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Convergence in compacta and linear Lindelöfness

Aleksander V. Arhangel'skii, Raushan Z. Buzyakova (1998)

Commentationes Mathematicae Universitatis Carolinae

Let X be a compact Hausdorff space with a point x such that X { x } is linearly Lindelöf. Is then X first countable at x ? What if this is true for every x in X ? We consider these and some related questions, and obtain partial answers; in particular, we prove that the answer to the second question is “yes” when X is, in addition, ω -monolithic. We also prove that if X is compact, Hausdorff, and X { x } is strongly discretely Lindelöf, for every x in X , then X is first countable. An example of linearly Lindelöf...

Convergence of conditional expectations for unbounded closed convex random sets

Charles Castaing, Fatima Ezzaki, Christian Hess (1997)

Studia Mathematica

We discuss here several types of convergence of conditional expectations for unbounded closed convex random sets of the form E n X n where ( n ) is a decreasing sequence of sub-σ-algebras and ( X n ) is a sequence of closed convex random sets in a separable Banach space.

Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point

K. P. R. Sastry, G. V. R. Babu (2005)

Czechoslovak Mathematical Journal

Existence of fixed points of multivalued mappings that satisfy a certain contractive condition was proved by N. Mizoguchi and W. Takahashi. An alternative proof of this theorem was given by Peter Z. Daffer and H. Kaneko. In the present paper, we give a simple proof of that theorem. Also, we define Mann and Ishikawa iterates for a multivalued map T with a fixed point p and prove that these iterates converge to a fixed point q of T under certain conditions. This fixed point q may be different from...

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