Stability results for Ekeland's \(\varepsilon\) variational principle for vector valued functions (Q1298736)

Under the assumption that the nonconvex vector valued function \(f\) satisfies some lower semicontinuity property and is bounded below, the nonconvex vector valued function sequence \(f_n\) satisfies the same lower semicontinuity property and is uniformly bounded below, and \(f_n\) converges to \(f\) in the generalized sense of Mosco, the authors obtain the relation: \[ \sqrt\varepsilon- \text{ext }f= \{\overline x: f(x)- f(\overline x)+ \sqrt\varepsilon\|x-\overline x\|e\not\in -C,\text{ when }x\neq \overline x\}\subseteq \varliminf_{n\to\infty} \sqrt\varepsilon- \text{ext }f_n, \] where \[ \sqrt\varepsilon- \text{ext }f_n= \{\overline x: f_n(x)- f_n(\overline x)+ \sqrt\varepsilon\|x-\overline x\|e\not\in -C,\text{ when }x\neq \overline x\}, \] \(C\) is the pointed closed convex dominating cone with nonempty interior \(\text{int }C\), \(e\in \text{int }C\). Under some conditions, the authors also prove the same result if \(f_n\) converges to \(f\) in the generalized sense of Painlevé-Kuratowski.