Analogs of the Schauder theorem that use anticompacta (Q325576)

According the author's definition in a previous paper [Din. Sist., Simferopol' 3(31), No. 3--4, 281--288 (2013; Zbl 1357.46035)], a closed convex symmetric set \(C\) in a normed space \(E\) is called anticompact if the linear hull span \(C\) of \(C\), equipped with the Minkowski functional of \(C\) as norm, contains all bounded subsets of \(E\), and these sets are precompact. One may show that \(E\) contains an anticompact set iff its dual \(E^*\) admits a countable set of functionals which separates the points of \(E\). Using the concept with this weird and unlucky name, the author proposes a fixed point theory for mappings \(f\) of a bounded closed subset \(B\) of a Banach space without requiring the precompactness of \(f(B)\) as in the classical Schauder theorem. A substantial advantage of the paper is that the author illustrates the abstract results and hypotheses throughout by illuminating examples; a serious drawback, however, is the poor English translation of the Russian original.