Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 2, Article 49, 2004
THE STABILITY OF SOME LINEAR FUNCTIONAL EQUATIONS
BELAID BOUIKHALENE
D EPARTMENT OF M ATHEMATICS
U NIVERSITY OF I BN T OFAIL
FACULTY OF S CIENCES BP 133
K ENITRA 14000, M OROCCO .
bbouikhalene@yahoo.fr
Received 14 January, 2004; accepted 25 April, 2004
Communicated by K. Nikodem
A BSTRACT. In this note, we deal with the Baker’s superstability for the following linear functional equations
m
X
f (x + y + ai ) = f (x)f (y),
x, y ∈ G,
i=1
m
X
[f (x + y + ai ) + f (x − y − ai )] = 2f (x)f (y),
x, y ∈ G,
i=1
where G is an abelian group, a1 , . . . , am (m ∈ N) are arbitrary elements in G and f is a
complex-valued function on G.
Key words and phrases: Linear functional equations, Stability, Superstability.
2000 Mathematics Subject Classification. 39B72.
1. I NTRODUCTION
Let G be an abelian group. The main purpose of this paper is to generalize the results obtained
in [4] and [5] for the linear functional equations
m
X
(1.1)
f (x + y + ai ) = f (x)f (y),
x, y ∈ G,
i=1
(1.2)
m
X
[f (x + y + ai ) + f (x − y − ai )] = 2f (x)f (y),
x, y ∈ G,
i=1
where a1 , . . . , am (m ∈ N), are arbitrary elements in G and f is a complex-valued function
on G. In the case where G is a locally compact group, the form of L∞ (G) solutions of (1.1)
(resp. (1.2)) are determined in [2] (resp. [6]). Some particular cases of these linear functional
equations are:
ISSN (electronic): 1443-5756
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012-04
2
B ELAID B OUIKHALENE
• The linear functional equations
x, y ∈ G,
(1.3)
f (x + y + a) = f (x)f (y),
(1.4)
f (x + y + a) + f (x − y − a) = 2f (x)f (y),
x, y ∈ G,
(1.5)
f (x + y + a) − f (x − y + a) = 2f (x)f (y),
x, y ∈ G,
(1.6)
f (x + y + a) + f (x − y + a) = 2f (x)f (y),
x, y ∈ G,
see [1], [2], [6], [7] and [8].
• Cauchy’s functional equation
f (x + y) = f (x)f (y),
(1.7)
x, y ∈ G,
• D’Alembert’s functional equation
f (x + y) + f (x + y) = 2f (x)f (y),
(1.8)
x, y ∈ G.
To complete our consideration, we give some applications.
We shall need the results below for later use.
2. G ENERAL P ROPERTIES
Proposition 2.1. Let δ > 0. Let G be an abelian group and let f be a complex-valued function
defined on G such that
m
X
(2.1)
f (x + y + ai ) − f (x)f (y) ≤ δ, x, y ∈ G,
i=1
then one of the assertions is satisfied
i) If f is bounded, then
√
m2 + 4δ
, x ∈ G.
2
ii) If f is unbounded, then there exists a sequence (zn )n∈N in G such that f (zn ) 6= 0 and
limn |f (zn )| = +∞ and that the convergence of the sequences of functions
|f (x)| ≤
(2.2)
m+
m
x→
(2.3)
1 X
f (zn + x + ai ), n ∈ N,
f (zn ) i=1
to the function
x → f (x),
m
(2.4)
1 X
x→
f (zn + x + y + aj + ai ), n ∈ N, 1 ≤ j ≤ m, y ∈ G,
f (zn ) i=1
to the function
x → f (x + y + aj ),
is uniform.
J. Inequal. Pure and Appl. Math., 5(2) Art. 49, 2004
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T HE S TABILITY
OF
S OME L INEAR F UNCTIONAL E QUATIONS
3
i) Let X = sup |f |, then for all x ∈ G we have
Proof.
|f (x)f (x)| ≤ mX + δ
from which we obtain that
X 2 − mX − δ ≤ 0
hence
√
m2 + 4δ
.
2
ii) Since f is unbounded then there exists a sequence (zn )n∈N in G such that f (zn ) 6= 0
and limn |f (zn )| = +∞. Using (2.1) one has
m
1 X
δ
f (zn + x + ai ) − f (x) ≤
, x ∈ G, n ∈ N,
|f (zn )|
f (zn )
i=1
m+
X≤
by letting n → ∞, we obtain
m
1 X
f (zn + x + ai ) = f (x)
lim
n f (zn )
i=1
and
m
1 X
lim
f (zn + x + y + aj + ai ) = f (x + y + aj ).
n f (zn )
i=1
Proposition 2.2. Let δ > 0. Let G be an abelian group and let f be a complex-valued function
defined on G such that
m
X
(2.5)
[f (x + y + ai ) + f (x − y − ai )] − 2f (x)f (y) ≤ δ, x, y ∈ G,
i=1
then one of the assertions is satisfied
i) If f is bounded, then
√
m2 + 2δ
, x ∈ G.
2
ii) If f is unbounded, then there exists a sequence (zn )n∈N ∈ G such that f (zn ) 6= 0 and
limn |f (zn )| = +∞ and that the convergence of the sequences of functions
|f (x)| ≤
(2.6)
m+
m
(2.7)
1 X
x→
[f (zn + x + ai ) + f (zn − x − ai )], n ∈ N,
f (zn ) i=1
to the function
x → 2f (x),
m
1 X
(2.8) x →
[f (zn + x + y + aj + ai ) + f (zn − x − y − aj − ai )],
f (zn ) i=1
n ∈ N, 1 ≤ j ≤ m, y ∈ G,
to the function
x → 2f (x + y + aj ),
J. Inequal. Pure and Appl. Math., 5(2) Art. 49, 2004
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B ELAID B OUIKHALENE
m
1 X
(2.9) x →
[f (zn + x − y − aj + ai ) + f (zn − x + y + aj − ai )],
f (zn ) i=1
n ∈ N, 1 ≤ j ≤ m, y ∈ G,
to the function
x → 2f (x − y − aj )
is uniform.
Proof. The proof is similar to the proof of Proposition 2.1.
i) Let X = sup |f |, then for all x ∈ G we have
δ
X 2 − mX − ≤ 0
2
hence
√
m + m2 + 2δ
X≤
.
2
ii) Follows from the fact that
m
1 X
δ
[f (zn + x + ai ) + f (zn − x − ai )] − 2f (x) ≤
, x ∈ G, n ∈ N.
f (zn )
|f (zn )|
i=1
3. T HE M AIN R ESULTS
The main results are the following theorems.
Theorem 3.1. Let δ > 0. Let G be an abelian group and let f be a complex-valued function
defined on G such that
m
X
(3.1)
f (x + y + ai ) − f (x)f (y) ≤ δ, x, y ∈ G,
i=1
then either
|f (x)| ≤
(3.2)
m+
√
m2 + 4δ
, x ∈ G,
2
or
m
X
(3.3)
f (x + y + ai ) = f (x)f (y), x, y ∈ G.
i=1
Proof. The idea is inspired by the paper [3].
If f is bounded, then from (2.2) we obtain the first case of the theorem. For the remainder, we
get by using the assertion ii) in Proposition 2.1, for all x, y ∈ G, n ∈ N
m
m
m
X
1 X
1 X
f (zn + x + y + aj + ai ) − f (x)
f (zn + y + aj )
f (zn ) i=1
f (zn ) j=1
j=1
(
)
m m
X
X
1
≤
f (zn + x + y + aj + ai ) − f (x)f (zn + y + aj ) f (zn )
j=1
i=1
≤
mδ
,
|f (zn )|
J. Inequal. Pure and Appl. Math., 5(2) Art. 49, 2004
http://jipam.vu.edu.au/
T HE S TABILITY
OF
S OME L INEAR F UNCTIONAL E QUATIONS
5
since the convergence is uniform, we have
m
X
f (x + y + ai ) − f (x)f (y) ≤ 0.
i=1
i.e. f is a solution of the functional equation (1.1).
Theorem 3.2. Let δ > 0. Let G be an abelian group and let f be a complex-valued function
defined on G such that
m
X
(3.4)
[f (x + y + ai ) + f (x − y − ai )] − 2f (x)f (y) ≤ δ, x, y ∈ G,
i=1
then either
|f (x)| ≤
(3.5)
m+
√
m2 + 2δ
, x ∈ G.
2
or
(3.6)
m
X
[f (x + y + ai ) + f (x − y − ai )] = 2f (x)f (y), x, y ∈ G.
i=1
Proof. By the assertion i) in Proposition 2.2 we get the first case of the theorem. For the second
case we have by the inequality (3.4) that
( m
)
m
X
X
1
[f (zn + x + y + aj + ai ) + f (zn − x − y − aj − ai )]
f
(z
)
n
j=1
i=1
)
( m
m
X
X
1
+
[f (zn + x − y − aj + ai ) + f (zn − x + y + aj − ai )]
f (zn ) i=1
j=1
m
1 X
− 2f (x)
[f (zn + y + aj ) + f (zn − y − aj )]
f (zn ) j=1
( m
m
X
X
1
=
[f (zn + x + y + aj + ai ) + f (zn − x + y + aj − ai )]
f (zn ) i=1
j=1
− 2f (x)f (zn + y + aj ) ( m
m
X
X
1
+
[f (zn + x − y − aj + ai ) + f (zn − x − y − aj − ai )]
f
(z
)
n
j=1
i=1
− 2f (x)f (zn − y − aj ) ( m
m X
X
1
≤
[f (zn + x + y + aj + ai ) + f (zn − x + y + aj − ai )]
f (zn )
j=1
i=1
− 2f (x)f (zn + y + aj ) J. Inequal. Pure and Appl. Math., 5(2) Art. 49, 2004
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6
B ELAID B OUIKHALENE
( m
m X
X
1
+
[f (zn + x − y − aj + ai ) + f (zn − x − y − aj − ai )]
f (zn )
j=1
i=1
− 2f (x)f (zn − y − aj ) ≤
2mδ
,
|f (zn )|
since the convergence is uniform, we have
m
X
[f (x + y + ai ) + f (x − y − ai )] − 4f (x)f (y) ≤ 0.
2
i=1
i.e. f is a solution of the functional equation (1.2).
4. A PPLICATIONS
From Theorems 3.1 and 3.2, we easily obtain .
Corollary 4.1. Let δ > 0. Let G be an abelian group and let f be a complex-valued function
defined on G such that
|f (x + y + a) − f (x)f (y)| ≤ δ, x, y ∈ G,
(4.1)
then either
|f (x)| ≤
(4.2)
1+
√
1 + 4δ
, x ∈ G.
2
or
f (x + y + a) = f (x)f (y) x, y ∈ G.
(4.3)
Remark 4.2. Taking a = 0 in Corollary 4.1, we find the result obtained in [4].
Corollary 4.3. Let δ > 0. Let G be an abelian group and let f be a complex-valued function
defined on G such that
(4.4)
|f (x + y + a) + f (x − y − a) − 2f (x)f (y)| ≤ δ, x, y ∈ G,
then either
|f (x)| ≤
(4.5)
1+
√
1 + 2δ
, x ∈ G,
2
or
f (x + y + a) + f (x − y − a) = 2f (x)f (y), x, y ∈ G.
(4.6)
Remark 4.4. Taking a = 0 in Corollary 4.3, we find the result obtained in [5].
Corollary 4.5. Let δ > 0. Let G be an abelian group and let f be a complex-valued function
defined on G such that
m
X
(4.7)
[f (x + y + ai ) − f (x − y + ai )] − 2f (x)f (y) ≤ δ, x, y ∈ G,
i=1
then either
(4.8)
|f (x)| ≤
J. Inequal. Pure and Appl. Math., 5(2) Art. 49, 2004
m+
√
m2 + 2δ
, x ∈ G,
2
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T HE S TABILITY
OF
S OME L INEAR F UNCTIONAL E QUATIONS
7
or
m
X
(4.9)
[f (x + y + ai ) + f (x − y − ai )] = 2f (x)f (y), x, y ∈ G.
i=1
Proof. Let f be a complex-valued function defined on G which satisfies the inequality (4.7),
then for all x, y ∈ G we have
2|f (x)||f (y) + f (−y)|
= |2f (x)f (y) + 2f (x)f (−y)|
m
X
= [f (x + y + ai ) − f (x − y + ai )]
i=1
−
[f (x + y + ai ) − f (x − y + ai )] + 2f (x)f (y) + 2f (x)f (−y)
i=1
m
X
≤ 2f (x)f (y) −
[f (x + y + ai ) − f (x − y + ai )]
i=1
m
X
+ 2f (x)f (−y) −
[f (x − y + ai ) − f (x + y + ai )]
m
X
i=1
≤ 2δ.
Since f is unbounded it follows that f (−y) = −f (y), for all y ∈ G. Consequently f satisfies
the inequality (3.4) and one has the remainder.
Corollary 4.6. Let δ > 0. Let G be an abelian group and let f be a complex-valued function
defined on G such that
m
X
(4.10)
[f
(x
+
y
+
a
)
+
f
(x
−
y
+
a
)]
−
2f
(x)f
(y)
≤ δ, x, y ∈ G,
i
i
i=1
then either
|f (x)| ≤
(4.11)
m+
√
m2 + 2δ
, x ∈ G,
2
or
(4.12)
m
X
[f (x + y + ai ) + f (x − y − ai )] = 2f (x)f (y) x, y ∈ G.
i=1
Proof. Let f be a complex-valued function defined on G which satisfies the inequality (4.10),
then for all x, y ∈ G we have
2|f (x)||f (y) − f (−y)| = |2f (x)f (y) − 2f (x)f (−y)|
m
X
= [f (x + y + ai ) + f (x − y + ai )]
i=1
−
m
X
[f (x + y + ai ) + f (x − y + ai )]
i=1
+ 2f (x)f (y) − 2f (x)f (−y)
J. Inequal. Pure and Appl. Math., 5(2) Art. 49, 2004
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B ELAID B OUIKHALENE
m
X
≤ [f (x − y + ai ) + f (x + y + ai )] − 2f (x)f (−y)
i=1
m
X
+ [f (x + y + ai ) + f (x − y + ai )] − 2f (x)f (y)
i=1
≤ 2δ.
Since f is unbounded it follows that f (−y) = f (y), for all y ∈ G. Consequently f satisfies the
inequality (3.4) and one has the remainder.
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(2000), 211–218.
[4] J. BAKER, J. LAWRENCE AND F. ZORZITTO, The stability of the equation f (x + y) = f (x)f (y),
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[7] B. NAGY, A sine functional equation in Banach algebras, Publ. Math. Debrecen, 24 (1977), 77–99.
[8] E.B. VAN VLECK, A functional equation for the sine, Ann. Math., 11 (1910), 161–165.
J. Inequal. Pure and Appl. Math., 5(2) Art. 49, 2004
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