Abstract and Applied Analysis

Stability of a Logarithmic Functional Equation in Distributions on a Restricted Domain

Jaeyoung Chung and Prasanna K. Sahoo

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Abstract

Let be the set of real numbers, + = { x x > 0 } , ϵ + , and f , g , h : + . As classical and L versions of the Hyers-Ulam stability of the logarithmic type functional equation in a restricted domain, we consider the following inequalities: | f ( x + y ) - g ( x y ) - h ( (1 / x) + (1 / y) ) | ϵ , and f ( x + y ) - g ( x y ) - h (1 / x) + (1 / y) L ( Γ d ) ϵ in the sectors Γ d = { ( x , y ) : x > 0 , y > 0 , (y / x) > d } . As consequences of the results, we obtain asymptotic behaviors of the previous inequalities. We also consider its distributional version u S - v Π - w R Γ d ϵ , where u , v , w 𝒟 ' ( + ) , S ( x , y ) = x + y , Π ( x , y ) = x y , R ( x , y ) = 1 / x + 1 / y , x , y + , and the inequality · Γ d ϵ means that | · , φ | ϵ φ L 1 for all test functions φ C c ( Γ d ) .

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 751680, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393448678

Digital Object Identifier
doi:10.1155/2013/751680

Mathematical Reviews number (MathSciNet)
MR3095363

Zentralblatt MATH identifier
07095327

Citation

Chung, Jaeyoung; Sahoo, Prasanna K. Stability of a Logarithmic Functional Equation in Distributions on a Restricted Domain. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 751680, 9 pages. doi:10.1155/2013/751680. https://projecteuclid.org/euclid.aaa/1393448678


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References

  • S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, NY, USA, 1960.
  • D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, Mass, USA, 1998.
  • 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.
  • T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.
  • D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, pp. 223–237, 1951.
  • D. G. Bourgin, “Multiplicative transformations,” Proceedings of the National Academy of Sciences of the United States of America, vol. 36, pp. 564–570, 1950.
  • T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
  • J. A. Baker, “On a functional equation of Aczél and Chung,” Aequationes Mathematicae, vol. 46, no. 1-2, pp. 99–111, 1993.
  • J. A. Baker, “The stability of the cosine equation,” Proceedings of the American Mathematical Society, vol. 80, no. 3, pp. 411–416, 1980.
  • J.-Y. Chung, “Stability of functional equations on restricted domains in a group and their asymptotic behaviors,” Computers & Mathematics with Applications, vol. 60, no. 9, pp. 2653–2665, 2010.
  • J. Chung, “Stability of approximately quadratic Schwartz distributions,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 67, no. 1, pp. 175–186, 2007.
  • J. Chung, “A distributional version of functional equations andtheir stabilities,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 62, no. 6, pp. 1037–1051, 2005.
  • S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Palm Harbor, Fla, USA, 2003.
  • Z. Daróczy, “On a functional equation of Hosszú type,” Mathematica Pannonica, vol. 10, no. 1, pp. 77–82, 1999.
  • K. J. Heuvers, “Another logarithmic functional equation,” Aequationes Mathematicae, vol. 58, no. 3, pp. 260–264, 1999.
  • K. J. Heuvers and P. Kannappan, “A third logarithmic functional equation and Pexider generalizations,” Aequationes Mathematicae, vol. 70, no. 1-2, pp. 117–121, 2005.
  • D. H. Hyers and T. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125–153, 1992.
  • S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001.
  • S.-M. Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic property,” Journal of Mathematical Analysis and Applications, vol. 222, no. 1, pp. 126–137, 1998.
  • S.-M. Jung, “Hyers-Ulam-Rassias stability of Jensen's equation and its application,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3137–3143, 1998.
  • K.-W. Jun and H.-M. Kim, “Stability problem for Jensen-type functional equations of cubic mappings,” Acta Mathematica Sinica, vol. 22, no. 6, pp. 1781–1788, 2006.
  • G. H. Kim, “On the stability of the Pexiderized trigonometric functional equation,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 99–105, 2008.
  • G. H. Kim and Y. W. Lee, “Boundedness of approximate trigonometric functions,” Applied Mathematics Letters, vol. 22, no. 4, pp.439–443, 2009.
  • C. Park, “Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras,” Bulletin des Sciences Mathématiques, vol. 132, no. 2, pp. 87–96, 2008.
  • J. M. Rassias and M. J. Rassias, “On the Ulam stability of Jensen and Jensen type mappings on restricted domains,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 516–524, 2003.
  • J. M. Rassias, “On the Ulam stability of mixed type mappings on restricted domains,” Journal of Mathematical Analysis and Applications, vol. 276, no. 2, pp. 747–762, 2002.
  • J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol.46, no. 1, pp. 126–130, 1982.
  • T. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264–284, 2000.
  • L. Schwartz, Théorie des Distributions, Hermann, Paris, France, 1966.
  • F. Skof, “On the approximation of locally $\delta $-additive mappings,” Atti della Accademia delle Scienze di Torino, vol. 117, no. 4–6, pp. 377–389, 1983.
  • F. Skof, “Proprieta' locali e approssimazione di operatori,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, no.1, pp. 113–129, 1983.
  • G. L. Forti, “The stability of homomorphisms and amenability, with applications to functional equations,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 57, pp. 215–226, 1987.
  • L. Hörmander, The Analysis of Linear Partial Differential Operator I, Springer, Berlin, Germany, 1983.