Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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STABILITY OF PEXIDERIZED JENSEN AND JENSEN TYPE FUNCTIONAL EQUATIONS ON RESTRICTED DOMAINS

  • Choi, Chang-Kwon (Department of Mathematics and Liberal Education Institute Kunsan National University)
  • Received : 2018.06.26
  • Accepted : 2019.02.07
  • Published : 2019.05.31
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Abstract

In this paper, using the Baire category theorem we investigate the Hyers-Ulam stability problem of pexiderized Jensen functional equation $$2f(\frac{x+y}{2})-g(x)-h(y)=0$$ and pexiderized Jensen type functional equations $$f(x+y)+g(x-y)-2h(x)=0,\\f(x+y)-g(x-y)-2h(y)=0$$ on a set of Lebesgue measure zero. As a consequence, we obtain asymptotic behaviors of the functional equations.

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References

  1. A. Bahyrycz and J. Brzdek, On solutions of the d'Alembert equation on a restricted domain, Aequationes Math. 85 (2013), no. 1-2, 169-183. https://doi.org/10.1007/s00010-012-0162-x
  2. B. Batko, Stability of an alternative functional equation, J. Math. Anal. Appl. 339 (2008), no. 1, 303-311. https://doi.org/10.1016/j.jmaa.2007.07.001
  3. J. Brzdek, On a method of proving the Hyers-Ulam stability of functional equations on restricted domains, Aust. J. Math. Anal. Appl. 6 (2009), no. 1, Art. 4, 10 pp.
  4. J. Brzdek, On the quotient stability of a family of functional equations, Nonlinear Anal. 71 (2009), no. 10, 4396-4404. https://doi.org/10.1016/j.na.2009.02.123
  5. J. Brzdek, W. Fechner, M. S. Moslehian, and J. Sikorska, Recent developments of the conditional stability of the homomorphism equation, Banach J. Math. Anal. 9 (2015), no. 3, 278-326. https://doi.org/10.15352/bjma/09-3-20
  6. J. Brzdek, D. Popa, I. Rasa, and B. Xu, Ulam Stability of Operators, Mathematical Analysis and Its Applications, Academic Press, London, 2018.
  7. J. Brzdek and J. Sikorska, A conditional exponential functional equation and its stability, Nonlinear Anal. 72 (2010), no. 6, 2923-2934. https://doi.org/10.1016/j.na.2009.11.036
  8. C.-K. Choi and B. Lee, Measure zero stability problem for Jensen type functional equations, Global J. Pure Appl. Math. 12 (2016), no. 4, 3673-3682.
  9. J. Chung, Stability of functional equations on restricted domains in a group and their asymptotic behaviors, Comput. Math. Appl. 60 (2010), no. 9, 2653-2665. https://doi.org/10.1016/j.camwa.2010.09.003
  10. J. Chung, Stability of a conditional Cauchy equation on a set of measure zero, Aequationes Math. 87 (2014), no. 3, 391-400. https://doi.org/10.1007/s00010-013-0235-5
  11. J. Chung, On the Drygas functional equation in restricted domains, Aequationes Math. 90 (2016), no. 4, 799-808. https://doi.org/10.1007/s00010-015-0388-5
  12. J. Chung and C.-K. Choi, Asymptotic behaviors of alternative Jensen functional equations - revisited, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 19 (2012), no. 4, 409-421.
  13. J. Chung, D. Kim, and J. M. Rassias, Stability of Jensen-type functional equations on restricted domains in a group and their asymptotic behaviors, J. Appl. Math. 2012 (2012), Art. ID 691981, 12 pp.
  14. J. Chung and J. M. Rassias, Quadratic functional equations in a set of Lebesgue measure zero, J. Math. Anal. Appl. 419 (2014), no. 2, 1065-1075. https://doi.org/10.1016/j.jmaa.2014.05.032
  15. J. Chung and J. M. Rassias, On a measure zero stability problem of a cyclic equation, Bull. Aust. Math. Soc. 93 (2016), no. 2, 272-282. https://doi.org/10.1017/S0004972715001185
  16. M. Fochi, An alternative functional equation on restricted domain, Aequationes Math. 70 (2005), no. 3, 201-212. https://doi.org/10.1007/s00010-005-2776-8
  17. R. Ger and J. Sikorska, On the Cauchy equation on spheres, Ann. Math. Sil. 11 (1997), 89-99.
  18. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  19. S.-M. Jung, Hyers-Ulam-Rassias stability of Jensen's equation and its application, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3137-3143. https://doi.org/10.1090/S0002-9939-98-04680-2
  20. S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), no. 1, 126-137. https://doi.org/10.1006/jmaa.1998.5916
  21. S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Springer Optimization and Its Applications, 48, Springer, New York, 2011.
  22. M. Kuczma, Functional equations on restricted domains, Aequationes Math. 18 (1978), no. 1-2, 1-34. https://doi.org/10.1007/BF01844065
  23. J. C. Oxtoby, Measure and Category, second edition, Graduate Texts in Mathematics, 2, Springer-Verlag, New York, 1980.
  24. S.-H. Park and C.-K. Choi, Measure zero stability problem for alternative Jensen functional equations, Global J. Pure Appl. Math. 13 (2017), no. 4, 1171-1182.
  25. J. M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl. 276 (2002), no. 2, 747-762. https://doi.org/10.1016/S0022-247X(02)00439-0
  26. J. M. Rassias and M. J. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl. 281 (2003), no. 2, 516-524. https://doi.org/10.1016/S0022-247X(03)00136-7
  27. J. M. Rassias and M. J. Rassias, Asymptotic behavior of alternative Jensen and Jensen type functional equations, Bull. Sci. Math. 129 (2005), no. 7, 545-558. https://doi.org/10.1016/j.bulsci.2005.02.001
  28. T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  29. J. Sikorska, On two conditional Pexider functional equations and their stabilities, Non-linear Anal. 70 (2009), no. 7, 2673-2684. https://doi.org/10.1016/j.na.2008.03.054
  30. F. Skof, Sull'approssimazione delle applicazioni localmente ${\delta}$-additive, Atii Accad. Sci.Torino Cl. Sci. Fis. Mat. Natur. 117 (1983), 377-389.
  31. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  32. S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York, 1960.