Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 03, pp. 1-10.

Title: Duality arguments for well-posedness of history-dependent variational inequalities
       
Authors: Rong Hu (Univ. of Electronic Science and Tech., Chengdu, Sichuan, China)
         Mircea Sofonea (Univ. of Perpignan Via Domitia, Perpignan, France)

Abstract:
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 06, 2022.
Math Subject Classifications: 35M86, 49J40, 47J20.
Key Words: History-dependent variational inequality; dual problem;
           history-dependent inclusion; Tykhonov well-posedness; convergence results.