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 mor- phism 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 mor- phisms 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.