Continuous selections as uniform limits of \(\delta\)-continuous \(\varepsilon\)-selections
The main result of the paper, Theorem C, gives sufficient conditions for the existence of continuous selections for a closed-valued, lower semicontinuous multifunction \(F\) from a paracompact space \(X\) into a Banach space \(Y\). The values of \(F\) need not be convex, but there are some assumptions on functions of nonconvexity of the sets \(F(x)\), \(x\in X\smallsetminus Z\), where \(Z\) is a zero-dimensional subset of the domain. This is a generalization of a nonconvex-valued selection theorem of the second author [Math. Notes 54, No. 6, 1236-1240 (1993); translation from Mat. Zametki 54, No. 6, 74-81 (1993; Zbl 0853.46008)]. In the proof the authors combine the methods of outside and inside approximations; the required continuous selection is constructed as the uniform limit of \(\delta\)-continuous \(\varepsilon\)-selections. Next they show that the main theorem holds, when the values \(F(x)\) are nonclosed for \(x\) from a countable subset of the domain. This result unifies many types of selection theorems; convex-valued, zero-dimensional, paraconvex-valued and countable selection theorems are special cases. Finally, the authors prove that if \(Y\) is separable, then under similar assumptions there exists a Michael representation for \(F\), i.e. a countable family \(\{f_w:w\in W\}\) of continuous selections of \(F\) such that the set \(\{f_w(x): w\in W\}\) is dense in \(F(x)\) for each \(x\in X\).
lower semicontinuous multivalued mapping
scientific article; zbMATH DE number 1418153