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

Ting Xie; Dapeng Li; Ting Xie; Dapeng Li

doi:10.3934/math.2020455

AIMS Mathematics

2020, Volume 5, Issue 6: 7107-7121. doi: 10.3934/math.2020455

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Research article

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

Ting Xie ^{1,2
,
,},
Dapeng Li ²

1.
College of Social Development and Public Administration, Northwest Normal University, Lanzhou 730070, China
2.
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China

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.
- generalized variational inequalities,
- hesitant fuzzy relations,
- level sets,
- projection methods,
- Lyapunov stability
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

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Abstract

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.

References

[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

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