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

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

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Additional resources for An Introduction to Metric Spaces and Fixed Point Theory

Sample text

17 Show that (Mc,dm) is complete. 18 Show that (M,d*) is dense in (Mc,d*). 11 is not complete. 15 is complete. 21 Show that the metric space consisting of all irrational numbers is separable. 22 Show that any subspace of a separable metric space is separable. 40 CHAPTER 2. 23 Show that the space of all nonempty compact subsets of a separable metric space M endowed with the Hausdorff metric is separable. 24 Let (M,d) be a complete metric space and φ : M —» [0, oo) an arbitrary nonnegative function.

It follows that there exists y EY and x G M such that lim/(a: a ) = y and a limx a = x. Since / is a closed mapping, f(x) = y, and lower semicontinuity of a φ yields φ(/(χ)) < r. Moreover, if a, β € / with β > a, then m a x { d ( i a , i ß ) , c d ( / ( i 0 ) , / ( i ß ) ) } < (/(*))· This proves that x > xa for each a £ I. Having thus shown that every chain in (M, >) has a upper bound we can appeal to Zorn's lemma to conclude that (M, >) has a maximal element, say x.

2) d(T(x),T(y))(d(x,y)). Then T has a unique fixed point z, and {Tn(x)} converges to z, for each x 6 M. Proof. This theorem is actually a special case of the previous theorem. First introduce the function φ : R —» [0,1) by setting ^>(0) = φ(0) and φ(ί) = Ά for t > 0. To see that φ is in the class S suppose φ{ίη) —» 1. Then {i„} must be bounded (otherwise, lim inf φ(ίη) = 0 ) . By passing to a subsequence we may assume n—>oo that tn —♦ to and we may assume further that either tn ]

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