Fixed point theory on extension-type spaces and essential maps on topological spaces
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We present several new fixed point results for admissible self-maps in extension-type spaces. We also discuss a continuation-type theorem for maps between topological spaces. 1. Introduction In Section 2, we begin by presenting most of the up-to-date results in the literature [3, 5, 6, 7, 8, 12] concerning fixed point theory in extension-type spaces. These results are then used to obtain a number of new fixed point theorems, one concerning approximate neighborhood extension spaces and another concerning inward-type maps in extensiontype spaces. Our first result was motivated by ideas in [12] whereas the second result is based on an argument of Ben-El-Mechaiekh and Kryszewski [9]. Also in Section 2 we present a new continuation theorem for maps defined between Hausdorff topological spaces, and our theorem improves results in [3]. For the remainder of this section we present some definitions and known results which will be needed throughout this paper. Suppose X and Y are topological spaces. Given a class ᐄ of maps, ᐄ(X,Y ) denotes the set of maps F : X → 2Y (nonempty subsets of Y ) belonging to ᐄ, and ᐄc the set of finite compositions of maps in ᐄ. We let
Ᏺ(ᐄ) = Z : Fix F = ∅ ∀F ∈ ᐄ(Z,Z) ,
(1.1)
where Fix F denotes the set of fixed points of F. The class Ꮽ of maps is defined by the following properties: (i) Ꮽ contains the class Ꮿ of single-valued continuous functions; (ii) each F ∈ Ꮽc is upper semicontinuous and closed valued; (iii) B n ∈ Ᏺ(Ꮽc ) for all n ∈ {1,2,...}; here Bn = {x ∈ Rn : x ≤ 1}. Remark 1.1. The class Ꮽ is essentially due to Ben-El-Mechaiekh and Deguire [7]. It includes the class of maps ᐁ of Park (ᐁ is the class of maps defined by (i), (iii), and (iv) each F ∈ ᐁc is upper semicontinuous and compact valued). Thus if each F ∈ Ꮽc is compact Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:1 (2004) 13–20 2000 Mathematics Subject Classification: 47H10 URL: http://dx.doi.org/10.1155/S1687182004311046
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Fixed point theorems
valued, the classes Ꮽ and ᐁ coincide and this is what occurs in Section 2 since our maps will be compact. The following result can be found in [7, Proposition 2.2] (see also [11, page 286] for a special case). Theorem 1.2. The Hilbert cube I ∞ (subset of l2 consisting of points (x1 ,x2 ,...) with |xi | ≤ 1/2i for all i) and the Tychonoff cube T (Cartesian product of copies of the unit interval) are in Ᏺ(Ꮽc ). We next consider the class ᐁcκ (X,Y ) (resp., Ꮽcκ (X,Y )) of maps F : X → 2Y such that for each F and each nonempty compact subset K of X, there exists a map G ∈ ᐁc (K,Y ) (resp., G ∈ Ꮽc (K,Y )) such that G(x) ⊆ F(x) for all x ∈ K. Theorem 1.3. The Hilbert cube I ∞ and the Tychonoff cube T are in Ᏺ(Ꮽcκ ) (resp., Ᏺ(ᐁcκ )). Proof. Let F ∈ Ꮽcκ (I ∞ ,I ∞ ). We must show that Fix F = ∅. Now, by definition, there exists G ∈ Ꮽc (I ∞ ,I ∞ ) with G(x) ⊆ F(x) for all x ∈ I ∞ , so Theorem 1.2 guarantees that there exists x ∈ I ∞ with x ∈ Gx. In particular, x ∈ Fx so Fix F = ∅. Thus I ∞ ∈ Ᏺ(Ꮽcκ ). Notice that ᐁcκ is closed under compositio
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