Transfer positive hemicontinuity and zeros, coincidences, and fixed points of maps in topological vector spaces
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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 whichtransfer 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. 1. Introduction One of the most important tools of investigations in nonlinear and convex analysis is the minimax inequality of Fan [11, Theorem 1]. There are many variations, generalizations, and applications of this result (see, e.g., Hu and Papageorgiou [16, 17], Ricceri and Simons [19], Yuan [21, 22], Zeidler [24] and the references therein). Using the partition of unity, his minimax inequality, introducing in [10, page 236] the concept of upper demicontinuity and giving in [11, page 108] the inwardness and outwardness conditions, Fan initiated a new line of research in coincidence and fixed point theory of set-valued maps in topological vector spaces, proving in [11] the general results ([11, Theorems 3–6]) which extend and unify several well-known theorems (e.g., Browder [7], [5, Theorems 1 and 2] Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 389–407 DOI: 10.1155/FPTA.2005.389
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Zeros, coincidences, and fixed points
and [6, Theorems 3 and 5], Fan [6, 9], [10, Theorem 5] and [8, Theorem 1], Glicksberg [14], Kakutani [18], Bohnenblust and Karlin [3], Halpern and Bergman [15], and others) concerning upper semicontinuous maps and, in particular, inward and outward maps (the condition of upper semicontinuity is stronger than that of upper demicontinuity). Let C be a nonempty compact convex subset of a real Hausdorff topological vector space E, let F : C → 2E and G : C → 2E be set-valued maps and let Φ : c∈C (F(c) × G(c)) → E be a single-valued map. The purpose of our pap
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