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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 $X(\cdot,\omega)$ and $F(\cdot,\omega)$ both be point-to-set mappings from the set $\Gamma\subset\bbfR\sp n$ into the subsets of $\bbfR\sp n$, where $\omega$ is a vector parameter in $W\subset\bbfR\sp r$. For each $\omega$, the problem GQVI$(X,F)$ is to find vectors $x\sp*\in X(x\sp*,\omega)$ and $y\sp*\in F(x\sp*,\omega)$ such that $\langle x- x\sp*, y\sp*\rangle \geq 0$ for any $x\in X(x\sp*,\omega)$. The purpose of the paper is to show upper semicontinuity of the solution set $I(\omega)$ of the above GQVI$(X,F)$. It is to show that, if $S$ is any open set with $I(\omega)\subset S$, then there exists $\delta >0$ such that, for all $\Delta\omega\in B(\omega,\delta)$, we have $I(\omega+\Delta\omega)\subset S$. A basic assumption is that $X(x,\omega)\neq \emptyset$, $X(x,\omega)\subset\Gamma$, and $F(x,\omega)\neq \emptyset$ 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 $\omega$.

49J40Variational methods including variational inequalities
49J45Optimal control problems involving semicontinuity and convergence; relaxation
49K40Sensitivity, stability, well-posedness of optimal solutions
26E25Set-valued real functions
Full Text: DOI
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