Fixed Point Technique: Hyers-Ulam Stability Results Deriving From Cubic Mapping in Fuzzy Normed Spaces
DOI:
https://doi.org/10.26713/cma.v15i2.2679Keywords:
Fuzzy normed spaces, Ulam stability, Cubic mappingAbstract
In this work, we introduce a novel finite-dimensional cubic functional equation
\begin{align*}
\phi\Bigg(\sum\limits_{a=1}^{l}a n_{a}\Bigg)&=\sum\limits_{1 \leq a < b < c \leq l}\phi(a n_{a}+b n_{b}+c n_{c})\\
&\quad +(3-l)\sum\limits_{1 \leq a < b \leq l}\phi(a n_{a}+b n_{b})\\
&\quad +\Bigg(\frac{(l^{2}-5l+6)}{2}\Bigg)\sum\limits_{a=0}^{l-1}(a+1)^{3}\phi(n_{a+1}),
\end{align*}
where $l \geq 4$ is an integer, and derive its general solution.\ The main purpose of this work is to examine the Hyers-Ulam stability of this functional equation in fuzzy normed spaces by means of direct approach and fixed point approach.
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References
T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear spaces, Journal of Fuzzy Mathematics 11 (2003), 687 – 705.
T. Bag and S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151(3) (2005), 513 – 547, DOI: 10.1016/j.fss.2004.05.004.
R. Biswas, Fuzzy inner product space and fuzzy norm functions, Information Sciences 53(1-2) (1991), 185 – 190, DOI: 10.1016/0020-0255(91)90063-Z
S. C. Cheng and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bulletin of the Calcutta Mathematical Society 86 (1994), 429 – 436.
J. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bulletin of the American Mathematical Society 74 (1968), 305 – 309, DOI: 10.1090/S0002-9904-1968-11933-0.
C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems 48(2) (1992), 239 – 248, DOI: 10.1016/0165-0114(92)90338-5.
A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12(2) (1984), 143 – 154, DOI: 10.1016/0165-0114(84)90034-4.
I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11(5) (1975), 336 – 344, URL: https://www.kybernetika.cz/content/1975/5/336/paper.pdf.
D. Mihe¸t and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, Journal of Mathematical Analysis and Applications 343(1) (2008), 567 – 572, DOI: 10.1016/j.jmaa.2008.01.100.
A. K. Mirmostafaee and M. S. Moslehian, Fuzzy almost quadratic functions, Results in Mathematics 52 (2008), 161 – 177, DOI: 10.1007/s00025-007-0278-9.
A. K. Mirmostafaee and M. S. Moslehian, Fuzzy approximately cubic mappings, Information Sciences 178(19) (2008), 3791 – 3798, DOI: 10.1016/j.ins.2008.05.032.
A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159(6) (2008), 720 – 729, DOI: 10.1016/j.fss.2007.09.016.
A. K. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159(6) (2008), 730 – 738, DOI: 10.1016/j.fss.2007.07.011.
A. Najati, Fuzzy stability of a generalized quadratic functional equation, Communications of the Korean Mathematical Society 25(3) (2010), 405 – 417, DOI: 10.4134/CKMS.2010.25.3.405.
V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4(1) (2003), 91 – 96.
B.-S. Shieh, Infinite fuzzy relation equations with continuous t-norms, Information Sciences 178(8) (2008), 1961 – 1967, DOI: 10.1016/j.ins.2007.12.006.
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