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Extended general variational inequalities. (English) Zbl 1163.49303
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.

MSC:
49J40Variational methods including variational inequalities
References:
[1]Baiocchi, C.; Capelo, A.: Variational and quasi-variational inequalities, (1984)
[2]Cristescu, G.; Lupsa, L.: Non-connected convexities and applications, (2002)
[3]Giannessi, F.; Maugeri, A.: Variational inequalities and network equilibrium problems, (1995)
[4]Glowinski, R.; Lions, J. L.; Trémolières, R.: Numerical analysis of variational inequalities, (1981)
[5]Jian, Jin-Bao: On (E,F) generalized convexity, Internat. J. Math. 2, 121-132 (2003) · Zbl 1165.90643
[6]Noor, M. Aslam: General variational inequalities, Appl. math. Lett. 1, 119-121 (1988) · Zbl 0655.49005 · doi:10.1016/0893-9659(88)90054-7
[7]Noor, M. Aslam: Wiener–Hopf equations and variational inequalities, J. optim. Theory appl. 79, 197-206 (1993) · Zbl 0799.49010 · doi:10.1007/BF00941894
[8]Noor, M. Aslam: Some recent advances in variational inequalities, part I, basic concepts, New Zealand J. Math. 26, 53-80 (1997) · Zbl 0886.49004
[9]Noor, M. Aslam: Some recent advances in variational inequalities, part II, other concepts, New Zealand J. Math. 26, 229-255 (1997) · Zbl 0889.49006
[10]Noor, M. Aslam: New approximation schemes for general variational inequalities, J. math. Anal. appl. 251, 217-229 (2000) · Zbl 0964.49007 · doi:10.1006/jmaa.2000.7042
[11]Noor, M. Aslam: Some developments in general variational inequalities, Appl. math. Comput. 152, 199-277 (2004) · Zbl 1134.49304 · doi:10.1016/S0096-3003(03)00558-7
[12]Noor, M. Aslam: Differentiable nonconvex functions and general variational inequalities, Appl. math. Comput. 199, 623-630 (2008) · Zbl 1147.65047 · doi:10.1016/j.amc.2007.10.023
[13]Noor, M. Aslam: Auxiliary principle technique for extended general variational inequalities, Banach J. Math. anal. 2, 33-39 (2008) · Zbl 1138.49016 · doi:emis:journals/BJMA/v2n1.html
[14]M. Aslam Noor, Projection iterative methods for extended general variational inequalities, J. Appl. Math. Comput. (2008) (in press)
[15]Rapcsak, T.: Smooth nonlinear optimization in rn, (1997)
[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 · doi:10.1023/A:1021792726715
[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 · doi:10.1016/S0362-546X(00)00154-1