Gap functions and error bounds for random extended generalized variational inequality problem. (English) Zbl 1459.47024
Gap functions for random extended generalized variational inequality problems in the fuzzy setting are developed and error bounds for the variational inequality problem with the help of residual vector are obtained. The results presented in this paper improve and generalize the corresponding known results in literature.
Reviewer: Hengyou Lan (Zigong)
MSC:
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 49J40 | Variational inequalities |
| 90C31 | Sensitivity, stability, parametric optimization |
| 47E05 | General theory of ordinary differential operators |
Keywords:
variational inequalities; nonlinear operators; sensitivity; stability; monotone operators; fuzzy differential operators; convergence analysisReferences:
| [1] | M.K. Ahmad, Salahuddin, R.U. Verma, Existence theorem for fuzzy mixed vector Fvariational inequalities, Adv. Nonlinear Var. Inequal. 16(1)(2013) 53-59. · Zbl 1275.49015 |
| [2] | G.A. Anastassiou, Salahuddin, Weakly set-valued generalized vector variational inequalities, J. Comput. Anal. Appl. 15(4)(2013) 622-632. · Zbl 1272.49011 |
| [3] | D. Aussel, R. Corea, M. Marechal, Gap functions for quasi variational inequalities and generalized Nash equilibrium problems, J. Optim. Theory Appl. 151(2011) 474-488. · Zbl 1250.90095 |
| [4] | D. Aussel, J. Dutta, On gap functions for set-valued Stampacchia variational inequalities, J. Optim. Theory Appl. 149(2011) 513-527. · Zbl 1267.90153 |
| [5] | D. Aussel, R. Gupta, A. Mehra, Gap functions and error bounds for inverse quasi-variational inequality problems, J. Math. Anal. Appl. 158(2013) 270-280. · Zbl 1311.49016 |
| [6] | H. Brezis, Operateurs maximaux monotone et semi-groupes de contractions dans les espace de Hilbert, North-Holland, Amsterdam (1973). · Zbl 0252.47055 |
| [7] | S.S. Chang, Y.G. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets Syst. 32(1989) 359-367. GAP FUNCTIONS AND ERROR BOUNDS23 · Zbl 0677.47037 |
| [8] | S.S. Chang,Fixed Point Theory with Applications,Chongquing Publishing House, Chongquing (1984). |
| [9] | H.Y. Lan, R.U. Verma, Iterative algorithms for non linear fuzzy variational inclusion with (A, η) acceretive mappings in Banach spaces, Adv. Nonlinear Var. Inequal. 11(1)(2008) 15-50. · Zbl 1176.49009 |
| [10] | S.S. Chang, G.M. Lee, B.S. Lee, Vector quasi variational inequalities for fuzzy mappings(II), Fuzzy Sets Syst. 102(1999) 333-344. · Zbl 0937.47062 |
| [11] | S.S. Chang, Salahuddin, Existence theorems for vector quasi variational-like inequalities for fuzzy mappings, Fuzzy Sets Syst. 233(2013) 89-95. · Zbl 1314.49003 |
| [12] | C. Charitha, J. Dutta, Regularized gap functions and error bounds for vector variational inequalities, Pac. J. Optim. 6(2010) 497-510. · Zbl 1228.90124 |
| [13] | A.H. Dar, Mohd. Sarfaraz, M.K. Ahmad, Gap functions and error bounds for generalized set-valued mixed variational inequality, Indian jounal of industrial and applied mathematics 7(2)(2016) 117-126. |
| [14] | X.P. Ding, M.K. Ahmad, Salahuddin, Fuzzy generalized vector variational inequalities and complementarity problem, Nonlinear Funct. Anal. Appl. 13(2)(2008) 253-263. · Zbl 1157.49011 |
| [15] | J.H. Fan, X.G. Wang, Gap functions and global error bouds for set-valued variational inequalities, J. Comput. Appl. Math. 233(2010) 2956-2965. · Zbl 1204.65079 |
| [16] | M. Fukushima, Merit functions for variational inequality and complementarity problems, Nonlinear Opt. and Appl. (1996) 155-170. · Zbl 0996.90082 |
| [17] | R. Gupta, A. Mehra, Gap functions and error bounds for quasi variational inequalities, J. Global Optim. 53(2012) 737-748. · Zbl 1281.90071 |
| [18] | N.J. Huang, Random generalized nonlinear variational inclusion for random fuzzy mappings, Fuzzy Sets Syst. 105(1999) 437-444. · Zbl 0961.49004 |
| [19] | L.R. Huang, K.F. Ng, Equivalent optimization formulations and error bounds for variational inequality problem, J. Optim.Theory Appl. 125(2005) 299-314. · Zbl 1071.49007 |
| [20] | S.A. Khan, J.W. Chen, Gap functions and global error bounds for generalized mixed quasi variational inequalities, Appl. Math. Comput. 260(2015) 71-81. · Zbl 1410.49010 |
| [21] | S.A. Khan, J.W. Chen, Gap functions and error bounds for generalized mixed vector equilibrium problems, J. Optim. Theory Appl. 166(2015) 767-776. · Zbl 1327.49021 |
| [22] | M.F. Khan, S. Hussain, Salahuddin, A fuzzy extension of generalized multi-valued η-mixed vector variational -like inequalities on locally convex Hausdorff topological vector spaces, Bull. Cal. Math. Soc. 100(1)(2008) 27-36. · Zbl 1388.47001 |
| [23] | S.A. Khan, J. Iqbal, Y. Shehu, Mixed quasi variational inequalities involving error bounds, J. Inequal. Appl. 417(2015) 1-19. · Zbl 1330.49007 |
| [24] | S.A. Khan, J. Iqbal, Y. Shehu, Gap functions and error bounds for random generalized variational inequality problems, J. Inequal. Appl. 47(2016) 1-17. · Zbl 1333.49020 |
| [25] | G.M. Lee, D.S. Kim, B.S. Lee, Vector variational inequality for fuzzy mappings, Non Linear Anal. Forum 4(1999) 119-129. · Zbl 1046.49004 |
| [26] | G. Li, K.F. Ng, On generalized D-gap functions for non smooth and non monotone variational inequality problems, SIAM J. Optim. 20(2009) 667-690. · Zbl 1194.65088 |
| [27] | J. Li, G. Mastroeni, Vector variational inequalities involving set-valued mappings via scalarization with applications to error bounds for gap functions, J. Optim. Theory Appl. 145(2010) 355-372. · Zbl 1205.90279 |
| [28] | M.A. Noor, Merit functions for general variational inequalities, J. Math. Anal. Appl. 316(2006) 736-752. · Zbl 1085.49011 |
| [29] | S. Park, B.S. Lee, G.M. Lee, A general vector valued variational inequality and its fuzzy extension, Int. J. Math. Sci. 21(1998) 637-642. · Zbl 0919.47057 |
| [30] | M.V. Solodov, Merit functions and error bounds for generalized variational inequalities, J. Math. Anal. Appl. 287(2003) 405-414. · Zbl 1036.49020 |
| [31] | L.L. Tan, Regularized gap functions for nonsmooth variational inequality problems, J. Math. Anal. Appl. 334(2007) 1022-1038. · Zbl 1114.49012 |
| [32] | G.J. Tang, N.J. Huang, Gap functions and global error bouds for set-valued mixed variational inequalites, Taiwanese J. Math. 17(2013) 1267-1286. · Zbl 1304.90207 |
| [33] | R.U. Verma, Salahuddin, A common fixed theorem for fuzzy mappings, Trans. Math. Program. Appl. 1(I)(2013) 59-68. |
| [34] | J.H. Wu, M. Horian, P. Marcotte, A general descent framework for the monotone variational inequality problem, Math. Program 61(1993) 281-300. · Zbl 0813.90111 |
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