Stability Results of the Additive-Quadratic Functional Equations in Random Normed Spaces by Using Direct and Fixed-Point Method

Authors

  • Asha Rani Department of Mathematics, Pandit Neki Ram Sharma Government College (Maharshi Dayanand University), Rohtak 124001, Haryana, India; Department of Mathematics, Baba Mastnath University, Asthal Bohar, Rohtak 124021, Haryana, India https://orcid.org/0009-0009-0373-4732
  • Sushma Devi Department of Mathematics, Kanya Mahavidyalaya Kharkhoda (Maharshi Dayanand University), Kharkhoda 131402, Haryana, India https://orcid.org/0000-0003-3722-860X
  • Manoj Kumar Antil Department of Mathematics, Baba Mastnath University, Asthal Bohar, Rohtak 124021, Haryana, India https://orcid.org/0000-0003-4455-8690

DOI:

https://doi.org/10.26713/cma.v14i2.2148

Keywords:

Hyers-Ulam stability, Additive functional equations, Quadratic functional equations, Random normed spaces, Fixed point method, Direct method

Abstract

In this paper, we prove the Hyers-Ulam stability of different additive-quadratic functional equations in Random Normed Space (RN-Space) by direct and fixed-point method.

Downloads

References

A. M. Alanazi, G. Muhiuddin, K. Tamilvanan, E. N. Alenze, A. Ebaid and K. Loganathan, Fuzzy stability results of finite variable additive functional equation: direct and fixed point methods, Mathematics 8(7) (2020), 1050, DOI: 10.3390/math8071050.

N. Alessa, K. Tamilvanan, G. Balasubramanian and K. Loganathan, Stability results of the functional equation deriving from quadratic function in random normed spaces, AIMS Mathematics 6(3) (2021), 2385 – 2397, DOI: 10.3934/math.2021145.

P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Journal of Mathematical Analysis and Applications 184(3) (1994), 431 – 436, DOI: 10.1006/jmaa.1994.1211.

M. E. Gordji, H. Khodaei and Th. M. Rassias, Fixed Points and stability for quadratic mappings in β-normed left Banach modules on Banach algebras, Results in Mathematics 61 (2012), 393 – 400, DOI: 10.1007/s00025-011-0123-z.

D. H. Hyers, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences 27(4)(1941), 222 – 224, DOI: 10.1073/pnas.27.4.222.

K.-W. Jun and Y.-H. Lee, On the Hyers-Ulam-Rassias stability of a pexiderized quadratic inequality, Mathematical Inequalities & Applications 4(1) (2001), 93 – 118, DOI: 10.7153/mia-04-08.

P. L. Kannappan, Quadratic functional equation and inner product spaces, Results in Mathematics 27 (1995), 368 – 372, DOI: 10.1007/BF03322841.

H. A. Kenary, C. Park, H. Rezaei and S. Y. Jang, Stability of a generalized quadratic functional equation in various spaces: a fixed point alternative approach, Advances in Difference Equations 2011 (2011), Article number: 62, DOI: 10.1186/1687-1847-2011-62.

D. Mihe¸t and R. Saadati, On the stability of the additive Cauchy functional equation in random normed spaces, Applied Mathematics Letters 24(12) (2011), 2005 – 2009, DOI: 10.1016/j.aml.2011.05.033.

C. Park, K. Tamilvanan, G. Balasubramanian, B. Noori and A. Najati, On a functional equation that has the quadratic-multiplicative property, Open Mathematics 18(1) (2020), 837 – 845, DOI: 10.1515/math-2020-0032.

M. Ramdoss, D. Pachaiyappan, C. Park and J. R. Lee, Stability of a generalized n-variable mixed-type functional equation in fuzzy modular spaces, Journal of Inequalities and Applications 2021 (2021), Article number: 61, DOI: 10.1186/s13660-021-02594-y.

J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Journal of Functional Analysis 46(1) (1982), 126 – 130, DOI: 10.1016/0022-1236(82)90048-9.

Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society 72 (1978), 297 – 300, DOI: 10.1090/S0002-9939-1978-0507327-1.

Th. M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Universitatis Babe¸s-Bolyai 43(3) (1998), 89 – 124.

Th. M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, Journal of Mathematical Analysis and Applications 228(1) (1998), 234 – 253, DOI: 10.1006/jmaa.1998.6129.

K. Ravi, M. Arunkumar and J. M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation, International Journal of Mathematics and Statistics 3(A08) (2008), 36 – 46.

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover Publications, Inc., New York (1983).

A. N. Serstnev, On the notion of a random normed space, Doklady Akademii Nauk SSSR 149 (1963), 280 – 283 (in Russian).

K. Tamilvanan, R. T. Alqahtani and S. A. Mohiuddine, Stability results of mixed type quadratic-additive functional equation in β-Banach modules by using fixed-point technique, Mathematics 10(3) (2022), 493, DOI: 10.3390/math10030493.

K. Tamilvanan, J. R. Lee and C. Park, Hyers-Ulam stability of a finite variable mixed type quadratic-additive functional equation in quasi-Banach spaces, AIMS Mathematics 5(6) (2020), 5993 – 6005, DOI: 10.3934/math.2020383.

S. M. Ulam, Problems in Modern Mathematics, 2nd editions, John Wiley & Sons, New York (1964).

T. Xu, J. M. Rassias and W. Xu, A fixed point approach to the stability of a general mixed additive-cubic equation on Banach modules, Acta Mathematica Scientia 32(3) (2012), 866 – 892, DOI: 10.1016/S0252-9602(12)60067-8.

Downloads

Published

18-09-2023
CITATION

How to Cite

Rani, A., Devi, S., & Antil, M. K. (2023). Stability Results of the Additive-Quadratic Functional Equations in Random Normed Spaces by Using Direct and Fixed-Point Method. Communications in Mathematics and Applications, 14(2), 827–843. https://doi.org/10.26713/cma.v14i2.2148

Issue

Vol. 14 No. 2 (2023)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.