Abstract and Applied Analysis

Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras

Faruk Polat

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Abstract

Badora (2002) proved the following stability result. Let ε and δ be nonnegative real numbers, then for every mapping f of a ring R onto a Banach algebra B satisfying | | f ( x + y ) - f ( x ) - f ( y ) | | ε and | | f ( x y ) - f ( x ) f ( y ) | | δ for all x , y R , there exists a unique ring homomorphism h : R B such that | | f ( x ) - h ( x ) | | ε , x R . Moreover, b ( f ( x ) - h ( x ) ) = 0 , ( f ( x ) - h ( x ) ) b = 0 , for all x R and all b from the algebra generated by h ( R ) . In this paper, we generalize Badora's stability result above on ring homomorphisms for Riesz algebras with extended norms.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 653508, 9 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495623

Digital Object Identifier
doi:10.1155/2012/653508

Mathematical Reviews number (MathSciNet)
MR2872311

Zentralblatt MATH identifier
1235.39030

Citation

Polat, Faruk. Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras. Abstr. Appl. Anal. 2012 (2012), Article ID 653508, 9 pages. doi:10.1155/2012/653508. https://projecteuclid.org/euclid.aaa/1355495623


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References

  • S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960.
  • D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941.
  • Z. Moszner, “On the stability of functional equations,” Aequationes Mathematicae, vol. 77, no. 1-2, pp. 33–88, 2009.
  • B. Paneah, “A new approach to the stability of linear functional operators,” Aequationes Mathematicae, vol. 78, no. 1-2, pp. 45–61, 2009.
  • D. G. Bourgin, “Approximately isometric and multiplicative transformations on continuous function rings,” Duke Mathematical Journal, vol. 16, pp. 385–397, 1949.
  • R. Badora, “On approximate ring homomorphisms,” Journal of Mathematical Analysis and Applications, vol. 276, no. 2, pp. 589–597, 2002.
  • W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces, vol. 1, North-Holland, Amsterdam, The Netherlands, 1971.
  • G.-L. Forti, “Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 127–133, 2004.
  • J. Brzdęk, “On a method of proving the Hyers-Ulam stability of functional equations on restricted domains,” The Australian Journal of Mathematical Analysis and Applications, vol. 6, pp. 1–10, 2009.