This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II which was published in 2002. In comparison, our present survey considers more restricted and specific areas of mathematics. We remark that we do not consider the theory of selectors (i.e. continuous choices of elements from subsets of topological spaces) since this topics is covered by another survey in this volume.
Let F 1 : X → Y 1 and F 2 : X → Y 2 be any convex-valued lower semicontinuous mappings and let L : Y 1 ⊕ Y 2 → Y be any linear surjection. The splitting problem is the problem of representation of any continuous selection f of the composite mapping L(F 1 ; F 2 ) in the form f = L(f 1 ; f 2 ), where f 1 and f 2 are some continuous selections of F 1 and F 2 , respectively. We prove that the splitting problem always admits an approximate solution with f i being an ε-selection (Theorem 2.1). We also propose a special case of finding exact splittings, whose occurrence is stable with respect to continuous variations of the data (Theorem 3.1) and we show that, in general, exact splittings do not exist even for the finite-dimensional range.
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