Partial solution for an open question on pseudomonotone variational inequalities

Abstract

This article gives a partial solution for the open question raised by Nguyen Thanh Hao [Tikhonov regularization algorithm for pseudomonotone variational inequalities, Acta Math. Vietnam., 31 (2006), 283–289] about uniqueness of the solution of the regularized problem VI(K, F _ϵ) of a pseudomonotone variational inequality VI(K, F) for sufficiently small parameter ϵ > 0. It is proved that, under certain additional assumptions, the desired solution uniqueness holds for some classes of pseudoaffine variational inequalities and pseudomonotone variational inequalities.

Keywords:

• variational inequality
• Tikhonov regularization
• small parameter
• solution uniqueness
• pseudomonotone operator
• pseudoaffine operator
• mean-value theorem for vector-valued functions

AMS Subject Classifications::

• 49J40
• 47J30

Acknowledgements

The author was supported by a grant from Institute of Mathematics, Vietnam Academy of Science and Technology, and the project ‘Joint research and training on Variational Analysis and Optimization Theory, with oriented applications in some technological areas’ (Vietnam-USA). He would like to thank Prof Nguyen Dong Yen and Dr Trinh Cong Dieu for the guidance, and the anonymous referee for insightful comments.