Electron. J. Differential Equations, Vol. 2022 (2022), No. 03, pp. 1-10.

Duality arguments for well-posedness of history-dependent variational inequalities

Rong Hu, Mircea Sofonea

In this article we introduce a concept of dual problems in metric spaces. Then we state and prove an equivalence result concerning their well-posedness with respect to appropriate Tykhonov triples. We exemplify this result in the study of a history-dependent variational inequality with time-dependent constraints, for which the dual problem is in a form of a history-dependent inclusion. This allows us to deduce a convergence result which provides the continuous dependence of the solution with respect to the data. We end this paper with an example which represents an evidence of our abstract results.

Submitted August 19, 2021. Published January 6, 2022.
Math Subject Classifications: 35M86, 49J40, 47J20.
Key Words: History-dependent variational inequality; dual problem; history-dependent inclusion; Tykhonov well-posedness; convergence results.
DOI: https://doi.org/10.58997/ejde.2022.03

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Rong Hu
School of Mathematical Sciences
University of Electronic Science and Technology of China
Chengdu, Sichuan 611731, China
email: hrong1130@foxmail.com
Mircea Sofonea
Laboratoire de Math\'ematiques et Physique
University of Perpignan Via Domitia
52 Avenue Paul Alduy, 66860 Perpignan, France
email: sofonea@univ-perp.fr

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