E
extracta mathematicae Vol. 28, Núm. 2, 157 – 167 (2013)
On the Approximate Solution of D’Alembert Type
Equation Originating from Number Theory
B. Bouikhalene1, , E. Elqorachi2, , A. Charifi1
1
Sultan Moulay Slimane University, Polydisciplinaire Faculty, Beni-Mellal, Morocco
2
Ibn Zohr University, Faculty of Sciences, Agadir, Morocco
[email protected],
[email protected],
[email protected]
Presented by David Yost
Received December 2, 2012
Abstract: We solve the functional equation
E(α) : f (x1 x2 + αy1 y2 , x1 y2 + x2 y1 ) + f (x1 x2 − αy1 y2 , x2 y1 − x1 y2 ) = 2f (x1 , y1 )f (x2 , y2 ),
where (x1 , y1 ), (x2 , y2 ) ∈ R2 , f : R2 → C and α is a real parameter, on the monoid R2 . Also
we investigate the stability of this equation in the following setting:
|f (x1 x2 + αy1 y2 , x1 y2 + x2 y1 ) + f (x1 x2 − αy1 y2 , x2 y1 − x1 y2 ) − 2f (x1 , y1 )f (x2 , y2 )|
≤ min{φ(x1 ), ψ(y1 ), ϕ(x2 ), ζ(y2 )}.
From this result, we obtain the superstability of this equation.
Key words: D’Alembert functional equation, monoid R2 , multiplicative function, stability,
superstability.
AMS Subject Class. (2010): 47D09, 22D10, 39B82.
1. Introduction
For any α ∈ R, Berrone and Dieulefait [5] equipped R2 with the multiplication rule ·α , defined by
(x1 , y1 ) ·α (x2 , y2 ) = (x1 x2 + αy1 y2 , x1 y2 + x2 y1 ),
(x1 , y1 ), (x2 , y2 ) ∈ R2 .
For α = −1, the multiplication is the usual product of complex numbers in
C = R2 . The rule makes R2 into a commutative monoid with neutral element
(1, 0) and σ(x, y) = (x, −y) (complex conjugation) as an involution.
Berrone and Dieulefait [5, Theorem 1] studied the homomorphisms
m : (R2 , ·α ) −→ (R, .), i.e., the multiplicative, real-valued functions on the
monoid (R2 , ·α ). We extend their investigations by finding the bigger set of all
multiplicative, complex-valued functions M : (R2 , ·α ) −→ (C, .). Combining
157
158
b. bouikhalene, e. elqorachi, a. charifi
this information with Davison’s work [9] about D’Alembert’s functional equation on monoids, we obtain an explicit description of the solutions f : R2 −→ C
of D’Alembert’s functional equation
E(α) : f (a ·α b) + f (a ·α σ(b)) = 2f (a)f (b),
a, b ∈ R2 ,
on the monoid (R2 , ·α ). The description falls into three different cases, according to whether α > 0 or α < 0. The equation E(α) is a common generalization
of many functional equations of type D’Alembert
f (ab) + f (aσ(b)) = 2f (a)f (b),
a, b ∈ R2
(1.1)
on the monoid R2 , like, e.g.,
1) If α = 0,
E(0) : f (x1 x2 , x1 y2 + x2 y1 ) + f (x1 x2 , x2 y1 − x1 y2 ) = 2f (x1 , y1 )f (x2 , y2 ),
for all (x1 , y1 ), (x2 , y2 ) ∈ R2 . Setting x1 = x2 = 1 and F (y) = f (1, y) for any
y ∈ R respectively y1 = y2 = 0 and m(x) = f (x, 0) for any x ∈ R in E(0), we
get the classical D’Alembert functional equation
F (y1 + y2 ) + F (y1 − y2 ) = 2F (y1 )F (y2 ),
y 1 , y2 ∈ R
(1.2)
on R (see [1], [4], [15] and [23]) respectively the classical Cauchy equation
m(x1 x2 ) = m(x1 )m(x2 ),
x1 , x2 ∈ R
(1.3)
on R. We call m a multiplicative function on R (see[1]).
2) If α = −1,
E(−1) : f (x1 x2 − y1 y2 , x1 y2 + x2 y1 ) + f (x1 x2 + y1 y2 , x2 y1 − x1 y2 )
= 2f (x1 , y1 )f (x2 , y2 ),
(x1 , y1 ), (x2 , y2 ) ∈ R2 . The equation E(−1) is in connection with the identity
(x1 x2 − y1 y2 )2 + (x1 y2 + x2 y1 )2 + (x1 x2 + y1 y2 )2 + (x2 y1 − x1 y2 )2
= 2(x21 + y12 )(x22 + y22 )
for any x1 , x2 , y1 , y2 ∈ R.
(1.4)
on the approximate solution of d’alembert type equation
159
{
}
√
√
3) If α ̸= 1 is a square free integer and Q( α) = x + y α : x, y ∈ Q is the
quadratic monoid equipped with the multiplicative rule
√
√
√
(x1 + y1 α)(x2 + y2 α) = (x1 x2 + αy1 y2 ) + (x1 y2 + x1 y1 ) α,
(1.5)
then E(α) reduces to D’Alembert functional equation (1.1) on the monoid
√
Q( α). In [9] Davison solved the D’Alembert functional equation with involution on a monoid A: any solution f : A −→ C has the general form
◦σ
f = M +M
, where M : A −→−→ C is a multiplicative function.
2
In 1940, Ulam [22] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems.
Among these was the following question concerning the stability of homomorphisms.
Question 1.1. Let (G1 , ∗) be a group and let (G1 , ⋄, d) be a metric group
with the metric d. Given ε > 0, does there exist δ(ε) > 0 such that if a
mapping h : G1 −→ G2 satisfies the inequality d(h(x ∗ y), h(x) ⋄ h(y)) <
δ for all x, y ∈ G1 , then there is a homomorphism H : G1 −→ G2 with
d(h(x), H(x)) < δ(ε) for all x ∈ G1 ?
In 1941, Hyers [12] answered this question for the case where G1 and G2
are Banach spaces. In 1978, Rassias [20] provided a generalization of Hyer’s
theorem which allows the Cauchy difference to be unbounded. The interested
reader may refer to the book by Hyers, Isac, Rassias [13] for an in depth
account on the subject of stability of functional equations. In 1982, Rassias
[19] solved the Ulam problem by involving a product of powers of norms.
Since then, the stability problems of various functional equations ha been
investigated by many authors (see [10], [11] and [14]). In [3] and [7] Baker et
al. and Bourgin respectively, introduced the notion that by now is frequently
referred to as superstability or Baker’s stability: if a function f satisfies the
stability inequality |E1 (f ) − E2 (f )| ≤ ε, then either f is bounded or E1 (f ) =
E2 (f ). The superstability of D’Alembert’s functional equation f (x + y) +
f (x − y) = 2f (x)f (y) was investigated by Baker [4] and Cholewa [8]. Badora
and Ger [2], and Kim ([16], [17] and [18]) proved its superstability under the
condition |f (x+ y)+ f (x− y)− 2f (x)f (y)| ≤ φ(x) or φ(y). In a previuos work,
Bouikhalene et al. [6] investigated the superstability of the cosine functional
equation on the Heisenberg group. Following this investigation we study the
superstability of the functional equation E(α) on the monoid (R2 , ·α ). Also
we say that a function f : R2 −→ C is of approximate a cosine type function,
160
b. bouikhalene, e. elqorachi, a. charifi
if there is δ > 0 such that
|f (a ·α b) + f (a ·α i(b)) − 2f (a)f (b)| < δ,
a, b ∈ R2 .
(1.6)
In the case where δ = 0, f satisfies the functional equation E(α). We call f
a cosine type function on R2 . The paper is organized as follows: In the first
section after this introduction we solve the functional equation E(α). In the
second section we study the superstability equation E(α).
2. Solution of equation E(α)
According to [9] we drive the following lemma.
Lemma 2.1. The solution f : R2 −→ C of E(α) is of the form
f=
M +M ◦σ
,
2
where M : (R2 , ·α ) −→ (C, ·) is a multiplicative function.
By extending Berrone-Dieulefait’s result [5] to complex-valued multiplicative functions, we get the following lemmas.
Lemma 2.2. The multiplicative functions M : (R2 , ·1 ) −→ (C, ·) are the
functions
M (x, y) = m1 (x + y)m2 (x − y), x, y ∈ R,
where m1 , m2 : R −→ C are multiplicative functions.
Lemma 2.3. The multiplicative functions M : (R2 , ·0 ) −→ (C, ·) are the
trivial function M = 1 and M (0, y) = 0 for any y ∈ R and M (x, y) =
m(x)γ( xy ) for any (x, y) ∈ R2 , with x ̸= 0, where m : R −→ C is a multiplicative function and γ : (R, +) −→ C is an arbitrary character.
Lemma 2.4. The multiplicative functions M : (C, ·−1 ) −→ (C, ·) are the
trivial functions M = 0 and M = 1 and
{
m(|z|)Γ(exp(iθ)),
e
for z = |z| exp(iθ) ̸= 0
M (z) =
0,
for z = 0.
where m
e : (R+ , ·) −→ C∗ and Γ : {exp(iθ), θ ∈ R} −→ C∗ are arbitrary
characters.
on the approximate solution of d’alembert type equation
161
Proof. When α = −1, the multiplicative rule ·−1 becomes the usual product numbers in C. By using the polar decomposition z = |z| exp(iθ) for any
z ∈ C∗ where θ = arg(z), we get
M (|z1 ||z2 |) = M (|z1 |)M (|z2 |),
z1 , z2 ∈ C∗
(2.1)
and
M (exp(i(θ1 + θ2 ))) = M (exp(iθ1 ))M (exp(iθ2 )),
θ1 , θ2 ∈ R.
(2.2)
By letting m(|z|)
e
= M (|z|), for any z ∈ C∗ , and Γ(exp(iθ)) = M (exp(iθ)) for
any θ ∈ R it follows that m
e : (R+ , ·) −→ C∗ and Γ : {exp(iθ), θ ∈ R} −→ C∗
are characters. If z = 0, we set M (z) = 0.
In the next corollary we give the set of all multiplicative complex-valued
functions M : (R2 , ·α ) −→ C.
Corollary 2.5. The multiplicative functions M : (R2 , ·α ) −→ (C, ·) are
given by the following list:
I) If α > 0, then
√
√
M (x, y) = m1 (x + y α)m2 (x − y α),
(x, y) ∈ R2 .
II) If α = 0, then
a) M (x, y) = 1, for any (x, y) ∈ R2 .
b) M (0, y) = 0, for any y ∈ R.
c) M (x, y) = m(x)γ( xy ), for any (x, y) ∈ R2 with x ̸= 0.
III) If α < 0, then
a) M (x, y) = 0, for any (x, y) ∈ R2 .
b) M (x, y) = 1, for any (x, y) ∈ R2 .
{ (√
)
m
e
x2 − αy 2 Γ(arg(x + iy)), for (x, y) ̸= (0, 0)
c) M (x, y) =
0,
for (x, y) = (0, 0).
where m1 , m2 , m : R −→ C are multiplicative functions, and m
e :
(R+ , ·) −→ C∗ , Γ : {exp(iθ), θ ∈ R} −→ C∗ and γ : (R, +) −→ C
are arbitrary characters.
162
b. bouikhalene, e. elqorachi, a. charifi
The next theorem is the main result of this section.
Theorem 2.6. The set of solutions of the functional equation E(α) consists of the following three cases:
A) If α > 0, then
f (x, y) =
√
√
√ }
√
m1 (x)m2 (y) {
m1 (y α)m2 (−y α) + m1 (−y α)m2 (y α) ,
2
for any (x, y) ∈ R2 .
B) If α = 0, then
a) f (x, y) = 1, for any (x, y) ∈ R2 .
b) f (0, y) = 0, for any y ∈ R.
{ y
}
2
c) f (x, y) = m(x)
γ( x ) + γ( −y
2
x ), (x, y) ∈ R , x ̸= 0.
C) If α < 0, then f (0, 0) = 0 and
f (x, y) =
m
e
(√
)
}
x2 − αy 2 {
Γ(arg(x + iy)), (x, y) ∈ R2 \(0, 0) ,
2
where m1 , m2 , m : R −→ C are multiplicative functions, and m
e : (R+ , ·)
−→ C∗ , Γ : {exp(iθ), θ ∈ R} −→ C∗ and γ : R −→ C are arbitrary
characters.
Proof. According to Lemma 2.1 and Corollary 2.5 we get the proof of
theorem.
3. Superstability of equation E(α)
In the next theorem we establish the stability of E(α).
Theorem 3.1. Let φ, ψ, ϕ, ζ : R −→ [0, +∞[ be functions and let
f : R2 −→ C be a function such that
f (x1 x2 +αy1 y2 , x1 y2 + x2 y1 ) + f (x1 x2 − αy1 y2 , x2 y1 − x1 y2 )
− 2f (x1 , y1 )f (x2 , y2 ) ≤ min{φ(x1 ), ψ(y1 ), ϕ(x2 ), ζ(y2 )}
(3.1)
on the approximate solution of d’alembert type equation
163
for all (x1 , y1 ), (x2 , y2 ) ∈ R2 and α is a real parameter. Then either f is
bounded or f satisfies the functional equation
E(α) : f (x1 x2 + αy1 y2 , x1 y2 + x2 y1 ) + f (x1 x2 − αy1 y2 , x2 y1 − x1 y2 )
= 2f (x1 , y1 )f (x2 , y2 )
for all (x1 , y1 ), (x2 , y2 ) ∈ R2 .
Proof. For all (x1 , y1 ), (x2 , y2 ) ∈ R2 and α a real parameter we get from
the inequality (3.1) that
f (x1 x2 + αy1 y2 , x1 y2 + x2 y1 ) + f (x1 x2 − αy1 y2 , x2 y1 − x1 y2 )
− 2f (x1 , y1 )f (x2 , y2 )
(3.2)
≤ φ(x1 ) or ψ(y1 ).
Since f is unbounded then we can choose a sequence (xn , yn )n≥3 in R2 such
that f (xn , yn ) ̸= 0 and limn→+∞ |f (xn , yn )| = +∞. Taking (x2 , y2 ) = (xn , yn )
in (3.2) we obtain
f (x1 xn + αy1 yn , x1 yn + xn y1 ) + f (x1 xn − αy1 yn , xn y1 − x1 yn )
− 2f (x1 , y1 )f (xn , yn )
≤ φ(x1 ) or ψ(y1 )
and
f (x1 xn + αy1 yn , x1 yn + xn y1 ) + f (x1 xn − αy1 yn , xn y1 − x1 yn )
− f (x1 , y1 )
2f (xn , yn )
≤
ψ(y1 )
φ(x1 )
or
.
2|f (xn , yn )|
2|f (xn , yn )|
That is we get
f (x1 , y1 )
f (x1 xn + αy1 yn , x1 yn + xn y1 ) + f (x1 xn − αy1 yn , xn y1 − x1 yn )
.
n→+∞
2f (xn , yn )
(3.3)
= lim
en = x2 xn − αy2 yn , Yen =
Setting Xn = x2 xn + αy2 yn , Yn = x2 yn + xn y2 , X
2
x2 yn − xn y2 . For any (x1 , y1 ), (x2 , y2 ) ∈ R it follows that
164
b. bouikhalene, e. elqorachi, a. charifi
(
f (x1 x2 + αy1 y2 )xn + α(x1 y2 + x2 y1 )yn ,
)
(x1 x2 + αy1 y2 )yn + xn (x1 y2 + x2 y1 )
(
+ f (x1 x2 + αy1 y2 )xn − α(x1 y2 + x2 y1 )yn ,
xn (x1 y2 + x2 y1 ) − (x1 x2 + αy1 y2 )yn
− 2f (x1 , y1 )f (x2 xn + αy2 yn , x2 yn + xn y2 )
(
+ f (x1 x2 − αy1 y2 )xn + α(x2 y1 − x1 y2 )yn ,
(x1 x2 − αy1 y2 )yn + xn (x2 y1 − x1 y2
)
)
(
+ f (x1 x2 − αy1 y2 )xn − α(x2 y1 − x1 y2 )yn ,
xn (x2 y1 − x1 y2 ) − (x1 x2 − αy1 y2 )yn
− 2f (x1 , y1 )f (x2 xn − αy2 yn , x2 yn − xn y2 )
(
≤ f (x1 x2 + αy1 y2 )xn + α(x1 y2 + x2 y1 )yn ,
)
)
(x1 x2 + αy1 y2 )yn + xn (x1 y2 + x2 y1 )
(
+ f (x1 x2 − αy1 y2 )xn − α(x2 y1 − x1 y2 )yn ,
xn (x2 y1 − x1 y2 ) − (x1 x2 − αy1 y2 )yn
− 2f (x1 , y1 )f (x2 xn + αy2 yn , x2 yn + xn y2 )
(
+ f (x1 x2 − αy1 y2 )xn + α(x2 y1 − x1 y2 )yn ,
)
)
(x1 x2 − αy1 y2 )yn + xn (x2 y1 − x1 y2 )
(
+ f (x1 x2 + αy1 y2 )xn − α(x1 y2 + x2 y1 )yn ,
xn (x1 y2 + x2 y1 ) − (x1 x2 + αy1 y2 )yn
− 2f (x1 , y1 )f (x2 xn − αy2 yn , x2 yn − xn y2 )
= f (x1 Xn + αy1 Yn , x1 Yn + Xn y1 ) + f (x1 Xn − αy1 Yn , Xn y1 − x1 Yn )
− 2f (x1 , y1 )f (Xn , Yn )
en + αy1 Yen , x1 Yen + X
en y1 ) + f (x1 X
en − αy1 Yen , X
en y1 − x1 Yen )
+ f (x1 X
en , Yen )
− 2f (x1 , y1 )f (X
≤ 2φ(x1 ) or 2ψ(y1 ).
)
on the approximate solution of d’alembert type equation
165
So that
(
f (x1 x2 + αy1 y2 )xn + α(x1 y2 + x2 y1 )yn ,
)
(x1 x2 + αy1 y2 )yn + xn (x1 y2 + x2 y1 )
f (xn , yn )
(
f (x1 x2 + αy1 y2 )xn − α(x1 y2 + x2 y1 )yn ,
)
xn (x1 y2 + x2 y1 ) − (x1 x2 + αy1 y2 )yn
+
f (xn , yn )
(
f (x1 x2 − αy1 y2 )xn + α(x2 y1 − x1 y2 )yn ,
)
xn (x2 y1 − x1 y2 ) + (x1 x2 − αy1 y2 )yn
+
f (xn , yn )
(
f (x1 x2 − αy1 y2 )xn − α(x2 y1 − x1 y2 )yn ,
)
xn (x2 y1 − x1 y2 ) − (x1 x2 − αy1 y2 )yn
+
f (xn , yn )


f (x2 xn + αy2 yn , x2 yn + xn y2 )






+f (x2 xn − αy2 yn , x2 yn − xn y2 ) − 2f (x1 , y1 )


f (xn , yn )




≤2
φ(x1 )
ψ(y1 )
or 2
.
|f (xn , yn )|
|f (xn , yn )|
for any (x1 , y1 ), (x2 , y2 ) ∈ R2 . Since |f (xn , yn )| −→ +∞ as n −→ +∞ we get
that f satisfies E(α).
By letting min{φ(x1 ), ψ(y1 ), ϕ(x2 ), ζ(y2 )} = δ we get the Baker’s stability
([3], [4]) for the functional equation E(α).
Corollary 3.2. Let δ > 0 and let f : R2 −→ C be a function such that
|f (x1 x2 + αy1 y2 , x1 y2 + x2 y1 ) + f (x1 x2 − αy1 y2 , x2 y1 − x1 y2 )
− 2f (x1 , y1 )f (x2 , y2 )| ≤ δ
2 and α is a real parameter. Then either f is
for all (x1 , y1 ), (x2 , y2 ) ∈ R
√
1+ 1+2δ
bounded and |f (x, y)| ≤
for all (x, y) ∈ R2 or f satisfies the functional
2
equation E(α).
166
b. bouikhalene, e. elqorachi, a. charifi
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