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Fixed points and additive \({\rho}\)-functional equations

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Abstract

In this paper, we solve the additive \({\rho}\)-functional equations

$$\begin{aligned} f(x+y)-f(x)-f(y)= & {} \rho(2f(\frac{x+y}{2})-f(x)-f(y)), \\ 2f(\frac{x+y}{2})-f(x)-f(y)= & {} \rho(f(x+y)-f(x)-f(y)), \end{aligned}$$

where \({\rho}\) is a fixed non-Archimedean number or a fixed real or complex number with \({\rho \neq 1}\). Using the fixed point method, we prove the Hyers–Ulam stability of the above additive \({\rho}\)-functional equations in non-Archimedean Banach spaces and in Banach spaces.

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Authors and Affiliations

  1. Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, South Korea

    Choonkil Park

  2. Department of Mathematics, University of Seoul, Seoul, 02504, South Korea

    Dong Yun Shin

  3. Department of Mathematics, Daejin University, Kyeonggi, 11159, South Korea

    Jung Rye Lee

Authors
  1. Choonkil Park
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  2. Dong Yun Shin
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  3. Jung Rye Lee
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Correspondence to Jung Rye Lee.

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Park, C., Shin, D.Y. & Lee, J.R. Fixed points and additive \({\rho}\)-functional equations. J. Fixed Point Theory Appl. 18, 569–586 (2016). https://doi.org/10.1007/s11784-016-0282-3

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