By Joel H. Shapiro
This textual content offers 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 progressively during the e-book, development from a easy requirement of undergraduate talent to graduate-level sophistication. Appendices supply an creation 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 worthwhile for self reliant research or as a complement to a graduate path 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 range of fascinating 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 countless measurement; 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
Every one-point space has it (trivially), and for each positive integer N the closed unit ball of RN has it (The Brouwer Fixed-Point Theorem). No circle has it (nontrivial rotations have no fixed point), hence no closed curve (homeomorphic image of a circle) has it. 3 allows us to exhibit more examples. Here is one that is “one dimensional,” but not homeomorphic to a closed interval (exercise). 4. The letter “X” has the fixed-point property. 48 4 Brouwer in Higher Dimensions Proof. a “the closed unit disc”).
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. This follows immediately from the N = 2 version of the Brouwer Fixed-Point Theorem (Chap. 2). Indeed, if there were a retraction P taking the closed unit disc onto the unit circle, then Q = −P would be a continuous mapping of the disc into itself (more precisely: onto the unit circle), that has no fixed point.
It remains to find the positive constant a. For this, let ξ j denote the j-th coordinate of the Perron vector x, and η j the j-th coordinate of y. We’re looking for a ∈ R such that aξ j ≥ η j for all j, and aξk = ηk for some k. Since no coordinate of x is zero, we can rewrite our criteria as: a ≥ η j /ξ j for all j, and a = ηk /ξk for some k; in other words the positive real number a = max j η j /ξ j does the job. 11. If A is a positive N × N stochastic matrix, then there is a positive vector x ∈ ΠN such that Ax = x.