Generalizations of a Restricted Stability Theorem of Losonczi on Cauchy Differences to Generalized Cocycles

Abstract

As a main result of this paper, we shall show that a natural generalization of a restricted stability theorem of László Losonczi on Cauchy differences to symmetric semi-cocycles can be derived from a similar generalization of an asymptotic stability theorem of Anna Bahyrycz, Zsolt Páles and Magdalena Piszczek.

For this, by using our former results, we shall prove that if \( F \) is a symmetric semi-cocycle on an unbounded commutative preseminormed group \( X \) to an arbitrary commutative preseminormed group \( Y \), and \( S \) is a relation on \( X \) such that the intersection of the domain and the range of \( S \) is bounded, then 
\[ \sup_{z \in S} \|F(z)\| \leq 5 \sup_{z \in S^c} \|F(z)\|, \] where \( S^c = X^2 \setminus S \).