Acta mathematica scientia,Series B ›› 2016, Vol. 36 ›› Issue (3): 791-801.doi: 10.1016/S0252-9602(16)30040-6

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SOLUTIONS AND STABILITY OF A GENERALIZATION OF WILSON'S EQUATION

Bouikhalene BELAID1, Elqorachi ELHOUCIEN2   

  1. 1. Bouikhalene Belaid, Polydisciplinary Faculty, University Sultan Moulay Slimane, Beni-Mellal, Morocco;
    2. Elqorachi Elhoucien, Functional Equations and Applications Team, Department of Mathematics, Faculty of Sciences, University Ibn Zohr, Agadir, Morocco
  • Received:2015-03-26 Revised:2015-07-15 Online:2016-06-25 Published:2016-06-25

Abstract:

In this paper we study the solutions and stability of the generalized Wilson's functional equation∫Gf(xty)dμ(t) +∫Gf(xtσ(y))dμ(t)=2f(x)g(y), x, yG, where G is a locally compact group, σ is a continuous involution of G and μ is an idempotent complex measure with compact support and which is σ-invariant. We show that∫Gg(xty)dμ(t)+∫Gg(xtσ(y))dμ(t)=2g(x)g(y) if f≠0 and∫Gf(t.)dμ(t)≠0, where[∫Gf(t.)dμ(t)](x)=∫Gf(tx)dμ(t). We also study some stability theorems of that equation and we establish the stability on noncommutative groups of the classical Wilson's functional equation f(xy)+χ(y)f((y))=2f(x)g(y)x, yG, where χ is a unitary character of G.

Key words: d'Alembert's functional equation, locally compact group, involution, character, complex measure, Wilson's functional equation, Hyers-Ulam stability

CLC Number: 

  • 39B82
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