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Non-Archimedean hyperstability of Cauchy–Jensen functional equations on a restricted domain

  • Iz-iddine EL-Fassi ORCID logo EMAIL logo

Abstract

Let X be a normed space, UX{0} a non-empty subset, and (G,+) a commutative group equipped with a complete ultrametric d that is invariant (i.e., d(x+z,y+z)=d(x,y) for x,y,zG). Under some weak natural assumptions on U and on the function γ:U3[0,), we study the new generalized hyperstability results when f:UG satisfies the inequality

d(αf(x+yα+z),αf(z)+f(y)+f(x))γ(x,y,z)

for all x,y,zU, where x+yα+zU and α2 is a fixed positive integer. The method is based on a quite recent fixed point theorem (Theorem 1 in [J. Brzdȩk and K. Ciepliński, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 2011, 18, 6861–6867]) (cf. [8, Theorem 1]) in some functions spaces.

Acknowledgements

The author would like to thank the anonymous referee for the careful reading and valuable suggestions to improve the quality of the paper.

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Received: 2017-08-16
Revised: 2018-02-02
Accepted: 2018-02-19
Published Online: 2018-11-13
Published in Print: 2018-12-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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