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Global stability result for the generalized quasivariational inequality problem. (English) Zbl 0737.49010
The paper studies some stability property for the generalized quasivariational problem (GQVI). Let $X\left(·,\omega \right)$ and $F\left(·,\omega \right)$ both be point-to-set mappings from the set ${\Gamma }\subset {ℝ}^{n}$ into the subsets of ${ℝ}^{n}$, where $\omega$ is a vector parameter in $W\subset {ℝ}^{r}$. For each $\omega$, the problem GQVI$\left(X,F\right)$ is to find vectors ${x}^{*}\in X\left({x}^{*},\omega \right)$ and ${y}^{*}\in F\left({x}^{*},\omega \right)$ such that $〈x-{x}^{*},{y}^{*}〉\ge 0$ for any $x\in X\left({x}^{*},\omega \right)$. The purpose of the paper is to show upper semicontinuity of the solution set $I\left(\omega \right)$ of the above GQVI$\left(X,F\right)$. It is to show that, if $S$ is any open set with $I\left(\omega \right)\subset S$, then there exists $\delta >0$ such that, for all ${\Delta }\omega \in B\left(\omega ,\delta \right)$, we have $I\left(\omega +{\Delta }\omega \right)\subset S$. A basic assumption is that $X\left(x,\omega \right)\ne \varnothing$, $X\left(x,\omega \right)\subset {\Gamma }$, and $F\left(x,\omega \right)\ne \varnothing$ for all $x$ and $\omega$. The above assertion is obtained under additional assumptions on compactness of the domain set and some semicontinuity properties of $X$ and $F$ relative to $x$ and $\omega$.

##### MSC:
 49J40 Variational methods including variational inequalities 49J45 Optimal control problems involving semicontinuity and convergence; relaxation 49K40 Sensitivity, stability, well-posedness of optimal solutions 26E25 Set-valued real functions
##### References:
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