Asymptotic behavior of alternative Jensen and Jensen type functional equations. (English) Zbl 1081.39028
Let \(X\) be a real normed space and \(Y\) be a real Banach space. A mapping \(A : X \to Y\) is called
(i) alternative additive of the first form if \(A(x + y) + A(x - y) = -2A(-x)\);
(ii) alternative additive of the second form if \(A(x + y) - A(x - y) = -2A(-y)\);
(iii) alternative Jensen if \(A( - \frac{x + y}{2}) = - \frac{1}{2}(A(x) + A(y))\);
(iv) alternative Jensen type if \(A( - \frac{x - y}{2}) = - \frac{1}{2}(A(x) - A(y))\);
The authors investigate the Hyers-Ulam stability of these 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 study the asymptotic aspects of these functional equations. Similar results for the classical Jensen equation \(A(\frac{x + y}{2}) = \frac{1}{2}(A(x) + A(y))\) may be found in the paper by S.-M. Jung [Proc. Am. Math. Soc. 126, No. 11, 3137–3143 (1998; Zbl 0909.39014)].
(i) alternative additive of the first form if \(A(x + y) + A(x - y) = -2A(-x)\);
(ii) alternative additive of the second form if \(A(x + y) - A(x - y) = -2A(-y)\);
(iii) alternative Jensen if \(A( - \frac{x + y}{2}) = - \frac{1}{2}(A(x) + A(y))\);
(iv) alternative Jensen type if \(A( - \frac{x - y}{2}) = - \frac{1}{2}(A(x) - A(y))\);
The authors investigate the Hyers-Ulam stability of these 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 study the asymptotic aspects of these functional equations. Similar results for the classical Jensen equation \(A(\frac{x + y}{2}) = \frac{1}{2}(A(x) + A(y))\) may be found in the paper by S.-M. Jung [Proc. Am. Math. Soc. 126, No. 11, 3137–3143 (1998; Zbl 0909.39014)].
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 |
| 39B72 | Systems of functional equations and inequalities |
Keywords:
Hyers-Ulam stability; alternative additive Jensen functional equation of the first form; alternative additive Jensen functional equation of the second form; Banach spaceCitations:
Zbl 0909.39014References:
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