Characterization of the Cone and Applications in Banach Spaces

Abstract

In this article, we first discuss the subduality and orthogonality of the cones and the dual cones when the norm is monotone in Banach spaces. Then, under different assumptions, the necessary and sufficient conditions for the ordering increasing property of the metric projection onto cones and order intervals are studied. Moreover, representations of the metric projection onto cones and order intervals are obtained. As applications, the solvability and approximation results of solutions to nonlinear discontinuous variational inequality and complementarity problems are proved by partial ordering methods.

Keywords:

• Cone
• Banach spaces
• metric projection
• variational inequality
• complementarity problem

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 41A65
• 47H07
• 06B30
• 47J20
• 47H10

Acknowledgments

The authors would like to thank the referee for his/her very important comments that improved the results and the quality of the article.

Disclosure statement

The authors declare that they have no competing interests.

Additional information

Funding

The authors were supported financially by the National Natural Science Foundation of China (11871302, 71773067), the Natural Science Foundation of Shandong Province of China (ZR2017MA034) and the China Postdoctoral Science Foundation (2017M612307). The support from the Australian Research Council for the research is also acknowledged.