Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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SOLUTION OF A VECTOR VARIABLE BI-ADDITIVE FUNCTIONAL EQUATION

  • Published : 2008.04.30
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Abstract

We investigate the relation between the vector variable bi-additive functional equation $f(\sum\limits^n_{i=1} xi,\;\sum\limits^n_{i=1} yj)={\sum\limits^n_{i=1}\sum\limits^n_ {j=1}f(x_i,y_j)$ and the multi-variable quadratic functional equation $$g(\sum\limits^n_{i=1}xi)\;+\;\sum\limits_{1{\leq}i<j{\leq}n}\;g(x_i-x_j)=n\sum\limits^n_{i=1}\;g(x_i)$$. Furthermore, we find out the general solution of the above two functional equations.

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References

  1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, Cambridge, 1989
  2. J.-H. Bae and K.-W. Jun, On the generalized Hyers-Ulam-Rassias stability of an ndimensional quadratic functional equation, J. Math. Anal. Appl. 258 (2001), 183-193 https://doi.org/10.1006/jmaa.2000.7372
  3. J.-H. Bae and W.-G. Park, On the generalized Hyers-Ulam-Rassias stability in Banach modules over a C-algebra, J. Math. Anal. Appl. 294 (2004), 196-205 https://doi.org/10.1016/j.jmaa.2004.02.009
  4. J.-H. Bae and W.-G. Park, On stability of a functional equation with n variables, Nonlinear Anal. 64 (2006), 856-868 https://doi.org/10.1016/j.na.2005.06.028
  5. S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137 https://doi.org/10.1006/jmaa.1998.5916
  6. C.-G. Park, Cauchy-Rassias stability of a generalized Trif 's mapping in Banach modules and its applications, Nonlinear Anal. 62 (2005), 595-613 https://doi.org/10.1016/j.na.2005.03.071
  7. S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1968, p.63

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