On the Ulam stability of Jensen and Jensen type mappings on restricted domains (Q1399339)

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 \textit{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.