Stability of Generalized Quartic Functional Equation in Random Normed Spaces
DOI:
https://doi.org/10.26713/cma.v14i5.2227Keywords:
Quartic functional equation, Hyers- Ulam stability, Random normed spaceAbstract
Aim of this paper is to investigate the Hyers-Ulam stability of generalized quartic functional equation
\begin{align*}
\sum^n_{i=1}\emptyset \bigg(-v_i+\sum^n_{j=1,i\neq j}{v_j}\bigg)
&=(n-8)\sum_{1=i<j<k<l=n}{\emptyset (v_i+v_j+v_k+v_l)}\nonumber\\&\quad -(n^2-12n+28)\sum_{1=i<j<k=n}{\emptyset (v_i+v_j+v_k)}\\
&\quad +\bigg(\frac{n^3-15n^2+60n-68}{2}\bigg)\sum_{1=i<j=n}{\emptyset (v_i+v_j)}\nonumber\\&\quad +2\sum_{1=i<j=n}{\emptyset (v_i-v_j)}+\sum^n_{i=1}{\emptyset}(3v_i)\\
&\quad-\bigg(\frac{{n^4-17n}^3+86n^2-148n+558}{6}\bigg)\sum^n_{i=1}{\emptyset (v_i)}
\end{align*}
in random normed space.
Downloads
References
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) (2020), 2385 – 2397, DOI: 10.3934/math.2021145.
T. Aoki, On the stability of the linear transformation in Banach spaces, Journal of the Mathematical Society of Japan 2(1-2) (1950), 64 – 66, DOI: 10.2969/jmsj/00210064.
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 420 pages (2002), DOI: 10.1142/4875.
Z. Gajda, On stability of additive mappings, International Journal of Mathematics and Mathematical Sciences 14 (1991), Article ID 817959, 4 pages, DOI: 10.1155/S016117129100056X.
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.
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.
D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications series, Birkhäuser, Boston, MA, (1998), vii + 318 pages, DOI: 10.1007/978-1-4612-1790-9.
S. S. Jin and Y.-H. Lee, On the stability of the functional equation deriving from quadratic and additive function in random normed spaces via fixed point method, Journal of the Chungcheong Mathematical Society 25(1) (2012), 51 – 63, DOI: 10.14403/jcms.2012.25.1.051.
K.-W. Jun and H.-M. Kim, On the Hyers-Ulam-Rassias stability of a generalized quadratic and additive type functional equation, Bulletin of the Korean Mathematical Society 42(1) (2005), 133 – 148, DOI: 10.4134/BKMS.2005.42.1.133.
K.-W. Jun and H.-M. Kim, On the stability of an n-dimensional quadratic and additive functional equation, Mathematical Inequalities & Applications 9(1) (2006), 153 – 165, DOI: 10.7153/mia-09-16.
S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor (2001).
S.-M. Jung, On the Hyers–Ulam stability of the functional equations that have the quadratic property, Journal of Mathematical Analysis and Applications 222(1) (1998), 126 – 137, DOI: 10.1006/jmaa.1998.5916.
S. Karthikeyan, J. M. Rassias, S. Vijayakumar and K. Sakthivel, Stability results of additivequadratic-dimensional functional equation: fixed point approach, Communications in Mathematics and Applications 13 (2022), 461 – 476, DOI: 10.26713/cma.v13i2.1757.
J. M. Rassias, On approximately of approximately linear mappings by linear mappings, Journal of Functional Analysis 46 (1982), 126 – 130, DOI: 10.1016/0022-1236(82)90048-9.
J. M. Rassias, P. Narasimman, R. Saadati and M. de la Sen, Approximation of mixed Euler-Lagrange σ-cubic-quartic functional equation in Felbin’s type f-NLS, Journal of Function Spaces 2021 (2021), Article ID 8068673, DOI: 10.1155/2021/8068673.
Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematica 62 (2000), 23 – 130, DOI: 10.1023/A:1006499223572.
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society 72(1978), 297 – 300, DOI: 10.2307/2042795.
Th. M. Rassias and P. Šemrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proceedings of the American Mathematical Society 114 (1992), 989 – 993, DOI: 10.2307/2159617.
K. Tamilvanan, J. R. Lee and C. Park, Ulam stability of a functional equation deriving from quadratic and additive mappings in random normed spaces, AIMS Mathematics 6(1) (2020), 908 – 924, DOI: 10.3934/math.2021054.
S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, Inc., New York (1964).
N. Uthirasamy, K. Tamilvanan, H. K. Nashine and R. George, Solution and stability of quartic functional equations in modular spaces by using Fatou property, Journal of Function Spaces 2022 (2022), Article ID 5965628, DOI: 10.1155/2022/5965628.
S. Vijayakumar, S. Karthikeyan, J. M. Rassias and B. Baskaran, A quartic functional equation originating from the sum of the medians of a triangle in fuzzy normed space, AIP Conference Proceedings 2282 (2020), 020006, DOI: 10.1063/5.0028290.
Downloads
Published
How to Cite
Issue
Section
License
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.
