## Abstract

The purpose of this article is to establish some new results on the Painlevé–Kuratowski convergence of the solution sets for controlled systems of fuzzy vector quasi-optimization problems with a sequence of mappings \(\varGamma _C\)-converging. First, we introduce a new class of controlled systems for fuzzy vector quasi-optimization problems and establish some conditions for the existence of approximate solutions to these problems using the Kakutani–Fan–Glicksberg fixed-point theorem. Then, we study the Painlevé–Kuratowski lower convergence, Painlevé–Kuratowski upper convergence and Painlevé–Kuratowski convergence of the solution sets for such problems. Finally, as a real-world application, we consider the special case of controlled systems of fuzzy traffic network problems. Existence conditions and the Painlevé–Kuratowski convergence of the solution sets for these problems are also investigated and studied. The results presented in the paper are new and extend the main results given by some authors in the literature.

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## Acknowledgements

This research was supported by Ministry of Education and Training of Vietnam under grant number B2021.SPD.03. The authors are grateful to the editor and three anonymous referees for their valuable remarks which improved the results and presentation of this article.

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Communicated by Marcos Eduardo Valle.

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Hung, N.V., Keller, A.A. Painlevé–Kuratowski convergence of the solution sets for controlled systems of fuzzy vector quasi-optimization problems with application to controlling traffic networks under uncertainty.
*Comp. Appl. Math.* **40, **28 (2021). https://doi.org/10.1007/s40314-021-01415-8

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### Keywords

- Kakutani–Fan–Glicksberg fixed-point theorem
- Painlevé–Kuratowski convergence
- Control systems of fuzzy vector quasi-optimization problems
- Control systems of fuzzy traffic network problems
- Existence conditions

### Mathematics Subject Classification

- 90C29
- 47J20
- 49J40