Generalized Additive Functional Equation: General Solution and Hyers-Ulam Stability in Banach Spaces via Alternative Fixed Point Theorem

Authors

  • Kandhasamy Tamilvanan Department of Mathematics, Faculty of Science & Humanities, R.M.K. Engineering College, Kavaraipettai, Tiruvallur 601 206, Tamil Nadu, India https://orcid.org/0000-0002-4900-7604
  • Syed Abdul Mohiuddine Department of General Required Courses, Mathematics, The Applied College, King Abdulaziz University, Jeddah 21589, Saudi Arabia https://orcid.org/0000-0002-9050-9104

DOI:

https://doi.org/10.56405/dngcrj.2023.08.01.01

Keywords:

Banach Space, Fixed Point, Hyers-Ulam Stability, Additive Functional Equation

Abstract

In this paper, we secure the general solution of the generalized additive functional equation

[Equation presented]

where r is a positive integer with N−{0,1,2,3,4}, and also examine Hyers-Ulam stability results by utilizing alternative fixed point for a generalized additive functional equation in Banach spaces.

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References

Aczel, J. (1996). Lectures on Functional Equations and Their Applications. Academic Press, New York, London.

Aczel, J. (1987). Short Course on Functional Equations. D. Reidel Publishing Company, Dordrecht, Holland.

Aoki, T. (1950). On the stability of the linear transformation in Banach Spaces. Journal of the Mathematical Society of Japan, 2, 64-66.

Cauchy, A.L. (1821). Cours d'Analyse de l'Ecole Polytechnique. Vol. 1, Analyse agebrique, Paris.

Czerwik, St. (1992). On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 62, 59-64.

Darboux, G. (1875). Sur la Composition des forces en statique. Bulletin des Sciences Mathématiques, 9, 281-288.

Eshaghi Gordji, M., Kaboli Gharetapeh, S., Rassias, J.M., Zolfaghari, S. (2009). Solution and Stability of a Mixed Type Additive, Quadratic, and Cubic Functional Equation. Advances in Difference Equations, 2009, 1-17.

Eshaghi Gordji, M., Khodabakhsh, R., Jung, S. -M., Khodaei, H. (2010). AQCQ-functional equation in non-Archimedean normed spaces. Abstract and Applied Analysis, 2010, Article ID 741942, 22 pages.

Eshaghi Gordji, M., Savadkouhi, M.B. (2010). Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Applied Mathematics Letters, 23, 1198-1202.

Gajda, Z. (1991). On Stability of Additive Mappings. International Journal of Mathematics and Mathematical Sciences, 14, 431-434.

Gavruta, P. (1994). A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings. Journal of Mathematical Analysis and Applications, 184, 431-436.

Hyers, D.H. (1941). On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of U.S.A., 27, 222-224.

Hyers, D.H., Isac, G., Rassias, Th. M. (1998). Stability of Functional Equations in Several Variables, Birkhauser, Boston.

Jung, S.M. (2001). Hyers-Ulam-Rassias stability of functional equations in Mathematical Analysis. Hadronic Press Inc., Palm Harbor, Florida.

Jung, S. -M. (1996). On the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications, 204 (1), 221-226.

Kannappan, Pl. (1995). Quadratic functional equation and inner product spaces. Results in Mathematics, 27 (3-4), 368-372.

Kuczma, M. (1985). An introduction to the Theory of Functional Equations and Inequalities, Uniwersytet Slaski, Warszawa-Krakow-Katowice.

Legendre, A.M. (1791). Elements de geometrie, Note II, Paris.

Lee, S., Im, S., Hwang, I. (2005). Quartic functional equations. Journal of Mathematical Analysis and Applications, 307 (2), 387-394.

Najati, A. (2007). Hyers-Ulam-Rassias stability of a cubic functional equation, Bulletin of Korean Mathematical Society, 44 (4), 825-840.

Najati, A., Park, C. (2008). On the stability of a cubic functional equation, Acta Mathematica Sinica, 24 (12), 1953-1964.

Najati, A., Moghimi, M.B. (2008). Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. Journal of Mathematical Analysis and Applications, 337 (1), 399-415.

Najati, A., Rassias, Th. M. (2010). Stability of a mixed functional equation in several variables on Banach modules. Nonlinear Analysis: Theory, Methods & Appl., 72 (3-4), 1755-1767.

Rassias, J. M. (1982). On approximation of approximately linear mappings by linear mappings, Journal of Functional Analysis,. 46 (1), 126-130.

Rassias, J. M. (1984). On approximation of approximately linear mappings by linear mappings. Bulletin des Science Mathmatiques, 108 (4), 445-446.

Rassias, J. M. (1989). Solution of a Problem of Ulam. Journal of Approximation Theory, 57 (3), 268-273.

Rassias, J.M. (1992). On the stability of the Euler-Lagrange functional equation. Chinese Journal of Mathematics, 20 (2),185-190.

Rassias, Th. M. (2000). On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications, 251, 264-284.

Rassias, Th. M. (1978). On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72 (2), 297-300.

Tamilvanan, K., Alanazi, A.M., Alshehri, M.G., Kafle, J. (2021). Hyers-Ulam stability of quadratic functional equation based on fixed point technique in Banach spaces and non-Archimedean Banach spaces. Mathematics, 9, 2575.

Tamilvanan, K., Alkhaldi, A.H., Jakhar, J., Chugh, R., Jakhar, J., Rassias, J. M. (2023). Ulam stability results of functional equations in modular spaces and 2-Banach spaces. Mathematics, 11, 371.

Tamilvanan, K., Lee, J.R., Park, C. (2020). Hyers-Ulam stability of a finite variable mixed type quadratic-additive functional equation in quasi-Banach spaces, AIMS Mathematics, 5 (6), 5993-6005.

Ulam, S.M. (1960). A Collection of Mathematical Problem, Interscience Tracts in Pure and Applied Mathematics, No. 8, Interscience, New York.

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Published

2023-12-26

How to Cite

Tamilvanan, K., & Mohiuddine, S. A. (2023). Generalized Additive Functional Equation: General Solution and Hyers-Ulam Stability in Banach Spaces via Alternative Fixed Point Theorem. Dera Natung Government College Research Journal, 8(1), 1–13. https://doi.org/10.56405/dngcrj.2023.08.01.01

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