Research article

On the stability of projected dynamical system for generalized variational inequality with hesitant fuzzy relation

  • Received: 29 June 2020 Accepted: 03 September 2020 Published: 10 September 2020
  • MSC : 26E50, 49K15

  • In this paper, we investigate the projected dynamical system associated with the generalized variational inequality with hesitant fuzzy relation. We establish the equivalence between the generalized variational inequality with hesitant fuzzy relation and the fuzzy fixed point problem. And we analyze the existence theorem and iterative algorithm of solutions to such problem. Furthermore, using the projection method, we propose a projection neural network for solving the generalized variational inequality with hesitant fuzzy relation and discuss the stability of the proposed projected dynamical system.

    Citation: Ting Xie, Dapeng Li. On the stability of projected dynamical system for generalized variational inequality with hesitant fuzzy relation[J]. AIMS Mathematics, 2020, 5(6): 7107-7121. doi: 10.3934/math.2020455

    Related Papers:

  • In this paper, we investigate the projected dynamical system associated with the generalized variational inequality with hesitant fuzzy relation. We establish the equivalence between the generalized variational inequality with hesitant fuzzy relation and the fuzzy fixed point problem. And we analyze the existence theorem and iterative algorithm of solutions to such problem. Furthermore, using the projection method, we propose a projection neural network for solving the generalized variational inequality with hesitant fuzzy relation and discuss the stability of the proposed projected dynamical system.


    加载中


    [1] B. H. Ahn, Computation of Market Equilibria for Policy Analysis: The Project Independence Evaluation Study (PIES) Approach, Garland Press, New York, 1979.
    [2] J. C. R. Alcantud, V. Torra, Decomposition theorems and extension principles for hesitant fuzzy set, Inform. Fusion, 41 (2018), 48-56. doi: 10.1016/j.inffus.2017.08.005
    [3] B. Bedregal, R. Reiser, H. Bustince, et al., Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms, Inform. Sciences, 255 (2014), 82-99. doi: 10.1016/j.ins.2013.08.024
    [4] R. Bellman, L. A. Zadeh, Decision making in a fuzzy environment, Manage. Sci., 17 (1970), 141- 164.
    [5] S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
    [6] T. R. Ding, C. Z. Li, Course of Ordinary Differential Equations (2nd ed.), Higher Education Press, Beijing, 2004.
    [7] B. C. Eaves, On the basic theorem of complementarity, Math. Program., 1 (1971), 68-75. doi: 10.1007/BF01584073
    [8] F. Facchinei, J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems I (1st ed.), Springer, New York, 2003.
    [9] S. C. Fang, E. L. Peterson, Generalized variational inequalities, J. Optimiza. Theory Appl., 38 (1982), 363-383. doi: 10.1007/BF00935344
    [10] T. L. Friesz, D. H. Bernstein, N. J. Mehta, et al., Day-to-day dynamic network disequilibria and idealized traveler information systems, Oper. Res., 42 (1994), 1120-1136. doi: 10.1287/opre.42.6.1120
    [11] P. T. Harker, J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Program., 48 (1990), 161-220. doi: 10.1007/BF01582255
    [12] P. Hartman, G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math., 115 (1966), 271-310. doi: 10.1007/BF02392210
    [13] C. F. Hu, Solving fuzzy variational inequalities over a compact set, J. Comput. Appl. Math., 129 (2001), 185-193. doi: 10.1016/S0377-0427(00)00549-5
    [14] C. F. Hu, Generalized variational inequalities with fuzzy relation, J. Comput. Appl. Math., 146 (2002), 47-56. doi: 10.1016/S0377-0427(02)00417-X
    [15] C. F. Hu, F. B. Liu, Solving mathematical programs with fuzzy equilibrium constraints, Comput. Math. Appl., 58 (2009), 1844-1851. doi: 10.1016/j.camwa.2009.08.037
    [16] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, Academic Press, New York, 1975.
    [17] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York and London, 1980.
    [18] E. Klein, A. Thompson, Theory of Correspondences, Wiley-Interscience, New York, 1984.
    [19] J. LaSalle, Some extensions of Liapunov's second method, Ire Transactions on Circuit Theory, 7 (1960), 520-527. doi: 10.1109/TCT.1960.1086720
    [20] L. W. Liu, Y. Q. Li, On Generalized set-valued variational inclusions, J. Math. Anal. Appl., 261 (2001), 231-240. doi: 10.1006/jmaa.2001.7493
    [21] L. Mathiesen, Computational experience in solving equilibrium models by a sequence of linear complementarity problems, Oper. Res., 33 (1985), 1225-1250. doi: 10.1287/opre.33.6.1225
    [22] L. Mathiesen, An algorithm based on a sequence of linear complementarity problems applied to a Walrasian equilibrium model: an example, Math. Program., 37 (1987), 1-18. doi: 10.1007/BF02591680
    [23] M. A. Noor, Variational inequalities for fuzzy mappings (I), Fuzzy Set. Syst., 55 (1993), 309-312. doi: 10.1016/0165-0114(93)90257-I
    [24] M. A. Noor, Implicit dynamical systems and quasi variational inequalities, Appl. Math. Comput., 134 (2003), 69-81.
    [25] M. J. Smith, The existence, uniqueness and stability of traffic equilibria, Transport. Res. B-Meth, 13 (1979), 295-304.
    [26] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst., 25 (2010), 529-539.
    [27] T. Xie, Z. T. Gong, A hesitant soft fuzzy rough set and its applications, IEEE Access, 7 (2019), 167766-167783. doi: 10.1109/ACCESS.2019.2954179
    [28] T. Xie, Z. T. Gong, Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization, Open Math., 17 (2019), 627-645. doi: 10.1515/math-2019-0050
    [29] X. B. Yang, X. N. Song, Y. S. Qi, et al., Constructive and axiomatic approaches to hesitant fuzzy rough set, Soft Comput., 18 (2014), 1067-1077. doi: 10.1007/s00500-013-1127-2
    [30] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2478) PDF downloads(100) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog