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Stability of Systems of General Functional Equations in the Compact-Open Topology

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Book cover Developments in Functional Equations and Related Topics

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 124))

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Abstract

We introduce a fairly general concept of functional equation for k-tuples of functions f 1, …, f k : X → Y between arbitrary sets. The homomorphy equations for mappings between groups and other algebraic systems, as well as various types of functional equations and recursion formulas occurring in mathematical analysis or combinatorics, respectively, become special cases (of systems) of such equations. Assuming that X is a locally compact and Y is a completely regular topological space, we show that systems of such functional equations, with parameters satisfying rather a modest continuity condition, are stable in the following intuitive sense: Every k-tuple of “sufficiently continuous,” “reasonably bounded” functions X → Y satisfying the given system with a “sufficient precision” on a “big enough” compact set is already “arbitrarily close” on an “arbitrarily big” compact set to a k-tuple of continuous functions solving the system. The result is derived as a consequence of certain intuitively appealing “almost-near” principle using the relation of infinitesimal nearness formulated in terms of nonstandard analysis.

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Acknowledgements

Research supported by grants no. 1/0608/13 and 1/0333/17 of the Slovak grant agency VEGA.

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Authors and Affiliations

  1. Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48, Bratislava, Slovakia

    Pavol Zlatoš

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  1. Pavol Zlatoš
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Correspondence to Pavol Zlatoš .

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Editors and Affiliations

  1. Department of Mathematics, Pedagogical University of Cracow, Kraków,, Poland

    Janusz Brzdęk

  2. Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków, Poland

    Krzysztof Ciepliński

  3. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Zlatoš, P. (2017). Stability of Systems of General Functional Equations in the Compact-Open Topology. In: Brzdęk, J., Ciepliński, K., Rassias, T. (eds) Developments in Functional Equations and Related Topics . Springer Optimization and Its Applications, vol 124. Springer, Cham. https://doi.org/10.1007/978-3-319-61732-9_15

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