Remarks on a fixed point problem of Ben-El-Mechaiekh (Q2753297)

The author and \textit{H. Kim} proved [J. Math. Anal. Appl. 197, No. 1, 173-187 (1993; Zbl 0851.54039)] the folloing coincidence theorem: NEWLINENEWLINENEWLINETheorem. Let \((X,D;\Gamma)\) be a compact \(G\)-convex space, and \(S: X\to D,\) \(T: X\to X\) maps such that NEWLINENEWLINENEWLINE(i) for each \(x\in X,\) \(M\in \langle Sx\rangle\) implies \(\Gamma_{M}\subset Tx\) and NEWLINENEWLINENEWLINE(ii) \(X=\cup\{\operatorname {Int} S^{-}y: y\in D\}.\) NEWLINENEWLINENEWLINEThen \(T\) has a fixed point \(x_{0}\in X;\) that is, \(x_{0}\in Tx_{0}.\) NEWLINENEWLINENEWLINE\textit{H. Ben-El-Mechaiekh} [Bull. Aust. Math. Soc. 41, No. 3, 421-434 (1990; Zbl 0685.54030); Quest. Answers Gen. Topology 10, No. 2, 153-156 (1992; Zbl 0803.54038)] raised for the case \(X=D\) is a convex subset of a topological vector space and \(\Gamma_{M}=\infty\) for \(M\in \langle X\rangle\) the following Problem: NEWLINENEWLINENEWLINEDoes Theorem hold if we assume \(T\) is compact instead of the compactness of \(X?\) NEWLINENEWLINENEWLINEThis problem is still open. In this paper the author gives partial solutions of this problem.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00060].