On the general solution of a nonsymmetric partial difference functional equation analogous to the wave equation |
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Authors: | Shigeru Haruki |
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Institution: | 1. Department of Applied Mathematics, Okayama University of Science, 1-1 Ridai-cho, 700, Okayama, Japan
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Abstract: | We give the general solution of the nonsymmetric partial difference functional equationf(x + t,y) + f(x – t,y) – 2f(x,y)/t
2 =f(x,y + s) + f(x,y – s) – 2f(x,y)/s
2 (N) analogous to the well-known wave equation (
2/ x
2
–
2/ y
2)f(x,y) = 0 with the aid of generalized polynomials when no regularity assumptions are imposed onf. The result is as follows.
Theorem.Let R be the set of all real numbers. A function f: R × R R satisfies the functional equation (N)for all x, y R, s, t R\{0}, and s t if and only if there exist
(i) |
additive functions A, B: R R
| (ii) |
a function C: R × R R which is additive in each variable, and
| (iii) |
polynomials
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Keywords: | AMS (1980) subject classification" target="_blank">AMS (1980) subject classification Primary 39B40 |
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