Infinite products of relations, set-valued series and uniform openness of multifunctions
The authors, using the notations of convergent series of sets and convergent products of relations, present general open mapping theorems. Let \(X\) be a topological vector space. A series of nonempty subsets of \(X\), \(\sum X_n\), is said to be Kuratowski (resp. Mosco) convergent if the sequence of partial sums \(S_n= X_0+ \dots+ X_n\) converges in the sense of Kuratowski (resp. Mosco). The uniform structure of \(X\) leads the authors to the use of hemiconvergence. \(\sum X_n\) is said to be hemiconvergent with sum \(S\) if the sequence \((S_n)\) hemiconverges to \(S\): \[ \forall U\in N_X (0)\;\exists n_U\in N:\;\forall n\geq n_U,\;S_n\subset S+U,\;S\subset S_n+U. \] \(N_X (0)\) denotes the family of neighborhoods of 0. \(\sum X_n\) is called selectionwise convergent (in short convergent) if for any selection \((x_n)\) of \((X_n)\) the series \(\sum x_n\) is convergent. \(\sum X_n\) is a Cauchy series if the following condition is satisfied: \[ \forall V\in N_X (0)\;\exists n_V\in N:\;p\geq n\geq n_V \Rightarrow X_n+ \dots+ X_p \subset V. \] It is proved that any convergent series is a Cauchy series. If the space \(X\) is sequentially complete the converse holds. It is also proved that any convergent series is hemiconvergent. Also a number of results valid for metric spaces extend to the framework of uniform spaces which is adapted to uniform notions which also appers when one deals with topological vector spaces or, more generally, with topological groups. To achieve this, the authors give the following definition: Given a sequence \((R_n )_{n\geq 0}\) of relations of a topological space \(X\) we say that the product \(\prod_{n\geq 0} R_n\) is pointwise convergent if the sequence \((x_n )_{n\geq 0}\) of \(X\) such that \(x_{n+1}\in R_n (x_n)\) for each \(n\geq 0\) is convergent. Then, the product relation \(\prod_{n\geq 0} R_n\) is given by: for \(x\in X\) the image \(R(x)= \prod_{n\geq 0} R_n (x)\) is the set of limits of such sequences with \(x_0 =x\). When \(X\) is a quasi-uniform space, the product \(\prod_{n\geq 0} R_n\) is said to be convergent if it is pointwise convergent and if for any entourage \(U\) of \(X\) there exists \(n_U\in \mathbb{N}\) such that \(\prod_{k\geq n} R_k \subset U\) for any \(n\geq n_U\). Then, the following interesting proposition is proved: If \(X\) is a topological vector space (or topological additive group), \((X_n)\) is a sequence of nonempty subsets of \(X\) and \(R_n= \{(x, x+u): x\in X\), \(u\in X_n\}\), then the product \(\prod R_n\) is convergent iff the series \(\sum X_n\) is convergent. The following definition is a useful extension of a notion given in metric spaces [the second author, Nonlinear Anal., Theory Methods Appl. 13, No. 6, 629-643 (1989; Zbl 0687.54015)]. Let \((X, U_X)\), \((Y,U_Y)\) be quasi-uniform spaces. A multifunction \(F: X\to Y\) is said to be uniformly open over \(C\subset F\) if for each \(U\in U_X\) there exists \(V\in U_Y\) such that for any \(c= (a, b)\in C\) we have \(V (b) \subset F(U (a))\). For \(C=F\), \(F\) is said to be uniformly open. The main theorem of this paper is the following: Let \((X, U_X)\), \((Y, U_Y)\), be quasi-uniform spaces and let \(F: X\to Y\) be a sequentially closed multifunction. Let \((R_n)\), \((S_n)\) be sequences of relations on \(X\) and \(Y\) respectively such that (a) for any \((x, y)\in F\), \(n\in \mathbb{N}\), \(S_n (y) \subset S_{n+1} (F(R_n (x)))\); (b) the product \(\prod_n R_n\) is convergent and each \(R_n\) is reflexive; (c) the family of relations \((S_n^{-1})\) convergences pointwise to the identity mapping of \(Y\): for each \(y\in Y\) and each sequence \((y_n)\) such that \(y_n\in S_n^{-1} (y)\) for each \(n\) one has \((y_n) \to y\); (d) the set of \(i\in \mathbb{N}\) such that \(S_i\in U_Y\) is infinite. Then \(F\) is uniformly open. Also some corollaries of the above theorem concerning metric spaces are proved in this paper.
scientific article; zbMATH DE number 773786
scientific article