Noor, Muhammad Aslam Extended general variational inequalities. (English) Zbl 1163.49303 Appl. Math. Lett. 22, No. 2, 182-186 (2009). Summary: We introduce and consider a new class of general variational inequalities involving three nonlinear operators, which is called the extended general variational inequalities. M. Aslam Noor [Appl. Math. Lett. 22, No. 2, 182–186 (2009; Zbl 1163.49303)] has shown that the minimum of nonconvex functions can be characterized via these variational inequalities. Using a projection technique, we establish the equivalence between the extended general variational inequalities and the general nonlinear projection equation. This equivalent formulation is used to discuss the existence of a solution of the extended general variational inequalities. Several special cases are also discussed. Cited in 3 ReviewsCited in 90 Documents MSC: 49J40 Variational inequalities Keywords:variational inequalities; nonconvex functions; fixed point problem; convergence; existence Citations:Zbl 1163.49303 PDFBibTeX XMLCite \textit{M. A. Noor}, Appl. Math. Lett. 22, No. 2, 182--186 (2009; Zbl 1163.49303) Full Text: DOI References: [1] Baiocchi, C.; Capelo, A., Variational and Quasi-Variational Inequalities (1984), J. Wiley and Sons: J. Wiley and Sons New York · Zbl 1308.49002 [2] Cristescu, G.; Lupsa, L., Non-connected Convexities and Applications (2002), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Holland · Zbl 1037.52008 [3] Giannessi, F.; Maugeri, A., Variational Inequalities and Network Equilibrium Problems (1995), Plenum Press: Plenum Press New York · Zbl 0834.00044 [4] Glowinski, R.; Lions, J. L.; Trémolières, R., Numerical Analysis of Variational Inequalities (1981), North-Holland: North-Holland Amsterdam · Zbl 0508.65029 [5] Jian, Jin-Bao, On \((E, F)\) generalized convexity, Internat. J. Math., 2, 121-132 (2003) · Zbl 1165.90643 [6] Aslam Noor, M., General variational inequalities, Appl. Math. Lett., 1, 119-121 (1988) · Zbl 0655.49005 [7] Aslam Noor, M., Wiener-Hopf equations and variational inequalities, J. Optim. Theory Appl., 79, 197-206 (1993) · Zbl 0799.49010 [8] Aslam Noor, M., Some recent advances in variational inequalities, Part I, basic concepts, New Zealand J. Math., 26, 53-80 (1997) · Zbl 0886.49004 [9] Aslam Noor, M., Some recent advances in variational inequalities, Part II, other concepts, New Zealand J. Math., 26, 229-255 (1997) · Zbl 0889.49006 [10] Aslam Noor, M., New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251, 217-229 (2000) · Zbl 0964.49007 [11] Aslam Noor, M., Some developments in general variational inequalities, Appl. Math. Comput., 152, 199-277 (2004) · Zbl 1134.49304 [12] Aslam Noor, M., Differentiable nonconvex functions and general variational inequalities, Appl. Math. Comput., 199, 623-630 (2008) · Zbl 1147.65047 [13] Aslam Noor, M., Auxiliary principle technique for extended general variational inequalities, Banach J. Math. Anal., 2, 33-39 (2008) · Zbl 1138.49016 [14] M. Aslam Noor, Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput. (2008) (in press); M. Aslam Noor, Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput. (2008) (in press) · Zbl 1158.49014 [15] Rapcsak, T., Smooth Nonlinear Optimization in \(R^n (1997)\), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Holland · Zbl 1009.90109 [16] Youness, E. A., \(E\)-convex sets, \(E\)-convex functions and \(E\)-convex programming, J. Optim. Theory Appl., 102, 439-450 (1999) · Zbl 0937.90082 [17] Zhao, Y.; Sun, D., Alternative theorems for nonlinear projection equations and applications to generalized complementarity problems, Nonl. Anal., 46, 853-868 (2001) · Zbl 1047.49014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.