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Báo cáo hóa học: " The stability of functional equation min{f(x + y), f (x - y)} = |f(x) - f(y)|"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: The stability of functional equation min{f(x + y), f (x - y)} = |f(x) - f(y)| | Przebieracz Journal of Inequalities and Applications 2011 2011 22 http www.journalofinequalitiesandapplications.eom content 2011 1 22 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access The stability of functional equation min f x y f x - y f x - f y Barbara Przebieracz Correspondence barbara. przebieracz@us.edu.pl Instytut Matematyki Uniwersytet Slạski Bankowa 14 Katowice Pl-40007 Poland SpringerOpen0 Abstract In this paper we prove the stability of the functional equation min f x y f x - y f x - f y in the class of real continuous functions of real variable. MSC2010 39B82 39B22 Keywords stability of functional equations absolute value of additive mappings 1. Introduction In the paper 1 Simon and Volkmann examined functional equations connected with the absolute value of an additive function that is max f x y f x - y f x f y x y e G 1.1 min f x y f x - y f x - f y x y e G 1.2 and max f x y f x - y f x f y x y e G 1.3 where G is an abelian group and f G R. The first two of them are satisfied by f x a x where a G R is an additive function moreover the first one characterizes the absolute value of additive functions. The solutions of Equation 1.2 are appointed by Volkmann during the Conference on Inequalities and Applications in Noszwaj Hungary 2007 under the assumption thatf R R is a continuous function. Namely we have Theorem 1.1 Jarczyk and Volkmann 2 . Let f R R be a continuous function satisfying Equation 1.2 . Then either there exists a constant c 0 such thatf x c x x e R or f is periodic with period 2p given by f x c x with some constant c 0 x e -p p . Actually it is enough to assume continuity at a point since this implies continuity on R see 2 . Moreover some measurability assumptions force continuity. Baron in 3 showed that if G is a metrizable topological group and f G R is Baire measurable and satisfies 1.2 then f is continuous. Kochanek and Lewicki see 4 proved that if G is metrizable locally compact group and f G R is .