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A result concerning additive maps on the set of quaternions and an application

Published online by Cambridge University Press:  17 April 2009

Damjan Kobal  and
Peter Šemrl
Damjan Kobal
Affiliation:
Department of Mathematics, University of Ljubljana, Jadranska 19 61000 Ljubljana, Yugoslavia
Peter Šemrl
Affiliation:
Department of Mathematics, University of Ljubljana, Jadranska 19 61000 Ljubljana, Yugoslavia
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Abstract

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We determine all additive F, G: ℍ → ℝ and multiplicative M: H → ℝ satisfying the functional equation F(λ) + M(λ)G(λ−1) = 0. As an application we generalise Kurepa's solution of one of Halperin's problem concerning quadratic functionals.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 3 , December 1991 , pp. 477 - 482
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Ebanks, B. R., ‘The equation F(x) + M(x)F(x −1) = 0 for additive F and multiplicative M on the positive cone Rn’, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), 247–252.Google Scholar
[2]Ebanks, B. R., ‘On the equation F(x) + M(x)G(x −1) = 0 on Kn’, Linear Algebra Appl. 125 (1989), 1–17.CrossRefGoogle Scholar
[3]Ebanks, B. R., Kannappan, Pl., and Ng, C. T., ‘Generalized fundamental equation of information of multiplicative type’, Aeqvationea Math. 32 (1987), 19–31.CrossRefGoogle Scholar
[4]Kannappan, Pl. and Ng, C. T., ‘On a generalized fundamental equation of information’, Canad. J. Math. 35 (1983), 862–872.CrossRefGoogle Scholar
[5]Kurepa, S., ‘The Cauchy functional equation and scalar product in vector spaces’, Glasnik Mat. Fiz.-Astr. 19 (1964), 23–36.Google Scholar
[6]Kurepa, S., ‘Quadratic and sesquilinear functionals’, Glasnik Mat. Fiz.-Astr. 20 (1965), 79–92.Google Scholar
[7]Kurepa, S., ‘On quadratic forms’, Aequationes Math. 34 (1987), 125–138.CrossRefGoogle Scholar
[8]Maksa, Gy., ‘Solution on the open triangle of the generalized fundamental equation of information with four unknown functions’, Utilitas Math. 21 (1982), 267–282.Google Scholar
[9]Ng, C. T., ‘The equation F(x) + M(x)G(x −1) = 0 and homogeneous biadditive forms’, Linear Algebra Appl. 93 (1987), 255–279.CrossRefGoogle Scholar
[10]Vrbová, P., ‘Quadratic functionals and bilinear forms’, Casopis Pest. Mat. 98 (1973), 159–161.CrossRefGoogle Scholar
[11]Vukman, J., ‘Some functional equations in Banach algebras and an application’, Proc. Amer. Math. Soc. 100 (1987), 133–136.CrossRefGoogle Scholar