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Transfer positive hemicontinuity and zeros, coincidences, and fixed points of maps in topological vector spaces
Fixed Point Theory and Applications volume 2005, Article number: 459128 (2005)
Abstract
Let be a real Hausdorff topological vector space. In the present paper, the concepts of the transfer positive hemicontinuity and strictly transfer positive hemicontinuity of set-valued maps in
are introduced (condition of strictly transfer positive hemicontinuity is stronger than that of transfer positive hemicontinuity) and for maps
and
defined on a nonempty compact convex subset
of
, we describe how some ideas of K. Fan have been used to prove several new, and rather general, conditions (in which transfer positive hemicontinuity plays an important role) that a single-valued map
has a zero, and, at the same time, we give various characterizations of the class of those pairs
and maps
that possess coincidences and fixed points, respectively. Transfer positive hemicontinuity and strictly transfer positive hemicontinuity generalize the famous Fan upper demicontinuity which generalizes upper semicontinuity. Furthermore, a new type of continuity defined here essentially generalizes upper hemicontinuity (the condition of upper demicontinuity is stronger than the upper hemicontinuity). Comparison of transfer positive hemicontinuity and strictly transfer positive hemicontinuity with upper demicontinuity and upper hemicontinuity and relevant connections of the results presented in this paper with those given in earlier works are also considered. Examples and remarks show a fundamental difference between our results and the well-known ones.
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Włodarczyk, K., Klim, D. Transfer positive hemicontinuity and zeros, coincidences, and fixed points of maps in topological vector spaces. Fixed Point Theory Appl 2005, 459128 (2005). https://doi.org/10.1155/FPTA.2005.389
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DOI: https://doi.org/10.1155/FPTA.2005.389