Degree Theory for Generalized Mixed Quasi-variational Inequalities and Its Applications
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Degree Theory for Generalized Mixed Quasi-variational Inequalities and Its Applications Zhong-bao Wang1,2 · Yi-bin Xiao2 · Zhang-you Chen1 Received: 13 March 2020 / Accepted: 8 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The present paper is devoted to building degree theory for a generalized mixed quasivariational inequality in finite dimensional spaces. Then, by employing the obtained results, we prove the existence and stability of solutions to the considered generalized mixed quasi-variational inequality. Keywords Degree theory · Generalized f -projection operator · Generalized mixed quasi-variational inequality · Existence · Stability Mathematics Subject Classification 49J40 · 90C26 · 49J45 · 90C25
1 Introduction Variational inequalities are a flexible and unifying framework, which incorporates optimization problems, fixed point problems, transportation problems, financial equilibrium problems, migration equilibrium problems, saddle point problems and so on; see, e.g., [1–7]. This is the reason why there is vast literature studying the theory and applications of variation inequalities. There are two kinds of important extensions of the classic variational inequalities. One is the mixed variational inequality (MVI, also
Communicated by Jen-Chih Yao.
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Zhong-bao Wang [email protected] Yi-bin Xiao [email protected] Zhang-you Chen [email protected]
1
Department of Mathematics, Southwest Jiaotong University, Chengdu 611756, Sichuan, People’s Republic of China
2
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, Sichuan, People’s Republic of China
123
Journal of Optimization Theory and Applications
known as Hemivariational inequality), which is characterized by the involvement of a proper, convex and lower semicontinuous function into the classic variational inequality. The mixed variational inequality has been studied by many scholars and has also been applied to many practical problems, such as circuits in electronics, power control problems in ad hoc networks and economic equilibrium problems (see, [8–13]). The other important extension of the classic variational inequalities is the quasi-variational inequality (QVI), whose constraint set depends on the decision variables. This characterization of the quasi-variational inequality allows one to model many complex problems, such as impulse control problems, frictional elastostatic contact problems, and power markets and generalized Nash equilibrium problems in game theory (see, [1,2,14–17]). However, the above-mentioned two kinds of extensions of variational inequalities are not enough for the applications of variational inequalities. In some practical situations, we need to consider more general extensions of variational inequalities to model more complicated problems, and thus, it is necessary for us to study them first from the theoretical point of view. The generalized mixed quasi-variational inequality (GMQVI) considered in
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