A variant of the quadratic functional equation on semigroups.
Keywords:
Symmetrized additive cauchy equation, Quadratic equation, Additive function, SemigroupAbstract
Let S be a semigroup, let H be an abelian group which is uniquely 2divisible, and let σ be an involutive automorphism of S. We express the solutions f : S → H of the following variant of the quadratic functional equation
f(xy) + f(σ(y)x) = 2f(x) + 2f(y), x, y ∈ S,
in terms of biadditive maps and solutions of the symmetrized additive Cauchy equation.
References
H. Drygas, Quasiinner products and their applications, in: Advances in Multivariate Statistical Analysis (ed. A. K. Gupta), D. Reidel Publishing Co., pp. 1330, (1987).
B. Fadli, D. Zeglami and S. Kabbaj, On a Gajda’s type quadratic equation on a locally compact abelian group, Indagationes Math. 26 (4), pp. 660668, (2015).
B. Fadli, D. Zeglami and S. Kabbaj, A variant of Wilson’s functional equation, Publ. Math. Debrecen 87 (34), pp. 415427, (2015).
B. Fadli, D. Zeglami and S. Kabbaj, A variant of Jensen’s functional equation on semigroups, Demonstratio Math. 49 (4), pp. 413420, (2016).
B. Fadli, A. Chahbi, Iz. ELFassi and S. Kabbaj, On Jensen’s and the quadratic functional equations with involutions, Proyecciones (Antofagasta) 35 (2), pp. 213223, (2016).
V. A. Faĭziev and P. K. Sahoo, Solution of Whitehead equation on groups, Math. Bohem. 138(2), pp. 171180, (2013).
PL. Kannappan, Functional equations and inequalities with applications, Springer, New York, (2009).
C.T. Ng, Jensen’s functional equation on groups, III, Aequationes Math. 62 (12), pp. 143159, (2001).
P. de Place Friis and H. Stetkær, On the quadratic functional equation on groups, Publ. Math. Debrecen 69 (12), pp. 6593, (2006).
P. Sinopoulos, Functional equations on semigroups, Aequationes Math. 59(3), pp. 255261, (2000).
H. Stetkær, Functional equations on abelian groups with involution, Aequationes Math. 54 (12), pp. 144172, (1997).
H. Stetkær, On a variant of Wilson’s functional equation on groups, Aequationes Math. 68 (3), pp. 160176, (2004).
H. Stetkær, Functional Equations on Groups, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2013).
H. Stetkær, A variant of d’Alembert’s functional equation, Aequationes Math. 89 (3), pp. 657662, (2015).
H. Stetkær, The kernel of the second order Cauchy difference on semigroups, Aequationes Math. (2016). DOI 10.1007/s0001001604538.
D. Zeglami, B. Fadli and S. Kabbaj, On a variant of μWilson’s functional equation on a locally compact group, Aequationes Math. 89 (5), pp. 12651280, (2015).
Published
How to Cite
Issue
Section

Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
 No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.