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Solution semicontinuity of parametric generalized vector equilibrium problems. (English) Zbl 1213.54028

The objective of a parametric generalized vector equilibrium problem (PGVEP) is defined as to find \(x\in A(\mu)\) such that
\[ F(x,y,\mu)\subset Y\backslash(-\text{int}\,C)\quad \forall y\in A(\mu) \]
Here \(F:\;B\times B\times\Lambda \rightarrow 2^Y\backslash\{\varnothing\}\) and \(A:\;\Lambda\rightarrow 2^X\backslash\{\varnothing\}\) are set-valued mappings, \(X\) and \(Y\) are Hausdorff topological vector spaces, \(Z\) is a real topological space, \(B\subset X\), \(\mu \in\Lambda\subset Z\), and \(C\subset Y\) is a convex cone with a nonempty interior. Let \(S(\mu)\) denote the set of all PGVEP solutions for a given \(\mu\). It is obtained that for \(S(\cdot)\) to be continuous on \(\Lambda\), it is sufficient that (1) the mapping \(A\) is compact-valued and continuous, (2) \(F\) is compact-valued and upper-semicontinuous, (3) \(F(\cdot,\cdot,\mu)\) is \(C\)-strictly monotone for each \(\mu\in\Lambda\), and (4) \(F(x,\cdot,\mu)\) is \(C\)-convexlike, that is for an arbitrary \(\rho\in[0,1]\) the inclusion
\[ \rho F(x,x_1,\mu)+(1-\rho) F(x,x_2,\mu)\subset F(x,x_3,\mu)+C \]
holds true for some \(x_3\in\Lambda\).

MSC:

54C60 Set-valued maps in general topology
26E25 Set-valued functions
47H04 Set-valued operators
58C06 Set-valued and function-space-valued mappings on manifolds
90C31 Sensitivity, stability, parametric optimization
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