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The Pexider Functional Equations in Distributions

Published online by Cambridge University Press:  20 November 2018

E. Y. Deeba  and
E. L. Koh
E. Y. Deeba
Affiliation:
University of Houston - Downtown, Houston, Texas
E. L. Koh
Affiliation:
University of Regina, Regina, Saskatchewan
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The Cauchy functional equations have been studied recently for Schwartz distributions by Koh in [3]. When the solutions are locally integrate functions, the equations reduce to the classical Cauchy equations (see [1]):

(1) f(x+y)=f﹛x)+f(y)

(2) f(x+y)=f(x)f(y)

(3) f(xy)=f(x)+f(y)

(4) f(xy)=f(x)f(y).

Earlier efforts to study functional equations in distributions were given by Fenyö [2]for the Hosszu’ equations

f(x + y - xy) +f(xy) =f(x) +f (y ),

by Neagu [4]for the Pompeiu equation

f(x+y+xy)=f(x)+f(y)+f(x)f(y)

and by Swiatak [6].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 2 , 01 April 1990 , pp. 304 - 314
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Aczel, J., Lectures on functional equations and their applications (Academic Press, New York, 1966).Google Scholar
2. Fenyö, I., On the general solution of a functional equation in the domain of distributions, Aequationes Math. 3 (1969), 236–246.Google Scholar
3. Koh, E.L., The Cauchy functional equations in distributions, Proc. Amer. Math. Soc. 106 (1989), 641–647.Google Scholar
4. Neagu, M., About the Pompeiu equation in distributions, Inst. Politehn. “Traran Vaia” Timis. Sem. Mat. Fiz. (1984), 62–66.Google Scholar
5. Schwartz, L., Theorie des distributions (Hermann, Paris, 1966).Google Scholar
6. Swiatak, H., On the regularity of the distributional and continuous solutions of the functional equations Aequationes Math. 1 (1968), 6–19.Google Scholar