Transfer positive hemicontinuity and zeros, coincidences, and fixed points of maps in topological vector spaces

Let E 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 E are introduced (condition of strictly transfer positive hemicontinuity is stronger than that of transfer positive hemicontinuity) and for maps F:C→2E and G:C→2E defined on a nonempty compact convex subset C of E, 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 Φ:∪c∈C(F(c)×G(c))→E has a zero, and, at the same time, we give various characterizations of the class of those pairs (F,G) and maps F 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