Mohamed A. Khamsi's An Introduction to Metric Spaces and Fixed Point Theory PDF

By Mohamed A. Khamsi

ISBN-10: 0471418250

ISBN-13: 9780471418252

ISBN-10: 1118033078

ISBN-13: 9781118033074

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” buildings 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 element conception (pages 197–241):
Chapter nine Banach house Ultrapowers (pages 243–271):

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Extra info for An Introduction to Metric Spaces and Fixed Point Theory

Example text

This is all we really need at this point. However it is possible to say more. Replacing T with T? in [Α;(Γ η )] 1/η < k(T) and taking the pth root of both sides gives \k(TPn)}l/pn < \k{Tp)]1/p , p = 1,2, · ■ · Also, a simple calculation shows that fen n—oo t f c ( r ")l 1 / n \k(Tn+1)]1^n+1 =l. 52 CHAPTER 3. METRIC CONTRACTION PRINCIPLES These two facts lead to the conclusion: *οο(Γ)= lim [fc(T n )] 1/n = inf{[fc(T")] 1 / n :n = l , 2 , · · - } . Now, by the Root Test for convergence of series, if / ^ ( T ) < 1 then Σ fc(Tl) < t=l oo.

Another approach is to consider the function / : [0,oo) —» [Ο,οο) defined by f(t) = ^ - t . Then /(0) = 0, and for t > 0, f'(t) = — l — ^ > 0 and f"(t) = j- -2 rz < 0. Thus / is increasing and concave downward. 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 each n let Dn — \%n> χη+ι, ' " · }· Then {Dn} is a descending sequence of nonempty closed subsets of M for which lim diam(Z) n ) = 0. n—»oo oo By assumption, there exists a point x € (~) Dn. n) < e. Then clearly for such n it must be the case that d(x,xn) < e. Hence lim xn — x, proving that M is comn—»oo plete. 1 Note that in the statement of the above theorem, the condition oo oo n—l n=l Pi Dn ψ 0 could have been replaced with the condition exactly one point. f] Dn consists of 28 CHAPTER 2.

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

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