Convergences for sequences of sets and linear mappings
Let \(E\) be a normed vector space. The authors work with several convergences of sequences of subsets of \(E\) such as \(X=K\)-\(\lim X_n\) and \(X=\text{AW-}\lim X_n\). These convergences are defined as follows: For a sequence \((X_n)\) of elements of \(E\) write \((x_n)\subset(X_n)\) if \(x_n\in X_n\) except for finitely many \(n\). The set \(\liminf X_n\) denotes the set of those \(x\in X\) for which \(x=\lim x_n\) for some sequence \((x_n)\subset(X_n)\), and \(\limsup X_n\) the set of those \(x\in E\) for which \(x=\lim x_{n_k}\) for some sequence \((x_{n_k})\subset(X_{n_k})\). Write \(X=K\)-\(\lim X_n\) if \(\liminf X_n= \limsup X_n=X\) and \(X=\text{AW-}\lim X_n\) if \(\limsup_{u\in B}|d(u,X_n)- d(u,X)|= 0\) for every non-empty bounded subset \(B\) of \(X\). (The letter \(K\) refers to Kurotowski and AW to Attouch and Wets.) The authors are interested in the following question: Let \(E\) and \(F\) be real normed vector spaces and let \(A_n:E\to F\) be continuous linear operators converging (weakly, strongly, uniformly) to a continuous linear operator \(A:E\to F\). Suppose \(X_n\subset E\) and that \((X_n)\) converges to \(X\subset E\) (in one of the two convergences introduced above or in some of the other convergences considered) or that \(Y_n\subset F\) and \((Y_n)\) converges to \(Y\subset F\). Does \(A_n(X_n)\) converge to \(A(X)\) or \(A^{-1}_n(Y_n)\) converge to \(A^{-1}(Y)\)? The authors generalize results of \textit{I. E. Schochetman} and \textit{R. L. Smith} [J. Math. Anal. Appl. 166, No. 1, 112-128 (1992; Zbl 0767.47008)] for Hilbert spaces and the case where the sets \(Y_n\) and \(Y\) are singletons to general normed spaces and arbitrary sets. They also change \(K\)-convergence to AW-convergence. Among results obtained are the following: 1) Suppose \(x_n\to x\) implies \(A_nx_n\to Ax\), that \(|A_nx|\geq\alpha|x|\) for some \(\alpha>0\) and all \(x\in E\) and that \(\limsup A_n(E)\subset A(E)\). Then \(X=K\)-\(\lim X_n\) implies that \(A(X)= K\)-\(\lim A_n(X_n)\). 2) Suppose \(A_n\to A\) uniformly, \(A_n(U_E)\supset\alpha U_F\) and \(A(U_E)\supset\alpha U_F\). (\(U\) is the closed unit ball.) Then \(Y=\text{AW-}\lim Y_n\) implies \(A^{-1}(Y)=\text{AW-}\lim A^{-1}_n(Y_n)\). The results are then applied to sets of solutions of optimization problems.
scientific article; zbMATH DE number 710344