An Introduction to Metric Spaces and Fixed Point Theory by Mohamed A. Khamsi

By Mohamed A. Khamsi

Content material:
Chapter 1 advent (pages 1–11):
Chapter 2 Metric areas (pages 13–40):
Chapter three Metric Contraction ideas (pages 41–69):
Chapter four Hyperconvex areas (pages 71–99):
Chapter five “Normal” constructions in Metric areas (pages 101–124):
Chapter 6 Banach areas: creation (pages 125–170):
Chapter 7 non-stop Mappings in Banach areas (pages 171–196):
Chapter eight Metric mounted aspect concept (pages 197–241):
Chapter nine Banach house Ultrapowers (pages 243–271):

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A;Q)} is nonincreasing and Ω is uncountable there exists c*o 6 Ω such that {^(χ α )}α>α 0 is constant. (xQü ) - φ(χαο+, ) = 0; thusö(z Q o ) = i Q 0 . ■ Next we illustrate a Zorn's lemma approach. The following theorem reduces to Caristi's theorem in the case that M = Y, f is the identity mapping, and c = 1. Here we need another definition. A mapping / of a subset A of metric space M into a metric space N is said to be closed if it has a closed graph; thus / : A —» N is closed if for {x n } Ç A the conditions lim xn = x and lim / ( i n ) n—»oo n—»oo = y imply x G A and f(x) = y.

Be a descending sequence of nonempty closed metrically convex subsets of a compact metric space (M,d). nonempty and metrically convex. oo Then f] Cn is n=l Proof. The fact that the intersection is nonempty is immediate from comoo pactness. Suppose x,y E f] Cn with x φ y. Then in each of the sets C„ there n=l exists a point zn such that d{x, zn) = d{y, Zn) = -d{x, y). 5. ) By compactness of M the sequence {zn} has a subsequence {zn„} which converges to a point z 6 M and since each of the sets Cn is closed, oo z G f) Cn- Since the metric d is continuous, 71 = 1 d(x,z) =d(y,z) = -d(x,y).

What if d is continuous? 3 Give an example of two different metric spaces, each of which is isometric with a subspace of the other. 4 Suppose (M,d) is a metric space and suppose {xn} is a sequence in M which converges to x 6 M. Show that {xn} is a Cauchy sequence. 5 Let M be the real unit interval [0,1] and for x,y £ M define d(x,y) — \x — y\ . Show that (M,d) is a semimetric space with continuous distance which is not a metric space. 6 Let S be any set of nonnegative real numbers which contains 0.

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