Stability results of mixed type functional equations in modular spaces and 2-Banach spaces ∗

Authors

  • Sushma Devi Kanya Mahavidyalaya, Kharkhoda, Haryana, India.
  • Asha Rani Pt. NRS Govt. College, Rohtak, Haryana, India.
  • Manoj Kumar Baba Mastnath University, Asthal Bohar, Rohtak, Haryana, India.

DOI:

https://doi.org/10.48165/bpas.2023.42E.2.1

Keywords:

additive and quartic functional equations, modular spaces, generalized Hyers-Ulam stability

Abstract

In this paper, we investigate the generalized Hyers-Ulam-Stability of additive and quartic functional equations in modular spaces with and without the △2-condition using the direct method and also in 2-Banach Spaces. 

References

Aoki, T. (1950). On the stability of the linear transformation in Banach spaces, Proc. Am. Math. Soc., 72, 64–66.

Gajda, Z. (1991). On stability of additive mappings, Internat. J. Math. Sci. 14, 431–434. [3] Gavruta, P. A. (1994). Generalization of the Hyers-Ulam-Rassias stability of approximately addi tive mappings, J. Math. Anal. Appl., 184, 431–436.

Gahler S. (1963). 2-metrische Rume und ihretopologischestruktur, Math. Nachr., 26, 115–148. [5] Gahler, S. (1964). Lineare 2-normierte Rumen, Math. Nachr., 28, 1–43.

Hyers, D. H. (1941). On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27, 222–224.

Khamsi, M. A. (2008). Quasicontraction mappings in modular spaces without D2-condition, Fixed Point Theory Appl., 916187, 6 pp. doi:10.1155/2008/916187

Kim, H. M., Chang, I. S. and Son E. (2013). Stability of Cauchy additive functional equation in fuzzy Banach spaces, Math. Inequal. Appl., 16, 1123–1136.

Kim, H. M. and Shin, H. Y. (2017). Refined stability of additive and quadratic functional equations in modular spaces, J. Inequal. Appl., 146. DOI10.1186/s13660-017-1422-z

Mohiuddine, S. A., Tamilvanan, K., Mursaleen, M. and Alotaibi, T. (2022). Stability of quartic functional equation in modular spaces via Hyers and fixed-point methods, Mathematics, 10, 1938. https://doi.org/10.3390/math10111938

Nakano, H. (1950). Modulared Semi-Ordered Linear Spaces, Maruzen Co., Ltd., Tokyo, Japan. [12] Park, C. (2013). Additive functional inequalities in 2-Banach spaces, J. Inequal. Appl., 447, 1–10. [13] Park, C. and Bodaghi, A. (2020). Two multi-cubic functional equations and some results on the

stability in modular spaces, J. Inequal. Appl., 6. https://doi.org/10.1186/s13660-019-2274-5 [14] Park, W. G. (2011). Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl., 376, 193–202.

Rassias, T. M. (1978). On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72, 297–300.

Sadeghi, G. (2014). A fixed-point approach to stability of functional equations in modular spaces, Bull. Malays. Math. Sci. Soc., 37, 333–344.

Senthil Kumar, B. V., Sabarinathan, S. and Chandrasekaran, A. D. (2019). Stability of a mixed type additive and quartic functional equation, AIP Conference Proceedings 2112, 1–7. [18] Tamilvanan, K., Alkhaldi, A. H., Jakhar, J. (Jyotsana), Chugh, R., Jakhar, J. (Jagjeet) and Rassias J. M. (2023). Ulam stability results of functional equations in Modular Spaces and 2- Banach Spaces, Mathematics, 11(371), 1–23.

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

Uthirasamy, N., Tamilvanan, K., Nashine, H. K. and George, R. (2022). Solution and stability of quartic functional equations in modular spaces by using Fatou property, J. Funct. Spaces, Vol. 2022, 5965628, 9 pp. https://doi.org/10.1155/2022/5965628

Sushma Devi, Asha Rani and Manoj Kumar

Wongkum, K., Chaipunya, P. and Kumam, P. (2015). On the generalized Ulam-Hyers-Rassias stability of quadratic mappings in modular spaces without D2-conditions, J. Funct. Spaces, Vol. 2015, 461719, 6pp. http://dx.doi.org/10.1155/2015/461719

Wongkum, K., Kumam, P. and Chaipunya, Y. (2017). On the generalized Ulam-Hyers-Rassias stability for quartic functional equation in modular spaces, J. Nonlinear Sci. Appl., 10, 1399– 1406.

Published

2023-12-25

How to Cite

Devi, S., Rani, A., & Kumar, M. (2023). Stability results of mixed type functional equations in modular spaces and 2-Banach spaces ∗ . Bulletin of Pure & Applied Sciences- Mathematics and Statistics, 42(2), 94–108. https://doi.org/10.48165/bpas.2023.42E.2.1