On the functional equation $\phi (x)=g(x)\phi (\beta (x))+u(x)$
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- by R. C. Buck PDF
- Proc. Amer. Math. Soc. 31 (1972), 159-161 Request permission
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.References
- R. C. Buck, On approximation theory and functional equations, J. Approximation Theory 5 (1972), 228–237. MR 377363, DOI 10.1016/0021-9045(72)90016-0
- 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 (1964/65), 351–363 (English, with Italian summary). MR 177317
- Marek Kuczma, Functional equations in a single variable, Monografie Matematyczne, Tom 46, Państwowe Wydawnictwo Naukowe, Warsaw, 1968. MR 0228862
Additional Information
- © Copyright 1972 American Mathematical Society
- 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