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DISTRIBUTIONAL SOLUTIONS OF WILSON'S FUNCTIONAL EQUATIONS WITH INVOLUTION AND THEIR ERDÖS' PROBLEM

  • Received : 2015.07.21
  • Published : 2016.07.31

Abstract

We find the distributional solutions of the Wilson's functional equations $$u{\circ}T+u{\circ}T^{\sigma}-2u{\otimes}v=0,\\u{\circ}T+u{\circ}T^{\sigma}-2v{\otimes}u=0,$$ where $u,v{\in}{\mathcal{D}}^{\prime}({\mathbb{R}}^n)$, the space of Schwartz distributions, T(x, y) = x + y, $T^{\sigma}(x,y)=x+{\sigma}y$, $x,y{\in}{\mathbb{R}}^n$, ${\sigma}$ an involution, and ${\circ}$, ${\otimes}$ are pullback and tensor product of distributions, respectively. As a consequence, we solve the $Erd{\ddot{o}}s$' problem for the Wilson's functional equations in the class of locally integrable functions. We also consider the Ulam-Hyers stability of the classical Wilson's functional equations $$f(x+y)+f(x+{\sigma}y)=2f(x)g(y),\\f(x+y)+f(x+{\sigma}y)=2g(x)f(y)$$ in the class of Lebesgue measurable functions.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. P. Sinopoulos, Functional equations on semigroups, Aequationes Math. 59 (2000), no. 3, 255-261. https://doi.org/10.1007/s000100050125
  2. H. Stetkaer, Functional equations on abelian groups with involution, Aequationes Math. 54 (1997), no. 1-2, 144-172. https://doi.org/10.1007/BF02755452
  3. G. van der Lyn, Sur l'equation fonctionnelle f(x + y) + f(x − y) = 2f(x)$\phi$(y), Mathematica (Cluj) 16 (1940), 91-96.
  4. D. V. Widder, The Heat Equation, Academic Press, New York, 1975.
  5. W. H. Wilson, On certain related functional equations, Bull. Amer. Math. Soc. 26 (1920), no. 7, 300-312. https://doi.org/10.1090/S0002-9904-1920-03310-0
  6. W. H. Wilson, Two general functional equations, Bull. Amer. Math. Soc. 31 (1925), no. 7, 330-333. https://doi.org/10.1090/S0002-9904-1925-04045-8
  7. J. K. Chung, B. R. Ebanks, C. T. Ng, and P. K. Sahoo, On a quadratic-trigonometric functional equation and some applications, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1131-1161. https://doi.org/10.1090/S0002-9947-1995-1290715-0
  8. I. Corovei, The cosine functional equation for nilpotent groups, Aequationes Math. 15 (1977), no. 1, 99-106. https://doi.org/10.1007/BF01837878
  9. I. Corovei, The functional equation f(xy) + f($xy^{-1}$) = 2f(x)g(y) for nilpotent groups, Mathematica (Cluj) 22(45) (1980), no. 1, 33-41.
  10. I. Corovei, The d'Alembert functional equation on metabelian groups, Aequationes Math. 57 (1999), no. 2-3, 201-205. https://doi.org/10.1007/s000100050077
  11. I. Corovei, Wilson's functional equation on metabelian groups, Mathematica (Cluj) 44(67) (2002), no. 2, 137-146.
  12. P. De Place Friis, D'Alembert's and Wilson's equations on Lie groups, Aequationes Math. 67 (2004), no. 1-2, 12-25. https://doi.org/10.1007/s00010-002-2665-3
  13. J. d'Alembert, Addition au Memoire sur la courbe que forme une corde tendue mise en vibration, Hist. Acad. Berlin (1750), 355-360.
  14. J. d'Alembert, Memoire sur les principes de mecanique, Hist. Acad. Sci. Paris, pages 278-286, 1769.
  15. P. Erdos, Problem P310, Colloq. Math. 7 (1960), 311.
  16. I. Fenyo, Uber eine Losungsmethode gewisser Funktionalgleichungen, Acta Math. Acad. Sci. Hungar. 7 (1956), 383-396. https://doi.org/10.1007/BF02020533
  17. I. M. Gelfand and G. E. Shilov, Generalized Functions II, Academic Press, New York, 1968.
  18. I. M. Gelfand and G. E. Shilov, Generalized Functions IV, Academic, Press, New York, 1968.
  19. L. Hormander, The analysis of linear partial differential operator I, Springer-Verlag, Berlin-New York, 1983.
  20. W. B. Jurkat, On Cauchy's functional equation, Proc. Amer. Math. Soc. 16 (1965), 683-686.
  21. S. Kaczmarz, Sur l'equation fonctionnelle f(x)+f(x+y) = $\phi$(y)f(x+y/2), Fund. Math. 6 (1924), 122-129. https://doi.org/10.4064/fm-6-1-122-129
  22. P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton, 2011.
  23. L. Schwartz, Theorie des Distributions, Hermann, Paris, 1966.
  24. J. Aczel, Lectures on Functional Equations and Their Applications, Dover Publications Inc., New York 2006.
  25. J. Aczel, J. K. Chung, and C. T. Ng, Symmetric second differences in product form on groups, Topics in mathematical analysis, 1-22, Ser. Pure Math., 11, World Sci. Publ., Teaneck, NJ, 1989.
  26. T. Angheluta, Asupra unei ecuatii functionale cu trei functii necunoscute, Lucr. Sti. Inst. Politehn. Astr. 5 (1960), 23-30.
  27. A. Bahyrycz and J. Brzdek, On solutions of the d'Alembert equation on a restricted domain, Aequationes Math. 85 (2013), no. 1-2, 169-183. https://doi.org/10.1007/s00010-012-0162-x
  28. N. G. De Brujin, On almost additive functions, Colloq. Math. 15 (1966), 59-63. https://doi.org/10.4064/cm-15-1-59-63
  29. J. Chung, A distributional version of functional equations and their stabilities, Nonlinear Anal. 62 (2005), no. 6, 1037-1051. https://doi.org/10.1016/j.na.2005.04.016
  30. J. Chung, Stability of exponential equations in Schwarz distributions, Nonlinear Anal. 69 (2008), no. 10, 3503-3511. https://doi.org/10.1016/j.na.2007.09.037
  31. J. Chung and P. K. Sahoo, Solution of several functional equations on nonunital semigroups using Wilson's functional equations with involution, Abstr. Appl. Anal. 2014 (2014), Art. ID 463918, 9 pp.
  32. J. Chung and P. K. Sahoo, Stability of Wilson's functional equations with involutions, Aequationes Math. 89 (2015), no. 3, 749-763. https://doi.org/10.1007/s00010-014-0262-x

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