Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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STABILITY OF THE MONOMIAL FUNCTIONAL EQUATION IN QUASI NORMED SPACES

  • Mirmostafaee, Alireza Kamel (DEPARTMENT OF PURE MATHEMATICS CENTER OF EXCELLENCE IN ANALYSIS ON ALGEBRAIC STRUCTURES FERDOWSI UNIVERSITY OF MASHHAD)
  • Received : 2009.02.17
  • Published : 2010.07.31
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Abstract

Let X be a linear space and Y be a complete quasi p-norm space. We will show that for each function f : X $\rightarrow$ Y, which satisfies the inequality ${\parallel}{\Delta}_x^nf(y)\;-\;n!f(x){\parallel}\;{\leq}\;\varphi(x,y)$ for suitable control function $\varphi$, there is a unique monomial function M of degree n which is a good approximation for f in such a way that the continuity of $t\;{\mapsto}\;f(tx)$ and $t\;{\mapsto}\;\varphi(tx,\;ty)$ imply the continuity of $t\;{\mapsto}\;M(tx)$.

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References

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