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Some functional inequalities and their Baire category properties. (English) Zbl 0860.39039
Motivated by problems related to the study of the iterative functional equation \(\varphi (f(x)) = g(x, \varphi (x))\), the author considers, from a topological point of view, the set of all continuous functions \(f:I \to I\) for which the unique continuous solution \(\psi: I\to [0,\infty]\) of the inequalities: \[ \psi \bigl(f(x) \bigr) \leq\beta \bigl(x,\psi (x)\bigr) \quad \text{and} \quad \alpha \bigl(x,\psi (x)\bigr) \leq\psi \bigl(f(x)\bigr) \leq\beta \bigl(x,\psi (x)\bigr) \] is the zero function.
39B72 Systems of functional equations and inequalities
39B12 Iteration theory, iterative and composite equations
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[1] Blokh, A. M.,The set of all iterates is nowhere dense in C([0, 1], [0, 1]). Trans. Amer. Math. Soc.333 (1992), 787–798. · Zbl 0762.26002 · doi:10.2307/2154062
[2] GaweŁ, B.,On the uniqueness of continuous solutions of functional equations. To appear in Ann. Polon. Math. · Zbl 0828.39018
[3] GaweŁ, B.,On the uniqueness of continuous solutions of an iterative functional inequality. InEuropean Conference on Iteration Theory (Lisboa, Portugal, 15–21 September 1991). World Scientific Publ., Singapore, 1992, pp. 126–135.
[4] Humke, P. D., Laczkovich, M.,The Borel structure of iterates of continuous functions. Proc. Edinburgh Math. Soc. (2)32 (1989), 483–494. · Zbl 0671.28003 · doi:10.1017/S0013091500004727
[5] Kuczma, M.,Functional equations in a single variable. [Monografie Matematyczne, Vol. 46]. PWN – Polish Scientific Publishers, Warszawa, 1968. · Zbl 0196.16403
[6] Kuczma, M., Choczewski B. W., Ger, R.,Iterative functional equations. [Encyclopedia of mathematics and its applications, Vol. 32]. Cambridge University Press, Cambridge, 1990. · Zbl 0703.39005
[7] Kuratowski, K.,Topology. Vol. II. Academic Press and PWN – Polish Scientific Publishers, Warsaw, 1968.
[8] Myjak, J.,Orlicz type category theorems for functional and differential equations. [Dissertationes Math. (Rozprawy Mat.) No. 206]. PWN – Polish Scientific Publishers, Warszawa, 1983. · Zbl 0523.34070
[9] Simon, K.,Some dual statements concerning Wiener measure and Baire category. Proc. Amer. Math. Soc.106 (1989), 455–463. · Zbl 0675.28005 · doi:10.1090/S0002-9939-1989-0961409-6
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