Journal of Global Optimization

, Volume 39, Issue 1, pp 101–111 | Cite as

On the solution stability of variational inequalities

  • B. T. KienEmail author
  • M. -M. Wong
Original Paper


In the present paper, we will study the solution stability of parametric variational conditions
$${{0 \in f(\mu, x)+ N_{K(\lambda)}(x)},}$$
where M and Λ are topological spaces, \({f : M \times R^n \to R^n}\) is a function, \({K : \Lambda\to 2^{R^n}}\) is a multifunction and N K(λ)(x) is the value at x of the normal cone operator associated with the set K(λ). By using the degree theory and the natural map we show that under certain conditions, the solution map of the problem is lower semicontinuous with respect to parameters (μ,λ). Our results are different versions of Robinson’s results [15] and proved directly without the homeomorphic result between the solution sets.


Solution stability Parametric variational conditions Variational inequality Degree theory Lower semicontinuity 


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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of applied MathematicNational Sun Yat-Sen UniversityKaohsiungTaiwan, Republic of China
  2. 2.Department of information TechnologyMeiho Institute of TechnologyPintaungTaiwan

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