Article ID: MTJPAM-D-22-00027

Title: On Some Pexider Type Sum Form Functional Equations


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-22-00027; Volume 4 / Issue 3 / Year 2022 (Special Issue), Pages 270-281

Document Type: Research Paper

Author(s): Dhiraj Kumar Singh a , Shveta Grover b , Surbhi Madan c

aDepartment of Mathematics, Zakir Husain Delhi College (University of Delhi), Jawaharlal Nehru Marg, Delhi 110002, India

bDepartment of Mathematics, University of Delhi, Delhi 110007, India

cDepartment of Mathematics, Shivaji College (University of Delhi), Raja Garden, Ring Road, New Delhi-110027, India

Received: 22 August 2022, Accepted: 21 December 2022, Published: 4 February 2023.

Corresponding Author: Dhiraj Kumar Singh (Email address: dhiraj426@rediffmail.com; dksingh@zh.du.ac.in)

Full Text: PDF

Abstract

Since its genesis, an equation of Pexider type has captivated the attention of the mathematical fraternity around the world. Over the decades, several Pexiderized forms of various functional equations have been studied meticulously. In comparison to the functional equations, such forms are less analysed for sum form functional equations and require substantial study. Taking lead from it, this paper is devoted to obtain the general solution of some Pexiderized forms of a sum form functional equation

\sum\limits\limits^n_{i=1}\sum\limits\limits^m_{j=1}T\left(p_iq_j\right)\!\!=\!\sum\limits\limits^n_{i=1}T\left(p_i\right)\! \sum\limits\limits^m_{j=1}T\left(q_j\right)\!+\!\!(m\!-\!n)T(0)\sum\limits\limits^m_{j=1}T\left(q_j\right)\!+\!m(n\!-\!1)T(0),

where T is a real-valued mapping with the domain I = [0, 1]; (p1, …, pn)∈Γn, (q1, …, qm)∈Γm and n ≥ 3, m ≥ 3 are fixed integers.

Keywords: Pexider’s equation, Sum form functional equation, Additive mapping, Multiplicative mapping, Entropy

References:
  1. J. Acźel, Lectures on functional equations and their applications, Academic Press, New York and London, 1966.
  2. J. Bečvář and A. Slavík, Jan Vilém Pexider (1874-1914), Matfyzpress, Prague, 2009.
  3. T. W. Chaundy and J. B. McLeod, On a functional equation, Edinburgh Math. Notes 43, 7–8, 1960.
  4. Z. Daróczy and L. Losonczi, Über die Erweiterung der auf einer Punktmenge additiven Funktionen, Publicationes Math. 14, 239–245, 1967.
  5. L. Losonzi and Gy. Maksa, On some functional equations of the information theory, Acta Math. Acad. Sci. Hung. 39 (1-3), 73–82, 1982.
  6. S. Madan, S. Grover and D. K. Singh, On the stability of a multiplicative type sum form functional equation, Ratio Math. 41, 7–19, 2021.
  7. P. Nath and D. K. Singh, On a multiplicative type sum form functional equation and its role in information theory, Appl. Math. 51 (5), 495–516, 2006.
  8. P. Nath and D. K. Singh, On a sum form functional equation and its role in information theory, In: Proc. of National Conference on Information Technology: Setting Trends In Modern Era’ and 8th Annual Conference of Indian Society of Information Theory and Applications, (March 18-20, 2006), 88–94, 2008.
  9. P. Nath and D. K. Singh, Some sum form functional equations containing at most two unknown mappings, In: Proc. of 9th National Conference of Indian Society of Information Theory and Applications (ISITA) on Information Technology and Operational Research Applications, (December 8-9, 2007), 54–71, 2008.
  10. P. Nath and D.K. Singh, A sum form functional equation and its relevance in information theory, Aust. J. Math. Anal. Appl. 5 (1), 1–18, 2008.
  11. P. Nath and D. K. Singh, On a sum form functional equation related to entropies of type (α, β), Asian-Eur. J. Math. 6 (2), 2013; Article ID: 1350036.
  12. P. Nath and D. K. Singh, On a sum form functional equation related to various nonadditive entropies in information theory, Tamsui Oxf. J. Inf. Math. Sci. 30, 23–43, 2014.
  13. P. Nath and D. K. Singh, Some functional equations and their corresponding sum forms, Sarajevo J. Math. 12 (24), 89–106, 2016.
  14. P. Nath and D.K. Singh, On a sum form functional equation containing five unknown mappings, Aequationes Math. 90, 1087–1101, 2016.
  15. P. Nath and D. K. Singh, On the stability of a functional equation, Palest. J. Math. 6 (2), 573–579, 2017.
  16. C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (3-4), 379–423; 623–656, 1948.
  17. D. K. Singh and P. Dass, On a functional equation related to some entropies in information theory, J. Discrete Math. Sci. Cryptogr. 21 (3), 713–726, 2018.
  18. D. K. Singh and S. Grover, On the stability and general solution of a sum form functional equation emerging from information theory, Bull. Math. Anal. Appl. 12 (3), 1–18, 2020.
  19. D.K. Singh and S. Grover, On the stability of a nonmultiplicative type sum form functional equation, Ital. J. Pure Appl. Math. 45, 360–374, 2021.
  20. D. K. Singh and S. Grover, On a sum form functional equation emerging from statistics and its applications, Results in Nonlinear Anal. 4 (2), 65–76, 2021.
  21. D. K. Singh and S. Grover, On the stability of a sum form functional equation related to entropies of type (α, β), J. Nonlinear Sci. Appl. 14 (3), 168–180, 2021.
  22. D. K. Singh and S. Grover, On the stability of a sum form functional equation related to nonadditive entropies, J. Math. Computer Sci. 23 (4), 328–340, 2021.