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Displaying similar documents to “Kuratowski convergence on compacta and Hausdorff metric convergence on compacta”

Best approximations and porous sets

Simeon Reich, Alexander J. Zaslavski (2003)

Commentationes Mathematicae Universitatis Carolinae

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Let D be a nonempty compact subset of a Banach space X and denote by S ( X ) the family of all nonempty bounded closed convex subsets of X . We endow S ( X ) with the Hausdorff metric and show that there exists a set ℱ ⊂ S ( X ) such that its complement S ( X ) ∖ ℱ is σ -porous and such that for each A ∈ ℱ and each x ˜ ∈ D , the set of solutions of the best approximation problem ∥ x ˜ - z ∥ → min , z ∈ A , is nonempty and compact, and each minimizing sequence has a convergent subsequence.

A β -normal Tychonoff space which is not normal

Eva Murtinová (2002)

Commentationes Mathematicae Universitatis Carolinae

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α -normality and β -normality are properties generalizing normality of topological spaces. They consist in separating dense subsets of closed disjoint sets. We construct an example of a Tychonoff β -normal non-normal space and an example of a Hausdorff α -normal non-regular space.

Non-compact perturbations of m -accretive operators in general Banach spaces

Mieczysław Cichoń (1992)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we deal with the Cauchy problem for differential inclusions governed by m -accretive operators in general Banach spaces. We are interested in finding the sufficient conditions for the existence of integral solutions of the problem x ' ( t ) ∈ - A x ( t ) + f ( t , x ( t ) ) , x ( 0 ) = x 0 , where A is an m -accretive operator, and f is a continuous, but non-compact perturbation, satisfying some additional conditions.