C. Bardos, J. M. Lasry, M. Schatzman's Bifurcation and nonlinear eigenvalue problems PDF

By C. Bardos, J. M. Lasry, M. Schatzman

ISBN-10: 3540097589

ISBN-13: 9783540097587

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Thus, there exists XELX such that \\mxn = x, and we claim that χΕ:Π^χΑη. Indeed, since xm E:An for m > n, we get x E:An for each n. But since each An is closed, An=An holds; therefore, x^An for each n9 and we are done. ■ A subset A of a metric space (X, d) is said to be nowhere dense if its closure has an empty interior, that is, if A=0. Since B = (B*)C holds for every subset B9 it is easy to see that a subset A is nowhere dense if and only if (A)c is dense in X. A classical nowhere dense subset of the real line is the so-called Cantor set.

Conversely, let (A, d) be a complete metric space. If a sequence {xn} of A satisfies \\mxn = x in X, then {xn} is a Cauchy sequence of A". But then {x n } is a Cauchy sequence of A, and hence, it must converge to a unique element of A. This element must be x. Thus, x G ^ , so that A is a closed subset of X. ■ The following very important result dealing with complete metric spaces is due to G. Cantor. 11. (Cantor). Let (X, d) be a complete metric space and let {An} be a sequence of closed, nonempty subsets of X such that An+l QAnfor each n and lim d(An) = 0.

Let C2 be the set remaining from Cx after these removals. That is, C2 = [ 0 , l / 9 ] u [ 2 / 9 , l / 3 ] u [ 2 / 3 , 7 / 9 ] u [ 8 / 9 , l ] ; note that C2 is the union of 2 2 = 4 disjoint closed intervals. The process of constructing Cn+l from C„ should be clear now. Trisect each of the 2" disjoint closed intervals of Cn, and remove from each one of them the middle open interval. What is left from Cn is then C n + 1 . Note that C„+x is the union of 2 n + 1 disjoint closed intervals. The graphs of the first few constructions are shown in Figure 1.

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Bifurcation and nonlinear eigenvalue problems by C. Bardos, J. M. Lasry, M. Schatzman


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