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Iterative algorithms for a new class of extended general nonconvex set-valued variational inequalities. (English) Zbl 1214.47065
Summary: Some new classes of extended general nonconvex set-valued variational inequalities and the extended general Wiener-Hopf inclusions are introduced. By the projection technique, the equivalence between the extended general nonconvex set-valued variational inequalities and fixed point problems as well as extended general nonconvex Wiener-Hopf inclusions is proved. Then, by using this equivalent formulation, we discuss the existence of solutions of the extended general nonconvex set-valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of extended general nonconvex set-valued variational inequalities. We also verify that the approximate solutions obtained by our algorithms converge to the solutions of extended general nonconvex set-valued variational inequalities. The results presented in this paper extend and improve some known results from the literature.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H05Monotone operators (with respect to duality) and generalizations
49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
47J22Variational and other types of inclusions
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References:
[1] Stampacchia, G.: Formes bilineaires coercitives sur LES ensembles convexes, C. R. Acad. sci. Paris 258, 4413-4416 (1964) · Zbl 0124.06401
[2] Lions, J. L.; Stampacchia, G.: Variational inequalities, Comm. pure appl. Math. 20, 493-512 (1967) · Zbl 0152.34601 · doi:10.1002/cpa.3160200302
[3] Shi, P.: Equivalence of variational inequalities with Wiener--Hopf equations, Proc. amer. Math. soc. 111, 339-346 (1991) · Zbl 0881.35049 · doi:10.2307/2048322
[4] Shi, P.: An iterative method for obstacles problems via Green’s functions, Nonlinear anal. 15, 339-344 (1990) · Zbl 0725.65068 · doi:10.1016/0362-546X(90)90142-4
[5] Robinson, S. M.: Normal maps induced by linear transformations, Math. oper. Res. 17, 691-714 (1992) · Zbl 0777.90063 · doi:10.1287/moor.17.3.691
[6] Robinson, S. M.: Sensitivity analysis of variational inequalities by normal-map techniques, Variational inequalities and network equilibrium problems, 257-276 (1995) · Zbl 0861.49009
[7] Sellami, H.; Robinson, S. M.: Implementation of a continuation method for normal maps, Math. program. 76, 563-578 (1997) · Zbl 0873.90093 · doi:10.1007/BF02614398
[8] Bounkhel, M.; Tadji, L.; Hamdi, A.: Iterative schemes to solve nonconvex variational problems, J. inequal. Pure appl. Math. 4, 1-14 (2003) · Zbl 1045.58014 · emis:journals/JIPAM/v4n1/index.html
[9] Clarke, F. H.; Ledyaev, Yu.S.; Stern, R. J.; Wolenski, P. R.: Nonsmooth analysis and control theory, (1998) · Zbl 1047.49500
[10] Clarke, F. H.; Stern, R. J.; Wolenski, P. R.: Proximal smoothness and the lower C2 property, J. convex anal. 2, No. 1/2, 117-144 (1995) · Zbl 0881.49008 · emis:journals/JCA/vol.2_no.1+2/
[11] Poliquin, R. A.; Rockafellar, R. T.; Thibault, L.: Local differentiability of distance functions, Trans. amer. Math. soc. 352, 5231-5249 (2000) · Zbl 0960.49018 · doi:10.1090/S0002-9947-00-02550-2
[12] Noor, M. A.: Iterative schemes for nonconvex variational inequalities, J. optim. Theory appl. 121, 385-395 (2004) · Zbl 1062.49009 · doi:10.1023/B:JOTA.0000037410.46182.e2
[13] M.A. Noor, Variational inequalities and applications, Lecture Notes, Mathematics Department, COMSATS Institute of information Technology, Islamabad, Pakistan, 2007--2009.
[14] Pang, L. P.; Shen, J.; Song, H. S.: A modified predictor--corrector algorithm for solving nonconvex generalized variational inequalities, Comput. math. Appl. 54, 319-325 (2007) · Zbl 1131.49010 · doi:10.1016/j.camwa.2006.07.010
[15] M.A. Noor, Projection methods for nonconvex variational inequalities, Optim. Lett. doi:10.1007/s11590-009-0121-1. · Zbl 1171.58307
[16] Noor, M. A.: Iterative methods for general nonconvex variational inequalities, Albanian J. Math. 3, No. 1, 117-127 (2009) · Zbl 1213.49017 · http://x.kerkoje.com/index.php/ajm/article/viewArticle/134
[17] Clarke, F. H.: Optimization and nonsmooth analysis, (1983) · Zbl 0582.49001
[18] Bounkhel, M.: Existence results of nonconvex differential inclusions, Port. math. (N.S.) 59, No. 3, 283-309 (2002) · Zbl 1022.34007
[19] Bounkhel, M.: General existence results for second order nonconvex sweeping process with unbounded perturbations, Port. math. (N.S.) 60, No. 3, 269-304 (2003) · Zbl 1055.34116
[20] Bounkhel, M.; Azzam, L.: Existence results on the second order nonconvex sweeping processes with perturbations, Set-valued anal. 12, No. 3, 291-318 (2004) · Zbl 1048.49002 · doi:10.1023/B:SVAN.0000031356.03559.91
[21] M. Bounkhel, L. Thibault, Further characterizations of regular sets in Hilbert spaces and their applications to nonconvex sweeping process, Centro de Modelamiento Matematico, CMM, Universidad de Chile, 2000, Preprint. · Zbl 0949.49014
[22] Canino, A.: On p-convex sets and geodesics, J. differential equations 75, 118-157 (1988) · Zbl 0661.34042 · doi:10.1016/0022-0396(88)90132-5
[23] Nadler, S. B.: Multi-valued contraction mappings, Pacific J. Math. 30, No. 2, 475-488 (1996) · Zbl 0187.45002
[24] Noor, M. A.: Sensitivity analysis of extended general variational inequalities, Appl. math. E-notes 9, 17-26 (2009) · Zbl 1158.49028 · emis:journals/AMEN/2009/2009.htm
[25] Noor, M. A.: Quasi variational inequalities, Appl. math. Lett. 1, 367-370 (1988) · Zbl 0708.49015 · doi:10.1016/0893-9659(88)90152-8
[26] Liu, L. S.: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. math. Anal. appl. 194, 114-125 (1995) · Zbl 0872.47031 · doi:10.1006/jmaa.1995.1289
[27] Noor, M. A.: Some developments in general variational inequalities, Appl. math. Comput. 152, 199-277 (2004) · Zbl 1134.49304 · doi:10.1016/S0096-3003(03)00558-7
[28] Lions, P. L.; Mercier, B.: Splitting algorithms for the sum of two nonlinear operators, SIAM, J. Numer. anal. 16, 964-979 (1979) · Zbl 0426.65050 · doi:10.1137/0716071
[29] Bnouhachem, A.; Noor, M. A.: Numerical methods for general mixed variational inequalities, Appl. math. Comput. 204, 27-36 (2008) · Zbl 1157.65037 · doi:10.1016/j.amc.2008.05.134
[30] Brezis, H.: Operateurs maximaux monotone, Mathematical studies 5 (1973) · Zbl 0257.46029
[31] Kinderlehrer, D.; Stampacchia, G.: An introduction to variational inequalities and their applications, (2000) · Zbl 0988.49003