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On the Ulam stability of Jensen and Jensen type mappings on restricted domains. (English) Zbl 1028.39011

The functional equations


are considered for functions mapping a real normed space X into a real Banach space Y. The stability of the equation (1) was proved by J. M. Rassias [Geometry, Analysis and Mechanics, World Scientific, 365-376 (1994; Zbl 0842.39014)]. For a fixed d>0 the stability of equations (1), (2) on a restricted domain x+yd is proved. As a consequence the following results are obtained.

Corollary 1. A mapping f:XY fulfils (1) iff the asymptotic condition

f ( x + y ) + f ( x - y ) - 2 f ( x )0asx+y


Corollary 2. A mapping f:XY fulfils (2) iff the asymptotic conditions

f ( - x ) + f ( x )0asx


f ( x + y ) - f ( x - y ) - 2 f ( y )0asx+y


39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges