On Hyers–Ulam Stability of Cauchy and Wilson Equations

“…In Corollaries 4.3 and 4.4 we give explicit formulas of solutions of (1.2) and (1.8) in terms of irreducible representations of G. Theses formulas generalize Euler's formula cos(x) = e ix +e −ix 2 on G = R. In the last section we study stability [48] and Baker's superstability (see [5] and [6]) of the functional equations (1.1), (1.2), (1.3), (1.4), (1.5), (1.6) and (1.7). For more information concerning the stability problem we refer to [3], [5], [6], [11], [12], [22], [40], [41], [42], [43], [44], [45], [46], [47] and [48]. The results of the last sections generalize the ones obtained in [12] and [21].…”

Section: Introductionmentioning

confidence: 82%

On the integral functional equations: On the integral d'Alembert's and Wilson's functional equations

Bouikhalene¹,

Elqorachi²

2016

Preprint

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Let G be a locally compact group, and let K be a compact subgroup of G. Let µ : G −→ C\{0} be a character of G. In this paper, we deal with the integral equationsandfor all x, y ∈ G where f, g : G −→ C, to be determined, are complex continuous functions on G. When K ⊂ Z(G), the center of G, Dµ(K) reduces to the new version of d'Almbert's functional equation f (xy) + µ(y)f (xy −1 ) = 2f (x)f (y), recently studied by Davison [18] and Stetkaer [35]. We derive the following link between the solutions of Wµ(K) and Dµ(K) in the following way :. This result is used to establish the superstability problem of Wµ(K). In the case where (G, K) is a central pair, we show that the solutions are expressed by means of K-spherical functions and related functions. Also we give explicit formulas of solutions of Dµ(K) in terms of irreducible representations of G. These formulas generalize Euler's formula cos(x) = e ix +e −ix 2 on G = R.

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“…Proof. The proof of the theorem is related to the one in [15], (see Theorem 3.1), where K = {Id, σ} and σ is a continuous involution of G. If f is unbounded, then by using the inequality (6.7), we get…”

Section: Hyers-ulam Stability Of Generalized Equations Of Stetkaer Typementioning

confidence: 92%

“…For the noncommutative case, some results for some particular equations of type (6.1) where obtained by Elqorachi and Akkouchi [14], [15], [17]. The stability of the classical examples f (x + y) = f (x)f (y), (6.5)…”

Section: Hyers-ulam Stability Of Generalized Equations Of Stetkaer Typementioning

confidence: 99%

On Stetkaer type functional equations and Hyers--Ulam stability

Bouikhalene¹,

Elqorachi²

2006

Publ. Math. Debrecen

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Let G be a locally compact group, K a compact subgroup of morphisms of G, χ : K −→ {z ∈ C | |z| = 1} a continuous homomorphism and µ a K-invariant bounded measure on G. In this paper we study functional equations of the formare unknown functions. These equations may be viewed as a generalization of the functional equations considered by Stetkaer in many of his works. We show how the solutions g and h are closely related to the solutions of Badora's functional equation solved in [4] and [13]. We treat examples and we give some applications. The case where G is a Lie group is considered. Furthermore, we investigate the Hyers-Ulam stability problem of these functional equations.

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On Hyers–Ulam Stability of Cauchy and Wilson Equations

Cited by 7 publications

References 8 publications

On Hyers--Ulam stability of the generalized Cauchy and Wilson equations

On Hyers--Ulam stability of the generalized Cauchy and Wilson equations

On the integral functional equations: On the integral d'Alembert's and Wilson's functional equations

On Stetkaer type functional equations and Hyers--Ulam stability

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