INTUITIONISTIC FUZZY STABILITY OF AN EULER-LAGRANGE TYPE QUARTIC FUNCTIONAL EQUATION

Ali Ebadian; Choonkil Park

doi:10.11948/20240334

2025 Volume 15 Issue 3

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Ali Ebadian, Choonkil Park. INTUITIONISTIC FUZZY STABILITY OF AN EULER-LAGRANGE TYPE QUARTIC FUNCTIONAL EQUATION[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1659-1673. doi: 10.11948/20240334

Citation:

Ali Ebadian, Choonkil Park. INTUITIONISTIC FUZZY STABILITY OF AN EULER-LAGRANGE TYPE QUARTIC FUNCTIONAL EQUATION[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1659-1673. doi: 10.11948/20240334

INTUITIONISTIC FUZZY STABILITY OF AN EULER-LAGRANGE TYPE QUARTIC FUNCTIONAL EQUATION

Ali Ebadian^1,,
Choonkil Park^2, ,

1.
Department of Mathematics, Urmia University, P. O. Box 165, Urmia, Iran
2.
Research Institute for Convergence of Basic Science, Hanyang University, Seoul 04763, Korea

Author Bio: Email: a.ebadian@urmia.ac.ir(A. Ebadian)
Corresponding author: Email: baak@hanyang.ac.kr (C. Park)

Abstract

Abstract

In this paper, we investigate the Hyers-Ulam stability of the following Euler-Lagrange type quartic functional equation
$\begin{array}{*{20}{c}}{f(ax+y)+f(x+ay)+\frac{1}{2}a(a-1)^{2}f(x-y)}\\{=(a^{2}-1)^{2}(f(x)+f(y))+\frac{1}{2}a(a+1)^{2}f(x+y)}\end{array}$
in intuitionistic fuzzy normed spaces, where $a\neq 0, a\neq\pm1$. Furthermore, we investigate intuitionistic fuzzy continuity through the existence of a certain solution of a fuzzy stability problem for approximately Euler-Lagrange quartic functional equation.
- Hyers-Ulam stability /
- Euler-Lagrange type quartic functional equation /
- intuitionistic fuzzy normed space /
- intuitionistic fuzzy continuity
MSC: 39B82, 39B52, 46S40, 47S40, 54A40

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References

[1]	T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 1950, 2, 245–251. Google Scholar
[2]	A. R. Aruldass, D. Pachaiyappan and C. Park, Kamal transform and Ulam stability of differential equations, J. Appl. Anal. Comput., 2021, 11(3), 1631–1639. Google Scholar
[3]	K. Atanassov, Intuitionistic fuzzy sets, Ⅶ ITKR'S, Session, Sofia, June 1983 (Deposed in Central Science-Technical Library of Bulg. Academy of Science, 1697/84)(in Bulgarian). Google Scholar
[4]	T. Bag and S. K. Samanta, Finite dimensional fuzzy normed linear space, J. Fuzzy Math., 2003, 11, 687–705. Google Scholar
[5]	T. Bag and S. K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets Syst., 2005, 151, 513–547. doi: 10.1016/j.fss.2004.05.004 CrossRef Google Scholar
[6]	R. Chaharpashlou and A. M. Lopes, Hyers-Ulam-Rassias stability of a nonlinear stochastic fractional Volterra integro-differential equation, J. Appl. Anal. Comput., 2023, 13(5), 2799–2808. Google Scholar
[7]	S. C. Cheng and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc., 1994, 86, 429–436. Google Scholar
[8]	Iz. EL-Fassi, E. El-hady and W. Sintunavarat, Hyperstability results for generalized quadratic functional equations in $ (2, \alpha) $-Banach spaces, J. Appl. Anal. Comput., 2023, 13(5), 2596–2612. $ (2, \alpha) $-Banach spaces" target="_blank">Google Scholar
[9]	C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Sets Syst., 1992, 48, 239–248. Google Scholar
[10]	P. Gǎvruta, A generalization of the Hyers-Ulam-Rassias stability of the approximately additive mappings, J. Math. Anal. Appl., 1994, 184, 431–436. Google Scholar
[11]	R. Ger, On alienation of two functional equations of quadratic type, Aequationes Math., 2021, 19, 1169–1180. Google Scholar
[12]	D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 1941, 27, 222–224. Google Scholar
[13]	K. Jun and H. Kim, On the stability of the Euler-Lagrange type cubic mappings in quasi-Banach spaces, J. Math. Anal. Appl., 2007, 33, 1335–1350. Google Scholar
[14]	O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets Syst., 1984, 12, 1–7. Google Scholar
[15]	D. Kang, On the stability of gneralized quadratic mappings in quasi $ \beta $-normed space, J. Inequal. Appl., 2010, 2010, Article ID 198098. $ \beta $-normed space" target="_blank">Google Scholar
[16]	D. Kang, H. Kim and B. Lee, Stability estimates for a radical functional equation with fixed-point approaches, J. Math. Inequal., 2022, 25(2), 433–446. Google Scholar
[17]	A. K. Katsaras, Fuzzy topological vector spaces, Fuzzy Sets Syst., 1984, 12, 143–154. Google Scholar
[18]	A. Zivari-Kazempour and M. Eshaghi Gordji, Generalized Hyers-Ulam stabilities of an Euler-Lagrange-Rassias quadratic functional equation, Asian-Eur. J. Math., 2012, 5, Article ID 1250014. Google Scholar
[19]	I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 1975, 11, 326–334. Google Scholar
[20]	S. V. Krishna and K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Sets Syst., 1994, 63, 207–217. Google Scholar
[21]	S. Lee, S. Im and I. Hwang, Quartic functional equation, J. Math. Anal. Appl., 2005, 307, 387–394. Google Scholar
[22]	S. A. Mohiuddine, Stability of Jensen functional equation in intuitionistic fuzzy normed space, Chaos Solitons Fract., 2009, 42, 2989–2996. Google Scholar
[23]	S. A. Mohiuddine, M. Cancan and H. Ševli, Intuitionistic fuzzy stability of a Jensen functional equation via fixed point technique, Math. Comput. Model., 2011, 54, 2403–2409. Google Scholar
[24]	S. A. Mohiuddine and Q. M. D. Lohani, On generalized statistical convergence in intuitionistic fuzzy normed spaces, Chaos Solitons Fract., 2009, 42, 1731–1737. Google Scholar
[25]	S. A. Mohiuddine and H. Ševli, Stability of pexiderized quadratic functional equation in intuitionistic fuzzy normed space, J. Comput. Appl. Math., 2011, 235, 2137–2146. Google Scholar
[26]	R. Murali, C. Park and A. Ponmana Selvan, Hyers-Ulam stability for an n^th order differential equation using fixed point approach, J. Appl. Anal. Comput., 2021, 11(2), 614–631. $ n^{th} $ order differential equation using fixed point approach" target="_blank">Google Scholar
[27]	M. Mursaleen and S. A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons Fract., 2009, 41, 2414–2421. Google Scholar
[28]	M. Mursaleen and S. A. Mohiuddine, On stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos Solitons Fract., 2009, 42, 2997–3005. Google Scholar
[29]	M. Mursaleen and S. A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed spaces, J. Comput. Appl. Math., 2009, 233, 141–149. Google Scholar
[30]	C. Park, Fuzzy stability of functional equation associated with inner product spaces, Fuzzy Sets Syst., 2009, 160, 1632–1642. Google Scholar
[31]	M. Ramdoss, D. Pachaiyappan, J. M. Rassias and C. Park, Stability of a generalized Euler-Lagrange radical multifarious functional equation, J. Appl., 2024, 32(6), 3185–3195. Google Scholar
[32]	J. M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese J. Math., 2004, 294, 196–205. Google Scholar
[33]	Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 1978, 72, 297–300. Google Scholar
[34]	R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos Solitons Fract., 2006, 27, 331–344. Google Scholar
[35]	S. M. Ulam, Problem in Modern Mathematics, Wiley, New York, 1964. Google Scholar
[36]	Z. Wang and Th. M. Rassias, Intuitionistic fuzzy stability of functional equations associated with inner product spaces, Abstr. Appl. Anal., 2011, 2011, Article ID 456182. Google Scholar
[37]	C. Wu and J. Fang, Fuzzy generalization of Kolmogoroff's theorem, J. Harbin Inst. Tech., 1984, 1984(1), 1–7. Google Scholar
[38]	J. Z. Xiao and X. H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets Syst., 2003, 133, 389–399. Google Scholar
[39]	T. Z. Xu, Approximate multi-Jensen, multi-Euler-Lagrange additive and quadratic mappings in n-Banach spaces, Abstr. Appl. Anal., 2013, 2013, Article ID 648709. Google Scholar
[40]	T. Z. Xu and J. M. Rassias, Stability of general multi-Euler-Lagrange quadratic functional equation in non-Archimedean fuzzy normed spaces, Adv. Difference Equ., 2012, 2012, Paper No. 119. Google Scholar
[41]	H. Yao, W. Jin and Q. Dong, Hyers-Ulam-Rassias stability of $ \kappa $-Caputo fractional differential equations, J. Appl. Anal. Comput., 2024, 14(5), 2903–2921. $ \kappa $-Caputo fractional differential equations" target="_blank">Google Scholar
[42]	A. Zada, L. Alam, J. Xu and W. Dong, Controllability and Hyers-Ulam stability of impulsive second order abstract damped differential systems, J. Appl. Anal. Comput., 2021, 11(3), 1222–1239. Google Scholar
[43]	D. Zhang, J. M. Rassias and Y. Li, On the Hyers-Ulam solution and stability problem for general set-valued Euler-Lagrange quadratic functional equations, Korean J. Math., 2022, 30(4), 571–592. Google Scholar