By Joel H. Shapiro
This textual content offers an creation to a few of the best-known fixed-point theorems, with an emphasis on their interactions with issues in research. the extent of exposition raises steadily through the booklet, development from a simple requirement of undergraduate skillability to graduate-level sophistication. Appendices offer an creation to (or refresher on) the various prerequisite fabric and routines 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 in particular priceless for self sufficient research or as a complement to a graduate path in fixed-point theory.
The fabric is divided into 4 elements: the 1st introduces the Banach Contraction-Mapping precept and the Brouwer Fixed-Point Theorem, in addition to a variety of fascinating functions; the second one specializes in Brouwer’s theorem and its program to John Nash’s paintings; the 3rd applies Brouwer’s theorem to areas of limitless size; and the fourth rests at the paintings of Markov, Kakutani, and Ryll–Nardzewski surrounding fastened issues for households of affine maps.
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Extra resources for A Fixed-Point Farrago
We’ll show in the next section that the No-Retraction Theorem implies the Brouwer Fixed-Point Theorem, after which we’ll give our “Brouwer-independent” proof of the No-Retraction Theorem. 2 “No-Retraction” ⇒ “Brouwer” We’ve already noted (Chap. 1, Sect. 6) that for N = 1 Brouwer’s Theorem follows from the Intermediate Value Theorem of elementary calculus, so now we’ll work in RN with N > 1, employing the notation | · | for the Euclidean norm in that space. , that there is a continuous map f : B → B that has no fixed point.
2) for the unit vector u, the function x − f (x) is C1 and never zero, hence the denominator |x − f (x)| is C1 and (thanks to the compactness of B) bounded away from zero on B. Thus u is C1 on B. The only issue left is the C1 nature of the parameter λ (x) on the right-hand side of Eq. 1), but this follows immediately from Eq. 2—strictly positive for each x ∈ B. (c) Proof of C1 -NRT. This is the heart of our proof of the BFPT. , suppose there exists a C1 retraction P taking B onto its boundary. We will show that this leads to a contradiction.
Subvertices lying in the interior of the original triangle can be labeled without restriction. We’ll call a subtriangle whose vertices have labels “1,” “2,” and “3” a completely labeled subtriangle. 2 shows a regular decomposition of a triangle into Sperner-labeled subtriangles, five of which (the shaded ones) are completely labeled. 3 1 3 1 1 2 3 1 2 1 3 2 2 1 2 Fig. 1 (Sperner’s Lemma for Dimension Two). Every Sperner-labeled regular decomposition of a triangle has an odd number of completely labeled subtriangles; in particular there is at least one.