Asymptotic behavior of Jensen and Jensen type functional equations. (English) Zbl 1082.39026
Let \(X\) be a real normed space and \(Y\) be a real Banach space. A mapping \(A : X \to Y\) is called
(I) additive of the first form if \(A(x + y) + A(x - y) = 2A(x)\);
(II) additive of the second form if \(A(x + y) - A(x - y) = 2A(y)\);
(III) Jensen if \(A(\frac{x + y}{2}) = \frac{1}{2}(A(x) + A(y))\);
(IV) Jensen type if \(A(\frac{x - y}{2}) = \frac{1}{2}(A(x) - A(y))\).
The authors investigate the Hyers-Ulam stability of the above equations by using the Hyers sequence \(\{2^{-n}f(2^nx)\}\) on the unrestricted domain \(X \times X\) and the restricted domain \(\{(x, y) : \| x\| + \| y\| \geq d\}\). They also deal with the asymptotic aspects of these functional equations. Similar results for the alternative type equations may be found in the paper of J. M. Rassias and M. J. Rassias [Bull. Sci. Math. 129, No. 7, 545–558 (2005; Zbl 1081.39028)].
(I) additive of the first form if \(A(x + y) + A(x - y) = 2A(x)\);
(II) additive of the second form if \(A(x + y) - A(x - y) = 2A(y)\);
(III) Jensen if \(A(\frac{x + y}{2}) = \frac{1}{2}(A(x) + A(y))\);
(IV) Jensen type if \(A(\frac{x - y}{2}) = \frac{1}{2}(A(x) - A(y))\).
The authors investigate the Hyers-Ulam stability of the above equations by using the Hyers sequence \(\{2^{-n}f(2^nx)\}\) on the unrestricted domain \(X \times X\) and the restricted domain \(\{(x, y) : \| x\| + \| y\| \geq d\}\). They also deal with the asymptotic aspects of these functional equations. Similar results for the alternative type equations may be found in the paper of J. M. Rassias and M. J. Rassias [Bull. Sci. Math. 129, No. 7, 545–558 (2005; Zbl 1081.39028)].
Reviewer: Mohammad Sal Moslehian (Leeds)
MSC:
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
