Generalized well-posedness results for a class of new mixed variational inequalities

Abstract

This paper is devoted to investigate a generalized type of mixed variational inequality (GMVI) in Banach space. Under the general assumptions, we first apply Minty's approach to deliver an equivalent result for GMVI and provide the existence condition of the solutions of GMVI. Then, the concepts of the strong and the weak well-posedness are introduced in the generalized sense, which are applied to discuss the essential relation between the metric characterizations and the generalized strong well-posedness as well as the weak well-posedness for GMVI. Moreover, the theorems are established to determine the generalized the strong and the weak well-posedness of GMVI, respectively. Furthermore, we consider a family of approximating problems corresponding to GMVI, which are dominated by the perturbation parameter ε, and a critical convergence result is obtained. Finally, an example is given to illustrate our main results.

Keywords:

• Mixed variational inequality
• generalized well-posedness
• approximating sequence
• perturbation
• convergence

2010 Mathematics Subject Classifications:

• 49J40
• 47J20
• 35J87
• 49J53
• 65Nxx

Acknowledgements

We would like to thank the anonymous referee's remarks which increase the content of the paper considerably.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This project has received funding from National Natural Science Foundation of China grant no. 11961006, NSF of Guangxi no. 2020GXNSFAA159100, International Cooperation Program of Chengdu City, 2020-GH02-00023-HZ, and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611.