Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 5, Article 104, 2003
ON THE STABILITY OF A CLASS OF FUNCTIONAL EQUATIONS
BELAID BOUIKHALENE
D ÉPARTEMENT DE M ATHÉMATIQUES ET I NFORMATIQUE
FACULTÉ DES S CIENCES BP 133,
14000 K ÉNITRA , M OROCCO .
bbouikhalene@yahoo.fr
Received 20 July, 2003; accepted 24 October, 2003
Communicated by K. Nikodem
A BSTRACT. In this paper, we study the Baker’s superstability for the following functional equation
XZ
(E (K))
f (xkϕ(y)k −1 )dωK (k) = |Φ|f (x)f (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 is a continuous
complex-valued function on G satisfying the Kannappan type condition, for all x, y, z ∈ G
Z Z
Z Z
(*)
f (zkxk −1 hyh−1 )dωK (k)dωK (h) =
f (zkyk −1 hxh−1 )dωK (k)dωK (h).
K
K
K
K
We treat examples and give some applications.
Key words and phrases: Functional equation, Stability, Superstability, Central function, Gelfand pairs.
2000 Mathematics Subject Classification. 39B72.
1. I NTRODUCTION , N OTATIONS AND P RELIMINARIES
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 7−→ G is a morphism of G if ϕ is a homeomorphism
of G onto itself which is either a group-homorphism, i.e (ϕ(xy) = ϕ(x)ϕ(y), x, y ∈ G), or a
group-antihomorphism, 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) of a K-invariant morphisms of G (i.e
ϕ(K) ⊂ K). The number of elements of a finite group Φ will be designated by |Φ|. The Banach
algebra of bounded measures on G with complex values is denoted by M (G) and the Banach
space of all complex measurable and essentially bounded functions on G by L∞ (G). C(G)
designates the Banach space of all continuous complex valued functions on G. We say that a
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
The author would like to greatly thank the referee for his helpful comments and remarks.
098-03
2
B ELAID B OUIKHALENE
function f is a K-central function on G if
f (kx) = f (xk), x ∈ G, k ∈ K.
(1.1)
In the case where G = K, a function f is central if
f (xy) = f (yx) x, y ∈ G.
(1.2)
See [2] for more information.
In this note, we are going to generalize the results obtained by J.A. Baker in [8] and [9]. As
applications, we discuss the following cases:
a) K ⊂ Z(G), (Z(G) is the center of G).
b) f (hxk) = f (x), h, k ∈ K, x ∈ G (i.e. f is bi-K-invariant (see [3] and [6])).
c) f (hxk) = χ(k)f (x)χ(h), x ∈ G, k, h ∈ K (χ is a unitary character of K) (see [11]).
d) (G, K) is a Gelfand pair (see [3], [6] and [11]).
e) G = K (see [2]).
In the next section, we note some results for later use.
2. G ENERAL P ROPERTIES
In what follows, we study general properties. Let G, K and Φ be given as above.
Proposition 2.1. For an arbitrary fixed τ ∈ Φ, the mapping
Φ −→ Φ,
ϕ −→ ϕ ◦ τ
is a bijection.
Proof. Follows from the fact that Φ is a finite group.
Proposition 2.2. Let ϕ ∈ Φ and f ∈ C(G), then we have:
R
R
i) K f (xkϕ(hy)k −1 )dωK (k) = K f (xkϕ(yh)k −1 )dωK (k), x, y ∈ G, h ∈ K.
ii) If f satisfy (*), the for all z, y, x ∈ G, we have
Z Z
Z Z
−1 −1
f (zhϕ(ykxk )h )dωK (h)dωK (k) =
f (zhϕ(xkyk −1 )h−1 )dωK (h)dωK (k).
K
Proof.
K
K
K
i) Let ϕ ∈ Φ and let x, y ∈ G, h ∈ K, then we have
Case 1: If ϕ is a group-homomorphism, we obtain, by replacing k by kϕ(h)−1
Z
Z
−1
f (xkϕ(hy)k )dωK (k) =
f (xkϕ(h)ϕ(y)k −1 )dωK (k)
K
K
Z
=
f (xkϕ(y)ϕ(h)k −1 )dωK (k)
K
Z
=
f (xkϕ(yh)k −1 )dωK (k).
K
Case 2: if ϕ is a group-antihomomorphism, we have, by replacing k by kϕ(h)
Z
Z
−1
f (xkϕ(hy)k )dωK (k) =
f (xkϕ(y)ϕ(h)k −1 )dωK (k)
K
K
Z
=
f (xkϕ(h)ϕ(y)k −1 )dωK (k)
K
Z
=
f (xkϕ(yh)k −1 )dωK (k).
K
J. Inequal. Pure and Appl. Math., 4(5) Art. 104, 2003
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O N THE S TABILITY
OF
A C LASS OF F UNCTIONAL E QUATIONS
3
ii) Follows by simple computation.
Proposition 2.3. For each τ ∈ Φ and x, y ∈ G, we have
XZ
XZ
−1
(2.1)
f (xkϕ(τ (y))k )dωK (k) =
f (xkψ(y)k −1 )dωK (k).
ϕ∈Φ
K
ψ∈Φ
K
Proof. By applying Proposition 2.1, we get that when ϕ is iterated over Φ, the morphism of the
form ϕ ◦ τ annihilates all the elements of Φ.
3. T HE M AIN R ESULTS
Theorem 3.1. Let G be a locally compact group; let K be a compact subgroup of G with the
normalized Haar measure ωK and let Φ given as above.
Let δ > 0 and let f ∈ C(G) such that f satisfies the condition (*) and the functional inequality
(3.1)
X Z
−1
f (xkϕ(y)k )dωK (k) − |Φ|f (x)f (y) ≤ δ, x, y ∈ G.
K
ϕ∈Φ
Then one of the assertions is satisfied:
(a) If f is bounded, then
|f (x)| ≤
(3.2)
|Φ| +
p
|Φ|2 + 4δ|Φ|
.
2|Φ|
(b) If f is unbounded, then
i) f is K-central,
ii) Rf ◦ τ = f , for all τ ∈ Φ, R
iii) K f (xkyk −1 )dωK (k) = K f (ykxk −1 )dωK (k), x, y ∈ G.
Proof.
a) Let X = sup |f |, then we get for all x ∈ G
|Φ||f (x)f (x)| ≤ |Φ|X + δ,
from which we obtain that
|Φ|X 2 − |Φ|X − δ ≤ 0,
such that
p
|Φ|2 + 4δ|Φ|
.
2|Φ|
b) i) Let x, y ∈ G, h ∈ K, then by using Proposition 2.2, we find
X≤
|Φ| +
|Φ||f (x)||f (hy) − f (yh)| = ||Φ|f (x)f (hy) − |Φ|f (x)f (yh)|
X Z
≤
f (xkϕ(hy)k −1 )dωK (k) − |Φ|f (x)f (hy)
ϕ∈Φ K
X Z
−1
+
f (xkϕ(yh)k )dωK (k) − |Φ|f (x)f (yh)
K
ϕ∈Φ
≤ 2δ.
Since f is unbounded it follows that f (yh) = f (hy), for all h ∈ K, y ∈ G.
J. Inequal. Pure and Appl. Math., 4(5) Art. 104, 2003
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4
B ELAID B OUIKHALENE
ii) Let τ ∈ Φ, by using Proposition 2.3, we get for all x, y ∈ G
|Φ||f (x)||f ◦ τ (y) − f (y)| = ||Φ|f (x)f (τ (y)) − |Φ|f (x)f (y)|
X Z
−1
≤
f (xkϕ(τ (y))k )dωK (k) − |Φ| f (x)f (τ (y))
ϕ∈Φ K
X Z
+
f (xkψ(y)k −1 )dωK (k) − |Φ|f (x)f (y)
K
ψ∈Φ
≤ 2δ.
Since f is unbounded it follows that f ◦ τ = f , for all τ ∈ Φ.
iii) Let f be an unbounded solution of the functional inequality (3.1), such that f satisfies the condition (*), then, for all x, y ∈ G, we obtain, by using Part i) of Proposition
2.2:
Z
Z
−1
−1
f (ykxk )dωK (k)
|Φ||f (z)| f (xkyk )dωK (k) −
K
K
Z
f (z)f (xkyk −1 )dωK (k)
= |Φ|
K
Z
−1
f (z)f (ykxk )dωK (k)
− |Φ|
K
Z
Z
f (zhϕ(xkyk −1 )h−1 )dωK (h)dωK (k)
≤ Σϕ∈Φ
K
K
Z
−1
f (z)f (xkyk )dωK (k)
− |Φ|
Z K Z
+ Σϕ∈Φ
f (zhϕ(ykxk −1 )h−1 )dωK (h)dωK (k)
K
K
Z
−|Φ|
f (z)f (ykxk −1 )dωK (k)
K
≤ 2δ.
Since f is unbounded we get
Z
Z
−1
f (xkyk )dωK (k) =
f (ykxk −1 )dωK (k), x, y ∈ G.
K
K
The main result is the following theorem.
Theorem 3.2. Let δ > 0 and let f ∈ C(G) such that f satisfies the condition (*) and the
functional inequality
X Z
f (xkϕ(y)k −1 )dωK (k) − |Φ|f (x)f (y) ≤ δ, x, y ∈ G.
(3.3)
K
ϕ∈Φ
Then either
(3.4)
|f (x)| ≤
J. Inequal. Pure and Appl. Math., 4(5) Art. 104, 2003
|Φ| +
p
|Φ|2 + 4δ|Φ|
, x ∈ G,
2|Φ|
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O N THE S TABILITY
OF
A C LASS OF F UNCTIONAL E QUATIONS
5
or
XZ
(E (K))
f (xkϕ(y)k −1 )dωK (k) = |Φ|f (x)f (y), x, y ∈ G.
K
ϕ∈Φ
Proof. The idea is inspired by the paper [1].
If f is bounded, by using Theorem 3.1, we obtain the first case of the theorem.
Now let f be an unbounded solution of the functional inequality (3.3), then there exists a
sequence (zn )n∈N in G such that f (zn ) 6= 0 and limn |f (zn )| = +∞.
For the second case we will use the following lemma.
Lemma 3.3. Let f be an unbounded solution of the functional inequality (3.3) satisfying the
condition (*) and let (zn )n∈N be a sequence in G such that f (zn ) 6= 0 and limn |f (zn )| = +∞.
It follows that the convergence of the sequences of functions:
i)
P
x 7−→
(3.5)
R
ϕ∈Φ
K
f (zn kϕ(x)k −1 )dωK (k)
, n ∈ N,
f (zn )
to the function
x 7−→ |Φ|f (x).
ii)
R
P
(3.6) x 7−→
ϕ∈Φ
K
f (zn hϕ(xkϕ(τ (y))k −1 )h−1 )dωK (h)
, n ∈ N, τ ∈ Φ, k ∈ K, y ∈ G
f (zn )
to the function
x 7−→ |Φ|f (xkτ (y)k −1 ) τ ∈ Φ, k ∈ K, y ∈ G,
is uniform.
By inequality (3.1), we have
P
R
−1
δ
ϕ∈Φ K f (zn kϕ(y)k )dωK (k)
− |Φ|f (y) ≤
,
|f (zn )|
f (zn )
then we have, by letting n 7−→ +∞, that
R
P
−1
ϕ∈Φ K f (zn kϕ(y)k )dωK (k)
lim
= |Φ|f (y),
n
f (zn )
and
f (zn hϕ (xkϕ (τ (y)) k −1 ) h−1 ) dωK (h)
lim
= |Φ|f (xkτ (y)k −1 ).
n
f (zn )
Since by Proposition 2.3, we have
R
P
−1 −1
XZ
ϕ∈Φ K f (zn hϕ(x)kϕ(τ (y))k h )dωK (h)
dωK (k)
f (zn )
τ ∈Φ K
R
P
−1 −1
XZ
ϕ∈Φ K f (zn hϕ(x)kψ(y)k h )dωK (h)
=
dωK (k),
f
(z
n)
K
ψ∈Φ
P
ϕ∈Φ
R
K
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6
B ELAID B OUIKHALENE
combining this and the fact that f satisfies the condition (*), we obtain
R
−1 −1
X Z P
ϕ∈Φ K f (zn hϕ(x)kϕ(τ (y))k h )dωK (h)
dωK (k)
f (zn )
τ ∈Φ K
R
P
−1
f
(z
kψ(y)k
)dω
(k)
δ
n
K
ψ∈Φ K
−|Φ|f (x)
.
≤
|f (zn )|
f (zn )
Since the convergence is uniform, we have
XZ
f (xkϕ(y)k −1 )dωK (k) − |Φ|2 f (x)f (y) ≤ 0,
|Φ|
K
ϕ∈Φ
thus (E (K)) holds and the proof is complete.
4. A PPLICATIONS
If K ⊂ Z(G), we obtain the following corollary.
Corollary 4.1. Let δ > 0 and let f be a complex-valued function on G satisfying the Kannappan
condition (see [10])
f (zxy) = f (zyx), x, y ∈ G,
(*)
and the functional inequality
X
f (xϕ(y)) − |Φ|f (x)f (y) ≤ δ, x, y ∈ G.
(4.1)
ϕ∈Φ
Then either
|f (x)| ≤
(4.2)
|Φ| +
p
|Φ|2 + 4δ|Φ|
, x ∈ G,
2|Φ|
or
X
(4.3)
f (xϕ(y)) = |Φ|f (x)f (y), x, y ∈ G.
ϕ∈Φ
If G is abelian, then the condition (*) holds and we have the following:
If Φ = {i} (resp. Φ = {i, σ}), where i(x) = x and σ(x) = −x, we find the Baker’s stability
see [8] (resp. [9]).
If f (kxh) = χ(k)f (x)χ(h), k, h ∈ K and x ∈ G, where χ is a character of K (see [11]),
then we have the following corollary.
Corollary 4.2. Let δ > 0 and let f ∈ C(G) such that f (kxh) = χ(k)f (x)χ(h), k, h ∈ K,
x ∈ G,
Z Z
Z Z
f (zkyhx)χ(k)χ(h)dωK (k)dωK (h)
(*)
f (zkxhy)χ(k)χ(h)dωK (k)dωK (h) =
K
K
K
K
and
(4.4)
X Z
f (xkϕ(y))χ(k)dωK (k) − |Φ|f (x)f (y) ≤ δ, x, y ∈ G.
K
ϕ∈Φ
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O N THE S TABILITY
OF
A C LASS OF F UNCTIONAL E QUATIONS
7
Then either
|f (x)| ≤
(4.5)
|Φ| +
p
|Φ|2 + 4δ|Φ|
, x ∈ G,
2|Φ|
or
XZ
(4.6)
ϕ∈Φ
f (xkϕ(y))χ(k)dωK (k) = |Φ|f (x)f (y), x, y ∈ G.
K
Proposition 4.3. If the algebra χωK ?M (G)?χωK is commutative then the condition (*) holds.
Proof. Since f (kxh) = χ(k)f (x)χ(h), k, h ∈ K, x ∈ G, then we have χωK ? f ? χωK = f .
Suppose that the algebra χωK ? M (G) ? χωK is commutative, then we get:
Z Z
Z Z
−1
−1
f (xkyk hzh )dωK (k)dωK (h) =
f (xkyhzh−1 k −1 )dωK (k)dωK (h)
K
K
K
K
= hδz ? χωK ? δy ? χωK ? δx , f i
= hδz ? χωK ? δy ? χωK ? δx , χωK ? f ? χωK i
= hχωK ? δz ? χωK ? δy ? χωK ? δx ? χωK , f i
= hχωK ? δz ? χωK ? δx ? χωK ? δy ? χωK , f i
Z Z
=
f (ykxk −1 hzh−1 )dωK (k)dωK (h).
K
K
Let f be bi-K-invariant (i.e f (hxk) = f (x), h, k ∈ K, x ∈ G), then we have:
Corollary 4.4. Let δ > 0 and let f ∈ C(G) be bi-K-invariant such that for all x, y, z ∈ G,
Z Z
Z Z
(*)
f (zkxhy)dωK (k)dωK (h) =
f (zkyhx)dωK (k)dωK (h),
K
K
K
K
and
(4.7)
X Z
f (xkϕ(y))dωK (k) − |Φ|f (x)f (y) ≤ δ, x, y ∈ G.
K
ϕ∈Φ
Then either
|f (x)| ≤
(4.8)
|Φ| +
p
|Φ|2 + 4δ|Φ|
, x ∈ G,
2|Φ|
or
(4.9)
XZ
ϕ∈Φ
f (xkϕ(y))dωK (k) = |Φ|f (x)f (y), x, y ∈ G.
K
Proposition 4.5. If the pair (G, K) is a Gelfand pair (i.e ωK ? M (G) ? ωK is commutative),
then the condition (*) holds.
Proof. We take χ = 1 (unit character of K) in Proposition 4.3 (see [3] and [6]).
In the next corollary, we assume that G = K is a compact group.
Lemma 4.6. If f is central, then f satisfies the condition (*). Consequently, we have
Z
Z
−1
(4.10)
f (xtyt )dt =
f (ytxt−1 )dt, x, y ∈ G.
G
J. Inequal. Pure and Appl. Math., 4(5) Art. 104, 2003
G
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8
B ELAID B OUIKHALENE
Corollary 4.7. Let δ > 0 and let f be a complex measurable and essentially bounded function
on G such that
X Z
f (xtϕ(y)t−1 )dt − |Φ|f (x)f (y) ≤ δ, x, y ∈ G.
(4.11)
G
ϕ∈Φ
Then
|f (x)| ≤
(4.12)
|Φ| +
p
|Φ|2 + 4δ|Φ|
, x ∈ G.
2|Φ|
Proof. Let f ∈ L∞ (G) be a solution of the inequality (4.11), then f is bounded, if not, then f
satisfies the second case of Theorem 3.2 which implies that f is central (i.e the condition (*)
holds) and f is a solution of the following functional equation
XZ
(4.13)
f (xtϕ(y)t−1 )dt = |Φ|f (x)f (y), x, y ∈ G.
ϕ∈Φ
G
In view of the proposition in [5], we have that f is continuous. Since G is compact, then the
proof is accomplished.
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J. Inequal. Pure and Appl. Math., 4(5) Art. 104, 2003
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