Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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STABILITY OF THE JENSEN TYPE FUNCTIONAL EQUATION IN BANACH ALGEBRAS: A FIXED POINT APPROACH

  • Received : 2011.02.09
  • Accepted : 2011.06.10
  • Published : 2011.06.30
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Abstract

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in Banach algebras and of derivations on Banach algebras for the following Jensen type functional equation: $$f({\frac{x+y}{2}})+f({\frac{x-y}{2}})=f(x)$$.

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References

  1. C. Baak, Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta Math. Sin. (Engl. Ser.) 22 (2006), 1789-1796. https://doi.org/10.1007/s10114-005-0697-z
  2. C. Baak, D. Boo and Th.M. Rassias, Generalized additive mapping in Banach modules and isomorphisms between $C^*$-algebras, J. Math. Anal. Appl. 314 (2006), 150-161. https://doi.org/10.1016/j.jmaa.2005.03.099
  3. L. Cadariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math. 4, no. 1 (2003), Art. ID 4.
  4. P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  5. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64. https://doi.org/10.1007/BF02941618
  6. J. Diaz and B. Margolis, A flxed point theorem of the alternative for contrac- tions on a generalized complete metric space, Bull. Amer. Math. Soc. (N.S) 74 (1968), 305-309. https://doi.org/10.1090/S0002-9904-1968-11933-0
  7. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211
  8. D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  9. D.H. Hyers, G. Isac and Th.M. Rassias, Topics in Nonlinear Analysis and Applications, World Scientific Publishing Co., Singapore, New Jersey, London, 1997.
  10. D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
  11. F.H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125-153. https://doi.org/10.1007/BF01830975
  12. G. Isac and Th.M. Rassias, Stability of $\psi$-additive mappings: appications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219-228. https://doi.org/10.1155/S0161171296000324
  13. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Florida, 2001.
  14. C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. 2007 (2007), Art. ID 50175. https://doi.org/10.1155/2007/50175
  15. C. Park and J. Hou, Homomorphisms between $C^*$-algebras associated with the Trif functional equation and linear derivations on $C^*$-algebras, J. Korean Math. Soc. 41 (2004), 461-477. https://doi.org/10.4134/JKMS.2004.41.3.461
  16. C. Park, J. Hou and S. Oh, Homomorphisms between J$C^*$-algebras and between Lie $C^*$-algebras, Acta Math. Sin. (Engl.Ser.) 21 (2005), 1391-1398. https://doi.org/10.1007/s10114-005-0629-y
  17. C. Park and Th.M. Rassias, On a generalized Trif 's mapping in Banach mod- ules over a $C^*$-algebra, J. Korean Math. Soc. 43 (2006), 323-356. https://doi.org/10.4134/JKMS.2006.43.2.323
  18. J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), 445-446.
  19. J.M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), 268-273. https://doi.org/10.1016/0021-9045(89)90041-5
  20. Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  21. Th.M. Rassias, On modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), 106-113. https://doi.org/10.1016/0022-247X(91)90270-A
  22. Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babes-Bolyai, Ser. Math. XLIII (1998), 89-124.
  23. Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352-378. https://doi.org/10.1006/jmaa.2000.6788
  24. Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264-284. https://doi.org/10.1006/jmaa.2000.7046
  25. Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 23-130. https://doi.org/10.1023/A:1006499223572
  26. Th.M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989-993. https://doi.org/10.1090/S0002-9939-1992-1059634-1
  27. Th.M. Rassias and P. Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), 325-338. https://doi.org/10.1006/jmaa.1993.1070
  28. Th.M. Rassias and K. Shibata, Variational problem of some quadratic func- tionals in complex analysis, J. Math. Anal. Appl. 228 (1998), 234-253. https://doi.org/10.1006/jmaa.1998.6129
  29. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  30. S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960.