Variational Analysis and Set Optimization

Using now the Aubin property of F_i around (a,b_i) (with constant L_Fi > 0) with $i\in \overline{2,k}$ one gets that for a neighborhood V of b_i and for every n sufficiently large

$b_i+t_n\left({\beta }_n^i+b_n^i\right)\in F_i\left(a+t_n\left({\alpha }_n^i+a_n\right)\right)\cap V\subset F_i\left(a+t_n\left({\alpha }_n+a_n\right)\right)+L_{F_i}{t}_n\Vert {\alpha }_n^i-{\alpha }_n\Vert D_Y,$

for all $i\in \overline{2,k}$

In conclusion, for every $i\in \overline{2,k}$, there exists ( $f_n^i$) ⊂ D_y such that ${\lambda }_n^i={\beta }_n^i+L_{F_i}\Vert {\alpha }_n^i-{\alpha }_n\Vert f_n^i\to 0$ and

$b_i+t_n\left(b_n^i+{\lambda }_n^i\right)\in F_i\left(a+t_n\left({\alpha }_n+a_n\right)\right).$

Since ${\displaystyle \sum _{i=1}^p{b}_n^i}\to w$, using (8.25), it follows that all the sequences (a_n),( $a_n^i$),( $b_n^j$) are bounded with $i\in \overline{k+1,p}$, $j\in \overline{1,p}$. Hence, for every n sufficiently large,

$t_n\left(\left({\alpha }_n,{\lambda }_n^1,{\lambda }_n^2,\mathrm{\dots },{\lambda }_n^k,{\alpha }_n^{k+1},{\lambda }_n^{k+1},\mathrm{\dots },{\alpha }_n^p,{\lambda }_n^p\right)+\left(a_n,b_n^1,b_n^2,\mathrm{\dots },b_n^k,a_n^{k+1},\mathrm{\dots },a_n^p,b_n^p\right)\right)+\left(a,b_1,b_2,\mathrm{\dots },b_k,a_{k+1},b_{k+1},\mathrm{\dots },a_p,b_p\right)\in \left[\text{Gr}\left(F_1,F_2,\mathrm{\dots },F_k\right)\times {\displaystyle \prod _{i=k+1}^p\text{Gr}{F}_i}\right]\cap W,$

(8.26)

as the above sequence converges ...