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An AQCQ-Functional Equation in Matrix Normed Spaces

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Abstract

In this paper, we prove the Hyers–Ulam stability of an additive–quadratic–cubic–quartic functional equation in matrix normed spaces.

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References

  1. Aczel J., Dhombres J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  2. Alimohammady M., Sadeghi A.: Some new results on the superstability of the Cauchy equation on semigroups. Results Math. 63, 705–712 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Almira J.M.: A note on classical and p-adic Fréchet functional equations with restrictions. Results Math. 63, 649–656 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aoki T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn 2, 64–66 (1950)

    Article  MATH  Google Scholar 

  5. Chang I., Eshaghi Gordji M., Khodaei H., Kim H.: Nearly quartic mappings in β-homogeneous F-spaces. Results Math. 63, 529–541 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Choi M.-D., Effros E.: Injectivity and operator spaces. J. Funct. Anal. 24, 156–209 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cholewa P.W.: Remarks on the stability of functional equations. Aequa. Math. 27, 76–86 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Czerwik S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg 62, 59–64 (1991)

    Article  MathSciNet  Google Scholar 

  9. Czerwik P.: Functional Equations and Inequalities in Several Variables. World Scientific Publishing Company, New Jersey (2002)

    Book  MATH  Google Scholar 

  10. Ebadian A., Ghobadipour H.: A fixed point approach to almost double derivations and Lie *-double derivations. Results Math. 63, 409–423 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Effros, E.: On multilinear completely bounded module maps. In: Contemporary Mathematics, vol. 62, pp. 479–501. American Mathematical Society. Providence (1987)

  12. Effros E., Ruan Z.-J.: On approximation properties for operator spaces. Int. J. Math. 1, 163–187 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. Effros E., Ruan Z.-J.: On the abstract characterization of operator spaces. Proc. Am. Math. Soc. 119, 579–584 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eshaghi Gordji, M., Abbaszadeh, S., Park, C.: On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces. J. Inequal. Appl. Article ID 153084 (2009)

  15. Eshaghi Gordji, M., Kaboli-Gharetapeh, S., Park, C., Zolfaghari, S.: Stability of an additive–cubic–quartic functional equation. Adv. Differ. Equa. Article ID 395693 (2009)

  16. Eshaghi Gordji M., Savadkouhi M.B.: Stability of a mixed type cubic–quartic functional equation in non-Archimedean spaces. Appl. Math. Lett. 23, 1198–1202 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gajda Z.: On stability of additive mappings. Int. J. Math. Math. Sci. 14, 431–434 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gǎvruta P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

    Article  MathSciNet  Google Scholar 

  19. Haagerup, U.: Decomposition of completely bounded maps. (Unpublished manuscript)

  20. Hyers D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. U.S.A. 27, 222–224 (1941)

    Article  MathSciNet  Google Scholar 

  21. Hyers D.H., Isac G., Rassias Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)

    Book  MATH  Google Scholar 

  22. Isac G., Rassias Th.M.: On the Hyers–Ulam stability of ψ-additive mappings. J. Approx. Theory 72, 131–137 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jun K., Kim H.: The generalized Hyers–Ulam–Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274, 867–878 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jun K., Lee Y.: A generalization of the Hyers–Ulam–Rassias stability of the pexiderized quadratic equations. J. Math. Anal. Appl. 297, 70–86 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jung S.: Hyers–Ulam–Rassias stability of functional equations in mathematical analysis. Hadronic Press lnc., Palm Harbor (2001)

    MATH  Google Scholar 

  26. Lee S., Im S., Hwang I.: Quartic functional equations. J. Math. Anal. Appl. 307, 387–394 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Park C.: Homomorphisms between Poisson JC*-algebras. Bull. Braz. Math. Soc. 36, 79–97 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pisier G.: Grothendieck’s Theorem for non-commutative C*-algebras with an appendix on Grothendieck’s constants. J. Funct. Anal. 29, 397–415 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rassias J.M.: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126–130 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rassias J.M.: Solution of a problem of Ulam. J. Approx. Theory 57, 268–273 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rassias Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    Article  MATH  Google Scholar 

  32. Rassias Th.M. (1990) Problem 16; 2, Report of the 27th international symposium on functional equations. Aequa. Math. 39, 292–293;309

  33. Rassias Th.M. (ed.): Functional Equations and Inequalities. Kluwer Academic, Dordrecht (2008)

  34. Rassias Th.M., Semrl P.: On the behaviour of mappings which do not satisfy Hyers–Ulam stability. Proc. Am. Math. Soc. 114, 989–993 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  35. Reich L., Tomaschek J.: Some remarks to the formal and local theory of the generalized Dhombresw functional equation. Results Math. 63, 377–395 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ruan Z.-J.: Subspaces of C*-algebras. J. Funct. Anal. 76, 217–230 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  37. Shin D., Lee S., Byun C., Kim S.: On matrix normed spaces. Bull. Korean Math. Soc. 27, 103–112 (1983)

    MathSciNet  Google Scholar 

  38. Skof F.: Proprietà locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53, 113–129 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ulam S.M.: A Collection of the Mathematical Problems. Interscience Publishers, New York (1960)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Daejin University, Kyeonggi, 487-711, Korea

    Jung Rye Lee

  2. Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea

    Choonkil Park

  3. Department of Mathematics, University of Seoul, Seoul, 130-743, Korea

    Dong Yun Shin

Authors
  1. Jung Rye Lee
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  2. Choonkil Park
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  3. Dong Yun Shin
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Correspondence to Dong Yun Shin.

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Lee, J.R., Park, C. & Shin, D.Y. An AQCQ-Functional Equation in Matrix Normed Spaces. Results. Math. 64, 305–318 (2013). https://doi.org/10.1007/s00025-013-0315-9

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