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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
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, Approximation theory and functional equations, J. Approximation Theory (to appear). MR 0377363 (51:13535)
  • [2] S. 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. MR 31 #1580. MR 0177317 (31:1580)
  • [3] M. Kuczma, Functional equations in a single variable, Monografie Mat., Tom 46, PWN, Warsaw, 1968. MR 37 #4441. MR 0228862 (37:4441)

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