Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 17, No. 2, pp. 107-112 (2001)

Hyperstability of a class of linear functional equations

Gyula Maksa and Zsolt Páles

Institute of Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary,

Abstract: The aim of this note is to offer hyperstability results for linear functional equations of the form
f(s)+f(t)=\frac{1}{n}\sum_{i=1}^n f(s\phi_i(t)) \qquad (s,t\in S),
where $S$ is a semigroup and where $\phi_1,\dots,\phi_n\colon S\to S$ are pairwise distinct automorphisms of $S$ such that the set $\{\phi_1,\dots,\phi_n\}$ is a group equipped with the composition as the group operation. The main results state that if $f$ satisfies a stability inequality related to the above equation then it is also a solution of this equation.

Keywords: Hyperstability of functional equations, cocyle equation, generalized cocycle equation.

Classification (MSC2000): 39B72

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