Integral Van Vleck’s and Kannappan’s functional equations on semigroups

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• 作者：Elqorachi Elhoucien
• 关键词：Semigroup ; d’Alembert’s equation ; Van Vleck’s equation ; Kannappan’s equation ; Involution ; Multiplicative function ; Complex measure
• 刊名：Aequationes mathematicae
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：91
• 期：1
• 页码：83-98
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Combinatorics;
• 出版者：Springer International Publishing
• ISSN：1420-8903
• 卷排序：91

文摘

In this paper we study the solutions of the integral Van Vleck’s functional equation for the sine $$\begin{aligned} \int _{S}f(x\tau (y)t)d\mu (t)-\int _{S}f(xyt)d\mu (t) =2f(x)f(y),\; x,y\in S \end{aligned}$$and the integral Kannappan’s functional equation $$\begin{aligned} \int _{S}f(xyt)d\mu (t)+\int _{S}f(x\tau (y)t)d\mu (t) =2f(x)f(y),\; x,y\in S, \end{aligned}$$where S is a semigroup, \(\tau \) is an involution of S and \(\mu \) is a measure that is a linear combination of Dirac measures \((\delta _{z_{i}})_{i\in I}\), such that for all \(i\in I\), \(z_{i}\) is contained in the center of S. We express the solutions of the first equation by means of multiplicative functions on S, and we prove that the solutions of the second equation are closely related to the solutions of d’Alembert’s classic functional equation with involution.KeywordsSemigroupd’Alembert’s equationVan Vleck’s equationKannappan’s equationInvolutionMultiplicative functionComplex measureMathematics Subject Classification39B3239B52References1.Akkouchi, M., Bakali, A.: Une généralisation des paires de Guelfand. Boll. Un. Math. Ital. 7, 759–822 (1992)Google Scholar2.Akkouchi, M., Bakali, A., Khalil, I.: A class of functional equations on a Locally compact group. J. Lond. Math. Soc. 57(3), 694–705 (1998)MathSciNetCrossRefMATHGoogle Scholar3.Davison, T.M.K.: D’Alembert’s functional equation on topological monoids. Publ. Math. Debrecen 75(1/2), 41–66 (2009)MathSciNetMATHGoogle Scholar4.Elqorachi, E., Akkouchi, M.: On generalized d’Alembert and Wilson functional equations. Aequationes Math. 66(3), 241–256 (2003)MathSciNetCrossRefMATHGoogle Scholar5.Elqorachi, E., Redouani, A.: Solutions and stability of generalized Kannappan’s and Van Vleck’s functional equations. (manuscript). (2016). arXiv: 1607.05166 6.Kannappan, Pl: A functional equation for the cosine. Can. Math. Bull. 2, 495–498 (1968)CrossRefMATHGoogle Scholar7.Perkins, A.M., Sahoo, P.K.: On two functional equations with involution on groups related to sine and cosine functions. Aequationes Math. (2014). doi:10.1007/s00010-014-0309-z 8.Van Stetkær, H.: Vleck’s functional equation for the sine. Aequationes Math. (2014). doi:10.1007/s00010-015-0349-z 9.Stetkær, H.: Kannappan’s functional equation on semigroups with involution. Semigroup Forum, First online 23, September 201510.Stetkær, H.: Functional Equations on Groups. World Scientific Publishing Co, Singapore (2013)CrossRefMATHGoogle Scholar11.Van Vleck, E.B.: A functional equation for the sine. Ann. Math. 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Second Ser. 13(1/4), 154 (1911–1912)13.Zeglami, D., Fadli, B.: Integral functional equations on locally compact groups with involution, Aequationes Math., pp. 1–16, First. online 21, March 2016Copyright information© Springer International Publishing 2016Authors and AffiliationsElqorachi Elhoucien1Email author1.Department of Mathematics, Faculty of SciencesIbn Zohr UniversityAgadirMorocco About this article CrossMark Publisher Name Springer International Publishing Print ISSN 0001-9054 Online ISSN 1420-8903 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; 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