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Let be a complex Banach space, a norming set for , and a bounded, closed, and convex domain such that its norm closure is compact in . Let lie strictly inside . We study convergence properties of infinite products of those self-mappings of which can be extended to holomorphic self-mappings of . Endowing the space of sequences of such mappings with an appropriate metric, we show that the subset consisting of all the sequences with divergent infinite products is -porous.
The goal of this paper is to provide an overview of results concerning, roughly speaking, the following issue: given a (topologized) class of minimum problems, “how many” of them are well-posed? We will consider several ways to define the concept of “how many,” and also several types of well-posedness concepts. We will concentrate our attention on results related to uniform convergence on bounded sets, or similar convergence notions, as far as the topology on the class of functions under investigation is concerned.
It is known that every subset of the plane containing a dense set of lines, even if it has measure zero, has the property that every real-valued Lipschitz function on has a point of differentiability in . Here we show that the set of points of differentiability of Lipschitz functions inside such sets may be surprisingly tiny: we construct a set containing a dense set of lines for which there is a pair of real-valued Lipschitz functions on having no common point of differentiability in , and there is a real-valued Lipschitz function on whose set of points of differentiability in is uniformly purely unrectifiable.
The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands which satisfy convexity and growth conditions. In 1996, the author obtained a generic existence and uniqueness result (with respect to variations of the integrand of the integral functional) without the convexity condition for a class of optimal control problems satisfying the Cesari growth condition. In this paper, we survey this result and its recent extensions, and establish several new results in this direction.
For a nonempty separable convex subset of a Hilbert space , it is typical (in the sense of Baire category) that a bounded closed convex set defines an -valued metric antiprojection (farthest point mapping) at the points of a dense subset of , whenever is a positive integer such that .