Order-Preservation Properties of Solution Mapping for Parametric Equilibrium Problems and Their Applications
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In this paper, we use some order-theoretic fixed point theorems to study the upper order-preservation properties of solution mapping for parametric equilibrium problems. In contrast to lots of existing works on the behaviors of solutions to equilibrium problems, the topic of the order-preservation properties of solutions is relatively new for equilibrium problems. It would be useful for us to analyze the changing trends of solutions to equilibrium problems. In order to show the applied value and theoretic value of this subject, we focus on a class of differential variational inequalities, which are currently receiving much attention. By applying the order-preservation properties of solution mapping to variational inequality, we investigate the existence of mild solutions to differential variational inequalities. Since our approaches are order-theoretic and the underlying spaces are Banach lattices, the results obtained in this paper neither require the bifunctions in equilibrium problems to be continuous nor assume the Lipschitz continuity for the involved mapping in ordinary differential equation.
KeywordsUpper order-preservation property Parametric equilibrium problems Order-theoretic fixed point theorems Differential variational inequality
Mathematics Subject Classification47H10 90C31 90C33
The authors are very grateful to two anonymous reviewers and the editor for their constructive comments and valuable suggestions. These comments have motivated us to develop an order-theoretic approach for differential variational inequalities and hence have broadened the results of this paper. This work is supported financially by the Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu Province (2017SJB0238), Natural Science Foundation in Jiangsu Provincial (BK20171041), Qinglan Project and Natural Science Foundation in colleges and universities in Jiangsu Province (16KJB110009).
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