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On the stability of the pexiderized trigonometric functional equation. (English) Zbl 1159.39013
Similar to his papers [Adv. Difference Equ. 2007, Article ID 90405 (2007; Zbl 1148.39024)] and [Banach J. Math. Anal. 1, No. 2, 227--236, electronic only (2007; Zbl 1129.39013)], the author investigates the superstability of the pexiderized trigonometric functional inequality $$|f(x+y)+f(x-y)-2g(x)h(y)|\leq \varphi(x) \quad \text{or}\quad \varphi(y)$$ and also stability of the Jensen type functional inequality $$|f(x+y)-f(x-y)-2f(y)|\leq \psi(x,y)$$ for each $x, y \in G$, where $(G,+)$ is an abelian group and the mappings $f, g,h :G\to \mathbb{C}$ are unknown and $\varphi:G \to \mathbb{R}$ and $ \psi:G\times G\to \mathbb{R}$ are known. He also extends his results to the Banach algebras.

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
39B62Functional inequalities, including subadditivity, convexity, etc. (functional equations)
Full Text: DOI
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