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Convergence of solutions to history‐dependent variational‐hemivariational inequalities
Author(s) -
Xiao Yibin,
Sofonea Mircea
Publication year - 2019
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201800292
Subject(s) - convergence (economics) , mathematics , inequality , mathematical analysis , economics , economic growth
In this paper, we state and prove two convergence results for the solution of a history‐dependent variational‐hemivariational inequality. The first one concerns the pointwise convergence of the solution with respect to the data, including the set of constraints, the nonlinear operator and the two functionals which govern the inequality. The second result, obtained under additional assumptions, concerns the uniform convergence of the solution with respect to the set of constraints. These convergence results allow us to consider two general optimization problems for which we prove the existence of minimizers. The mathematical tools developed in this paper are useful in the analysis and control of a large class of boundary value problems which, in a weak formulation, lead to history‐dependent variational‐hemivariational inequalities. To provide an example, we illustrate our results in the study of a nonlinear problem with unilateral constraints and subdifferential boundary conditions.