Georgian Mathematical Journal Volume 10 (2003), Number 3, 503–508

THE SUPERSTABILITY OF THE GENERALIZED D’ALEMBERT FUNCTIONAL EQUATION

ELHOUCIEN ELQORACHI AND MOHAMED AKKOUCHI

Abstract. We generalize the well-known Baker’s superstability result for the d’Alembert functional equation with values in the field of complex numbers to the case of the integral equation

∫_{G} f(xty)dμ(t) + ∫_{G} f(xtσ(y))dμ(t) = 2f(x)f(y) x, y ∈ G,

where G is a locally compact group, μ is a generalized Gelfand measure and σ is a continuous involution of G.

2000 Mathematics Subject Classification: 39B72.

Key words and phrases: D’Alembert equation, Gelfand measure, Gelfand pair, stability, superstability.

1. Introduction

Let G be a locally compact group. We denote by M(G) the Banach algebra of bounded measures on G with complex values. It is the topological dual of C_{0}(G), the Banach space of continuous functions vanishing at infinity (cf. 13.1.2 of [5]). σ is a continuous involution of G, i.e. ,(σ ∘ σ)(x) = x and σ(xy) = σ(y)σ(x) for all x, y ∈ G.

Let μ ∈ M(G) be a compactly supported measure on G. We say that μ is σ-invariant if ≺ μ, f ∘σ ≻=≺ μ, f ≻ for all continuous functions f on G, where ≺ μ, f ≻= ∫_{G} f(x)dμ(x).

Throughout this paper we assume that μ is a generalized Gelfand measure on G with compact support. This means that the following conditions are satisfied

(i) μ ∗ μ = μ and

(ii) μ ∗ M(G) ∗ μ is a commutative Banach algebra under the convolution product (see [1] for more information).

In a previous work [6], complex continuous solutions of the generalized d’Alembert functional equation

∫_{G} f(xty)dμ(t) + ∫_{G} f(xtσ(y))dμ(t) = 2f(x)f(y), x, y ∈ G,  (1)

are determined.

There is an important particular case of the integral equation (1) : μ = δ_{e} and σ(x) = −x, where δ_{e} denotes the Dirac measure concentrated at the identity element of G. In this setting G is an abelian group and (1) reduces to the classical d’Alembert functional equation

f(x + y) + f(x − y) = 2f(x)f(y), x, y ∈ G.  (2)

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In the paper [2] the superstability theorem of the d’Alembert functional equation (2) appears. More precisely, J. A. Baker proved the following result in [2] (Theorem 5).

Let G be an abelian group and δ > 0. Let f be a complex function such that

|f(x + y) + f(x − y) − 2f(x)f(y)| ≤ δ, x, y ∈ G.

Then either

|f(x)| ≤ \frac{1 + \sqrt{1 + 2δ}}{2} , x ∈ G,

or

f(x + y) + f(x − y) = 2f(x)f(y), x, y ∈ G.

The aim of this note is to extend the above Baker’s stability theorem to the case of the generalized d’Alembert functional equation (1) in which μ is a generalized σ-invariant Gelfand measure with compact support.

2. The Main Results

Theorem. Let δ > 0 and let f be a continuous complex-valued function on G such that

| ∫_{G} f(xty)dμ(t) + ∫_{G} f(xtσ(y))dμ(t) − 2f(x)f(y)| ≤ δ, x, y ∈ G. (3)

Then either

|f(x)| ≤ \frac{∥ μ ∥ +\sqrt{∥ μ ∥^{2} +2δ}}{2} , x ∈ G,

or

∫_{G} f(xty)dμ(t) + ∫_{G} f(xtσ(y))dμ(t) = 2f(x)f(y), x, y ∈ G.

The following lemma will be useful in the proof of the main results.

Lemma. If f is a continuous and bounded solution of the functional inequality (3) then

sup |f| ≤ \frac{∥ μ ∥ +\sqrt{∥ μ ∥^{2} +2δ}}{2} .

Proof. Let M = sup |f|. Using the inequality of the lemma we find that

2|f(x)f(x)| ≤ M ∥ μ ∥ +M ∥ μ ∥ +δ,

from which we conclude that M = sup |f| satisfies M^{2} ≤ M ∥ μ ∥ +\frac{δ}{2}.

The rest of the proof consists in finding the roots of the second order polynomial x^{2} − x ∥ μ ∥ −\frac{δ}{2}. □

Proof of Theorem. If f is bounded, then according to the lemma we are in the first case of the theorem. So we may from now on assume that f is unbounded.

Step one. First we show that

∫_{G} f(xt)dμ(t) = ∫_{G} f(tx)dμ(t) = f(x),

for all x ∈ G in the manner as follows.

For any x, y ∈ G,

|2f(x)| | ∫_{G} f(yt)dμ(t) − f(y)| = |2f(x) ∫_{G} f(yt)dμ(t) − 2f(x)f(y)|
≤ | ∫_{G} f(ytx)dμ(t) + ∫_{G} f(ytσ(x))dμ(t) − 2f(x)f(y)|
+| ∫_{G} f(ytx)dμ(t) + ∫_{G} f(ytσ(x))dμ(t) − 2f(x) ∫_{G} f(yt)dμ(t)|.

Since μ ∗ μ = μ, we get that

| ∫_{G} f(ytx)dμ(t) + ∫_{G} f(ytσ(x))dμ(t) − 2f(x) ∫_{G} f(yt)dμ(t)|
= | ∫_{G} ∫_{G} f(ytsx)dμ(t)dμ(s) + ∫_{G} ∫_{G} f(ytsσ(x))dμ(t)dμ(s)−
−2f(x) ∫_{G} f(yt)dμ(t)|

≤ ∫_{G} | ∫_{G} f(ytsx)dμ(s) + ∫_{G} f(ytsσ(x))dμ(s) − 2f(x)f(yt)|d|μ|(t)≤δ ∥ μ ∥ .

It follows that

|2f(x)|| ∫_{G} f(yt)dμ(t) − f(y)| ≤ δ + δ ∥ μ ∥ .

Since f is unbounded, we have

∫_{G} f(yt)dμ(t) = f(y), y ∈ G.

On the other hand,

|2f(x)||f(y) − f(σ(y))| = |2f(x)f(y) − 2f(x)f(σ(y))|
≤ | ∫_{G} f(xty)dμ(t) + ∫_{G} f(xtσ(y))dμ(t) − 2f(x)f(y)|
+| ∫_{G} f(xtσ(y))dμ(t) + ∫_{G} f(xty)dμ(t) − 2f(x)f(σ(y))| ≤ 2δ.

Since f is unbounded, we have f(σ(y)) = f(y), for all y ∈ G. By using the above results we will prove that

∫_{G} f(ty)dμ(t) = f(y), y ∈ G.

Since μ is σ-invariant, we get for any y ∈ G that

∫_{G} f(ty)dμ(t) = ∫_{G} f(σ(y)σ(t))dμ(t) = ∫_{G} f(σ(y)t)dμ(t) = f(σ(y)) = f(y).

Now μ is a generalized Gelfand measure and therefore then we have

∫_{G} f(xty)dμ(t) = ∫_{G} ∫_{G} ∫_{G}(kxtys)dμ(k)dμ(t)dμ(s)

=≺ μ ∗ δ_{x} ∗ μ ∗ δ_{y} ∗ μ, f ≻=≺ μ ∗ δ_{x} ∗ μ ∗ μ ∗ δ_{y} ∗ μ, f ≻ =≺ μ ∗ δ_{y} ∗ μ ∗ μ ∗ δ_{x} ∗ μ, f ≻=≺ μ ∗ δ_{y} ∗ μ ∗ δ_{x} ∗ μ, f ≻

= ∫_{G} ∫_{G} ∫_{G} f(kytxs)dμ(k)dμ(t)μ(s) = ∫_{G} f(ytx)dμ(t) (4)

for all x, y ∈ G.

On the other hand, if we replace f by Ψ(x) = ∫_{G} f(zsx)dμ(s) in the previous formula (4), we get

∫_{G} ∫_{G} f(zsxty)dμ(s)dμ(t) = ∫_{G} ∫_{G} f(zsytx)dμ(s)dμ(t)

for all x, y, z ∈ G.

Step two. By using the ideas from the paper by Badora [3] we will prove that f is a solution of the integral equation (1). f is unbounded, so there exists a sequence (a_{n})_{n∈ℕ} in G such that

f(a_{n}) ≠ 0 and lim_{n→+∞}|f(a_{n})| = +∞.

By inequality (3), for x = a_{n} we have

|\frac{∫_{G} f(a_{n}ty)dμ(t) + ∫_{G} f(a_{n}tσ(y))dμ(t)}{f(a_{n})} − 2f(y)| ≤ \frac{δ}{|f(a_{n})|}

for all y ∈ G and n ∈ ℕ.

It follows that the convergence of the sequence of functions

x ↦ \frac{∫_{G} f(a_{n}tx)dμ(t) + ∫_{G} f(a_{n}tσ(x))dμ(t)}{f(a_{n})}, n ∈ ℕ,  (5)

to the function

x ↦ 2f(x)

is uniform.

For any x, y ∈ G and n ∈ ℕ it is easily seen that

| ∫_{G} ∫_{G} f(a_{n}tysx)dμ(t)dμ(s) + ∫_{G} ∫_{G} f(a_{n}tysσ(x))dμ(t)dμ(s)
−2f(x) ∫_{G} f(a_{n}ty)dμ(t)|

≤ ∫_{G} | ∫_{G} f(a_{n}tysx)dμ(s) + ∫_{G} f(a_{n}tysσ(x))dμ(s) − 2f(x)f(a_{n}ty)| d|μ|(t)

≤ δ ∥ μ ∥ .

Similarly, we get

| ∫_{G} ∫_{G} f(a_{n}tσ(y)sx)dμ(t)dμ(s) + ∫_{G} ∫_{G} f(a_{n}tσ(y)sσ(x))dμ(t)dμ(s)

−2f(x) ∫_{G} f(a_{n}tσ(y))dμ(t)| ≤ δ ∥ μ ∥ .

Combining this and

∫_{G} ∫_{G} f(zsxty)dμ(s)dμ(t) = ∫_{G} ∫_{G} f(zsytx)dμ(s)dμ(t),

we obtain

| ∫_{G} ∫_{G} f(a_{n}tσ(y)sx)dμ(t)dμ(s) + ∫_{G} ∫_{G} f(a_{n}tσ(x)sy)dμ(t)dμ(s)
+ ∫_{G} ∫_{G} f(a_{n}tσ(y)sσ(x))dμ(t)dμ(s) + ∫_{G} ∫_{G} f(a_{n}txsy)dμ(t)dμ(s)
−2f(x)[ ∫_{G} f(a_{n}ty)dμ(t) + ∫_{G} f(a_{n}tσ(y))dμ(t)]| ≤ 2δ ∥ μ ∥ .

After dividing both sides of this inequality by |f(a_{n})| we get

| ∫_{G} \frac{∫_{G} f(a_{n}tσ(y)sx)dμ(t) + ∫_{G} f(a_{n}tσ(x)σ(s)y)dμ(t)}{f(a_{n})} dμ(s)

+ ∫_{G} \frac{∫_{G} f(a_{n}txsy)dμ(t) + ∫_{G} f(a_{n}tσ(y)σ(s)σ(x))dμ(t)}{f(a_{n})}dμ(s)

−2f(x)[\frac{∫_{G} f(a_{n}ty)dμ(t) + ∫_{G} f(a_{n}tσ(y))dμ(t)}{f(a_{n})}]| ≤ \frac{2δ ∥ μ ∥}{|f(a_{n})|}.

In view of (5), we get by letting n → +∞ that

lim_{n→+∞} \frac{∫_{G} f(a_{n}tσ(y)sx)dμ(t) + ∫_{G} f(a_{n}tσ(x)σ(s)y)dμ(t)}{f(a_{n})} = 2f(σ(y)sx),

lim_{n→+∞} \frac{∫_{G} f(a_{n}txsy)dμ(t) + ∫_{G} f(a_{n}tσ(y)σ(s)σ(x))dμ(t)}{f(a_{n})} = 2f(xsy),

and

lim_{n→+∞} \frac{∫_{G} f(a_{n}ty)dμ(t) + ∫_{G} f(a_{n}tσ(y))dμ(t)}{f(a_{n})} = 2f(y).

Moreover, since the convergence is uniform, we have

|2 ∫_{G} f(σ(y)sx)dμ(s) + 2 ∫_{G} f(xsy)dμ(s) − 4f(x)f(y)| ≤ 0,

for all x, y ∈ G.

In view of (4) ∫_{G} f(xty)dμ(t) = ∫_{G} f(ytx)dμ(t), x, y ∈ G, and thus we conclude that f is a solution of the functional equation (1). □

Corollary. Let (G, K) be a compact Gelfand pair (see [4]) with σ(K) ⊂ K. Let δ > 0 and let f be a continuous complex-valued function on G such that

| ∫_{K} f(xky)dk + ∫_{K} f(xkσ(y))dk − 2f(x)f(y)| ≤ δ, x, y ∈ G,

where dk denotes the normalized Haar measure on K.

Then either

|f(x)| ≤ \frac{1 + \sqrt{1 + 2δ}}{2} , x ∈ G,

or

∫_{K} f(xky)dk + ∫_{K} f(xkσ(y))dk = 2f(x)f(y), x, y ∈ G.

Remarks 1. (1) In the theorem, we can replace the condition that μ is a generalized Gelfand measure by a weaker condition that f satisfies the following version of Kannappan’s condition :

∫_{G} ∫_{G} f(zsxty)dμ(s)dμ(t) = ∫_{G} ∫_{G} f(zsytx)dμ(s)dμ(t), x, y, z ∈ G.

(2) If μ is a complex measure with finite support, the complex function f in the theorem need not be assumed to be continuous.

Acknowledgement

We would like to express our thanks to the referee for his helpful comments and remarks.

References

1. M. Akkouchi and A. Bakali, Une généralisation des paires de Guelfand. Boll. Un. Mat. Ital. B (7) 6(1992), No. 4, 795–822.

2. J. A. Baker, The stability of the cosine equation. Proc. Amer. Math. Soc. 80(1980), No. 3, 411–416.

3. R. Badora, On Hyers-Ulam stability of Wilson’s functional equation. Aequationes Math. 60(2000), No. 3, 211–218.

4. J. Dieudonné, Eléments d’analyse. Tome 6. Gauthier-Villars, Éditeur, Paris, 1968.

5. J. Dieudonné, Eléments d’analyse. Tome 2. Gauthier-Villars, Éditeur, Paris, 1968.

6. E. Elqorachi and M. Akkouchi, On generalized d’Alembert and Wilson functional equations. Aequationes Math. (to appear).

(Received 18.03.2002; revised 24.09.2002)

Author’s addresses:

E. Elqorachi Department of Mathematics, Faculty of Sciences University of Ibnou Zohr, Agadir Morocco E-mail: h elqorachi@caramail.com

M. Akkouchi Department of Mathematics Faculty of Sciences Semlalia University of Cadi Ayyad, Marrakech Morocco E-mail: makkouchi@hotmail.com