Solutions and stability of a variant of Wilson’s functional equation.

• Elqorachi, Elhoucien ^[1] ; Redouani, Ahmed ^[1]
  1. [1] Université Ibn Zohr

     Université Ibn Zohr

     Agadir, Marruecos

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 37, Nº. 2, 2018, págs. 317-344
• Idioma: inglés
• DOI: 10.4067/s0716-09172018000200317
• Enlaces
  • Texto completo
• Resumen

  • In this paper we will investigate the complex-valued solutions and stability of the generalized variant of Wilson’s functional equation (E) : f(xy) + χ(y)f(σ(y)x) = 2f(x)g(y), x, y ∈ G, where G is a group, σ is an involutive morphism of G and χ is a character of G. (a) We solve (E) when σ is an involutive automorphism, and we obtain some properties about solutions of (E) when σ is an involutive anti-automorphism. (b) We obtain the Hyers Ulam stability of equation (E). As an application, we prove the superstability of the functional equation f(xy) + χ(y)f(σ(y)x) = 2f(x)f(y), x, y ∈ G.

• Referencias bibliográficas
  • J. Aczél and J. Dhombres, Functional equations in several variables. With applications to mathematics, information theory and to the natural...
  • R. Badora, Stability Properties of Some Functional Equations. In: Themistocles Rassias, Janusz Brzdek (ed.) Functional Equations in Mathematical...
  • R. Badora, On the stability of a functional equation for generalized trigonometric functions. In: Th. M. Rassias (ed.) Functional Equations...
  • J. A. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc., 80, pp. 411-416, (1980).
  • B. Bouikhalene and E. Elqorachi, Stability of the spherical functions, Georgian Math. J., – DOI: https://doi.org/10.1515/gmj-2015-0052, 23...
  • B. Bouikhalene, E. Elqorachi and J. M. Rassias, The superstability of d’Alembert’s functional equation on the Heisenberg group, App. Math....
  • T.M.K. Davison, D’Alembert’s functional equation on topological groups. Aequationes Math., 76, pp. 33-53. (2008).
  • T.M.K. Davison, D’Alembert’s functional equation on topological monoids. Publ. Math. Debrecen, 75, 1/2, pp. 41-66, (2009).
  • Y. Dilian, Factorization of cosine functions on compact connected groups. Math. Z., 254, No. 4, pp. 655-674, (2006).
  • Y. Dilian, Functional equations and Fourier analysis, Canadian Mathematical Bulletin, 56, pp. 218, (2011).
  • Y. Dilian, Cosine functions revisited. Banach J. Math. Anal., 5, pp. 126-130, (2011).
  • R. Ebanks, Bruce and H. Stetkær, On Wilson’s functional equations, Aequationes Math., 89, pp. 339-354, (2015).
  • E. Elqorachi and M. Akkouchi, The superstability of the generalized dAlembert functional equation, Georgian Math. J., 10, pp. 503-508, (2003).
  • G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math., 50, pp. 143-190, (1995).
  • F. Peter de Place, d’Alembert’s and Wilson’s functional equations on Lie groups, Aequationes Math., 67, pp. 12-25, (2004).
  • R. Ger, Superstability is not natural, Rocznik Nauk.-Dydakt. Prace Mat., 159, No. 13, pp. 109-123, (1993).
  • R. Ger and P. Šemrl, The stability of the exponential equation, Proc. Amer. Soc., 124, pp. 779-787, (1996).
  • D. H. Hyers, G. I. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel,, (1998).
  • S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Inc., Palm Harbor, Florida,, (2003).
  • PL. Kannappan, The functional equation f(xy)+f(xy¯¹) = 2f(x)f(y) for groups, Proc. Amer. Math. Soc., 19, pp. 69-74, (1968).
  • G. H. Kim, On the stability of trigonometric functional equations, Advances in Difference Equations, Vol. (2007), Article ID 90405, 10 pages.
  • G. H. Kim, On the stability of the Pexiderized trigonometric functional equation, Applied Mathematics and Computation, 203, No. 1, pp. 99-105,...
  • A. Redouani, E. Elqorachi and M. Th. Rassias, The superstability of d’Alembert’s functional equation on step 2-nilpotent groups, Aequationes...
  • H. Stetkær, A link between Wilson’s and d’Alembert’s functional equations, Aequationes Math., 90, pp. 407-409, (2016).
  • H. Stetkær, A variant of d’Alemberts functional equation, Aequationes Math., 89, pp. 657-662, (2015).
  • H. Stetkær, D’Alembert’s functional equation on groups, Banach Center Publ., 99, pp. 173-191, (2013).
  • H. Stetkær, Functional equations on groups, World Scientific, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai,...
  • H. Stetkær, Properties of d’Alembert functions, Aequationes Math., 77, pp. 281-301, (2009).
  • H. Stetkær, d’Alembert’s and Wilson’s functional equations on step 2 nilpotent groups, Aequationes Math., 67, No. 3, pp. 241-261, (2004).
  • H. Stetkær, On multiplicative maps, Semigroup Forum, 63, pp. 466-468, (2001).
  • H. Stetkær, On a variant of Wilson’s functional equation on groups, Aequationes Math., 68, No. 3, pp. 160-176, (2004).
  • L. Székelyhidi, On a stability theorem, C. R. Math. Acad. Sc. Canada, 3, pp. 253-255, (1981).
  • L. Székelyhidi, On a theorem of Baker, Lawrence and Zorzitto, Proc. Amer. Math. Soc., 84, pp. 95-96, (1982).
  • L. Székelyhidi, The stability of dAlembert-type functional equations, Acta Sci. Math. Szeged 44, pp. 313-320, (1982).
  • L. Székelyhidi, The stability of the sine and cosine functional equations, Proc. Amer. Math. Soc., 110, pp. 109-115, (1990).
  • L. Székelyhidi, Fréchet equation and Hyers’s theorem on noncommutative semigroups, Ann. Polon. Math., 48, pp. 183-189, (1988).