Well-posedness of a class generalized split quasi-inverse tensor variational inequalities

Year 2025, Volume: 54 Issue: 6, 2168 - 2181, 30.12.2025

Qing Nie , Vo Minh Tam , Boling Chen

https://doi.org/10.15672/hujms.1605434

Abstract

This paper aims to study a generalized split quasi-inverse tensor variational inequality (GSQITVI) in tensor spaces. Building on the concept of well-posedness, we establish several metric-based features that provide necessary and sufficient conditions for the well-posedness of the GSQITVI. By utilizing the measure of non-compactness and the correlation theorem, we also derive results concerning the well-posedness of the problem. These findings emphasize the key properties of the GSQITVI and offer an analysis of the convergence of its solutions.

Keywords

Generalized split quasi-inverse tensor variational inequality , well-posedness , measure of non-compactness , convergence analysis

References

• [1] F. Anceschi, A. Barbagallo and S. G. Lo Bianco, Inverse tensor variational inequalities and applications, J. Optim. Theory Appl. 196, 570–589, 2023.
• [2] A. Barbagallo and P. Mauro, Inverse variational inequality approach and applications, Numer. Funct. Anal. Optim. 35, 851–867, 2014.
• [3] L. Cao, H. Kong and S. D. Zeng, On the well-posedness of the generalized split quasiinverse variational inequalities, J. Nonlinear Sci. Appl. 9, 5497–5509, 2016.
• [4] G. P. Crespi, A. Guerraggio and M. Rocca, Well posedness in vector optimization problems and vector variational inequalities, J. Optim. Theory Appl. 132, 213–226, 2007.
• [5] H. He, C. Ling and H. K. Xu, A relaxed projection method for split variational inequalities, J. Optim. Theory Appl. 166, 213–233, 2015.
• [6] R. Hu and Y. P. Fang, Levitin–Polyak well-posedness by perturbations of inverse variational inequalities, Optim. Lett. 7, 343–359, 2013.
• [7] R. Hu and Y. P. Fang, Well-posedness of the split inverse variational inequality problem, Bull. Malays. Math. Sci. Soc. 40, 1733–1744, 2017.
• [8] N. V. Hung, Generalized Levitin–Polyak well-posedness for controlled systems of FMQHI-fuzzy mixed quasi-hemivariational inequalities of Minty type, J. Comput. Appl. Math. 386, 113263, 2021.
• [9] N. V. Hung, V. M. Tam and D. Baleanu, Regularized gap functions and error bounds for split mixed vector quasivariational inequality problems, Math. Methods Appl. Sci. 43, 4614–4626, 2020.
• [10] N. V. Hung, V. M. Tam and D. O’Regan, Error bound analysis for split weak vector mixed quasi-variational inequality problems in fuzzy environment, Appl. Anal. 102, 1874–1888, 2023.
• [11] C. Kanzow and D. Steck, Quasi-variational inequalities in Banach spaces: theory and augmented Lagrangian methods, SIAM J. Optim. 29, 3174–3200, 2019.
• [12] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, SIAM, 2000.
• [13] K. Kuratowski, Topology, New edition, revised and augmented, Academic Press, 1966.
• [14] M. B. Lignola, Well-posedness and L-well-posedness for quasivariational inequalities, J. Optim. Theory Appl. 128, 119–138, 2006.
• [15] M. B. Lignola and J. Morgan, Vector quasi-variational inequalities: approximate solutions and well-posedness, J. Convex Anal. 13, 373, 2006.
• [16] M. A. Noor, General variational inequalities, Appl. Math. Lett. 1, 119–122, 1988.
• [17] M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152, 199–277, 2004.
• [18] M. A. Noor, Differentiable non-convex functions and general variational inequalities, Appl. Math. Comput. 199, 623–630, 2008.
• [19] V. M. Tam, N. V. Hung, Z. Liu and J. C. Yao, Levitin–Polyak well-posedness by perturbations for the split hemivariational inequality problem on Hadamard manifolds, J. Optim. Theory Appl. 195, 684–706, 2022.
• [20] Y. Wang, Z. H. Huang and L. Qi, Global uniqueness and solvability of tensor variational inequalities, J. Optim. Theory Appl. 177, 137–152, 2018.