Generalizations of a Restricted Stability Theorem of Losonczi on Cauchy Differences to Generalized Cocycles - Árpád Száz (Scientia, 2025)
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 \[ _z S \|F(z)\| ≤ 5 _z S^c \|F(z)\|, \] where \( S^c = X^2 S \).