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Some Fundamental Topological Fixed Point Theorems for Set-Valued Maps

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Topics in Fixed Point Theory

Abstract

This chapter introduces the reader to two of the most fundamental topological fixed point theorems for set-valued maps: the Browder–Ky Fan and the Kakutani–Ky Fan theorems. It provides a concise discussion including motivations, techniques, as well as some most important applications. The exposition is driven by clarity and simplicity. Generality of statements is deliberately sacrificed to the benefit of conceptual significance. Generalizations based on technicalities or artificial definitions which, with little effort, can be reduced to classical settings are set aside, unless they are motivated by convincing applications. Rather, the treatment here is reduced to the classical convex case, which is—we firmly believe—where the essence belongs. The arguments are kept elementary, as to allow the use of this chapter in a first course in topological fixed point theory and its applications.

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Notes

  1. 1.

    Let X be a non-empty convex compact subset of a topological vector space and let A be a subset of X × X disjoint from the diagonal. If for each fixed x ∈ X, the section \(\{y \in X: (x,y) \in A\}\) is convex (or empty) and for each fixed y ∈ X, the section \(\{x \in X: (x,y) \in A\}\) is open in X, then \(\{x_{0}\} \times X \not\subseteq A\) for some x 0 ∈ X. To see the equivalence with the Browder–Ky Fan fixed point theorem, set \(\varPhi (x):=\{ y \in X: (x,y) \in A\}\).

  2. 2.

    A mapping f: X → Y with values in a topological space Y is said to be compact if its image is relatively compact in Y, that is, \(f(X) \subset K\) compact \(\subset Y.\)

  3. 3.

    The terminology “applications de Ky Fan” appeared first in Ben-El-Mechaiekh et al. [11].

  4. 4.

    We have: [A: X ⇉ Y is u.s.c. with closed values and Y is regular ]⇒A has closed graph. Conversely, [A is locally compact and has closed graph] ⇒ A is u.s.c. (with compact values).

  5. 5.

    A set-valued map \(A: X \rightrightarrows Y\) between two topological spaces is said to be lower semicontinuous (l.s.c.) at x 0 ∈ X if for any open subset V of Y such that \(V \cap A(x_{0})\neq \varnothing,\) the upper inverse set \({A}^{-}(V ) =\{ x \in X: A(x) \cap V \neq \varnothing \}\) is an open subset of X containing x 0. A is said to be l.s.c. on X if it is l.s.c. at every point of X. Also, lower semicontinuity coincides with continuity for single-valued mappings.

  6. 6.

    Clearly, if the map is u. s. c. on X in the ordinary sense, then it is \(\mathcal{V}\)-u. s. c on X. The converse holds true in the case where Φ is compact-valued. In the case where Y is a subset of a topological vector space F, the concept of \(\mathcal{V}\)-upper semicontinuity (\(\mathcal{V}\) being the uniformity generated by a fundamental basis of neighborhoods of the origin in F) is known as Hausdorff upper semicontinuity.

  7. 7.

    KKM stands for Knaster, Kuratowski, and Mazurkiewicz. Using the Sperner Lemma, the three famous Polish topologists established in 1929 which became known as the KKM Lemma: if X consists of the set of vertices of a simplex in \({\mathbb{R}}^{n}\)  and \(\varGamma: X \rightrightarrows {R}^{n}\)  is a set-valued map with non-empty compact values verifying: \(\forall \{x_{1},\ldots,x_{k}\} \subset X,\) \(conv(\{x_{1},\ldots,x_{k}\}) \subset \bigcup _{i=1}^{k}\varGamma (x_{i}),\) then \(\bigcap _{x\in X}\varGamma (x)\neq \varnothing.\)

  8. 8.

    He provided the characterization: f is quasiconvex on a convex set X, if and only if \(f(\mu x_{1} + (1-\mu )x_{2}) \leq \max \{ f(x_{1}),f(x_{2})\}\) for all \(x_{1},x_{2} \in X\) and all \(\mu \in [0, 1].\)

  9. 9.

    A topological vector space E has separating dual if for each x ∈ E, x ≠ 0, there exists a bounded linear form ℓ ∈ E ′, the topological dual of E, such that ℓ(x) ≠ 0. Locally convex topological vector spaces have separating duals. Sequence spaces ℓ p, 0 < p < 1, and Hardy spaces H p, 0 < p < 1, are instances of non-locally convex spaces with separating duals.

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Authors and Affiliations

  1. Department of Mathematics, Brock University, St. Catharines, Ontario, Canada, L2S 3A1

    Hichem Ben-El-Mechaiekh

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  1. Hichem Ben-El-Mechaiekh
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Correspondence to Hichem Ben-El-Mechaiekh .

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Editors and Affiliations

  1. Department of Mathematics, University of Tabuk, Tabuk, Saudi Arabia

    Saleh Almezel

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, Uttar Pradesh, India

    Qamrul Hasan Ansari

  3. Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas, USA

    Mohamed Amine Khamsi

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Ben-El-Mechaiekh, H. (2014). Some Fundamental Topological Fixed Point Theorems for Set-Valued Maps. In: Almezel, S., Ansari, Q., Khamsi, M. (eds) Topics in Fixed Point Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-01586-6_7

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