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**Lower semicontinuity of the solution map to a parametric generalized variational inequality in reflexive Banach spaces.**
*(English)*
Zbl 1210.47088

In this paper, the authors discuss the stability of the solution set for the following parametric generalized variational inequality, denoted by \(GVI(F(\mu,\cdot),K(\lambda))\), which consists in finding \(x(\mu,\lambda)\) such that

\[ 0 \in F(\mu,x)+N_{K(\lambda)}(x), \]

where \(E\) is a normed space with the dual \(E^*\), \((M, d)\) and \((\Lambda, d)\) are metric spaces, \(F : M \times E \rightarrow 2^{E^*}\) is a set-valued operator, and \(K: \Lambda\rightarrow 2^E\) is a multifunction with closed convex values.

They prove that the solution map of \(GVI(F(\mu,\cdot),K(\lambda))\) is lower semicontinuous when the operator of the unperturbed problem is of the class \((S)_+\) and strictly monotone, the operators of the perturbed problems are pseudo-monotone and demicontinuous, and the perturbed constraint set \(K(\cdot)\) has the Aubin property. The obtained results are proved without conditions related to degree theory and the metric projection. Thus, they do not have to verify the conditions via the metric projection and computing of the degree of the normal map which were required by B.T.Kien and J.-C.Yao [Set-Valued Anal.16, No.4, 399–412 (2008; Zbl 1210.47089)].

\[ 0 \in F(\mu,x)+N_{K(\lambda)}(x), \]

where \(E\) is a normed space with the dual \(E^*\), \((M, d)\) and \((\Lambda, d)\) are metric spaces, \(F : M \times E \rightarrow 2^{E^*}\) is a set-valued operator, and \(K: \Lambda\rightarrow 2^E\) is a multifunction with closed convex values.

They prove that the solution map of \(GVI(F(\mu,\cdot),K(\lambda))\) is lower semicontinuous when the operator of the unperturbed problem is of the class \((S)_+\) and strictly monotone, the operators of the perturbed problems are pseudo-monotone and demicontinuous, and the perturbed constraint set \(K(\cdot)\) has the Aubin property. The obtained results are proved without conditions related to degree theory and the metric projection. Thus, they do not have to verify the conditions via the metric projection and computing of the degree of the normal map which were required by B.T.Kien and J.-C.Yao [Set-Valued Anal.16, No.4, 399–412 (2008; Zbl 1210.47089)].

Reviewer: Nan-Jing Huang (Chengdu)

### MSC:

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

49J40 | Variational inequalities |

49J53 | Set-valued and variational analysis |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

### Keywords:

parametric generalized variational inequality; lower semicontinuity; pseudo-monotone mapping; Aubin property### Citations:

Zbl 1210.47089
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\textit{B. T. Kien}, Set-Valued Anal. 16, No. 7--8, 1089--1105 (2008; Zbl 1210.47088)

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