A System of Generalized Quadratic Functional Equations and Fuzzy Difference Results
Abstract
1. Introduction
- M-QMs defined by the QFE with an involution [30].
- M-QMs defined through the Euler–Lagrange QFE , where with are fixed [33].
- M-QMs defined via the QFE , where with is fixed [34].
- M-QMs by using the generalized quadratic functional equation, or briefly, GQFE:
2. Characterization of GM-QMs
- (H1) Ψ has zero condition; that is, for any with at least one variable equal to zero.
- (H2) Ψ satisfies the quadratic condition if for any we havefor all , where is the fixed integers, with .
- (i)
- Ψ is a GM-QM;
- (ii)
3. Stability Results for Equation (10)
- (T1)
- for ;
- (T2)
- for all ;
- (T3)
- if ;
- (T4)
- ;
- (T5)
- is a non-decreasing function on and ;
- (T6)
- For , is upper semi-continuous on .
3.1. Stability Results: Direct Method
3.2. Stability Results: The FP Approach
- for all ;
- there exists a such that the following hold:
- (1)
- for all ;
- (2)
- the sequence converges to an element of ;
- (3)
- is the unique fixed point of ;
- (4)
- for all .
4. Conclusions and Future Works
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Zadeh, L. Fuzzy sets. Inform. Control 1965, 8, 338–353. [Google Scholar] [CrossRef]
- Katsaras, A.K. Fuzzy topological vector spaces II. Fuzzy Sets Syst. 1984, 12, 143–154. [Google Scholar] [CrossRef]
- Wu, C.; Fang, J. Fuzzy generalization of Klomogoroff’s theorem. J. Harbin Inst. Technol. 1984, 1, 1–7, (In Chinese, English abstract). [Google Scholar]
- Biswas, R. Fuzzy inner product spaces and fuzzy norm functions. Inform. Sci. 1991, 53, 185–190. [Google Scholar] [CrossRef]
- Bag, T.; Samanta, S.K. Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 2003, 11, 687–705. [Google Scholar]
- Felbin, C. Finite dimensional fuzzy normed linear spaces. Fuzzy Sets Syst. 1992, 48, 239–248. [Google Scholar] [CrossRef]
- Krishna, S.V.; Sarma, K.K.M. Separation of fuzzy normed linear spaces. Fuzzy Sets Syst. 1994, 63, 207–217. [Google Scholar] [CrossRef]
- Xiao, J.Z.; Zhu, X.H. Fuzzy normed spaces of operators and its completeness. Fuzzy Sets Syst. 2003, 133, 389–399. [Google Scholar] [CrossRef]
- Cheng, S.C.; Mordeson, J.M. Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 1994, 86, 429–436. [Google Scholar]
- Kramosil, I.; Michalek, J. Fuzzy metric and statistical metric spaces. Kybernetica 1975, 11, 326–334. [Google Scholar]
- Bag, T.; Samanta, S.K. Fuzzy bounded linear operators. Fuzzy Sets Syst. 2005, 151, 513–547. [Google Scholar] [CrossRef]
- Ulam, S.M. Problems in Modern Mathematic, Science Editions; John Wiley & Sons, Inc.: New York, NY, USA, 1964. [Google Scholar]
- Hyers, D.H. On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27, 222–224. [Google Scholar] [CrossRef] [PubMed]
- Czerwik, S. On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg. 1992, 62, 59–64. [Google Scholar] [CrossRef]
- Găvruţa, P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184, 431–436. [Google Scholar] [CrossRef]
- Rassias, T.M. On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72, 297–300. [Google Scholar] [CrossRef]
- Mirmostafaee, A.K.; Mirzavaziri, M.; Moslehian, M.S. Fuzzy stability of the Jensen functional equation. Fuzzy Sets Syst. 2008, 159, 730–738. [Google Scholar]
- Miheţ, D.; Radu, V. On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 2008, 343, 567–572. [Google Scholar]
- Mirmostafaee, A.K.; Moslehian, M.S. Fuzzy approximately cubic mappings. Inf. Sci. 2008, 178, 3791–3798. [Google Scholar] [CrossRef]
- Mirmostafaee, A.K.; Moslehian, M.S. Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets Syst. 2008, 159, 720–729. [Google Scholar] [CrossRef]
- Saheli, M.; Goraghani, H.S. Hyers-Ulam-Rassias stability of Jensen’s functional equation on fuzzy normed linear spaces. Int. J. Nonlinear Anal. Appl. 2023, 14, 3011–3023. [Google Scholar]
- Wang, Z. Stability of a mixed type additive-quadratic functional equation with a parameter in matrix intuitionistic fuzzy normed spaces. AIMS Math. 2023, 8, 25422–25442. [Google Scholar] [CrossRef]
- Park, C. Stability of the Cauchy-Jensen functional equation in fuzzy Banach algebras. Appl. Math. Lett. 2011, 24, 2024–2029. [Google Scholar] [CrossRef]
- Azadi Kenari, H. Ulam-Hyers-Rassias-stability of a Cauchy-Jensen additive mapping in fuzzy Banach space. Int. J. Nonlinear Anal. Appl. 2025; in press. [Google Scholar]
- Skof, F. Proprieta locali e approssimazione di operatori. Rend. Semin. Mat. Fis. Milano 1983, 53, 113–129. [Google Scholar] [CrossRef]
- Amir, D. Characterizations of Inner Product Spaces; Birkhäuser-Verlag: Basel, Switzerland, 1986. [Google Scholar]
- Park, C.-G. Multi-quadratic mappings in Banach spaces. Proc. Am. Math. Soc. 2002, 131, 2501–2504. [Google Scholar] [CrossRef]
- Ciepliński, K. On the generalized Hyers-Ulam stability of multi-quadratic mappings. Comput. Math. Appl. 2011, 62, 3418–3426. [Google Scholar] [CrossRef]
- Zhao, X.; Yang, X.; Pang, C.-T. Solution and stability of the multiquadratic functional equation. Abstr. Appl. Anal. 2013, 2013, 415053. [Google Scholar] [CrossRef]
- Bodaghi, A. Multi-quadratic mappings with an involution. J. Anal. 2022, 30, 859–870. [Google Scholar] [CrossRef]
- Bodaghi, A. Functional inequalities for generalized multi-quadratic mappings. J. Inequal. Appl. 2021, 2021, 145. [Google Scholar] [CrossRef]
- Bodaghi, A.; Park, C.; Yun, S. Almost multi-quadratic mappings in non-Archimedean spaces. AIMS Math. 2020, 5, 5230–5239. [Google Scholar] [CrossRef]
- Bodaghi, A.; Moshtagh, H.; Mousivand, A. Characterization and stability of multi-Euler-Lagrange quadratic functional equations. J. Funct. Spaces 2022, 2022, 3021457. [Google Scholar] [CrossRef]
- Bodaghi, A.; Salimi, S.; Abbasi, G. Approximation for multi-quadratic mappings in non-Archimedean spaces. Annal. Univ. Craiova-Math. Comput. Sci. Ser. 2021, 48, 88–97. [Google Scholar] [CrossRef]
- Bodaghi, A.; Moshtagh, H.; Dutta, H. Characterization and stability analysis of advanced multi-quadratic functional equations. Adv. Differ. Equ. 2021, 2021, 380. [Google Scholar] [CrossRef]
- Saadati, S.; Vaezpour, S.M. Some results on fuzzy Banach spaces. J. Appl. Math. Comput. 2005, 17, 475–484. [Google Scholar] [CrossRef]
- Tuinici, M. Sequentially iterative processes and applications to Volterra functional equations. Ann. Uni. Mariae Curie-Sklodowska Sect.-A 1978, 32, 127–134. [Google Scholar]
- Diaz, J.B.; Margolis, B. A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 1968, 74, 305–309. [Google Scholar] [CrossRef]
- Hadžić, O.; Pap, E.; Radu, V. Generalized contraction mapping principles in probabilistic metric spaces. Acta Math. Hungar. 2003, 101, 131–148. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Alsahli, G.; Bodaghi, A. A System of Generalized Quadratic Functional Equations and Fuzzy Difference Results. Symmetry 2025, 17, 866. https://doi.org/10.3390/sym17060866
Alsahli G, Bodaghi A. A System of Generalized Quadratic Functional Equations and Fuzzy Difference Results. Symmetry. 2025; 17(6):866. https://doi.org/10.3390/sym17060866
Chicago/Turabian StyleAlsahli, Ghaziyah, and Abasalt Bodaghi. 2025. "A System of Generalized Quadratic Functional Equations and Fuzzy Difference Results" Symmetry 17, no. 6: 866. https://doi.org/10.3390/sym17060866
APA StyleAlsahli, G., & Bodaghi, A. (2025). A System of Generalized Quadratic Functional Equations and Fuzzy Difference Results. Symmetry, 17(6), 866. https://doi.org/10.3390/sym17060866