(|a(e)| + |a(x0 )|)
,
(|a(x0 )| − |a(e)|)
x ∈ G.
Corollary 4.3. Let δ > 0. Let f, a be a complex-valued functions on G such
that
(4.8)
|f (x + y) + f (x − y) − 2a(x)a(y)| ≤ δ,
x, y ∈ G.
Then either
r
δ
sup |f | + ,
x ∈ G,
2
r
δ δ
|f (x)| ≤ |a(e)| sup |f | + + ,
x ∈ G,
2 2
or there exist x0 ∈ G such that |a(x0 )| > |a(e)| and a unique function F :
G −→ C such that
(4.9)
|a(x)| ≤
a)
a(e)F (x + y) + a(e)F (x − y) = 2F (x)ǎ(y),
x, y ∈ G,
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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b) F − f is bounded and one has
δ(|a(e)| + |a(x0 )|)
|F (x) − f (x)| ≤
,
2(|a(x0 )| − |a(e)|)
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
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x ∈ G.
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
If f (kxh) = χ(k)f (x)χ(h), k, h ∈ K and x ∈ G, where χ is a character of
K, then we have
Theorem 4.4. Let δ > 0 and let χ be a character of K. Assume that (f, a) ∈
C(G) satisfy f (kxh) = χ(k)f (x)χ(h), k, h ∈ K, x ∈ G,
Z Z
(4.10)
K
K
f (zkxhy)χ(k)χ(h)dωK (k)dωK (h)
Z Z
=
f (zkyhx)χ(k)χ(h)dωK (k)dωK (h)
K
K
and
X Z
(4.11) f (xkϕ(y −1 ))χ(k)dωK (k) − |Φ|a(x)a(y) ≤ δ,
K
x, y ∈ G.
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
ϕ∈Φ
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Then either
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s
(4.12)
|a(x)| ≤
sup |f | +
s
(4.13)
|f (x)| ≤ |a(e)| sup |f | +
δ
,
|Φ|
x ∈ G,
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δ
δ
+
,
|Φ| |Φ|
x ∈ G,
or there exist x0 ∈ G such that |a(x0 )| > |a(e)| and a unique continuous
function F : G −→ C such that
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
a)
a(e)
XZ
ϕ∈Φ
F (xkϕ(y))χ(k)dωK (k) = |Φ|F (x)ǎ(y),
x, y ∈ G,
K
b) F − f is bounded and one has
|F (x) − f (x)| ≤
δ(|a(e)| + |a(x0 )|)
,
|Φ|(|a(x0 )| − |a(e)|)
x ∈ G.
Corollary 4.5. Let δ > 0 and let χ be a character of K. Assume that (f, a) ∈
C(G) satisfy f (kxh) = χ(k)f (x)χ(h), k, h ∈ K, x ∈ G,
Z Z
f (zkxhy)χ(k)χ(h)dωK (k)dωK (h)
K K
Z Z
=
f (zkyhx)χ(k)χ(h)dωK (k)dωK (h)
K
Z
Z
−1
f (xky)χ(k)dωK (k) +
f
(xky
))χ(k)dω
(k)
−
2a(x)a(y)
K
K
K
≤ δ,
Then either
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x, y ∈ G.
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r
(4.15)
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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(4.14)
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
|a(x)| ≤
δ
sup |f | + ,
2
x ∈ G,
Page 29 of 38
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
r
|f (x)| ≤ |a(e)| sup |f | +
(4.16)
δ δ
+ ,
2 2
x ∈ G,
or there exist x0 ∈ G such that |a(x0 )| > |a(e)| and a unique continuous
function F : G −→ C such that
a)
Z
Z
a(e)
F (xky)χ(k)dωK (k) + a(e)
K
F (xky −1 ))χ(k)dωK (k)
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
K
= 2F (x)ǎ(y),
x, y ∈ G,
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
b) F − f is bounded and one has
δ(|a(e)| + |a(x0 )|)
|F (x) − f (x)| ≤
,
2(|a(x0 )| − |a(e)|)
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x ∈ G.
Corollary 4.6. Let δ > 0 and let χ be a character of K. Assume that (f, a) ∈
C(G) satisfy f (kxh) = χ(k)f (x)χ(h), k, h ∈ K, x ∈ G, and
Z
−1
f (xky ))χ(k)dωK (k) − a(x)a(y) ≤ δ,
(4.17)
x, y ∈ G.
K
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Then either
(4.18)
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|a(x)| ≤
p
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sup |f | + δ,
x ∈ G,
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
p
|f (x)| ≤ |a(e)| sup |f | + δ + δ, x ∈ G,
or there exist x0 ∈ G such that |a(x0 )| > |a(e)| and a unique continuous
function F : G −→ C such that
a)
Z
a(e)
F (xky)χ(k)dωK (k) = F (x)ǎ(y),
x, y ∈ G,
K
b) F − f is bounded and one has
δ(|a(e)|+ | a(x0 ) |)
,
|F (x) − f (x)| ≤
(|a(x0 )| − |a(e)|)
x ∈ G.
Remark 1. If the algebra χωK ∗M (G)∗χωK is commutative then the condition
(∗) holds [4]. Furthermore in the case where Φ = {I}, we do not need the
condition (∗).
In the next theorem we assume that f is bi-K-invariant (i.e. f (hxk) =
f (x), h, k ∈ K, x ∈ G), then we have
Theorem 4.7. Let δ > 0 and assume that (f, a) ∈ C(G) satisfy f (kxh) = f (x),
k, h ∈ K, x ∈ G,
Z Z
Z Z
(4.19)
f (zkxhy)dωK (k)dωK (h) =
f (zkyhx)dωK (k)dωK (h)
K
K
K
K
(4.20)
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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and
X Z
f (xkϕ(y −1 ))dωK (k) − |Φ|a(x)a(y) ≤ δ,
K
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Special Case of O’Connor’s and
Gajda’s Functional Equations
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x, y ∈ G.
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ϕ∈Φ
J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
Then either
s
|a(x)| ≤
(4.21)
sup |f | +
s
|f (x)| ≤ |a(e)| sup |f | +
(4.22)
δ
,
|Φ|
x ∈ G,
δ
δ
+
,
|Φ| |Φ|
x ∈ G,
or there exist x0 ∈ G such that |a(x0 )| > |a(e)| and a unique continuous
function F : G −→ C such that
a)
a(e)
XZ
ϕ∈Φ
F (xkϕ(y))dωK (k) = |Φ|F (x)ǎ(y),
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b) F − f is bounded and one has
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δ(|a(e)| + |a(x0 )|)
,
|Φ|(|a(x0 )| − |a(e)|)
x ∈ G.
Corollary 4.8. Let δ > 0 and assume that (f, a) ∈ C(G) satisfy f (kxh) =
f (x), k, h ∈ K, x ∈ G,
Z Z
Z Z
f (zkxhy)dωK (k)dωK (h) =
f (zkyhx)dωK (k)dωK (h)
K
K
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
x, y ∈ G,
K
|F (x) − f (x)| ≤
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
K
K
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and
(4.23)
Z
Z
−1
f (xky)dωK (k) +
f (xky ))dωK (k) − 2a(x)a(y)
K
K
≤ δ,
x, y ∈ G.
Then either
r
δ
sup |f | + ,
x ∈ G,
2
r
δ δ
|f (x)| ≤ |a(e)| sup |f | + + ,
x ∈ G,
2 2
or there exist x0 ∈ G such that |a(x0 )| > |a(e)| and a unique continuous
function F : G −→ C such that
|a(x)| ≤
(4.24)
a)
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
Title Page
Contents
Z
a(e)
Z
F (xky)dωK (k)+a(e)
K
F (xky −1 )dωK (k) = 2F (x)ǎ(y), x, y ∈ G,
K
δ(|a(e)| + |a(x0 )|)
,
2(|a(x0 )| − |a(e)|)
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b) F − f is bounded and one has
|F (x) − f (x)| ≤
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
Corollary 4.9. Let δ > 0 and assume that (f, a) ∈ C(G) satisfy f (kxh) =
f (x), k, h ∈ K, x ∈ G, and
Z
−1
≤ δ,
(4.25)
f
(xky
))dω
(k)
−
a(x)a(y)
x, y ∈ G.
K
K
Then either
|a(x)| ≤
p
sup |f | + δ,
x ∈ G,
p
|f (x)| ≤ |a(e)| sup |f | + δ + δ,
x ∈ G,
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
or there exist x0 ∈ G such that |a(x0 )| > |a(e)| and a unique continuous
function F : G −→ C such that
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
(4.26)
a)
Z
a(e)
F (xky)dωK (k) = F (x)ǎ(y),
x, y ∈ G,
K
Contents
b) F − f is bounded and one has
|F (x) − f (x)| ≤
Title Page
δ(|a(e)| + |a(x0 )|)
,
(|a(x0 )| − |a(e)|)
x ∈ G.
Remark 2. If the algebra ωK ∗ M (G) ∗ ωK is commutative then the condition
(∗) holds [4].
In the next corollary, we assume that G = K is a compact group.
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J. Ineq. Pure and Appl. Math. 6(2) Art. 32, 2005
Theorem 4.10. Let δ > 0 and let f, a be complex measurable and essentially
bounded functions on G such that f is a central function and (f, a) satisfy the
inequality
X Z
−1
f
(xtϕ(y)t
)dt
−
|Φ|a(x)a(y)
(4.27)
x, y ∈ G.
≤ δ,
G
ϕ∈Φ
Then
s
(4.28)
and
sup |f | +
|a(x)| ≤
s
|f (x)| ≤ |a(e)| sup |f | +
δ
,
|Φ|
On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
δ
δ
+
,
|Φ| |Φ|
for all x ∈ G.
Proof. Let f, a ∈ L∞ (G). Since f is central, then it satisfies the condition (∗)
([4], [6]). If f is unbounded then a is a solution of the functional equation (3.2).
In view of [15], we get the fact that a is continuous. Since G is compact then a
is bounded. Consequently f is bounded, which is the desired property.
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References
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On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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On Hyers-Ulam Stability of a
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Gajda’s Functional Equations
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Ahmed Redouani
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Title Page
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On Hyers-Ulam Stability of a
Special Case of O’Connor’s and
Gajda’s Functional Equations
Belaid Bouikhalene,
Elhoucien Elqorachi and
Ahmed Redouani
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