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Fundamentals of mixed quasi variational inequalities. (English) Zbl 1059.49018
Summary: Mixed quasi variational inequalities are very important and significant generalizations of variational inequalities involving the nonlinear bifunction. It is well known that a large class of problems arising in various branches of pure and applied sciences can be studied in the general framework of mixed quasi variational inequalities. Due to the presence of the bifunction in the formulation of variational inequalities, the projection method and its variant forms cannot be used to suggest and analyze iterative methods for mixed quasi variational inequalities. In this paper, we suggest and analyze various iterative schemes for solving these variational inequalities using resolvent methods, resolvent equations and auxiliary principle techniques. We discuss the sensitivity analysis, stability of the dynamical systems and well-posedness of the mixed quasi variational inequalities. We also consider and investigate various classes of mixed quasi variational inequalities in the setting of invexity, \(g\)-convexity and uniformly prox-regular convexity. The concepts of invexity, \(g\)-convexity and uniformly prox-regular convexity are generalizations of the classical convexity in different directions. Some classes of equilibrium problems are introduced and studied. We also suggest several open problems with sufficient information and references.

MSC:
49J40 Variational inequalities
90C30 Nonlinear programming
35A15 Variational methods applied to PDEs
47J25 Iterative procedures involving nonlinear operators
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