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On the functional equation $ \phi(x)=g(x)\phi(\beta (x))+u(x)$


Author: R. C. Buck
Journal: Proc. Amer. Math. Soc. 31 (1972), 159-161
MSC: Primary 39A15
DOI: https://doi.org/10.1090/S0002-9939-1972-0308632-9
MathSciNet review: 0308632
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Abstract: The linear functional equation of the title is one that has been studied extensively for real or complex $ x$, and for restricted choices of the functions $ g$ and $ \beta $. (See Kuczma [3].) In this paper, we use results of ours [1], combined with an idea due to Diaz and Chu [2], to obtain a powerful existence theorem for continuous solutions of this equation in a generalized form where the domain is an arbitrary compact space, the solutions are vector valued functions, and $ \beta $ is unspecialized, except for continuity.


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  • [1] R. C. Buck, On approximation theory and functional equations, J. Approximation Theory 5 (1972), 228–237. Collection of articles dedicated to J. L. Walsh on his 75th birthday, III (Proc. Internat. Conf. Approximation Theory, Related Topics and their Applications, Univ. Maryland, College Park, Md., 1970). MR 0377363
  • [2] Sherwood C. Chu and J. B. Diaz, A fixed point theorem for “in the large” application of the contraction principle, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 99 (1964/1965), 351–363 (English, with Italian summary). MR 0177317
  • [3] Marek Kuczma, Functional equations in a single variable, Monografie Matematyczne, Tom 46, Państwowe Wydawnictwo Naukowe, Warsaw, 1968. MR 0228862

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DOI: https://doi.org/10.1090/S0002-9939-1972-0308632-9
Article copyright: © Copyright 1972 American Mathematical Society