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A fixed point approach to the hyperstability of the general linear equation in \(\beta\)-Banach spaces. (English) Zbl 1398.39016

The authors firstly reformulate the fixed point theorem of J. Brzdęk et al. (see Theorem 1 of [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 6728–6732 (2011; Zbl 1236.39022)]) in \(\beta\)-Banach spaces.
Their main results state that, under some assumptions, functions satisfying the general linear equation \[ f(ax+by)=rf(x)+sf(y) \qquad \left(x, y\in X\right) \] only approximately (in some sense), must actually be solutions to it.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B62 Functional inequalities, including subadditivity, convexity, etc.
47H14 Perturbations of nonlinear operators
47H10 Fixed-point theorems

Citations:

Zbl 1236.39022
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Full Text: DOI

References:

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