## 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

Zbl 0842.39014
Full Text:

### References:

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