On a sum form functional equation containing five unknown mappings

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• 作者：P. Nath ; D. K. Singh
• 关键词：Functional equation ; Additive mapping ; Multiplicative mapping ; Logarithmic mapping
• 刊名：Aequationes mathematicae
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：90
• 期：6
• 页码：1087-1101
• 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：90

文摘

The general solutions of the functional equation$$\sum\limits_{i=1}^n \sum\limits_{j=1}^m F(p_{i}q_{j}) = \sum\limits_{i=1}^n G(p_{i})\sum\limits_{j=1}^m H(q_{j}) + \sum\limits_{i=1}^n K(p_{i}) \sum_{j=1}^m L(q_{j})$$in which F,G,H,K and L are real-valued mappings with domain I = [0, 1]; L is a multiplicative mapping with L(0) =  0, L(1) =  1; \({(p_{1},\ldots,p_{n}) \in \Gamma_{n}}\), \({(q_{1},\ldots,q_{m}) \in \Gamma_{m}}\); \({n \ge 3}\); \({m \ge 3}\) being fixed integers; have been obtained.KeywordsFunctional equationAdditive mappingMultiplicative mappingLogarithmic mappingMathematics Subject Classification39B2239B52References1.Behara M., Nath P.: Information and entropy of countable measurable partitions I. Kybernetika 10, 491–503 (1974)MathSciNetMATHGoogle Scholar2.Chaundy T.W., Mcleod J.B.: On a functional equation. Edinburgh Math. Notes 43, 7–8 (1960)MathSciNetCrossRefMATHGoogle Scholar3.Daróczy Z., Losonczi L.: Über die Erweiterung der auf einer Punktmenge additiven Funktionen. Publ. Math. (Debrecen) 14, 239–245 (1967)MathSciNetMATHGoogle Scholar4.Havrda J., Charvat F.: Quantification method of classification process. Concept of structural \({\alpha}\)-entropy. Kybernetika (Prague) 3, 30–35 (1967)MathSciNetMATHGoogle Scholar5.Losonczi L., Maksa Gy.: On some functional equations of the information theory. Acta Math. Acad. Sci. Hung. 39, 73–82 (1982)MathSciNetCrossRefMATHGoogle Scholar6.Kocsis I.: On the stability of a sum form functional equation of multiplicative type. Acta Acad. Paed. Agriensis Sect. Math. 28, 43–53 (2001)MathSciNetMATHGoogle Scholar7.Nath P., Singh D.K.: On a multiplicative type sum form functional equation and its role in information theory. Appl. Math. 51(5), 495–516 (2006)MathSciNetCrossRefMATHGoogle Scholar8.Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423; 623–656 (1948)Copyright information© Springer International Publishing 2016Authors and AffiliationsP. Nath1Email authorD. K. Singh21.Department of MathematicsUniversity of DelhiDelhiIndia2.Department of MathematicsZakir Husain Delhi College (University of Delhi)DelhiIndia 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; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; 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