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JENSEN TYPE QUADRATIC-QUADRATIC MAPPING IN BANACH SPACES

  • Published : 2006.11.30

Abstract

Let X, Y be vector spaces. It is shown that if an even mapping $f:X{\rightarrow}Y$ satisfies f(0) = 0 and $$(0.1)\;f(\frac {x+y} 2+z)+f(\frac {x+y} 2-z)+f(\frac {x-y} 2+z)+f(\frac {x-y} 2-z)=f(x)+f(y)+4f(z)$$ for all x, y, z ${\in}$X, then the mapping $f:X{\rightarrow}Y$ is quadratic. Furthermore, we prove the Cauchy-Rassias stability of the functional equation (0.1) in Banach spaces.

Keywords

Cauchy-Rassias stability;quadratic mapping;functional equation

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  1. Stability of a Bi-Additive Functional Equation in Banach Modules Over aC⋆-Algebra vol.2012, 2012, https://doi.org/10.1155/2012/835893
  2. A FIXED POINT APPROACH TO THE CAUCHY-RASSIAS STABILITY OF GENERAL JENSEN TYPE QUADRATIC-QUADRATIC MAPPINGS vol.47, pp.5, 2010, https://doi.org/10.4134/BKMS.2010.47.5.987