Finally, error bounds for the HDVI controlled by the regularized gap function and the $\mathcal{D}$-gap function are derived under suitable conditions.
📌 3 Key Takeaways
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Objective & Hypothesis:In the present paper, we are concerned with investigating error bounds for history-dependent variational inequalities controlled by the difference gap (for brevity, $\mathcal{D}$-gap) functions.
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Key Findings & Data:First, we recall a class of elliptic variational inequalities involving the history-dependent operators (for brevity, HDVI).
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Conclusion & Impact:Finally, error bounds for the HDVI controlled by the regularized gap function and the $\mathcal{D}$-gap function are derived under suitable conditions.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:History-dependent operator · Gap function · Error bound
Abstract
In the present paper, we are concerned with investigating error bounds for history-dependent variational inequalities controlled by the difference gap (for brevity, $\mathcal{D}$-gap) functions. First, we recall a class of elliptic variational inequalities involving the history-dependent operators (for brevity, HDVI). Then, we introduce a new concept of gap functions to the HDVI and propose the regularized gap function for the HDVI via the optimality condition for the concerning minimization problem. Consequently, the $\mathcal{D}$-gap function for the HDVI depends on these regularized gap functions is established. Finally, error bounds for the HDVI controlled by the regularized gap function and the $\mathcal{D}$-gap function are derived under suitable conditions.
CHEN, B., TAM, V. M. (2023). Error bounds for a class of history-dependent variational inequalities controlled by $\mathcal{D}$-gap~functions. Hacettepe Journal of Mathematics and Statistics, Cilt: 52 Sayı: 6 - Special Issue: Nonlinear Evolution Problems with Applications, 1550–1566. https://ajindex.com/articles/971785
G. Bigi and M. Passacantando, D-gap functions and descent techniques for solving equilibrium problems, J. Global Optim. 62 (1), 183–203, 2015
J.X. Cen, A.A. Khan, D. Motreanu and S.D. Zeng, Inverse problems for generalized quasi-variational inequalities with application to elliptic mixed boundary value systems, Inverse Problems 38, 065006, 2022
J.X. Cen, V.T. Nguyen and S.D. Zeng, Gap functions and global error bounds for history-dependent variational-hemivariational inequalities, J. Nonlinear Var. Anal. 6, 461–481, 2022
C. Charitha, A note on D-gap functions for equilibrium problems, Optimization, 62 (2), 211–226, 2013
Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003
Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003
M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Program. 53 (4), 99–110, 1992
J. Haslinger, M. Miettinen and P.D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications. Kluwer Academic Publishers, Boston, 1999
N.V. Hung, S. Migórski, V.M. Tam and S. Zeng, Gap functions and error bounds for variational-hemivariational inequalities, Acta. Appl. Math. 169, 691–709, 2020
N.V. Hung and V.M. Tam, Error bound analysis of the D-gap functions for a class of elliptic variational inequalities with applications to frictional contact mechanics, Z. Angew. Math. Phys. 72, 173, 2021
N.V. Hung, V.M. Tam and B. Dumitru, Regularized gap functions and error bounds for split mixed vector quasivariational inequality problems, Math. Methods Appl. Sci. 43, 4614–4626, 2020
N.V. Hung, V.M. Tam and Y. Zhou, A new class of strong mixed vector GQVIP- generalized quasi-variational inequality problems in fuzzy environment with regularized gap functions based error bounds, J Comput Appl Math. 381, 113055, 2021
N.V. Hung, X. Qin, V.M. Tam and J.C. Yao, Difference gap functions and global error bounds for random mixed equilibrium problems, Filomat 34, 2739–2761, 2020
I.V. Konnov and O.V. Pinyagina, D-gap functions for a class of equilibrium problems in Banach spaces, Comput. Methods Appl. Math. 3 (2), 274–286, 2003. [1