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

The functional equations \[ \begin{aligned} f(x+y)+f(x-y)&=2f(x),\tag{1}\\ f(x+y)-f(x-y)&=2f(y)\tag{2}\end{aligned} \] 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 \(\|x\|+\|y\|\geq d\) is proved. As a consequence the following results are obtained.
Corollary 1. A mapping \(f:X\to Y\) fulfils (1) iff the asymptotic condition \[ \bigl\|f(x+y)+f(x-y)-2f(x)\bigr\|\to 0\quad\text{as } \|x\|+\|y\|\to\infty \] holds.
Corollary 2. A mapping \(f:X\to Y\) fulfils (2) iff the asymptotic conditions \[ \bigl\|f(-x)+f(x)\bigr\|\to 0\quad\text{ as }\|x\|\to\infty \] and \[ \bigl\|f(x+y)-f(x-y)-2f(y)\bigr\|\to 0\quad\text{ as } \|x\|+\|y\|\to\infty \] hold.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges

Citations:

Zbl 0842.39014
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References:

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