Stable approximations of set-valued maps
The aim of this paper is an extension of the Lax' principle: ``stability and consistency implies convergence'' to the case of general inclusion \(y_ 0\in F(x_ 0)\) where F is a multifunction from a Banach space X to a Banach space Y, \(y_ 0\in Y\). If \(X_ n\), \(Y_ n\) are some approximations of the space X and Y respectively, \(y_ n\in Y_ n\) approximates \(y_ 0\) and if \(F_ n\) are ``consistent with F'', an adequate stability property of the sequence \(\{F_ n\}\) implies the convergence of some sequence \(\{x_ n\}\) of solutions of the inclusions \(y_ n\in F(x_ n)\), \(n=1,2,..\). to \(x_ 0\). This stability is based on the properties of contingent derivative of multifunctions. An extension of the Banach-Steinhaus theorem to the set-valued analogues of linear, continuous operators is also proved.
scientific article; zbMATH DE number 4115509
scientific article