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Stability of normal regions for linear homogeneous functional equations. (English) Zbl 0664.39003

The solution of the functional equation (1) \(\phi [f(x)]=g(x)\phi (x)\) is known to depend on an arbitrary function \(\phi_ 0(x)\) if certain simple conditions on f and g are satisfied. Starting from the notion of set stability for difference equations, introduced by G. A. Shanholt [Int. J. Control, I. Ser. 19, 309-314 (1974; Zbl 0291.93036)], the author defines stability in the context of (1) and proceeds to give a number of results on stability, in particular also about continuous dependence on initial conditions.
Reviewer: Gy.Targonski

MSC:

39B12 Iteration theory, iterative and composite equations

Citations:

Zbl 0291.93036
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References:

[1] Brydak, D.,On functional inequalities in a single variable. Dissertationes Math. 150 (1979), 1–48. · Zbl 0405.39011
[2] Czerni, M.,Interval stability for a linear homogeneous functional equation. Annl. Math. Sil. (to appear). · Zbl 0664.39003
[3] Kuczma, M.,Functional equations in a single variable. Monografie Mat. 46, Polish Scientific Publishers, Warszawa, 1968. · Zbl 0196.16403
[4] Shanholt, G. A.,Set stability for difference equation. Int. J. Control19 (1974), 309–314. · Zbl 0291.93036 · doi:10.1080/00207177408932630
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