Generalized differentiation and fixed points sets behaviors with respect to Fisher convergence (Q645409)

Stability of the (strict) \(H\)-differentiability of set-valued maps is discussed. The question is whether the limit of a sequence of differentiable set-valued mappings (in the sense of \textit{C. H. J. Pang}'s definition [Math. Oper. Res. 36, No. 3, 377--397 (2011; \url{doi:10.1287/moor.1110.0497})]) is differentiable? First, positively homogeneous mappings are studied. Then the authors propose a suitable topology from which the notion of convergence (so-called Fisher convergence) is derived. Given a converging sequence \(F_n\) of closed-valued mappings and a sequence \(H_n\) of homogeneous set-valued mappings, it is proved that under the assumption of uniform \(H_n\)-differentiability of the sequence \(F_n\), the Fisher limit \(F\) of \(F_n\) is (strictly) \(H\)-differentiable whenever \(H_n\) converges to a homogeneous mapping \(H\) with finite outer norm.