## Abstract and Applied Analysis

### Functional Inequalities Associated with Additive Mappings

#### Abstract

The functional inequality $\Vert f(x)+2f(y)+2f(z)\Vert \leq \Vert 2f(x/2+y+z)\Vert +\phi \,\,(x,y,z)\text{\,}(x,y,z\in G)$ is investigated, where $G$ is a group divisible by $2,f:G \rightarrow X$ and $\phi :{G}^{3} \rightarrow [0,\infty )$ are mappings, and $X$ is a Banach space. The main result of the paper states that the assumptions above together with (1) $\phi (2x,-x,0)=0=\phi (0,x,-x)$ $(x\in G)$ and (2) ${\lim }_{n\rightarrow \infty }(1/{2}^{n})\phi ({2}^{n+1}x,{2}^{n}y,{2}^{n}z)=0$, or ${\lim }_{n\rightarrow \infty }{2}^{n}\phi (x/{2}^{n-1},y/{2}^{n},z/{2}^{n})=0\,\,(x,y,z\in G)$, 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

#### 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