On the behavior of mappings which do not satisfy Hyers-Ulam stability. (English) Zbl 0761.47004

The main result of the paper is the following
Theorem. There exists a continuous function \(f:R\to R\), satisfying \[ | f(x+y)-f(x)-f(y)|\leq| x|+| y|, \] for any \(x,y\in R\), with \(\lim_{x\to\infty}(f(x)/x)=\infty\).
This theorem gives an example to show that a stability theorem of Hyers- Rassias-Gajda-Ulam cannot be proved for \(p=1\).


47A58 Linear operator approximation theory
41A35 Approximation by operators (in particular, by integral operators)
47J05 Equations involving nonlinear operators (general)
Full Text: DOI


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