## Abstract

In this paper, two new existence theorems of solutions to inverse variational and quasi-variational inequality problems are proved using the Fan-Knaster-Kuratowski-Mazurkiewicz (KKM) theorem and the Kakutani-Fan-Glicksberg fixed point theorem. Upper semicontinuity and lower semicontinuity of the solution mapping and the approximate solution mapping to the parametric inverse variational inequality problem are also discussed under some suitable conditions. An application to a road pricing problem is given.

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## Acknowledgements

The authors are grateful to the editor and the referees for their valuable comments and suggestions.

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## Additional information

Project supported by the National Natural Science Foundation of China (No. 11671282), the Joint Foundation of the Ministry of Education of China and China Mobile Communication Corporation (No.MCM20150505), the China Postdoctoral Science Foundation (No. 2015T80967), the Applied Basic Project of Sichuan Province (No. 2016JY0170), the Open Foundation of State Key Laboratory of Electronic Thin Films and Integrated Devices (No.KFJJ201611), the Key Program of Education Department of Sichuan Province (No. 16ZA0007), and the Fundamental Research Funds for the Central Universities (No. ZYGX2015J098)

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### Cite this article

Han, Y., Huang, N., Lu, J. *et al.* Existence and stability of solutions to inverse variational inequality problems.
*Appl. Math. Mech.-Engl. Ed.* **38, **749–764 (2017) doi:10.1007/s10483-017-2191-9

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### Keywords

- inverse variational inequality
- Fan-Knaster-Kuratowski-Mazurkiewicz (KKM) theorem
- Kakutani-Fan-Glicksberg fixed point theorem
- upper semicontinuity
- lower semicontinuity

### Chinese Library Classification

- O224
- O177

### 2010 Mathematics Subject Classification

- 49J40
- 47J20