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A Unified Fixed Point Theory in Generalized Convex Spaces

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Abstract

Let \({\mathfrak {B}}\)be the class of ‘better’ admissible multimaps due to the author. We introduce new concepts of admissibility (in the sense of Klee) and of Klee approximability for subsets of G-convex uniform spaces and show that any compact closed multimap in \({\mathfrak {B}}\) from a G-convex space into itself with the Klee approximable range has a fixed point. This new theorem contains a large number of known results on topological vector spaces or on various subclasses of the class of admissible G-convex spaces. Such subclasses are those of Φ-spaces, sets of the Zima–Hadžić type, locally G-convex spaces, and LG-spaces. Mutual relations among those subclasses and some related results are added.

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Authors and Affiliations

  1. The National Academy of Sciences, Republic of Korea

    Sehie Park

  2. School of Mathematical Sciences, Seoul National University, Seoul, 151–747, Korea

    Sehie Park

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Park, S. A Unified Fixed Point Theory in Generalized Convex Spaces. Acta Math Sinica 23, 1509–1526 (2007). https://doi.org/10.1007/s10114-007-0947-3

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