A Fixed-Point Farrago by Joel H. Shapiro

By Joel H. Shapiro

This textual content presents an advent to a few of the best-known fixed-point theorems, with an emphasis on their interactions with themes in research. the extent of exposition raises steadily in the course of the ebook, development from a uncomplicated requirement of undergraduate talent to graduate-level sophistication. Appendices supply an advent to (or refresher on) the various prerequisite fabric and workouts are built-in into the textual content, contributing to the volume’s skill for use as a self-contained textual content. Readers will locate the presentation specifically worthy for autonomous examine or as a complement to a graduate direction in fixed-point theory.

The fabric is divided into 4 components: the 1st introduces the Banach Contraction-Mapping precept and the Brouwer Fixed-Point Theorem, besides a variety of attention-grabbing functions; the second one makes a speciality of Brouwer’s theorem and its software to John Nash’s paintings; the 3rd applies Brouwer’s theorem to areas of endless measurement; and the fourth rests at the paintings of Markov, Kakutani, and Ryll–Nardzewski surrounding mounted issues for households of affine maps.

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Thanks to the work just done in Sect. 2, an algorithm for finding a completely labeled subtriangles will do the trick. Here’s an alternate proof of Sperner’s Lemma that speaks to this issue. Imagine our triangle Δ to be a house, and that the subtriangles of a regular subdivision are its rooms. Given a Sperner labeling of the subvertices that arise from this decomposition, think of each {1, 2}-labeled segment of a subtriangle boundary as a door; these are the only doors. , those with no subvertex labeled “2”); the completely labeled subtriangles are those rooms with exactly one door.

Suppose x ∈ RN .

Now P is a 45◦ projection onto the horizontal axis. Here’s an example more relevant to our immediate purposes. Consider a closed annulus in R2 centered at the origin, having outer radius 1 and some positive inner radius. For x in this annulus let P(x) = x/|x|, where | · | denotes the Euclidean norm on R2 . Then P is a continuous map taking the annulus onto its outer boundary, the unit circle, upon which its restriction is the identity map. Thus the unit circle is a retract of the annulus. This example is of interest to us because no such mapping exists for the unit disc: The unit circle not a retract of the closed unit disc.

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