# Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems

## Abstract

In this paper, we consider vector quasi-equilibrium problems under perturbation in terms of suitable asymptotically solving sequences, not embedding given problems into a parameterized family. By employing some types of convergences for mapping and set sequences, we obtain the Painlevé–Kuratowski upper convergence of solution sets for the reference problems. Then, using nonlinear scalarization functions, we propose gap functions for such problems, and later employing these functions, we study necessary and sufficient conditions for Painlevé–Kuratowski lower convergence and Painlevé–Kuratowski convergence. As an application, we discuss the special case of vector quasi-variational inequality.

- Authors:

- Can Tho University, Department of Mathematics, Teacher College (Viet Nam)
- Valaya Alongkorn Rajabhat University under the Royal Patronage (Thailand)
- Ton Duc Thang University, Department for Management of Science and Technology Development (Viet Nam)
- Dong Thap University, Department of Mathematics (Viet Nam)
- Naresuan University, Department of Mathematics, Faculty of Science (Thailand)

- Publication Date:

- OSTI Identifier:
- 22769255

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 3; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; CONVERGENCE; FUNCTIONS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; PERTURBATION THEORY; VARIATIONAL METHODS; VECTORS

### Citation Formats

```
Anh, Lam Quoc, E-mail: quocanh@ctu.edu.vn, Bantaojai, Thanatporn, Hung, Nguyen Van, E-mail: nguyenvanhung2@tdt.edu.vn, Tam, Vo Minh, E-mail: vmtam@dthu.edu.vn, and Wangkeeree, Rabian.
```*Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems*. United States: N. p., 2018.
Web. doi:10.1007/S40314-017-0548-4.

```
Anh, Lam Quoc, E-mail: quocanh@ctu.edu.vn, Bantaojai, Thanatporn, Hung, Nguyen Van, E-mail: nguyenvanhung2@tdt.edu.vn, Tam, Vo Minh, E-mail: vmtam@dthu.edu.vn, & Wangkeeree, Rabian.
```*Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems*. United States. doi:10.1007/S40314-017-0548-4.

```
Anh, Lam Quoc, E-mail: quocanh@ctu.edu.vn, Bantaojai, Thanatporn, Hung, Nguyen Van, E-mail: nguyenvanhung2@tdt.edu.vn, Tam, Vo Minh, E-mail: vmtam@dthu.edu.vn, and Wangkeeree, Rabian. Sun .
"Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems". United States. doi:10.1007/S40314-017-0548-4.
```

```
@article{osti_22769255,
```

title = {Painlevé–Kuratowski convergences of the solution sets for generalized vector quasi-equilibrium problems},

author = {Anh, Lam Quoc, E-mail: quocanh@ctu.edu.vn and Bantaojai, Thanatporn and Hung, Nguyen Van, E-mail: nguyenvanhung2@tdt.edu.vn and Tam, Vo Minh, E-mail: vmtam@dthu.edu.vn and Wangkeeree, Rabian},

abstractNote = {In this paper, we consider vector quasi-equilibrium problems under perturbation in terms of suitable asymptotically solving sequences, not embedding given problems into a parameterized family. By employing some types of convergences for mapping and set sequences, we obtain the Painlevé–Kuratowski upper convergence of solution sets for the reference problems. Then, using nonlinear scalarization functions, we propose gap functions for such problems, and later employing these functions, we study necessary and sufficient conditions for Painlevé–Kuratowski lower convergence and Painlevé–Kuratowski convergence. As an application, we discuss the special case of vector quasi-variational inequality.},

doi = {10.1007/S40314-017-0548-4},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 3,

volume = 37,

place = {United States},

year = {2018},

month = {7}

}

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