On Implicit Algorithms for Solving Variational Inequalities

Abstract

This paper presents new implicit algorithms for solving the variational inequality and shows that the proposed methods converge under certain conditions. Some special cases are also discussed.

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E. Al-Shemas, "On Implicit Algorithms for Solving Variational Inequalities," Applied Mathematics, Vol. 4 No. 1, 2013, pp. 102-106. doi: 10.4236/am.2013.41018.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] G. Stampacchia, “Formes Bilineaires Coercitives Sur Les Ensembles Convexes,” Académie des Sciences de Paris, Vol. 258, 1964, pp. 4413-4416.
[2] J. L. Lions and G. Stampacchia, “Variational Inequalities,” Communications on Pure and Applied Mathematics, Vol. 20, No. 3, 1967, pp. 493-512. doi:10.1002/cpa.3160200302
[3] G. M. Korpelevich, “An Extragradient Method for Finding Saddle Points and for Other Problems,” Ekonomika i Matematicheskie Metody, Vol. 12, No. 4, 1976, pp. 747-756.
[4] M. A. Noor, K. I. Noor and E. Al-Said, “On New Proximal Methods for Solving the Variational Inequalities,” Journal of Applied Mathematics, 2012, pp. 1-7.
[5] M. A. Noor, K. I. Noor, E. Al-Said and S. Zainab, “Study on Unified Implicit Methods for Solving Variational Inequalities,” International Journal of Physics, Vol. 7, No. 2, 2012, pp. 222-225.
[6] M. A. Noor, “Some Developments in General Variational Inequalities,” Applied Mathematics and Computation, Vol. 152, No. 1, 2004, pp. 199-277. doi:10.1016/S0096-3003(03)00558-7
[7] M. A. Noor, K. I. Noor and T. M. Rassias, “Some Aspects of Variational Inequalities,” Journal of Computational and Applied Mathematics, Vol. 47, No. 3, 1993, pp. 285-312.
[8] D. Kinderlehrer and G. Stampacchia, “An Introduction to Variational Inequalities and Their Applications,” Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2000. doi:10.1137/1.9780898719451
[9] E. Al-Shemas, “Wiener-Hopf Equations Technique for Multi-Valued General Variational Inequalities,” Journal of Advanced Mathematical Studies, Vol. 2, No. 2, 2009, pp. 01-08.
[10] E. Al-Shemas and S. Billups, “An Iterative Method for Generalized Set-Valued Nonlinear Mixed Quasi-Variational Inequalities,” Journal of Applied Mathematics, Vol. 170, No. 2, 2004, pp. 423-432. doi:10.1016/j.cam.2004.01.028
[11] E. Al-Shemas, “Projection Iterative Methods for Multi-Valued General Variational Inequalities,” Applied Mathematics & Information Sciences, Vol. 3, No. 2, 2009, pp. 177-184.
[12] E. Al-Shemas, “Resolvent Operator Method for General Variational Inclusions,” Journal of Mathematical Inequalities, Vol. 3, No. 3, 2009, pp. 455-462. doi:10.7153/jmi-03-45

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