Abstract and Applied Analysis

Functional Inequalities Associated with Additive Mappings

Jaiok Roh and Ick-Soon Chang

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Abstract

The functional inequality f ( x ) + 2 f ( y ) + 2 f ( z ) 2 f ( x / 2 + y + z ) + ϕ   ( x , y , z ) ( x , y , z G ) is investigated, where G is a group divisible by 2 , f : G X and ϕ : G 3 [ 0 , ) are mappings, and X is a Banach space. The main result of the paper states that the assumptions above together with (1) ϕ ( 2 x , x , 0 ) = 0 = ϕ ( 0 , x , x ) ( x G ) and (2) lim n ( 1 / 2 n ) ϕ ( 2 n + 1 x , 2 n y , 2 n z ) = 0 , or lim n 2 n ϕ ( x / 2 n 1 , y / 2 n , z / 2 n ) = 0   ( x , y , z G ) , imply that f is additive. In addition, some stability theorems are proved.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 136592, 11 pages.

Dates
First available in Project Euclid: 10 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1234298987

Digital Object Identifier
doi:10.1155/2008/136592

Mathematical Reviews number (MathSciNet)
MR2438261

Zentralblatt MATH identifier
1161.26012

Citation

Roh, Jaiok; Chang, Ick-Soon. Functional Inequalities Associated with Additive Mappings. Abstr. Appl. Anal. 2008 (2008), Article ID 136592, 11 pages. doi:10.1155/2008/136592. https://projecteuclid.org/euclid.aaa/1234298987


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References

  • S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, 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, no. 4, 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.
  • Th. 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.
  • L. Maligranda, ``A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions---a question of priority,'' Aequationes Mathematicae, vol. 75, no. 3, pp. 289--296, 2008.
  • Z. Gajda, ``On stability of additive mappings,'' International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431--434, 1991.
  • Th. M. Rassias and P. Šemrl, ``On the behavior of mappings which do not satisfy Hyers-Ulam stability,'' Proceedings of the American Mathematical Society, vol. 114, no. 4, pp. 989--993, 1992.
  • J. M. Rassias, ``On approximation of approximately linear mappings by linear mappings,'' Journal of Functional Analysis, vol. 46, no. 1, pp. 126--130, 1982.
  • J. M. Rassias, ``On approximation of approximately linear mappings by linear mappings,'' Bulletin des Sciences Mathématiques, vol. 108, no. 4, pp. 445--446, 1984.
  • J. M. Rassias, ``Solution of a problem of Ulam,'' Journal of Approximation Theory, vol. 57, no. 3, pp. 268--273, 1989.
  • D. G. Bourgin, ``Classes of transformations and bordering transformations,'' Bulletin of the American Mathematical Society, vol. 57, pp. 223--237, 1951.
  • D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol. 34 of Progress in Nonlinear Differential Equations and Their Applications, Birkh"a"user, Boston, Mass, USA, 1998.
  • K.-W. Jun and H.-M. Kim, ``Stability problem of Ulam for generalized forms of Cauchy functional equation,'' Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 535--547, 2005.
  • K.-W. Jun and Y.-H. Lee, ``A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations,'' Journal of Mathematical Analysis and Applications, vol. 297, no. 1, pp. 70--86, 2004.
  • S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001.
  • S.-M. Jung, ``Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients,'' Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 549--561, 2006.
  • C.-G. Park, ``Homomorphisms between Poisson $JC^\ast $-algebras,'' Bulletin of the Brazilian Mathematical Society, vol. 36, no. 1, pp. 79--97, 2005.
  • C. Park, ``Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras,'' Bulletin des Sciences Mathématiques, vol. 132, no. 2, pp. 87--96, 2008.
  • C.-G. Park and Th. M. Rassias, ``Hyers-Ulam stability of a generalized Apollonius type quadratic mapping,'' Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 371--381, 2006.
  • W. Fechner, ``Stability of a functional inequality associated with the Jordan-von Neumann functional equation,'' Aequationes Mathematicae, vol. 71, no. 1-2, pp. 149--161, 2006.
  • S.-M. Jung and J. M. Rassias, ``A fixed point approach to the stability of a functional equation of the spiral of Theodorus,'' Fixed Point Theory and Applications, vol. 2008, Article ID 945010, 7 pages, 2008.
  • H.-M. Kim and J. M. Rassias, ``Generalization of Ulam stability problem for Euler-Lagrange quadratic mappings,'' Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 277--296, 2007.
  • Y.-S. Lee and S.-Y. Chung, ``Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions,'' Applied Mathematics Letters, vol. 21, no. 7, pp. 694--700, 2008.
  • P. Nakmahachalasint, ``On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations,'' International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 63239, 10 pages, 2007.
  • C.-G. Park, ``Stability of an Euler-Lagrange-Rassias type additive mapping,'' International Journal of Applied Mathematics & Statistics, vol. 7, no. Fe07, pp. 101--111, 2007.
  • A. Pietrzyk, ``Stability of the Euler-Lagrange-Rassias functional equation,'' Demonstratio Mathematica, vol. 39, no. 3, pp. 523--530, 2006.
  • J. M. Rassias, ``On a new approximation of approximately linear mappings by linear mappings,'' Discussiones Mathematicae, vol. 7, pp. 193--196, 1985.
  • J. M. Rassias, ``On the stability of the Euler-Lagrange functional equation,'' Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185--190, 1992.
  • J. M. Rassias, ``Alternative contraction principle and alternative Jensen and Jensen type mappings,'' International Journal of Applied Mathematics & Statistics, vol. 4, no. M06, pp. 1--10, 2006.
  • J. M. Rassias and M. J. Rassias, ``Refined Ulam stability for Euler-Lagrange type mappings in Hilbert spaces,'' International Journal of Applied Mathematics & Statistics, vol. 7, no. Fe07, pp. 126--132, 2007.
  • J. Tabor, ``Stability of the Fischer-Muszély functional equation,'' Publicationes Mathematicae Debrecen, vol. 62, no. 1-2, pp. 205--211, 2003.
  • J. Tabor and J. Tabor, ``19. remark, (solution of the 7. Problem by K. Nikodem), report of meeting,'' Aequationes Mathematicae, vol. 61, no. 3, pp. 307--309, 2001.
  • J. Roh and H. J. Shin, ``Approximation of Cauchy additive mappings,'' Bulletin of the Korean Mathematical Society, vol. 44, no. 4, pp. 851--860, 2007.
  • P. Fischer and Gy. Muszély, ``On some new generalizations of the functional equation of Cauchy,'' Canadian Mathematical Bulletin, vol. 10, pp. 197--205, 1967.
  • Gy. Maksa and P. Volkmann, ``Characterization of group homomorphisms having values in an inner product space,'' Publicationes Mathematicae Debrecen, vol. 56, no. 1-2, pp. 197--200, 2000.
  • P. Volkmann, ``Pour une fonction réelle $f$ l'inéquation $|f(x)+f(y)|\leq |f(x+y)|$ et l'équation de Cauchy sont équivalentes,'' in Proceedings of the 23rd International Symposium on Functional Equations, University of Waterloo, Gargnano, Italy, June 1985. \endthebibliography