AIMS Mathematics, 2020, 5(6): 7161-7174. doi: 10.3934/math.2020458

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Fuzzy normed spaces and stability of a generalized quadratic functional equation

1 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
2 Department of Mathematics, Government Arts College for Men, Krishnagiri, Tamilnadu 635001, India
3 Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran

In this paper, we acquire the general solution of the generalized quadratic functional equation \[ \begin{aligned} \sum_{1 \leq a < b < c \leq m}\varphi\left(r_{a}+r_{b}+r_{c}\right)&=(m-2)\sum_{1\leq a < b\leq m}\varphi\left(r_{a}+r_{b}\right) \\ &\quad-\left(\frac{m^{2}-3m+2}{2}\right)\nonumber \sum_{a=1}^{m}\frac{\varphi\left(r_{a}\right)+\varphi\left(-r_{a}\right)}{2} \end{aligned} \] where $m\geqslant 3$ is an integer. We also investigate Hyers-Ulam stability results by means of using alternative fixed point theorem for this generalized quadratic functional equation.
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1. D. Amir, Characterizations of Inner Product spaces, Birkhäuser, Basel, 1986.

2. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.

3. T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11 (2003), 687-706.

4. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Hamburg, 62 (1992), 59-64.

5. J. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305-309.    

6. M. E. Gordji, A. Najati, Approximately $J^*$-homomorphisms: A fixed point approach, J. Geom. Phys., 60 (2010), 809-814.

7. P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.

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

9. P. Jordan, J. Neumann, On inner products in linear metric spaces, Ann. Math., 36 (1935), 719-723.

10. S. M. Jung, P. K. Sahoo, Hyers-Ulam stability of the quadratic equation of Pexider type, J. Korean Math. Soc., 38 (2001), 645-656.

11. P. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27 (1995), 368-372.

12. D. Miheţ, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl., 343 (2008), 567-572.

13. A. K. Mirmostafaee, M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Set. Syst., 159 (2008), 720-729.    

14. A. Najati, Fuzzy stability of a generalized quadratic functional equation, Commun. Korean Math. Soc., 25 (2010), 405-417.

15. A. Najati, M. B. Moghimi, Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces, J. Math. Anal. Appl., 337 (2008), 399-415.

16. A. Najati, C. Park, Fixed points and stability of a generalized quadratic functional equation, J. Inequal. Appl., 2009 (2009), 193035.

17. A. Najati, T. M. Rassias, Stability of a mixed functional equation in several variables on Banach modules, Nonlinear Anal., 72 (2010), 1755-1767.

18. C. Park, Fuzzy stability of a functional equation associated with inner product spaces, Fuzzy Set. Syst., 160 (2009), 1632-1642.

19. C. Park, J. R. Lee, X. Zhang, Additive s-functional inequality and hom-derivations in Banach algebras, J. Fix. Point Theory Appl., 21 (2019), 18.

20. V. Radu, The fixed point alternative and the stability of functional equations, Sem. Fix. Point Theory, 4 (2003), 91-96.

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

22. T. M. Rassias, New characterization of inner product spaces, Bull. Sci. Math., 108 (1984), 95-99.

23. S. Pinelas, V. Govindan, K. Tamilvanan, Stability of a quartic functional equation, J. Fix. Point Theory Appl., 20 (2018), 148.

24. S. Pinelas, V. Govindan, K. Tamilvanan, Solution and stability of an n-dimensional functional equation, Analysis (Berlin), 39 (2019), 107-115.

25. F. Skof, Proprietá locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129.

26. S. M. Ulam, A collection of the mathematical problems, Interscience Publ., New York, 1960.

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