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

By Mohamed A. Khamsi

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

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Example text

14 (The metric transform φ) Let (M,d) be a metric space, and define the metric space (M, άφ) by taking for x,y € M άφ(χ,ν) = Φ(ά(χ,ν)), where φ : [0, oo) —» [0, oo) is increasing, concave downward, and satisfies φ(0) — 0. 15 (The Hausdorff metric) Let (M,d) be a metric space and let ΛΊ denote the family of all nonempty bounded closed subsets of M. For A € M. and ε > 0 define the ε-neighborhood of A to be the set Νε(Α) = {x € M : dist(x, A) < ε}. where dist(x, A) = inf d(x,y). A Now for Α,ΒΕΜ H(A, B) = inf{e > 0 : A C Ne(B) Then (M,H) set and B Ç Ne(A)}.

Hence if |i — to\ < δ, 1/(0 - /(*o)l < 1/(0 - fN(t)\ + \fN(t) - fN(to)\ + |/jv(io) - f(to)\ < e. This proves continuity of / . To see that lim d(fn,f) = 0, let ε > 0 and n—*oo observe that since {/„} is a Cauchy sequence there is an integer TV such that if m,n> N then sup | / „ ( 0 - / m ( 0 l < e . «€[0,1] that is, / n ( 0 - e < / m ( 0 < / n ( 0 + e · Letting m —» oo we see that for any t € [0,1] and n > N, / n ( 0 ~e< / ( 0 < / n ( 0 + e; hence 1/(0 - /n(0l < ε. from which d(/„, / ) < ε. Since ε > 0 is arbitrary we conclude lim d(fn, f) = 0.

Let {xa}ael D e a n y chain in (M, >), and for α,β € / set β > a <=> Χβ > xa. ι is a nonincreasing net in R + so there exists r > 0 such that = r. \imip(f(xa)) a Let ε > 0. Then there exists ao 6 / such that a > ao implies r < tp(f(xa)) a > ao, m a x i d ^ ^ ^ ) ^ ^ / ^ ) , / ^ ) ) } <

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