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Some existence theorems for nonconvex variational inequalities problems. (English) Zbl 1206.49012
Summary: By using nonsmooth analysis knowledge, we provide conditions for the existence of solutions of variational inequalities problems in nonconvex setting. We also show that the strongly monotonic assumption of the mapping is not necessary for the existence of solutions. Consequently, the results presented in this paper can be viewed as an improvement and refinement of some known results from the literature.

49J40 Variational inequalities
49J52 Nonsmooth analysis
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[1] G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” Comptes Rendus de l’Académie des Sciences, vol. 258, pp. 4413-4416, 1964. · Zbl 0124.06401
[2] Y. J. Cho and X. Qin, “Systems of generalized nonlinear variational inequalities and its projection methods,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4443-4451, 2008. · Zbl 1153.49009
[3] Y. J. Cho, S. M. Kang, and X. Qin, “On systems of generalized nonlinear variational inequalities in Banach spaces,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 214-220, 2008. · Zbl 1185.65105
[4] N. Petrot, “Existence and algorithm of solutions for general set-valued Noor variational inequalities with relaxed (\mu ,\nu )-cocoercive operators in Hilbert spaces,” Journal of Applied Mathematics and Computing, vol. 32, no. 2, pp. 393-404, 2010. · Zbl 1229.47102
[5] S. Suantai and N. Petrot, “Existence and stability of iterative algorithms for the system of nonlinear quasi-mixed equilibrium problems,” Applied Mathematics Letters, vol. 24, no. 3, pp. 308-313, 2011. · Zbl 1220.47125
[6] F. H. Clarke, Y. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1998. · Zbl 1047.49500
[7] F. H. Clarke, R. J. Stern, and P. R. Wolenski, “Proximal smoothness and the lower-C2 property,” Journal of Convex Analysis, vol. 2, no. 1-2, pp. 117-144, 1995. · Zbl 0881.49008
[8] R. A. Poliquin, R. T. Rockafellar, and L. Thibault, “Local differentiability of distance functions,” Transactions of the American Mathematical Society, vol. 352, no. 11, pp. 5231-5249, 2000. · Zbl 0960.49018
[9] A. Moudafi, “Projection methods for a system of nonconvex variational inequalities,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 1-2, pp. 517-520, 2009. · Zbl 1162.49013
[10] R. U. Verma, “General convergence analysis for two-step projection methods and applications to variational problems,” Applied Mathematics Letters, vol. 18, no. 11, pp. 1286-1292, 2005. · Zbl 1099.47054
[11] R. P. Agarwal and R. U. Verma, “General system of A(\eta )-maximal relaxed monotone variational inclusion problems based on generalized hybrid algorithms,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 238-251, 2010. · Zbl 1221.49005
[12] M. A. Noor, “On a system of general mixed variational inequalities,” Optimization Letters, vol. 3, no. 3, pp. 437-451, 2009. · Zbl 1171.58308
[13] N. Petrot, “A resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems,” Applied Mathematics Letters, vol. 23, no. 4, pp. 440-445, 2010. · Zbl 1185.49010
[14] M. A. Noor, “Iterative schemes for nonconvex variational inequalities,” Journal of Optimization Theory and Applications, vol. 121, no. 2, pp. 385-395, 2004. · Zbl 1062.49009
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