Open Access

Generalized orthogonal stability of some functional equations

Journal of Inequalities and Applications20062006:12404

DOI: 10.1155/JIA/2006/12404

Received: 19 November 2005

Accepted: 2 July 2006

Published: 21 September 2006


We deal with a conditional functional inequality , where is a given orthogonality relation, is a given nonnegative number, and is a given real number. Under suitable assumptions, we prove that any solution of the above inequality has to be uniformly close to an orthogonally additive mapping , that is, satisfying the condition . In the sequel, we deal with some other functional inequalities and we also present some applications and generalizations of the first result.


Authors’ Affiliations

Institute of Mathematics, University of Silesian


  1. Birkhoff G: Orthogonality in linear metric spaces. Duke Mathematical Journal 1935,1(2):169–172. 10.1215/S0012-7094-35-00115-6MathSciNetView ArticleMATHGoogle Scholar
  2. Drewnowski L, Orlicz W: On orthogonally additive functionals. Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 1968, 16: 883–888.MathSciNetMATHGoogle Scholar
  3. Fochi M: Functional equations on-orthogonal vectors. Aequationes Mathematicae 1989,38(1):28–40. 10.1007/BF01839491MATHMathSciNetView ArticleGoogle Scholar
  4. Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991,14(3):431–434. 10.1155/S016117129100056XMATHMathSciNetView ArticleGoogle Scholar
  5. Ger R, Sikorska J: Stability of the orthogonal additivity. Bulletin of the Polish Academy of Sciences, Mathematics 1995,43(2):143–151.MATHMathSciNetGoogle Scholar
  6. Gudder S, Strawther D: Orthogonally additive and orthogonally increasing functions on vector spaces. Pacific Journal of Mathematics 1975,58(2):427–436.MATHMathSciNetView ArticleGoogle Scholar
  7. Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941,27(4):222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
  8. James RC: Orthogonality in normed linear spaces. Duke Mathematical Journal 1945,12(2):291–302. 10.1215/S0012-7094-45-01223-3MATHMathSciNetView ArticleGoogle Scholar
  9. James RC: Inner product in normed linear spaces. Bulletin of the American Mathematical Society 1947, 53: 559–566. 10.1090/S0002-9904-1947-08831-5MATHMathSciNetView ArticleGoogle Scholar
  10. James RC: Orthogonality and linear functionals in normed linear spaces. Transactions of the American Mathematical Society 1947,61(2):265–292. 10.1090/S0002-9947-1947-0021241-4MathSciNetView ArticleGoogle Scholar
  11. Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MATHMathSciNetView ArticleGoogle Scholar
  12. Rätz J: On orthogonally additive mappings. Aequationes Mathematicae 1985,28(1–2):35–49.MATHMathSciNetView ArticleGoogle Scholar
  13. Sikorska J: Stability of the orthogonal additivity, doctoral dissertation. , University of Silesia, Katowice; 1998.Google Scholar
  14. Sundaresan K: Orthogonality and nonlinear functionals on Banach spaces. Proceedings of the American Mathematical Society 1972,34(1):187–190. 10.1090/S0002-9939-1972-0291835-XMATHMathSciNetView ArticleGoogle Scholar
  15. Szabó Gy: On mappings, orthogonally additive in the Birkhoff-James sense. Aequationes Mathematicae 1986,30(1):93–105. 10.1007/BF02189914MATHMathSciNetView ArticleGoogle Scholar
  16. Szabó Gy: A conditional Cauchy equation on normed spaces. Publicationes Mathematicae Debrecen 1993,42(3–4):265–271.MATHMathSciNetGoogle Scholar
  17. Szabó Gy: Isosceles orthogonally additive mappings and inner product spaces. Publicationes Mathematicae Debrecen 1995,46(3–4):373–384.MATHMathSciNetGoogle Scholar
  18. Ulam SM: Problems of Modern Mathematics. Interscience, New York; 1960.MATHGoogle Scholar
  19. Ulam SM: A Collection of Mathematical Problems, Science Editions. John Wiley & Sons, New York; 1968.Google Scholar
  20. Vajzović F: Über das Funktionalmit der Eigenschaft:. Glasnik Matematički. Serija III 1967, 2 (22): 73–81.MATHGoogle Scholar


© Sikorska 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.