On convergence of a sequence of parameterized closed convex sets and its applications (Q2704336)

The paper is concerned with the convergence of parametrized closed convex sets in the so-called Mosco sense and/or local gap sense.NEWLINENEWLINENEWLINEThe main results obtained are of abstract type and deal with the limits in the Mosco sense and/or local gap sense in reflexive Banach spaces of sequences of closed convex sets depending on two indexes. It is proved a result ensuring the independence on the order of the parameters in which the limits are taken.NEWLINENEWLINENEWLINEThen the abstract results are applied to some concrete cases of the closed convex sets. If for every \(n\in{\mathbf N}\) the closed convex set \({\mathbf K}_n\) is given by NEWLINE\[NEWLINE{\mathbf K}_n=\{v\in W : T_n(v-\psi)\in X_n\},NEWLINE\]NEWLINE where \(W\) is a Banach space, \(X_1,\ldots, X_n,\ldots\) are closed convex sets in a Banach space \(V\), \(T_1,\ldots, T_n,\ldots\) are continuous operators from \(W\) to \(V\), and if \(T_n\to T_\infty\) in \(L(W,V)\), and \(X_n\to X_\infty\) in the Mosco sense and/or local gap sense, then \({\mathbf K}_n\to{\mathbf K}_\infty\) in the Mosco sense and/or local gap sense.NEWLINENEWLINENEWLINETo prove such result, it is first assumed that each \(X_n\) contains a ball centred at the origin, and then the abstract result is exploited to get rid of this condition in some cases.NEWLINENEWLINENEWLINETwo applications are also given when \(T_n\) are differential and integrodifferential operators.