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Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. (English) Zbl 1229.47104
Summary: We introduce and consider a new generalized system of nonconvex variational inequalities with different nonlinear operators. We establish the equivalence between the generalized system of nonconvex variational inequalities and the fixed point problems using the projection technique. This equivalent alternative formulation is used to suggest and analyze a general explicit projection method for solving the generalized system of nonconvex variational inequalities. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.

MSC:
 47J20 Variational and other types of inequalities involving nonlinear operators (general) 47J25 Iterative procedures involving nonlinear operators
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References:
 [1] Stampacchia, G., Formes bilinearies coercivities sur LES ensembles convexes, C.R. acad. sci. Paris, 258, 4413-4416, (1964) · Zbl 0124.06401 [2] Noor, M.A., Some developments in general variational inequalities, Appl. math. comput., 152, 199-277, (2004) · Zbl 1134.49304 [3] Bounkhel, M.; Tadj, L.; Hamdi, A., Iterative schemes to solve nonconvex variational problems, J. inequal. pure appl. math., 4, 1-14, (2003) · Zbl 1045.58014 [4] Noor, M.A., New approximation schemes for general variational inequalities, J. math. anal. appl., 251, 217-229, (2000) · Zbl 0964.49007 [5] 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 [6] Verma, R.U., Generalized system for relaxed cocoercive variational inequalities and projection methods, J. optim. theory appl., 121, 1, 203-210, (2004) · Zbl 1056.49017 [7] Noor, M.A., Iterative schemes for nonconvex variational inequalities, J. optim. theory appl., 121, 385-395, (2004) · Zbl 1062.49009 [8] Clarke, F.H.; Ledyaev, Y.S.; Wolenski, P.R., Nonsmooth analysis and control theory, (1998), Springer Berlin · Zbl 1047.49500 [9] Verma, R.U., General convergence analysis for two-step projection methods and applications to variational problems, Appl. math. lett., 18, 1286-1292, (2005) · Zbl 1099.47054 [10] Huang, Z.; Noor, M.A., An explicit projection method for a system of nonlinear variational inequalities with different ($$\gamma$$, $$r$$)-cocoercive mappings, Appl. math. comput., 190, 356-361, (2007) · Zbl 1120.65080 [11] Noor, M.A., Differentiable nonconvex functions and general variational inequalities, Appl. math. comput., 199, 623-630, (2008) · Zbl 1147.65047 [12] Noor, M.A., Projection methods for nonconvex variational inequalities, Optim. lett., 3, 411-418, (2009) · Zbl 1171.58307 [13] Noor, M.A., Extended general variational inequalities, Appl. math. lett., 22, 182-186, (2009) · Zbl 1163.49303 [14] Noor, M.A.; Noor, K.I., Projection algorithms for solving system of general variational inequalities, Nonlinear anal., 70, 2700-2706, (2009) · Zbl 1156.49010 [15] Poliquin, R.A.; Rockafellar, R.T.; Thibault, L., Local differentiability of distance functions, Trans. amer. math. soc., 352, 5231-5249, (2000) · Zbl 0960.49018
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