Relaxed upper stability of Ekeland's variational principle for fuzzy-valued functions

Document Type : Research Paper

Author

Dong Thap University

Abstract

We study the relaxed upper stability of Ekeland's variational principle for fuzzy functions under fuzzy order relations induced by the lower and upper set orders of $\alpha$-level sets. Under a D-convergence assumption for fuzzy-valued  functions, we prove that every Painleve--Kuratowski outer limit point of exact fuzzy Ekeland variational points for perturbed problems belongs to the relaxed variational set of the limit problem. We also clarify the relation between exact  and relaxed variational sets by explicit examples, show that exact upper stability fails in general and prove that it can be  recovered under an additional separation condition on the limit problem. Several open questions concerning lower and  exact upper stability are discussed. 

Keywords

Main Subjects

References

[1] Y. Araya, Ekeland’s variational principle and its equivalent theorems in vector optimization, Journal of Mathematical Analysis and Applications, 346(1) (2008), 9-16. https://doi.org/10.1016/j.jmaa.2008.04.055
[2] H. Attouch, H. Riahi, Stability results for Ekeland’s ε-variational principle and cone extremal solutions, Mathematics of Operations Research, 18(1) (1993), 173-201. https://doi.org/10.1287/moor.18.1.173
[3] J. P. Aubin, I. Ekeland, Applied nonlinear analysis, Wiley, New York, 1984.
[4] Y. Bai, S. Migórski, S. D. Zeng, Generalized vector complementarity problem in fuzzy environment, Fuzzy Sets and Systems, 347 (2018), 142-151. https://doi.org/10.1016/j.fss.2017.09.010
[5] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013), 119-141. https://doi.org/10.1016/j.fss.2012.10.003
[6] M. Bianchi, G. Kassay, R. Pini, Ekeland’s principle for vector equilibrium problems, Nonlinear Analysis: Theory, Methods and Applications, 66(7) (2007), 1454-1464. https://doi.org/10.1016/j.na.2006.02.003
[7] M. Castellani, M. Giuli, Ekeland’s principle for cyclically antimonotone equilibrium problems, Nonlinear Analysis: Real World Applications, 32 (2016), 213-228. https://doi.org/10.1016/j.nonrwa.2016.04.011
[8] S. S. L. Chang, L. A. Zadeh, On fuzzy mappings and control, IEEE Transactions on Systems, Man, and Cybernetics, SMC-2(1) (1972), 30-34. https://doi.org/10.1109/TSMC.1972.5408553
[9] G. Y. Chen, X. X. Huang, Stability results for Ekeland’s ε-variational principle for vector valued functions, Mathematical Methods of Operations Research, 48 (1998), 97-103. https://doi.org/10.1007/s001860050014
[10] G. Y. Chen, X. X. Huang, X. Q. Yang, Vector optimization: Set-valued and variational analysis, Springer, Berlin, 2005. https://doi.org/10.1007/3-540-28445-1
[11] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets: Theory and applications, World Scientific, Singapore, 1994.
[12] I. Ekeland, On the variational principle, Journal of Mathematical Analysis and Applications, 47(2) (1974), 324-353. https://doi.org/10.1016/0022-247X(74)90025-0
[13] A. Ghodousian, S. Zal, A two-phase-ACO algorithm for solving nonlinear optimization problems subjected to fuzzy relational equations, Iranian Journal of Fuzzy Systems, 21(5) (2024), 151-174. https://doi.org/10.22111/ijfs. 2024.49652.8760
[14] A. Göpfert, H. Riahi, C. Tammer, C. Zălinescu, Variational methods in partially ordered spaces, Springer, Berlin, 2023. https://doi.org/10.1007/978-3-031-36534-8
[15] C. Gutièrrez, G. Kassay, V. Novo, J. L. Ródenas-Pedregosa, Ekeland variational principles in vector equilibrium problems, SIAM Journal on Optimization, 27 (2017), 2405-2425. https://doi.org/10.1137/17M111883X
[16] T. X. D. Ha, Some variants of the Ekeland variational principle for a set-valued map, Journal of Optimization Theory and Applications, 124 (2005), 187-206. https://doi.org/10.1007/s10957-004-6472-y
[17] B. Hazarika, Pointwise ideal convergence and uniformly ideal convergence of sequences of fuzzy valued functions, Journal of Intelligent and Fuzzy Systems, 32(3) (2017), 2665-2677. https://doi.org/10.3233/JIFS-16662
[18] X. X. Huang, New stability results for Ekeland’s ε-variational principles for vector-valued and set-valued maps, Journal of Mathematical Analysis and Applications, 262(1) (2001), 12-23. https://doi.org/10.1006/jmaa.2000. 7357
[19] X. X. Huang, Stability results for Ekeland’s ε-variational principle for set-valued mappings, Optimization, 51 (2002), 31-45. https://doi.org/10.1080/02331930211982
[20] N. V. Hung, N. H. V. Duy, J. C. Yao, S. D. Zeng, Convergence analysis of solutions to fuzzy weak vector quasi equilibrium problems and applications, Journal of Computational and Applied Mathematics, 484 (2026), 117502. https://doi.org/10.1016/j.cam.2026.117502
[21] N. V. Hung, V. M. Tam, N. H. Tuan, D. O’Regan, Convergence analysis of solution sets for fuzzy optimization problems, Journal of Computational and Applied Mathematics, 369 (2020), 112615. https://doi.org/10.1016/ j.cam.2019.112615
[22] Y. Jia, Existence and uniqueness of mild solution for initial value problem of a class of fuzzy evolution equations with delay, Iranian Journal of Fuzzy Systems, 22(6) (2025), 67-81. https://doi.org/10.22111/ijfs.2025.49723. 8779
[23] D. Kuroiwa, Existence theorems of set optimization with set-valued maps, Journal of Information and Optimization Sciences, 24(1) (2003), 73-84. https://doi.org/10.1080/02522667.2003.10699556
[24] Y. B. Li, W. S. Jia, Existence and well-posedness of the α-core for generalized fuzzy games, Fuzzy Sets and Systems, 458 (2023), 108-117. https://doi.org/10.1016/j.fss.2022.06.018
[25] J. Liu, Necessary and sufficient conditions for the existence of fuzzy doubly strong equilibria in generalized fuzzy games, Fuzzy Sets and Systems, 481 (2024), 108879. https://doi.org/10.1016/j.fss.2024.108879
[26] J. H. Qiu, F. He, Set-valued pseudo-metric families and Ekeland’s variational principles in fuzzy metric spaces, Fuzzy Sets and Systems, 300 (2016), 1-23. https://doi.org/10.1016/j.fss.2016.02.007
[27] R. T. Rockafellar, R. J. B. Wets, Variational analysis, Springer, Berlin, 1998. https://doi.org/10.1007/ 978-3-642-02431-3
[28] V. M. Tam, Stability analysis of a class of fuzzy Caputo fractional differential inclusions driven by fuzzy mixed variational inequalities, Iranian Journal of Fuzzy Systems, 20(3) (2023), 43-59. https://doi.org/10.22111/ijfs. 2023.7638
[29] J. P. Tian, W. S. Jia, L. Zhou, Existence and stability of fuzzy slightly altruistic equilibrium for a class of generalized multiobjective fuzzy games, Journal of Optimization Theory and Applications, 203 (2024), 111-125. https://doi. org/10.1007/s10957-024-02471-w
[30] X. Wang, K. L. Teo, Generalized Nash equilibrium problem over a fuzzy strategy set, Fuzzy Sets and Systems, 434 (2022), 172-184. https://doi.org/10.1016/j.fss.2021.06.006
[31] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338-353. https://doi.org/10.1016/ S0019-9958(65)90241-X
[32] S. D. Zeng, J. Cen, V. T. Nguyen, Generalized fractional evolution equations driven by fuzzy variational inequalities, Fuzzy Sets and Systems, 429 (2022), 60-73. https://doi.org/10.1016/j.fss.2021.01.005
[33] C. L. Zhang, N. J. Huang, On Ekeland’s variational principle for interval-valued functions with applications, Fuzzy Sets and Systems, 436 (2022), 152-174. https://doi.org/10.1016/j.fss.2021.10.003
[34] C. L. Zhang, N. J. Huang, D. O’Regan, Stability of solutions for fuzzy set optimization problems with applications, Fuzzy Sets and Systems, 466 (2023), 108470. https://doi.org/10.1016/j.fss.2023.01.005
[35] C. L. Zhang, S. D. Zeng, Generalized set-valued Ekeland variational principles via the null set concept with applications to some set-valued optimization problems, Journal of Computational and Applied Mathematics, 450 (2024), 116011. https://doi.org/10.1016/j.cam.2024.116011
[36] C. L. Zhang, S. D. Zeng, Ekeland’s variational principle for fuzzy functions with applications, Journal of Compu tational and Applied Mathematics, 485 (2026), 117486. https://doi.org/10.1016/j.cam.2026.117486
[37] L. Zhou, W. S. Jia, Generalized unified well-posedness of equilibrium solutions for fuzzy multi-leader-multi-follower games under bounded rationality, Fuzzy Sets and Systems, 518 (2025), 109512. https://doi.org/10.1016/j. fss.2025.109512
[38] L. Zhou, W. S. Jia, Existence and continuity theorems of equilibrium solutions of multi-objective leader-follower games with fuzzy payoffs, Journal of Computational and Applied Mathematics, 484 (2026), 117518. https://doi. org/10.1016/j.cam.2026.117518
Iranian Journal of Fuzzy Systems
Volume 23, Issue 6
November and December 2026
Pages 51-66

Files

Share

How to cite

Statistics