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  • Research Article
  • Open Access

Generalized orthogonal stability of some functional equations

Journal of Inequalities and Applications20062006:12404

  • Received: 19 November 2005
  • Accepted: 2 July 2006
  • Published:


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.


  • Real Number
  • Functional Equation
  • Additive Mapping
  • Orthogonality Relation
  • Suitable Assumption


Authors’ Affiliations

Institute of Mathematics, University of Silesian, Bankowa 14, Katowice, 40-007, Poland


  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

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