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ON A FUNCTIONAL EQUATION ARISING FROM PROTH IDENTITY

  • Chung, Jaeyoung ;
  • Sahoo, Prasanna K.
  • Received : 2015.06.08
  • Published : 2016.01.31

Abstract

We determine the general solutions $f:\mathbb{R}^2{\rightarrow}\mathbb{R}$ of the functional equation f(ux-vy, uy+v(x+y)) = f(x, y)f(u, v) for all x, y, u, $v{\in}\mathbb{R}$. We also investigate both bounded and unbounded solutions of the functional inequality ${\mid}f(ux-vy,uy+v(x+y))-f(x,y)f(u,v){\mid}{\leq}{\phi}(u,v)$ for all x, y, u, $v{\in}\mathbb{R}$, where ${\ph}:\mathbb{R}^2{\rightarrow}\mathbb{R}_+$ is a given function.

Keywords

exponential type functional equation;general solution;multiplicative function;Proth identity;stability;bounded solution

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