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volume 5, issue 2, article 49, 2004.

Received 14 January, 2004;

accepted 25 April, 2004.

Communicated by:Kazimierz Nikodem

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Journal of Inequalities in Pure and Applied Mathematics

THE STABILITY OF SOME LINEAR FUNCTIONAL EQUATIONS

BELAID BOUIKHALENE

Department of Mathematics University of Ibn Tofail Faculty of Sciences BP 133 Kenitra 14000, Morocco.

EMail:bbouikhalene@yahoo.fr

c

2000Victoria University ISSN (electronic): 1443-5756 012-04

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The Stability of Some Linear Functional Equations

Belaid Bouikhalene

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J. Ineq. Pure and Appl. Math. 5(2) Art. 49, 2004

http://jipam.vu.edu.au

Abstract

In this note, we deal with the Baker’s superstability for the following linear functional equations

m

X

i=1

f(x+y+a_i) =f(x)f(y), x, y∈G,

m

X

i=1

[f(x+y+ai) +f(x−y−ai)] = 2f(x)f(y), x, y∈G,

whereGis an abelian group,a₁, . . . , a_m(m ∈N) are arbitrary elements inG andfis a complex-valued function onG.

2000 Mathematics Subject Classification:39B72.

Key words: Linear functional equations, Stability, Superstability.

1. Introduction

LetGbe 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

i=1

f(x+y+a_i) =f(x)f(y), x, y ∈G, (1.1)

m

X

i=1

[f(x+y+a_i) +f(x−y−a_i)] = 2f(x)f(y), x, y ∈G, (1.2)

where a₁, . . . , a_m (m ∈ N), are arbitrary elements in G and f is a complex- valued function onG. In the case whereGis a locally compact group, the form ofL^∞(G)solutions of (1.1) (resp. (1.2)) are determined in [2] (resp. [6]). Some particular cases of these linear functional equations are:

• The linear functional equations

f(x+y+a) = f(x)f(y), x, y ∈G, (1.3)

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

see [1], [2], [6], [7] and [8].

• Cauchy’s functional equation

(1.7) f(x+y) =f(x)f(y), x, y ∈G,

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• D’Alembert’s functional equation

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

To complete our consideration, we give some applications.

We shall need the results below for later use.

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2. General Properties

Proposition 2.1. Letδ >0. LetGbe an abelian group and letfbe a complex- valued function defined onGsuch that

(2.1)

m

X

i=1

f(x+y+a_i)−f(x)f(y)

≤δ, x, y ∈G, then one of the assertions is satisfied

i) Iff is bounded, then

(2.2) |f(x)| ≤ m+√

m²+ 4δ

2 , x∈G.

ii) If f is unbounded, then there exists a sequence (z_n)n∈N in G such that f(z_n) 6= 0 andlim_n|f(z_n)| = +∞ and that the convergence of the se- quences of functions

(2.3) x→ 1

f(z_n)

m

X

i=1

f(z_n+x+a_i), n∈N, to the function

x→f(x),

(2.4) x→ 1

f(z_n)

m

X

i=1

f(z_n+x+y+a_j +a_i),

n ∈N, 1≤j ≤m, y ∈G,

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to the function

x→f(x+y+aj), is uniform.

Proof. i) LetX = sup|f|, then for allx∈Gwe have

|f(x)f(x)| ≤mX+δ from which we obtain that

X²−mX−δ ≤0 hence

X ≤ m+√

m²+ 4δ

2 .

ii) Sincef is unbounded then there exists a sequence(z_n)n∈NinGsuch that f(zn)6= 0andlimn|f(zn)|= +∞. Using (2.1) one has

1 f(z_n)

m

X

i=1

f(z_n+x+a_i)−f(x)

≤ δ

|f(z_n)|, x∈G, n∈N, by lettingn → ∞, we obtain

limn

1 f(z_n)

m

X

i=1

f(z_n+x+a_i) =f(x)

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and

limn

1 f(z_n)

m

X

i=1

f(z_n+x+y+a_j+a_i) = f(x+y+a_j).

Proposition 2.2. Letδ >0. LetGbe an abelian group and letfbe a complex- valued function defined onGsuch that

(2.5)

m

X

i=1

[f(x+y+a_i) +f(x−y−a_i)]−2f(x)f(y)

≤δ, x, y∈G,

then one of the assertions is satisfied i) Iff is bounded, then

(2.6) |f(x)| ≤ m+√

m²+ 2δ

2 , x∈G.

ii) If f is unbounded, then there exists a sequence (z_n)n∈N ∈ G such that f(z_n) 6= 0 andlim_n|f(z_n)| = +∞ and that the convergence of the se- quences of functions

(2.7) x→ 1

f(z_n)

m

X

i=1

[f(z_n+x+a_i) +f(z_n−x−a_i)], n∈N, to the function

x→2f(x),

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(2.8) x→ 1 f(zn)

m

X

i=1

[f(z_n+x+y+a_j +a_i) +f(z_n−x−y−a_j−a_i)],

n ∈N, 1≤j ≤m, y ∈G, to the function

x→2f(x+y+a_j),

(2.9) x→ 1 f(z_n)

m

X

i=1

[f(z_n+x−y−a_j +a_i) +f(z_n−x+y+a_j −a_i)],

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 Proposition2.1.

i) LetX = sup|f|, then for allx∈Gwe have X²−mX− δ

2 ≤0 hence

X ≤ m+√

m²+ 2δ

2 .

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ii) Follows from the fact that

1 f(z_n)

m

X

i=1

[f(z_n+x+a_i) +f(z_n−x−a_i)]−2f(x)

≤ δ

|f(z_n)|, x∈G, n∈N.

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3. The Main Results

The main results are the following theorems.

Theorem 3.1. Let δ > 0. Let Gbe an abelian group and let f be a complex- valued function defined onGsuch that

(3.1)

m

X

i=1

≤δ, x, y∈G,

then either

(3.2) |f(x)| ≤ m+√

m²+ 4δ

2 , x∈G, or

(3.3)

m

X

i=1

f(x+y+ai) =f(x)f(y), x, y ∈G.

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 Proposition2.1, for allx, y ∈ G, n∈N

m

X

j=1

1 f(z_n)

m

X

i=1

f(z_n+x+y+a_j +a_i)−f(x) 1 f(z_n)

m

X

j=1

f(z_n+y+a_j)

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≤

m

X

j=1

1 f(zn)

( _m X

i=1

f(z_n+x+y+a_j+a_i)−f(x)f(z_n+y+a_j) )

≤ mδ

|f(z_n)|,

since the convergence is uniform, we have

m

X

i=1

≤0.

i.e. f is a solution of the functional equation (1.1).

Theorem 3.2. Let δ > 0. Let Gbe an abelian group and let f be a complex- valued function defined onGsuch that

(3.4)

m

X

i=1

≤δ, x, y∈G,

then either

(3.5) |f(x)| ≤ m+√

m²+ 2δ

2 , x∈G.

or

(3.6)

m

X

i=1

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

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Proof. By the assertion i) in Proposition2.2we get the first case of the theorem.

For the second case we have by the inequality (3.4) that

m

X

j=1

1 f(z_n)

( _m X

i=1

[f(z_n+x+y+a_j +a_i) +f(z_n−x−y−a_j −a_i)]

)

+

m

X

j=1

1 f(z_n)

( _m X

i=1

[f(z_n+x−y−a_j +a_i) +f(z_n−x+y+a_j−a_i)]

)

− 2f(x) 1 f(z_n)

m

X

j=1

[f(z_n+y+a_j) +f(z_n−y−a_j)]

=

m

X

j=1

1 f(z_n)

( _m X

i=1

[f(zn+x+y+aj+ai) +f(zn−x+y+aj −ai)]

−2f(x)f(z_n+y+a_j)

+

m

X

j=1

1 f(z_n)

( _m X

i=1

[f(z_n+x−y−a_j +a_i) +f(z_n−x−y−a_j −a_i)]

−2f(x)f(z_n−y−a_j)

≤

m

X

j=1

1 f(z_n)

( _m X

i=1

[f(z_n+x+y+a_j +a_i) +f(z_n−x+y+a_j−a_i) ]

−2f(x)f(z_n+y+a_j)

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+

m

X

j=1

1 f(zn)

( _m X

i=1

[f(z_n+x−y−a_j +a_i) +f(z_n−x−y−a_j −a_i)]

−2f(x)f(z_n−y−a_j)

≤ 2mδ

|f(z_n)|,

since the convergence is uniform, we have

2

m

X

i=1

≤0.

i.e. f is a solution of the functional equation (1.2).

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4. Applications

From Theorems3.1and3.2, we easily obtain .

Corollary 4.1. Let δ > 0. LetGbe an abelian group and letf be a complex- valued function defined onGsuch that

(4.1) |f(x+y+a)−f(x)f(y)| ≤δ, x, y∈G, then either

(4.2) |f(x)| ≤ 1 +√

1 + 4δ

2 , x∈G.

or

(4.3) f(x+y+a) = f(x)f(y) x, y ∈G.

Remark 4.1. Takinga= 0in Corollary4.1, we find the result obtained in [4].

(4.4) |f(x+y+a) +f(x−y−a)−2f(x)f(y)| ≤δ, x, y ∈G, then either

(4.5) |f(x)| ≤ 1 +√

1 + 2δ

2 , x∈G, or

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

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Remark 4.2. Takinga= 0in Corollary4.2, we find the result obtained in [5].

(4.7)

m

X

i=1

[f(x+y+ai)−f(x−y+ai)]−2f(x)f(y)

≤δ, x, y∈G,

then either

(4.8) |f(x)| ≤ m+√

m²+ 2δ

2 , x∈G, or

(4.9)

m

X

i=1

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

Proof. Let f be a complex-valued function defined on G which satisfies the inequality (4.7), then for allx, y ∈Gwe have

2|f(x)||f(y) +f(−y)|

=|2f(x)f(y) + 2f(x)f(−y)|

=

m

X

i=1

[f(x+y+ai)−f(x−y+ai)]

−

m

X

i=1

[f(x+y+a_i)−f(x−y+a_i)] + 2f(x)f(y) + 2f(x)f(−y)

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≤

2f(x)f(y)−

m

X

i=1

[f(x+y+a_i)−f(x−y+a_i)]

+

2f(x)f(−y)−

m

X

i=1

[f(x−y+a_i)−f(x+y+a_i)]

≤2δ.

Since f is unbounded it follows that f(−y) = −f(y), for ally ∈ G. Conse- quentlyf satisfies the inequality (3.4) and one has the remainder.

(4.10)

m

X

i=1

[f(x+y+a_i) +f(x−y+a_i)]−2f(x)f(y)

≤δ, x, y∈G,

then either

(4.11) |f(x)| ≤ m+√

m²+ 2δ

2 , x∈G, or

(4.12)

m

X

i=1

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

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Proof. Let f be a complex-valued function defined on G which satisfies the inequality (4.10), then for allx, y ∈Gwe have

2|f(x)||f(y)−f(−y)|

=|2f(x)f(y)−2f(x)f(−y)|

=

m

X

i=1

[f(x+y+a_i) +f(x−y+a_i)]

−

m

X

i=1

[f(x+y+a_i) +f(x−y+a_i)]

+ 2f(x)f(y)−2f(x)f(−y)

≤

m

X

i=1

[f(x−y+a_i) +f(x+y+a_i)]−2f(x)f(−y)

+

m

X

i=1

[f(x+y+a_i) +f(x−y+a_i)]−2f(x)f(y)

≤2δ.

Since f is unbounded it follows that f(−y) = f(y), for all y ∈ G. Conse- quentlyf satisfies the inequality (3.4) and one has the remainder.

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References

[1] J. ACZÉL, Lectures on Functional Equations and their Applications, Aca- demic Press, New York-Sain Francisco-London, 1966.

[2] R. BADORA, On a joint generalization of Cauchy’s and d’Alembert func- tional equations, Aequations Math., 43 (1992), 72–89.

[3] R. BADORA, On Heyers-Ulam stability of Wilson’s functional equation, Aequations Math., 60 (2000), 211–218.

[4] J. BAKER, J. LAWRENCE ANDF. ZORZITTO, The stability of the equa- tionf(x+y) = f(x)f(y), Proc. Amer. Math. Soc., 74 (1979), 242–246.

[5] J. BAKER, The stability of the cosine equation, Proc. Amer. Math. Soc., 80(3) (1980), 411–416.

[6] Z. GAJDA, A generalization of d’Alembert’s functional equation, Funkcial.

Evac., 33 (1990), 69–77.

[7] B. NAGY, A sine functional equation in Banach algebras, Publ. Math. De- brecen, 24 (1977), 77–99.

[8] E.B. VAN VLECK, A functional equation for the sine, Ann. Math., 11 (1910), 161–165.