On the Stability of a Class of Functional Equations

Journal of Inequalities in Pure and Applied Mathematics

ON THE STABILITY OF A CLASS OF FUNCTIONAL EQUATIONS

volume 4, issue 5, article 104, BELAID BOUIKHALENE 2003. Département de Mathématiques et Informatique Received 20 July, 2003; Faculté des Sciences BP 133, accepted 24 October, 2003. 14000 Kénitra, Morocco. Communicated by: K. Nikodem EMail: [email protected]

Abstract Contents JJ II J I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 Quit 098-03 Abstract

In this paper, we study the Baker’s superstability for the following functional equation Z X −1 (E (K)) f(xkϕ(y)k )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 On the Stability of A Class of G and f is a continuous complex-valued function on G satisfying the Kannap- Functional Equations pan type condition, for all x, y, z ∈ G Belaid Bouikhalene Z Z Z Z −1 −1 −1 −1 (*) f(zkxk hyh )dωK (k)dωK (h)= f(zkyk hxh )dωK (k)dωK (h). K K K K Title Page We treat examples and give some applications. Contents 2000 Mathematics Subject Classification: 39B72. Key words: Functional equation, Stability, Superstability, Central function, Gelfand JJ II pairs. The author would like to greatly thank the referee for his helpful comments and re- J I marks. Go Back Contents Close 1 Introduction, Notations and Preliminaries ...... 3 Quit 2 General Properties ...... 5 Page 2 of 21 3 The Main Results ...... 8 4 Applications ...... 15 J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 References http://jipam.vu.edu.au 1. Introduction, Notations and Preliminaries 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 Mor(G) the group of morphisms of G and Φ a finite subgroup of Mor(G) of a K-invariant morphisms of G (i.e ϕ(K) ⊂ K). The number of elements of a finite group Φ will be des- On the Stability of A Class of ignated by |Φ|. The Banach algebra of bounded measures on G with complex Functional Equations values is denoted by M(G) and the Banach space of all complex measurable Belaid Bouikhalene and essentially bounded functions on G by L∞(G). C(G) designates the Ba- nach space of all continuous complex valued functions on G. We say that a function f is a K-central function on G if Title Page (1.1) f(kx) = f(xk), x ∈ G, k ∈ K. Contents In the case where G = K, a function f is central if JJ II J I (1.2) f(xy) = f(yx) x, y ∈ G. Go Back See [2] for more information. Close 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: Quit a) K ⊂ Z(G),(Z(G) is the center of G). Page 3 of 21

b) f(hxk) = f(x), h, k ∈ K, x ∈ G (i.e. f is bi-K-invariant (see [3] and J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 [6])). http://jipam.vu.edu.au 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.

On the Stability of A Class of Functional Equations

Belaid Bouikhalene

Title Page Contents JJ II J I Go Back Close Quit Page 4 of 21

J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au 2. General Properties 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. On the Stability of A Class of Functional Equations Proof. Follows from the fact that Φ is a ﬁnite group. Belaid Bouikhalene Proposition 2.2. Let ϕ ∈ Φ and f ∈ C(G), then we have: Title Page R −1 R −1 i) K f(xkϕ(hy)k )dωK (k) = K f(xkϕ(yh)k )dωK (k), x, y ∈ G, h ∈ K. Contents ii) If f satisfy (*), the for all z, y, x ∈ G, we have JJ II Z Z J I −1 −1 f(zhϕ(ykxk )h )dωK (h)dωK (k) Go Back K K Z Z −1 −1 Close = f(zhϕ(xkyk )h )dωK (h)dωK (k). K K Quit Proof. i) Let ϕ ∈ Φ and let x, y ∈ G, h ∈ K, then we have Page 5 of 21

J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au Case 1: If ϕ is a group-homomorphism, we obtain, by replacing k by kϕ(h)−1 Z Z −1 −1 f(xkϕ(hy)k )dωK (k) = f(xkϕ(h)ϕ(y)k )dωK (k) K K Z −1 = f(xkϕ(y)ϕ(h)k )dωK (k) K Z = f(xkϕ(yh)k−1)dω (k). K On the Stability of A Class of K Functional Equations Case 2: if ϕ is a group-antihomomorphism, we have, by replacing k by Belaid Bouikhalene kϕ(h) Z Z −1 −1 Title Page f(xkϕ(hy)k )dωK (k) = f(xkϕ(y)ϕ(h)k )dωK (k) K K Z Contents −1 = f(xkϕ(h)ϕ(y)k )dωK (k) K JJ II Z −1 J I = f(xkϕ(yh)k )dωK (k). K Go Back ii) Follows by simple computation. Close Quit Proposition 2.3. For each τ ∈ Φ and x, y ∈ G, we have Page 6 of 21 Z Z X −1 X −1 (2.1) f(xkϕ(τ(y))k )dωK (k) = f(xkψ(y)k )dωK (k). J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 ϕ∈Φ K ψ∈Φ K http://jipam.vu.edu.au 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.

On the Stability of A Class of Functional Equations

Belaid Bouikhalene

Title Page Contents JJ II J I Go Back Close Quit Page 7 of 21

J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au 3. The Main Results 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

Z X −1 (3.1) f(xkϕ(y)k )dωK (k) − |Φ|f(x)f(y) ≤ δ, x, y ∈ G. ϕ∈Φ K On the Stability of A Class of Functional Equations

Then one of the assertions is satisﬁed: Belaid Bouikhalene (a) If f is bounded, then Title Page |Φ| + p|Φ|2 + 4δ|Φ| (3.2) |f(x)| ≤ . Contents 2|Φ| JJ II (b) If f is unbounded, then J I i) f is K-central, Go Back f ◦ τ = f τ ∈ Φ ii) , for all , Close iii) R f(xkyk−1)dω (k) = R f(ykxk−1)dω (k), x, y ∈ G. K K K K Quit Proof. Page 8 of 21 a) Let X = sup |f|, then we get for all x ∈ G J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 |Φ||f(x)f(x)| ≤ |Φ|X + δ, http://jipam.vu.edu.au from which we obtain that

|Φ|X2 − |Φ|X − δ ≤ 0,

such that |Φ| + p|Φ|2 + 4δ|Φ| X ≤ . 2|Φ| b) i) Let x, y ∈ G, h ∈ K, then by using Proposition 2.2, we ﬁnd On the Stability of A Class of |Φ||f(x)||f(hy) − f(yh)| Functional Equations = ||Φ|f(x)f(hy) − |Φ|f(x)f(yh)| Belaid Bouikhalene

Z X −1 ≤ f(xkϕ(hy)k )dωK (k) − |Φ|f(x)f(hy) Title Page ϕ∈Φ K Contents Z X −1 + f(xkϕ(yh)k )dωK (k) − |Φ|f(x)f(yh) JJ II ϕ∈Φ K J I ≤ 2δ. Go Back Since f is unbounded it follows that f(yh) = f(hy), for all h ∈ K, y ∈ G. Close ii) Let τ ∈ Φ, by using Proposition 2.3, we get for all x, y ∈ G Quit

|Φ||f(x)||f ◦ τ(y) − f(y)| Page 9 of 21 = ||Φ|f(x)f(τ(y)) − |Φ|f(x)f(y)| J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au

Z X −1 ≤ f(xkϕ(τ(y))k )dωK (k) − |Φ| f(x)f(τ(y)) ϕ∈Φ K

Z X −1 + f(xkψ(y)k )dωK (k) − |Φ|f(x)f(y) ψ∈Φ K ≤ 2δ.

Since f is unbounded it follows that f ◦ τ = f, for all τ ∈ Φ. On the Stability of A Class of iii) Let f be an unbounded solution of the functional inequality (3.1), such Functional Equations that f satisﬁes the condition (*), then, for all x, y ∈ G, we obtain, by using Belaid Bouikhalene Part i) of Proposition 2.2:

Z Z Title Page −1 −1 |Φ||f(z)| f(xkyk )dωK (k) − f(ykxk )dωK (k) K K Contents Z −1 = |Φ| f(z)f(xkyk )dωK (k) JJ II K Z J I −1 − |Φ| f(z)f(ykxk )dωK (k) K Go Back Z Z −1 −1 Close ≤ Σϕ∈Φ f(zhϕ(xkyk )h )dωK (h)dωK (k) K K Z Quit −1 − |Φ| f(z)f(xkyk )dωK (k) Page 10 of 21 K

J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au Z Z −1 −1 + Σϕ∈Φ f(zhϕ(ykxk )h )dωK (h)dωK (k) K K Z −1 −|Φ| f(z)f(ykxk )dωK (k) K ≤ 2δ. Since f is unbounded we get Z Z −1 −1 f(xkyk )dωK (k) = f(ykxk )dωK (k), x, y ∈ G. On the Stability of A Class of K K Functional Equations

Belaid Bouikhalene

The main result is the following theorem. Title Page Theorem 3.2. Let δ > 0 and let f ∈ C(G) such that f satisﬁes the condition (*) and the functional inequality Contents

Z JJ II X −1 (3.3) f(xkϕ(y)k )dωK (k) − |Φ|f(x)f(y) ≤ δ, x, y ∈ G. ϕ∈Φ K J I Then either Go Back |Φ| + p|Φ|2 + 4δ|Φ| Close (3.4) |f(x)| ≤ , x ∈ G, 2|Φ| Quit or Page 11 of 21 Z X −1 (E (K)) f(xkϕ(y)k )dωK (k) = |Φ|f(x)f(y), x, y ∈ G. J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 ϕ∈Φ K http://jipam.vu.edu.au 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) On the Stability of A Class of satisfying the condition (*) and let (zn)n∈N be a sequence in G such that f(zn) 6= Functional Equations 0 and limn |f(zn)| = +∞. It follows that the convergence of the sequences of Belaid Bouikhalene functions:

i) Title Page

P R −1 Contents ϕ∈Φ K f(znkϕ(x)k )dωK (k) (3.5) x 7−→ , n ∈ N, f(zn) JJ II to the function J I x 7−→ |Φ|f(x). Go Back ii) Close Quit P R −1 −1 ϕ∈Φ K f(znhϕ(xkϕ(τ(y))k )h )dωK (h) (3.6) x 7−→ , Page 12 of 21 f(zn)

n ∈ N, τ ∈ Φ, k ∈ K, y ∈ G J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au 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 f(znkϕ(y)k )dωK (k) δ ϕ∈Φ K − |Φ|f(y) ≤ , f(zn) |f(zn)| On the Stability of A Class of Functional Equations then we have, by letting n 7−→ +∞, that Belaid Bouikhalene

P R −1 ϕ∈Φ K f(znkϕ(y)k )dωK (k) lim = |Φ|f(y), Title Page n f(zn) Contents and P R f (z hϕ (xkϕ (τ(y)) k−1) h−1) dω (h) JJ II ϕ∈Φ K n K −1 lim = |Φ|f(xkτ(y)k ). J I n f(zn) Go Back Since by Proposition 2.3, we have Close Z P R −1 −1 X ϕ∈Φ K f(znhϕ(x)kϕ(τ(y))k h )dωK (h) Quit dωK (k) f(zn) τ∈Φ K Page 13 of 21 Z P R −1 −1 X ϕ∈Φ K f(znhϕ(x)kψ(y)k h )dωK (h) = dωK (k), J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 f(zn) ψ∈Φ K http://jipam.vu.edu.au combining this and the fact that f satisﬁes the condition (*), we obtain

Z P R −1 −1 X ϕ∈Φ K f(znhϕ(x)kϕ(τ(y))k h )dωK (h) dωK (k) f(zn) τ∈Φ K P R −1 f(znkψ(y)k )dωK (k) δ ψ∈Φ K −|Φ|f(x) ≤ . f(zn) |f(zn)|

Since the convergence is uniform, we have On the Stability of A Class of Functional Equations

Z X −1 2 Belaid Bouikhalene |Φ| f(xkϕ(y)k )dωK (k) − |Φ| f(x)f(y) ≤ 0, ϕ∈Φ K Title Page thus (E (K)) holds and the proof is complete. Contents JJ II J I Go Back Close Quit Page 14 of 21

J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au 4. Applications 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 On the Stability of A Class of Functional Equations

X (4.1) f(xϕ(y)) − |Φ|f(x)f(y) ≤ δ, x, y ∈ G. Belaid Bouikhalene ϕ∈Φ

Then either Title Page |Φ| + p|Φ|2 + 4δ|Φ| Contents (4.2) |f(x)| ≤ , x ∈ G, 2|Φ| JJ II or J I X (4.3) f(xϕ(y)) = |Φ|f(x)f(y), x, y ∈ G. Go Back ϕ∈Φ Close If G is abelian, then the condition (*) holds and we have the following: Quit If Φ = {i} (resp. Φ = {i, σ}), where i(x) = x and σ(x) = −x, we ﬁnd the Page 15 of 21 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 J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 K (see [11]), then we have the following corollary. http://jipam.vu.edu.au 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 (*) f(zkxhy)χ(k)χ(h)dωK (k)dωK (h) K K Z Z = f(zkyhx)χ(k)χ(h)dωK (k)dωK (h) K K and On the Stability of A Class of Functional Equations Z X (4.4) f(xkϕ(y))χ(k)dωK (k) − |Φ|f(x)f(y) ≤ δ, x, y ∈ G. Belaid Bouikhalene ϕ∈Φ K

Then either Title Page Contents |Φ| + p|Φ|2 + 4δ|Φ| (4.5) |f(x)| ≤ , x ∈ G, 2|Φ| JJ II or J I X Z Go Back (4.6) f(xkϕ(y))χ(k)dωK (k) = |Φ|f(x)f(y), x, y ∈ G. Close ϕ∈Φ K Quit Proposition 4.3. If the algebra χωK ?M(G) ? χωK is commutative then the condition (*) holds. Page 16 of 21

Proof. Since f(kxh) = χ(k)f(x)χ(h), k, h ∈ K, x ∈ G, then we have χωK ? J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au f ? χωK = f. Suppose that the algebra χωK ?M(G) ? χωK is commutative, then we get: Z Z −1 −1 f(xkyk hzh )dωK (k)dωK (h) K K Z Z −1 −1 = f(xkyhzh k )dωK (k)dωK (h) K K

= hδz ? χωK ? δy ? χωK ? δx, fi

= hδz ? χωK ? δy ? χωK ? δx, χωK ? f ? χωK i On the Stability of A Class of = hχωK ? δz ? χωK ? δy ? χωK ? δx ? χωK , fi Functional Equations

= hχωK ? δz ? χωK ? δx ? χωK ? δy ? χωK , fi Belaid Bouikhalene Z Z = f(ykxk−1hzh−1)dω (k)dω (h). K K Title Page K K Contents Let f be bi-K-invariant (i.e f(hxk) = f(x), h, k ∈ K, x ∈ G), then we JJ II have: J I Corollary 4.4. Let δ > 0 and let f ∈ C(G) be bi-K-invariant such that for all Go Back x, y, z ∈ G, Z Z Z Z Close (*) f(zkxhy)dω (k)dω (h) = f(zkyhx)dω (k)dω (h), K K K K Quit K K K K and Page 17 of 21

X Z (4.7) f(xkϕ(y))dωK (k) − |Φ|f(x)f(y) ≤ δ, x, y ∈ G. J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au ϕ∈Φ K Then either |Φ| + p|Φ|2 + 4δ|Φ| (4.8) |f(x)| ≤ , x ∈ G, 2|Φ| or X Z (4.9) f(xkϕ(y))dωK (k) = |Φ|f(x)f(y), x, y ∈ G. ϕ∈Φ K On the Stability of A Class of Functional Equations Proposition 4.5. If the pair (G, K) is a Gelfand pair (i.e ωK ?M(G) ? ωK is commutative), then the condition (*) holds. Belaid Bouikhalene Proof. We take χ = 1 (unit character of K) in Proposition 4.3 (see [3] and [6]). Title Page In the next corollary, we assume that G = K is a compact group. Contents Lemma 4.6. If f is central, then f satisﬁes the condition (*). Consequently, we JJ II have J I Z Z Go Back (4.10) f(xtyt−1)dt = f(ytxt−1)dt, x, y ∈ G. G G Close Corollary 4.7. Let δ > 0 and let f be a complex measurable and essentially Quit bounded function on G such that Page 18 of 21 Z X −1 (4.11) f(xtϕ(y)t )dt − |Φ|f(x)f(y) ≤ δ, x, y ∈ G. J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au ϕ∈Φ G Then

|Φ| + p|Φ|2 + 4δ|Φ| (4.12) |f(x)| ≤ , 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 On the Stability of A Class of Z Functional Equations X −1 (4.13) f(xtϕ(y)t )dt = |Φ|f(x)f(y), x, y ∈ G. Belaid Bouikhalene ϕ∈Φ G

In view of the proposition in [5], we have that f is continuous. Since G is Title Page compact, then the proof is accomplished. Contents JJ II J I Go Back Close Quit Page 19 of 21

J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au References [1] R. BADORA, On Hyers-Ulam stability of Wilson’s functional equation, Aequations Math., 60 (2000), 211–218. [2] J.L. CLERC, Les représentations des groupes compacts, Analyse Har- moniques, les Cours du CIMPA, Université de Nancy I, 1980. [3] J. FARAUT, Analyse harmonique sur les paires de Guelfand et les espaces

hyperboliques, les Cours du CIMPA, Université de Nancy I, 1980. On the Stability of A Class of Functional Equations [4] W. FORG-ROB AND J. SCHWAIGER, The stability of some functional equations for generalized hyperbolic functions and for the generalized Belaid Bouikhalene hyperbolic functionsand for the generalized cosine equation, Results in Math., 26 (1994), 247–280. Title Page [5] Z. GAJDA, On functional equations associated with characters of unitary Contents representations of groups, Aequationes Math., 44 (1992), 109–121. JJ II [6] S. HELGASON, Groups and Geometric Analysis, Academic Press, New J I York-London, 1984. Go Back [7] E. HEWITT AND K.A. ROSS, Abstract Harmonic Analysis, Vol. I and II., Springer-Verlag, Berlin-Gottingen-Heidelberg, 1963. Close Quit [8] J. BAKER, J. LAWRENCE AND F. ZORZITTO, The stability of the equation f(x + y) = f(x)f(y), Proc. Amer. Math. Soc., 74 (1979), 242–246. Page 20 of 21

[9] J. BAKER, The stability of the cosine equation, Proc. Amer. Math. Soc., J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 80(3) (1980), 411–416. http://jipam.vu.edu.au [10] Pl. KANNAPPAN, The functional equation f(xy) + f(xy−1) = 2f(x)f(y), for groups, Proc. Amer. Math. Soc., 19 (1968), 69–74.

[11] R. TAKAHASHI, SL(2, R), Analyse Harmoniques, les Cours du CIMPA, Université de Nancy I, 1980.

On the Stability of A Class of Functional Equations

Belaid Bouikhalene

Title Page Contents JJ II J I Go Back Close Quit Page 21 of 21

J. Ineq. Pure and Appl. Math. 4(5) Art. 104, 2003 http://jipam.vu.edu.au