Yet another variant of the Drygas functional equation on groups
DOI:
https://doi.org/10.4067/S0716-09172017000100002Keywords:
Drygasfunctional equation, group, Fréchet’s functional equation, involution, semigroup, Whiteheadfunctional equationAbstract
Let G be a group and C the field of complex numbers. Suppose σ1, σ 2 : G → G are endomorphisms satisfying the condition σi(σi(x)) = x for all x in G and for i = 1, 2. In this paper, we find the central solution f : G → C of the equation f (xy) + f (σi(y)x) =2f (x) + f (y) + f (σ2(y)) for all x,y ∈ G which is a variant of the Drygas functional equation with two involutions. Further, we present a generalization the above functional equation and determine its central solutions. As an application, using the solutions ofthe generalized equation, we determine the solutions f, g, h, k : GxG → C ofthefunc-tional equation f (pr, qs) + g(sp, rq) = 2f (p, q) + h(r, s) + k(s, r) when f satisfies the condition f (pr, qs) = f (rp, sq) for all p, q, r, s ∈ G.References
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[2] J. Aczél and E. Vincze, Über eine gemeinsame Verallgemeinerung zweier Funktionalgleichungen von Jensen, Publ. Math. Debrecen, 10, pp. 326-344, (1963).
[3] J. K. Chung, Pl. Kannappan, C. T. Ng, and P. K. Sahoo, Measures of distance between probability distributions, J. Math. Anal. Appl., 139, pp. 280-292, (1989).
[4] H. H. Elfen, T. Riedel and P.K. Sahoo, A variant of quadratic functional equation on groups. Submitted, (2016).
[5] H. H. Elfen, T. Riedel and P.K. Sahoo, A variant of a generalized quadratic functional equation on groups, To appear in Results in Math., (2017).
[6] V. A. Faiziev and P. K. Sahoo, Solution of Whitehead equation on groups, Math. Bohemica, (2) 138 (2013), 171-180.
[7] G. L. Forti, Stability of quadratic and Drygas functional equations, with an application for solving alternative quadratic equation, T.M. Rassias (ed.), Handbook of Functional Equations, Springer Optimization and Its Applications 96, DOI 10.1007/978-1-4939-1286-5-8.
[8] J. L. W. V. Jensen, On the solution of fundamental equations by elementary means (Danish), Tidsskr. Math., 4, pp. 149-155, (1878).
[9] J. L. W. V. Jensen, On the solution of functional equations with the minimal numbers of suppositions (Danish), Mat. Tidsskr. B, pp. 25-28, (1897).
[10] P. Jordan and J. von Neumann, On the inner products in linear metric spaces, Ann. Math., 36, pp. 719-723, (1935).
[11] Pl. Kannappan, Functional Equations and Inequalities with Applications, Singapore, pp. 249, (2009).
[12] Pl. Kannappan, On inner product spaces, I, Math. Jpn., (2) 45, pp. 289-296, (1997).
[13] Pl. Kannappan, On quadratic functional equation, Int. J. Math. Statist. Sci., 9, pp. 35-60, (2000).
[14] S. Kurepa, On the quadratic functional, Publ. Inst. Math. Acad. Serbe Sci., 13, pp. 58-78, (1959).
[15] C. T. Ng and H. Y. Zhao, Kernel of the second order Cauchy difference on groups, Aequat. Math., 86, pp. 155-170, (2013).
[16] T. Riedel and P. K. Sahoo, On two functional equations connected with the characterizations of the distance measures, Aequat. Math., 54, pp. 242-263, (1997).
[17] T. Riedel and P. K. Sahoo, On a generalization of a functional equation associated with the distance between the probability distributions, Publ. Math. Debrecen, 46, pp. 125-135, (1995).
[18] P. K. Sahoo, The Drygas functional equation with involution on groups, Submitted, (2016).
[19] P. K. Sahoo, On a functional equation associated with stochastic distance measures, Bull. Korean Math. Soc., 36, pp. 287-303, (1999).
[20] P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, Chapman and Hall/CRC, pp. 269-290, (2011).
[21] P. Sinopoulos, Functional equations on semigroups, Aequat. Math., 59, pp. 255-261, (2000).
[22] H. Stetkaer, Functional Equations on Groups, World Scientific Publishing, Singapore (2013).
[23] H. Stetkaer, A variant of d’Alembert’s functional equation, Aequat. Math., 89, pp. 657-662, (2015).
[24] D. Yang, The quadratic functional equation on groups, Publ. Math. Debrecen, (3) 66, pp. 327-348, (2004).
[25] J. H. C. Whitehead, A certain exact sequence, Ann. Math., (2), 52, pp. 51-110, (1950).
Published
2017-04-06
How to Cite
[1]
P. K. Sahoo, “Yet another variant of the Drygas functional equation on groups”, Proyecciones (Antofagasta, On line), vol. 36, no. 1, pp. 13-27, Apr. 2017.
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